In the Tradition of Thurston III: Geometry and Dynamics (In the Tradition of Thurston, 3) 303143501X, 9783031435010

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Ken’ichi Ohshika Athanase Papadopoulos  Eds.

In the Tradition of Thurston III Geometry and Dynamics

In the Tradition of Thurston III

Ken’ichi Ohshika • Athanase Papadopoulos Editors

In the Tradition of Thurston III Geometry and Dynamics

Editors Ken’ichi Ohshika Department of Mathematics Gakushuin University Tokyo, Japan

Athanase Papadopoulos Institut de Recherche Mathématique Avancée Université de Strasbourg et CNRS Strasbourg, France

ISBN 978-3-031-43501-0 ISBN 978-3-031-43502-7 https://doi.org/10.1007/978-3-031-43502-7

(eBook)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Cover illustration: William Thurston at the Clay Research Conference on the Poincaré conjecture, Institut Océanographique, Paris, June 8, 2010. Photo © François Tisseyre, Atelier EcoutezVoir. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Preface

Since the beginning of the 1970s, the works of William Thurston (1946–2012) have exerted a decisive impact on mathematics, and today they continue to serve as guidelines for an important part of research in geometry, topology, and dynamics. The present volume, the third in the series In the Tradition of Thurston, contains a collection of articles whose aim is to help the reader enter the new worlds of ideas discovered by Thurston. Each of these articles surveys an important aspect of Thurston’s work. The topics considered in this volume include Thurston’s and Weil–Petersson’s metrics of Teichmüller space, Kleinian groups, holomorphic motions, earthquakes, 3-manifolds, geometric structures, homeomorphism groups of 2-manifolds, dynamics, and orbifolds together with their relations with Vmanifolds in algebraic geometry. We would like to thank Elena Griniari for her kind support and care for this project, and the reviewers of the various chapters for their valuable anonymous work. Our warm thanks go to the authors of the various chapters, for a fruitful and friendly collaboration. Tokyo, Japan Strasbourg, France February, 2024

Ken’ichi Ohshika Athanase Papadopoulos

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Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ken’ichi Ohshika and Athanase Papadopoulos

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The Geometry of the Thurston Metric: A Survey . . . . . . . . . . . . . . . . . . . . . . Huiping Pan and Weixu Su 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Extremal Lipschitz Maps and Geodesic Rays . . . . . . . . . . . . . . . . . . . . . 2.4 Harmonic Stretch Lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Thurston Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Isometry Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Coarse Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Counting Lattice Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Shearing Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Generalization of the Thurston Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Thurston’s Metric on the Teichmüller Space of Flat Tori . . . . . . . . . . . . . . Binbin Xu 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Teichmüller Space of Flat Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Affine Maps between Flat Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Geodesic Foliation on a Flat Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Lipschitz Constants of Homeomorphisms between Flat Tori . . . . . 3.6 Curve Length Ratios for Two Flat Structures on S . . . . . . . . . . . . . . . . 3.7 Thurston’s Metric on T (S) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Further Discussions on Extremal Lipschitz Constants and Singular Values of Affine Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 9 13 23 26 28 30 35 37 39 40 45 45 51 53 55 57 59 61 63 66

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The Anti-de Sitter Proof of Thurston’s Earthquake Theorem . . . . . . . . 67 Farid Diaf and Andrea Seppi 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.2 Earthquake Maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.3 Anti-de Sitter Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.4 Convexity Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.5 The Case of Two Spacelike Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.6 Proof of the Earthquake Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

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Homeomorphism Groups of Self-Similar 2-Manifolds . . . . . . . . . . . . . . . . . Nicholas G. Vlamis 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Overview of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Topology of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Stable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Freudenthal Subsurfaces and Anderson’s Method . . . . . . . . . . . . . . . . 5.6 Topology of Homeomorphism Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Equivalent Notions of Self-Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Normal Generation and Purity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Strong Distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Coarse Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Rokhlin Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12 Automatic Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13 Commutator Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Weil–Petersson Teichmüller Theory of Surfaces of Infinite Conformal Type. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eric Schippers and Wolfgang Staubach 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Three Models of Teichmüller Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Complex Manifold and Tangent Space Structure . . . . . . . . . . . . . . . . . 6.5 The Weil–Petersson Universal Teichmüller Space . . . . . . . . . . . . . . . . 6.6 The Period Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 A Brief Overview of Other Refinements of Teichmüller Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Weil–Petersson Teichmüller Spaces of General Surfaces . . . . . . . . . 6.9 Kähler Geometry and Global Analysis of the Weil–Petersson Teichmüller Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Weil–Petersson Beyond Teichmüller Theory . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105 106 108 120 124 129 135 145 147 149 151 154 158 164 165 169 170 175 179 185 192 201 208 210 222 232 241

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Kleinian Groups and Geometric Function Theory . . . . . . . . . . . . . . . . . . . . . Hiroshige Shiga and Toshiyuki Sugawa 7.1 Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Function Theory on the Components of Kleinian Groups . . . . . . . . 7.3 Teichmüller Spaces and Univalent Functions . . . . . . . . . . . . . . . . . . . . . 7.4 Holomorphic Motions, Quasiconformal Motions and Kleinian Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

249

Thurston’s Broken Windows Only Theorem Revisited . . . . . . . . . . . . . . . . Ken’ichi Ohshika 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Counter-Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 A Weaker Version of Thurston’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . 8.5 The Bounded Image Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Geometric Structures in Topology, Geometry, Global Analysis and Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Christoforos Neofytidis 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Domination, Monotonicity and Anosov Maps . . . . . . . . . . . . . . . . . . . . 9.3 The Gromov Order for Thurston Geometries in Dimensions ≤ 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Geometric Kodaira Dimension, Monotonicity, and Simplicial Volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Anosov Diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Counting Problems for Invariant Point Processes . . . . . . . . . . . . . . . . . . . . . . Jayadev S. Athreya 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 SL(2, R)-Invariant Point Processes on C . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Counting Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Counting for Poisson Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Ergodic Theory of SL(2, R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Circles and Sectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8 Hyperbolas and Geodesic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.9 Triangles and Horocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.10 Further Questions/Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

249 254 264 270 272

275 278 284 287 293 296 299 299 300 307 318 328 336 339 339 340 342 348 351 352 353 356 358 359 361

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Orbifolds and the Modular Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Juan Martín Pérez and Florent Schaffhauser 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Complex Elliptic Curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Moduli Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 The Moduli Stack of Elliptic Curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some Footnotes on Thurston’s Notes The Geometry and Topology of 3-Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Athanase Papadopoulos 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Non-Euclidean Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Excursus: Dante . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Geometric Structures: From Ehresmann and Haefliger to Thurston . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Spheres and Horospheres in Hyperbolic Space: Lobachevsky’s Insight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Polyhedra: Andreev’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7 Volumes of Polyhedra: Lobachevsky and Milnor . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

365 365 368 381 389 397 404 421 423 424 425 428 435 438 439 441 444

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

Chapter 1

Introduction Ken’ichi Ohshika and Athanase Papadopoulos

Abstract In this introductory chapter, we describe the various topics covered in the volume In the tradition of Thurston, III. Keywords Thurston metric · Teichmüller space · Quasisymmetric map · Quasiconformal map · Weil–Petersson metric · Period embedding · Asymptotic counting problems · Complex-analytic moduli problems · Stacks and moduli problems · Aspherical manifold · Domination · Gromov order · Monotone invariant · Kodaira dimension · Anosov diffeomorphism AMS Codes 14D23, 20F34, 22E25, 30C62, 30F30, 30F35, 30F40, 30F60, 32G13, 37D20, 51F30, 53A15, 55R10, 58J52, 57M05, 57-06, 57M10, 57M50, 57K32, 81T40

Since his works of the 1970s, William Thurston (1946–2012) have changed the landscape of mathematics, especially the fields of geometry, topology, geometric group theory and dynamics. The aim of the collection In the tradition of Thurston, of which the present volume is the third, is to provide the reader with an overview of important aspects of Thurston’s heritage, including ideas he introduced and developments that have taken place in fields closely related to his work. The topics considered in this volume include Thurston’s asymmetric metric of the Teichmüller spaces of hyperbolic surfaces, with its analogue on the Teichmüller space of Euclidean surfaces, the Weil–Petersson geometry of Teichmüller space, Kleinian groups, holomorphic motions, earthquakes, 3-manifolds, geometric struc-

K. Ohshika Gakushuin University, Tokyo, Japan e-mail: [email protected] A. Papadopoulos () Institut de Recherche Mathématique Avanvée, Strasbourg, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 K. Ohshika, A. Papadopoulos (eds.), In the Tradition of Thurston III, https://doi.org/10.1007/978-3-031-43502-7_1

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K. Ohshika and A. Papadopoulos

tures, homeomorphism groups of 2-manifolds, dynamics, and orbifolds together with their relations with V-manifolds in algebraic geometry. Let us review in some detail the contents of each chapter. The first two chapters that follow this introduction are concerned with Thurston’s asymmetric metric on Teichmüller space. Here, the distance between two hyperbolic metrics on a given surface is defined to be the logarithm of the smallest global Lipschitz constant of a homeomorphism between them which is in the homotopy class of the identity. This metric was introduced by Thurston in his paper Minimal stretch maps between hyperbolic surfaces, first circulated in 1985. Thurston’s metric and its generalizations to various settings have been the object of intense and growing research in the last two decades by several authors, and having good surveys on such a topic should be useful to the geometers. Chapter 2, by Huiping Pan and Weixu Su, is titled The geometry of the Thurston metric: a survey. It is a panorama of old and new constructions and results, due to several authors, related to Thurston’s metric. Some of Thurston’s work from his original paper is also reviewed. A central element of this chapter is the construction of extremal Lipschitz maps, called stretch maps, between hyperbolic surfaces. These maps, by varying the stretching parameter, give rise to one-parameter families called stretch lines. The authors review the behavior at infinity of these lines, making relations between Thurston’s metric and Thurston’s compactification of Teichmülller space. They also discuss the coarse geometry of this metric, describing recent rigidity results concerning its isometry group as well as developments and generalizations of this metric to other settings. Chapter 3, by Binbin Xu, titled Thurston’s metric on Teichmüller space of flat tori, is a survey of results by various authors on Thurston’s metric extended to the case of the Teichmüller space of flat tori, reviewing in particular the relation between best Lipschitz maps and affine maps between flat tori. Chapter 4 is built on another important theory developed by Thurston in his study of surfaces, namely, the earthquake theorem. Earthquakes are deformations of hyperbolic surfaces which generalize the Fenchel–Nielsen twists along weighted simple closed curves to deformations associated with measured geodesic laminations. Thurston’s earthquake theorem, in its original form, states that any two points in the Teichmüller space of a closed hyperbolic surface can be joined by a unique left earthquake. This theorem was used in an essential way by S. Kerckhoff in his 1983 paper The Nielsen realization problem. Kerckhoff also provided the first written proof of this theorem. In his paper Earthquakes in two-dimensional hyperbolic geometry, published in 1986, Thurston gave another proof of the earthquake theorem in the setting of the universal Teichmüller space (from which one can also recover the case of closed surfaces). In this context, the earthquake theorem says that every orientation-preserving homeomorphism of the circle (seen as the boundary of the hyperbolic disc) admits an extension to the hyperbolic disc which is a left (or right) earthquake. In Chap. 4 of the present volume, titled The Anti-de Sitter proof of Thurston’s earthquake theorem, Farid Diaf and Andrea Seppi give a new proof, in the setting of anti-de Sitter geometry, of the universal Teichmüller space version of the earthquake theorem. This proof is based on the bi-invariant geometry of the

1 Introduction

3

Lie group .PSL(2, R), which is also the anti-de Sitter three-space. Diaf and Seppi also recover the original earthquake theorem for closed surfaces that we find in Kerckhoff’s paper. Chapter 5, by Nicholas Vlamis, is concerned with the geometry of infinite-type surfaces. It is titled Homeomorphism groups of self-similar 2-manifolds. Mapping class groups of infinite-type surfaces (sometimes called big mapping class groups) have been a subject of intense research in the last few years. In Chap. 5, Vlamis studies homeomorphism groups and mapping class groups of a class of infinite type surfaces which he describes as self-similar 2-manifolds. These are surfaces that exhibit a certain type of homogeneity akin to the 2-sphere and the Cantor set. The theory developed in this chapter is a natural extension of that of homeomorphism groups of the 2-sphere and of the Cantor set. In doing so, Vlamis generalizes recent results on mapping class groups to the setting of homeomorphism groups. At the same time, he introduces the tools needed for the study of end spaces of 2manifolds, and he provides the necessary background for the study of the topology of homeomorphism groups and mapping class groups of self-similar 2-manifolds. In particular, he develops the structure theory of stable sets and he obtains strong versions of several recent results on this topic. Chapter 6, by Eric Schippers and Wolfgang Staubach, is titled Weil–Petersson Teichmüller theory of surfaces of infinite conformal type. The Weil–Petersson geometry of Teichmüller space is a rich theory that involves algebraic geometry, complex geometry and analysis, and it has applications in mathematical physics. The topics surveyed by Schippers and Staubach include complex Hilbert manifolds, Kähler geometry, global analysis and generalizations of the period mapping. The authors also discuss the motivations of the theory which originated, at the beginning of the 1980s, in relation with representation theory, and they explain how these developments led to the Kähler geometry of the moduli spaces of surfaces of infinite conformal type, making connections with geometry, analysis, representation theory and physics, in particular fluid mechanics and two-dimensional conformal field theory. Chapter 7, by Hiroshige Shiga and Toshiyuki Sugawa, is titled Kleinian groups and geometric function theory. Here, the authors use the term “geometric function theory” to denote the geometric study of analytic functions and domains in the complex plane and on Riemann surfaces. In particular, they are interested in the complex analytic properties of the region of discontinuity and the limit set of a Kleinian group. They review several kinds of domains in .Rn , with the Bers embedding of Teichmüller space in the background. This is an embedding which realizes this space as a bounded domain in a complex Banach space. After this review, Shiga and Sugawa discuss several versions of the so-called modified Bers conjecture for Fuchsian groups, which concerns the closure of the Bers embedding and which was extended by Sullivan and Thurston to the so-called density conjecture. The latter says that every finitely generated Kleinian group is an algebraic limit of geometrically finite Kleinian groups. The conjecture has been proved by Namazi and Souto, and by Ohshika independently. Several hints and developments of these

4

K. Ohshika and A. Papadopoulos

works are given. Results of Sullivan–Thurston, Bers–Royden and Slodkowski on holomorphic motions are also discussed. Chapter 8, by Ken’ichi Ohshika, is titled Thurston’s broken windows only theorem revisited. The title refers to a theorem which appears in Thurston’s preprint, Hyperbolic structures on 3-manifolds III: Deformations of 3-manifolds with incompressible boundary, which was circulated in 1986 and which constitutes the third of a series of papers in which Thurston gave a proof of his uniformization theorem for Haken manifolds. The broken windows only theorem is the main result of that third paper, and, as Ohshika notes, it was intended to be the key part of the proof of the so-called bounded image theorem, one of the main steps of the proof of the Haken uniformization theorem. The broken windows only theorem consists of two statements, and Ohshika shows that the second of them is false. Indeed, he gives a counter-example to this statement. He provides a proof for a weaker version of it, which, however, is not sufficient for the proof of the bounded image theorem. At the same time, Ohshika provides a short summary of the intricate history of the bounded image theorem, for which a weak version, known as the bounded orbit theorem, which appears in Thurston’s introductory paper of the series we mentioned, is sufficient for the proof of his uniformization theorem. A proof of the bounded orbit theorem was given by M. Kapovich in his book Hyperbolic manifolds and discrete groups published in 2001. Ohshika also speculates on how the second statement in Thurston’s broken windows only theorem was intended to constitute a proof of the bounded image theorem. Motivated by Thurston’s geometrization picture in dimension three, Christoforos Neofytidis, in Chap. 9, titled Geometric structures in topology, geometry, global analysis and dynamics, addresses several questions on manifolds of arbitrary dimension. Some of these questions concern geometric structures in the sense of Thurston, and others involve non-geometric manifolds. The author’s goal is to provide a unified treatment of several problems arising in topology, geometry, global analysis and dynamics. The main problems with which he is concerned are: (i) The existence of an order on the set of homotopy classes of manifolds of a given dimension, which he calls “Gromov order” or “domination relation”: here, a manifold M dominates a manifold N if there exists a map .f : M → N of non-zero degree; (ii) the Gromov– Thurston monotonicity problem for numerical homotopy invariants with respect to the domination relation: here, the author introduces a notion of geometric Kodaira dimension; (iii) the existence of Anosov diffeomorphisms, in relation with the Anosov–Smale conjecture stating that all Anosov diffeomorphisms are conjugate to hyperbolic automorphisms of nilmanifolds. Neofytidis shows that Thurston’s geometrization gives a unified approach to these questions. Chapter 10, by Jayadev Athreya, is titled Counting problems for invariant point processes. In this chapter, using linear algebra and the ergodic theory of .SL(2, R)actions, the author shows how to solve several natural asymptotic counting problems concerning discrete subsets of the plane. The problems arise from the geometry of surfaces. The point of view is axiomatic, and the goal is to give the general important ideas in this theory, with a focus on some key motivating examples. Poisson point processes, which are famous examples of .SL(2, R)-probability invariant measures

1 Introduction

5

on the space of discrete subsets of the complex plane, act as motivating examples. Connections with Thurston’s work arise from the latter’s theory of geometric structures on surfaces, where counting problems emanate naturally, in relation with the study of interval exchange maps and quadratic differentials. Athreya also discusses applications of these ideas to the problems of counting holonomies of saddle connections, lattice points, and fine scale distribution in various contexts. Besides being inspired by Thurston’s work, his perspective is also motivated by works of Veech, Eskin–Masur, Marklof, Athreya–Ghosh and others. Chapter 11, by Juan Martín Pérez and Florent Schaffhauser, is titled Orbifolds and the modular curve. The authors start from the observation that the ramification points which appear in the geometric concept of orbifold, developed by Satake and then Thurston, have an analytic and algebraic counterpart, in a setting where one deals with morphisms that are not étale. They recall that in algebraic geometry, there is a theory which is parallel to Thurston’s theory of orbifolds and their fundamental groups, and they survey this theory, developing the idea that the two points of view, the algebro-geometric and the Satake–Thurston, can enrich the scope and methods of each other. In particular, they present a construction of the moduli stack of elliptic curves and show that it is in fact an analytic orbifold. Chapter 12, the last chapter of this volume, by Athanase Papadopoulos, is titled Footnotes on Thurston’s Notes on the Geometry and topology of three-manifolds. In this chapter, the author provides some historical remarks, addenda and references with comments on some topics tackled by Thurston in his famous notes, The geometry and topology of three-manifolds. The various sections in this chapter, considered as footnotes to Thurston’s notes, mostly concern the parts of Thurston’s notes on hyperbolic geometry, geometric structures, the computation of volumes of hyperbolic polyhedra and the so-called Koebe–Andreev–Thurston theorem. The author discusses in particular some works of Lobachevsky, Andreev and Milnor, with an excursus in Dante’s cosmology, based on the insight of Pavel Florensky. The present volume will have a sequel, naturally titled In the tradition, of Thurston, IV. We all hope that the collection of articles that constitute this collection will be useful for students and researchers who are interested in Thurston’s mathematics.

Chapter 2

The Geometry of the Thurston Metric: A Survey Huiping Pan and Weixu Su

Abstract This chapter is a survey about the Thurston metric on the Teichmüller space. The central issue is the construction of extremal Lipschitz maps between hyperbolic surfaces. We review several constructions, including the original work of Thurston. Coarse geometry and isometry rigidity of the Thurston metric, relation between the Thurston metric and the Thurston compactification are discussed. Some recent generalizations and developments of the Thurston metric are sketched. Keywords Teichmüller space · Thurston metric · Hyperbolic surfaces · Lipschitz maps · Harmonic maps · Thurston boundary · Isometry rigidity · Shearing coordinates Mathematical Classification (2010) 32G15, 30F45, 30F60

2.1 Introduction This survey aims to give a brief exposition of recent results on the Thurston metric. This asymmetric metric, defined by Thurston in his 1986 preprint [74], has a lot of beautiful geometric and analytic properties. Thurston’s original investigation, just like his many other masterpieces, are based on first principles. Until now, that preprint [74] together with Chapter 8 and Chapter 9 of [73], are excellent references for learning hyperbolic geometry. Aiming for a geometric understanding of the Teichmüller space from the point of view of hyperbolic surfaces and Lipschitz maps, in parallel with the point of

H. Pan () School of Mathematics, South China University of Technology, Guangzhou, China e-mail: [email protected] W. Su School of Mathematics, Sun Yat-Sen University, Guangzhou, China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 K. Ohshika, A. Papadopoulos (eds.), In the Tradition of Thurston III, https://doi.org/10.1007/978-3-031-43502-7_2

7

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H. Pan and W. Su

view of Riemann surfaces and quasiconformal maps, Thurston [74] considered the following problem: Given any two hyperbolic surfaces X and Y , what is the least possible value of the global Lipschitz constant .L(φ)

:= sup

p/=q

d(φ(p)), φ(q) d(p, q)

for a homeomorphism .φ : X → Y in a given homotopy class?

Thurston gave a geometric construction of extremal Lipschitz homeomorphisms realizing the extremal Lipschitz constant .Lip(X, Y ) in the homotopy class of the identity map, and proved that the extremal Lipschitz constant is the same as another geometric quantity: K(X, Y ) := sup

.

α

𝓁α (Y ) 𝓁α (X)

where .𝓁α (·) is the length of the geodesic representative of .α on the corresponding surface and .α ranges over all nontrivial homotopy classes of simple closed curves. For any two points X and Y in the Teichmüller space, the Thurston metric is defined as: dT h (X, Y ) := log Lip(X, Y ).

.

Among others, Thurston proved that this metric is Finslerian, asymmetric, geodesic but not uniquely geodesic [74]. Through the efforts of many mathematicians, we now have a fairly clear picture of this metric: the horofunction boundary, the isometry group, the coarse geometry, the analytic construction of extremal Lipschitz maps and geodesic rays, to name a few. Due to its geometric inspiration, the Thurston metric has been generalized to deformation spaces of other geometric structures and provides new perspectives there. In this survey about the Thurston metric, we restrict our attention to recent research. We will look more carefully at several constructions of extremal Lipschitz maps. Relevant developments will be briefly sketched or summarized without proofs. Other surveys on the Thurston metric include [53, 54, 68]. In particular, the article [54] contains foundational background material, both on the Teichmüller metric and the Thurston metric. As for prerequisites, the reader is expected to be familiar with hyperbolic geometry and Teichmüller theory. We recommend [23, 35]. Here is the outline of this paper. In Sect. 2.2, we review some of the standard facts on Teichmüller space, including the Thurston norm. In Sect. 2.3, we provide various constructions of extremal Lipschitz maps and geodesic rays of the Thurston metric. In Sect. 2.4, we deal with a special type of geodesics, called harmonic stretch lines, and discuss two versions of the Thurston (co-)geodesic flows. In Sect. 2.5, we describe the Thurston boundary and its identification with the horofunction

2 The Geometry of the Thurston Metric: A Survey

9

boundary. In Sect. 2.6, we illustrate the isometry rigidity of the Thurston metric, both globally and infinitesimally. In Sect. 2.7 we discuss the coarse geometry of the Thurston metric. In Sect. 2.8, we are concerned with the lattice counting problem. In Sect. 2.9, we discuss the shearing coordinates. In Sect. 2.10, we sketch several generalizations of the Thurston metric.

2.2 Background 2.2.1 Teichmüller Space Let S be an orientable closed surface of genus at least two. The Teichmüller space T (S) is the space of equivalence classes of complex structures on S, where two complex structures X and Y are said to be equivalent if there exists a conformal map .X → Y which is homotopic to the identity map. By the Uniformization Theorem, the Teichmüller space .T (S) is also the space of equivalence classes of hyperbolic structures on S, where two hyperbolic structures X and Y are said to be equivalent if there exists an isometry .X → Y which is homotopic to the identity map. For simplicity, we denote the equivalence class of the complex/hyperbolic structure X by X itself. .

2.2.2 Geodesic Laminations and Measured Geodesic Laminations Let .X ∈ T (S) be a hyperbolic surface. A geodesic lamination .λ on X is a closed subset of X which can be decomposed as a disjoint union of simple geodesics. Typical examples are simple closed geodesics. A multicurve is a disjoint union of distinct simple closed geodesics. Let .λ be a geodesic lamination. Then • .λ is called maximal if it cuts X into ideal triangles, i.e. .X\λ is a union of ideal triangles; • .λ is called chain-recurrent if it is the Hausdorff limit of multicurves. A chain-recurrent geodesic lamination is said to be maximal if it is not contained in any other chain-recurrent geodesic lamination. In the sequel, there will be no confusion between these two notions of maximality, because of the context. One can equip the set of geodesic laminations on X with the Hausdorff metric and the corresponding Hausdorff topology. With this topology, the set of chain-recurrent geodesic laminations is a closed subset of the space of geodesic laminations [74, Proposition 6.2].

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H. Pan and W. Su

Notice that the space of geodesic laminations is independent of the choice of hyperbolic metrics in the following sense. Let .X' ∈ T (S) be another hyperbolic surface. There is a natural one-to-one correspondence between the space of (resp. chain-recurrent) geodesic laminations on X and the space of (resp. chain-recurrent) geodesic laminations on .X' . Let .λ be a geodesic lamination. A transverse measure .μ on .λ is an assignment of a Radon measure .dμτ on each arc .τ transverse to .λ, subject to the following conditions: • if the arc .τ ' is a subarc of .τ then . dμτ ' = dμτ |τ ' , • if two arcs .τ and .τ are homotopic through a family of arcs transverse to .λ, then the homotopy sends .dμτ ' to .dμτ . A geodesic lamination .λ with a transverse measure .dμ is called a measured geodesic lamination. Typical examples of measured laminations are weigthed simple closed curves and weighted multicurves. Usually, we will simply denote a measured geodesic lamination .(λ, dμ) by .λ. For any simple closed curve .γ , the intersection number .i(λ, γ ) is defined as:  i(λ, γ ) := inf

.

γ'

dμ

where .γ ' ranges over all simple closed curves homopotic to .γ . As we mentioned above, the space of geodesic laminations is independent of the choice of the hyperbolic metrics. So is the space of measured geodesic laminations. Let .ML(S) be the space of measured geodesic laminations, equipped with the weak-.∗ topology: a sequence of measured laminations .λn converges to .λ if .i(λn , γ ) → i(λ, γ ) for every simple closed curve .α. With this topology, .ML(S) is homeomorphic to .R6g−6+2n , where g and n are respectively the genus and the number of punctures of S (see for instance [58]). Let .PML(S) := ML(S)/R+ be the space of projective classes of measured laminations. The set of weighted simple closed curves is dense in both .ML(S) and .PML(S).

2.2.3 The Thurston Metric and the Maximally Stretched Lamination For any two hyperbolic surfaces X and Y in .T (S), the extremal Lipschitz constant from X to Y is defined as Lip(X, Y ) := inf{L(f )}

.

f

where .L(f ) is the Lipschitz constant of f and where f ranges over all Lipschitz homeomorphisms from X to Y in the homotopy class of the identity map. The

2 The Geometry of the Thurston Metric: A Survey

11

Thurston metric .dT h is defined as dT h (X, Y ) := log Lip(X, Y )

(2.1)

.

Thurston [74, Theorem 8.5] gave another characterization of this metric in terms of simple closed curves: dT h (X, Y ) = log sup

.

α

𝓁α (Y ) 𝓁α (X)

(2.2)

where .𝓁α (·) represents the length of the (unique) geodesic representative of .α and where .α ranges over all homotopically nontrivial simple closed curves. Since weighted simple closed curves are dense in .ML(S), one can rewrite the above identity as: dT h (X, Y ) = log

.

max

𝓁α (Y )

α∈ML(S) 𝓁α (X)

.

In general, the measured laminations realizing the maximal ratio of length are not unique. For instance, if .λ is a non-uniquely ergodic measured lamination which realizes the maximal ratio of length from X to Y , then any measured lamination with the same support as .λ also realizes the maximal ratio from X to Y . Nevertheless, Thurston introduced the notion of maximal ratio-maximizing chainrecurrent geodesic lamination (the maximally stretched lamination for short) from X to Y , which is unique. Given X and Y in .T (S), the maximally stretched lamination .Λ(X, Y ) from X to Y is the largest chain recurrent geodesic lamination .λ with following property: there exists an .exp(dT h (X, Y ))-Lipschitz map, homotopic to the identity map, from a neighbourhood of .λ on X to a neighbourhood of .λ on Y which takes leaves of .λ on X to corresponding leaves of .λ on Y by multiplying arc length by a factor of .exp(dT h (X, Y )). It is also the union of all chain recurrent geodesic laminations with the aforementioned property [74, Theorem 8.2]. Equivalently, the maximally stretched lamination is also the union of chain-recurrent laminations to which the restriction of every Lipschitz map from X to Y with the extremal Lipschitz constant, which is homotopic to the identity, takes leaves of .λ on X to corresponding leaves of .λ on Y by multiplying arclength by a factor of .exp(dT h (X, Y )). (For the equivalence between these two descriptions, we refer to [28, Section 9], see also [17, Section 5] for a related discussion.) Regarding the behavior of maximally stretched laminations, we have: Theorem 2.2.1 ([74], Theorem 8.4) Let .X, Y ∈ T (S) be any two distinct hyperbolic surfaces. If .Xi and .Yi are sequences of hyperbolic surfaces converging to X and Y respectively, then .Λ(X, Y ) contains any lamination in the limit set of .Λ(Xi , Yi ) in the Hausdorff topology.

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Any Lipschitz map f from X to Y realizing the extremal Lipschitz constant Lip(X, Y ) is called an extremal Lipschitz map. If moreover, f maximally stretches exactly along the maximally stretched lamination .Λ(X, Y ), then it is called an optimal Lipschitz map from X to Y . Before closing this subsection, let us mention the following result:

.

Theorem 2.2.2 ([74], Theorem 8.5) Let .X, Y ∈ T (S). (1) The extremal Lipschitz constant .Lip(X, Y ) can always be realized by a homeomorphism. (2) There exists a Thurston geodesic from X to Y . Remark 2.2.3 (i) For various constructions of “extremal Lipschitz maps” and Thurston geodesic rays, see Sect. 2.3. (ii). In general, there may be more than one geodesic from X to Y . The union of all such geodesics is called the envelope from X to Y . For more information about envelopes, we refer to [6, 19], and [48].

2.2.4 The Thurston Norm Let .X ∈ T (S). The Thurston norm on the tangent space .TX T (S) is defined as: ‖v‖Th :=

.

sup

λ∈PML(S)

(dX log 𝓁λ )[v],

∀ v ∈ TX T (Sg,n ),

(2.3)

where .PML(S) = ML(S)/R>0 is the space of projective measured laminations and .ML(S) is the space of measured laminations on S. The Thurston norm is the infinitesimal norm of the Thurston metric ([74, p. 20], see also [53, Theorem 2.3]). The unit sphere in the cotangent bundle has a very nice description as follows. Theorem 2.2.4 ([74], Theorem 5.1) For any hyperbolic surface X of finite type, the map .

d log 𝓁 : PML(S) −→ TX∗ T (S) [μ] I−→ dX log 𝓁μ

embeds .PML(S) as the boundary of a convex neighbourhood of the origin. This convex neighbourhood is dual to the unit ball .{v ∈ TX T (S) : ‖v‖Th ≤ 1}. There are flat places on the unit sphere .TX1 T (S) of .TX T (S). A facet is a maximal flat portion of .TX1 T (S) which has maximum possible dimension so that it has interior.

2 The Geometry of the Thurston Metric: A Survey

13

Theorem 2.2.5 ([74], Theorem 10.1) There is a bijection between the set of facets on the unit sphere .TX1 T (S) and the set of simple closed curves. In other words, every facet is contained in a plane .dX log 𝓁α = 1 for some simple closed curve .α. For each simple closed curve .α, let FX (α) := {v ∈ TX1 T (S) : (dX log 𝓁α )(v) = 1}

.

be the corresponding facet obtained in Theorem 2.2.5. Lemma 2.2.6 ([47], Lemma 4.2; [34], Corollary 6.13) Let .α, β be two simple closed curves. Then i(α, β) = 0 ⇐⇒ ∂FX (α) ∩ ∂FX (β) /= ∅.

.

The unit sphere of the Thurston norm in the tangent space encodes not only the intersection patterns of simple closed curves, but also their hyperbolic lengths. In [19], Dumas–Lenzhen–Rafi–Tao proved that one can recognize lengths and intersection numbers of curves on X from the lengths of facets in the unit sphere in .TX T (S1,1 ), based on the observation that extremal points of any facet in the unit sphere in .TX T (S1,1 ) are exactly those vectors tangent to the Thurston stretch lines directed by the two canonical completions of the corresponding simple closed curve. This idea is further exploited and extended to surfaces of higher complexity by Huang–Ohshika–Papadopoulos [34]. Huang–Ohshika–Papadopoulos first observed that stretch vectors with respect to maximal chain-recurrent laminations necessarily arise as the Hausdorff limit of a sequence of shrinking faces of the unit spheres in .TX T (S), and then proceeded to prove that one can recognize lengths of simple closed curves on X using such stretch vectors. Remark 2.2.7 (i) In his thesis, Bar-Natan [6] proved that the set of tangent vectors to the stretch lines corresponding to completions of maximal chain-recurrent laminations is precisely the set of extreme points in the unit sphere of .TX T (S). (ii) For more information about the unit sphere in .TX T (S), we refer to [34]. (iii) For the comparison of the Thurston norm with the Teichmüller norm, the Weil–Petersson norm, and the earthquake norm, we refer to [33].

2.3 Extremal Lipschitz Maps and Geodesic Rays In this section, we shall sketch various constructions of extremal Lipschitz maps between hyperbolic surfaces, and the corresponding geodesic rays in the Teichmüller space.

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2.3.1 Deforming Polygons via Explicit Homeomorphisms One way of constructing extremal Lipschitz maps between surfaces is to consider a pair of orthogonal measured (partial) foliations and to control the effect of the map on the leaves of these foliations. Following Thurston’s idea, one can deform hyperbolic surfaces by cutting the surface into polygons, deforming each polygon individually, and then gluing the deformed polygons to get new surfaces. One of the advantages of this construction is that the involved deformations/homeomorphisms can be explicitly written down. In the following, we shall introduce this idea with more details, starting with Thurston’s original construction.

2.3.1.1

Thurston’s Construction

Let X be a complete hyperbolic surface of finite area. Let .λ be a maximal geodesic lamination on X, that is, the complementary region .X\λ is the union of ideal triangles. Suppose further that .λ admits at least one leaf which does not go to a cusp on both sides. Let .Δ be an arbitrary ideal triangle in .X\λ. Notice that each corner of .Δ can be foliated by horocycles. Extend these foliations until they fill all but the region in the center bounded by three horocycles. Let F be the extended partial foliation, called the horocycle foliation of .Δ. Let G be the partial foliation whose leaves are geodesics rays orthogonal to the leaves of F . Consider the .et Lipschitz (self)homeomorphism .ft : Δ → Δ which fixes the central region, and maps a horocycle which has distance r from the central region to the horocycle with distance .et r linearly with respect to arc length on the horocycles, see Fig. 2.1. In particular, .ft sends leaves of F to (other) leaves of F and keeps each leaf of G invariant (as a set). Thurston proved that the collection of pullback metrics on .X\λ by all such .{ft } extends to a new hyperbolic metric on X, denoted by .stretch(X, λ, t).

ft

r

et r

Fig. 2.1 In both pictures, the black arcs are leaves of the horocycle foliation F while the red dashed arcs are leaves of G

2 The Geometry of the Thurston Metric: A Survey

15

Theorem 2.3.1 ([74]) For any complete hyperbolic surface X of finite area, for any maximal geodesic lamination .λ not all of whose leaves go to a cusp on both sides, there exists a new hyperbolic surface .stretch(X, λ, t) depending analytically on .t > 0, such that (1) the identity map .X → stretch(X, λ, t) is Lipschitz with Lipschitz constant .et , and (2) the identity map exactly expands arc length of .λ by the constant factor .et . The family .{stretch(X, λ, t) : t > 0} is called the Thurston stretch ray directed by .λ. It is a geodesic under the Thurston metric. For the proof of Theorem 2.3.1, we refer to [74, Section 4] and [54, Chapter 2, Section 3.5]. Remark 2.3.2 Thurston stretch lines are rare in the sense that at any point .X ∈ T (S), the set of unit vectors tangent to Thurston stretch lines through X is of Hausdorff dimension zero [74, Theorem 10.5]. However, for any pair of distinct points in .T (S), there exists a geodesic from the first point to the second point which is a concatenation of finitely many stretch segments [74, Theorem 8.5] Remark 2.3.3 Calderon and Farre [13, Lemma 15.11 and Proposition 15.12] generalized Thurston’s construction to the case where .X\λ is a union of regular ideal polygons.

2.3.1.2

The Construction of Papadapoulos-Théret

A right-angled hyperbolic hexagon is said to be symmetric if it has three nonadjacent edges of equal length. Using symmetric right-angled hyperbolic hexagons, Papadopoulos and Théret [55] constructed the first examples of Thurston geodesics which are also geodesics in the reversed direction (up to reparametrization). Their construction starts by constructing extremal Lipschitz homeomorphisms between symmetric right-angled hyperbolic hexagons with controlled Lipschitz constant. By doubling the hexagons, they then obtain extremal Lipschitz homeomorphisms between symmetric hyperbolic pairs of pants, that is, hyperbolic pairs of pants which have three geodesic boundary components of equal length. By gluing pairs of pants along their boundary components without twist and by combining the resulting Lipschitz maps between pairs of pants, they obtain new stretch lines in the Teichmüller space of any hyperbolic surface of finite type which are also geodesics (up to reparametrization) in the opposite direction. The Lipschitz homeomorphisms between symmetric right-angled hyperbolic hexagons is constructed as follows. Consider a symmetric right-angled hyperbolic hexagon H . There are two triples of pairwise non-consecutive edges, one of which is called long with common length L while the other is called short with common length l. Similarly as in the case of ideal triangles, one can equip H with two orthogonal partial measured foliations .(F, G) defined as follows. The leaves of F are the loci of equidistant points from the short edges. These leaves foliate H except a triangle-shaped central region. The transverse measure on F is induced by

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H. Pan and W. Su

ft

r

et r

Fig. 2.2 Each of the hexagons has three long sides of equal length (red) and three short sides of equal length (black). The black arcs are leaves of F , which are loci of equidistant points from the short edges. The red dashed arcs are leaves of G, which are geodesics orthogonal to the leaves of F

the arclength of the long edges of H . The leaves of G are geodesics orthogonal to the leaves of F and the transverse measure of G is induced by the arclength of short edges of H (see Fig. 2.2) Now Let .Ht be a family of symmetric right-angled hyperbolic hexagons with long edges being of length .t > 0. For any .t > 1, Papadopoulos and Théret constructed a homeomorphism .ft : H → Ht such that • .ft sends the central region on H to the central region on .Ht , • .ft sends affinely the leaves of F at distance r from the central region on H to the leaves at distance tr from the central region on .Ht , • .ft sends leaves of G on H to the corresponding leaves of G on .Ht by multiplying arclength by the factor t, • .ft is a t-Lipschitz homeomorphism. Doubling the symmetric right-angled hexagons along short edges and then gluing the resulting pairs of pants without twist, Papadopoulos and Théret constructed a family of hyperbolic surfaces .{Xt }t>0 which is a Thurston geodesic in the forward direction. Notice that the set of short edges of hexagons are glued together to form a pants decomposition of .Xt while the set of long edges are glued together to form several closed geodesics orthogonal to the aforementioned pants decomposition. Interchanging the role of short edges and long edges, we see that .{Xt }t>0 is also a Thurston geodesic in the backward direction.

2.3.1.3

The Construction of Papadopoulos-Yamada

Starting with an arbitrary right-angled hyperbolic hexagon, Papadopoulos and Yamada constructed a one-parameter family of right-angled hexagons with a Lipschitz map between any two elements in this family, realizing the smallest

2 The Geometry of the Thurston Metric: A Survey

17

Lipschitz constant in the homotopy class of this map relative to the boundary [56]. Let H be an arbitrary right-angled hexagon. Notice that there are two triples of nonconsecutive edges on H , one of which called long while the other is called short. Let .α1 , α2 , α3 be the three long edges of H . Considering the equidistant curves to the long edges of H and the corresponding orthogonal geodesic arcs, Papadopoulos and Yamada constructed a pair of orthogonal measured (partial) foliations .(F, G) on H , which extends the construction in [51]. (The construction here is more involved depending on whether the lengths of short edges satisfy the triangle inequality or not, see [56, Section 2].) Let .Ht be a family of right-angled hyperbolic hexagons such that .

cosh 𝓁Ht (αi ) = t. cosh 𝓁H (αi )

Let ki (t) :=

.

𝓁Ht (αi ) , 𝓁H (αi )

k(t) = max{ki (t)}.

For any .t > 1, Papadopoulos and Yamada constructed a homeomorphism .ft : H → Ht such that • • • •

ft ft .ft .ft . .

sends the central region on H to the central region on .Ht , linearly sends the leaves of F on H to the corresponding leaves on .Ht , (nonlinearly) sends leaves of G on H to the corresponding leaves of G on .Ht , is a .k(t)-Lipschitz homeomorphism.

Notice that each long side .αi is a leaf of F , and hence is stretched under f by the factor .ki (t). Unlike the construction of Thurston and Papadopoulos-Théret, where the constructed homeomorphism linearly stretches the leaves of G with a common factor and linearly shrinks the leaves of F , the map here linearly stretches leaves of F with varying factors and non-linearly shrinks leaves of G. In particular, the map .f : H → Ht is linear on each of the three long sides of H but not linear on any of the three short sides of H . Now we consider an arbitrary hyperbolic surface X of genus g with b geodesic boundary components. Let .{γi }1≤i≤6g−6+3n be a collection of pairwise disjoint geodesic arcs orthogonal to the geodesic boundary components of X, i.e. .{γi }1≤i≤6g−6+3n is a triangulation of X. The complementary region .X\(∪i γi ) is a union of right-angled hexagons .{H i }. Viewing those .{γi } as long sides of .H i and applying the previously constructed Lipschitz homeomorphisms to .H i , we arrive at a family of hyperbolic surfaces .Xt and a family of Lipschitz homeomorphisms .ft : X → Xt attaining the optimal Lipschitz constant on some of the .{γi }. Doubling .Xt along the geodesic boundary components, we get a family of closed hyperbolic surfaces .{Xtd } and a family of Lipschitz homeomorphisms .ftd : Xd → Xtd attaining the optimal Lipschitz constant on some of the double of .{γi }.

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2.3.1.4

The Construction of Huang-Papadopoulos

Since the optimal Lipschitz map is not as rigid outside the maximally stretched loci, one may hope to construct Thurston geodesics which keep some non-trivial subsurface invariant. This is partially done by Huang and Papadopoulos under some conditions. Extending the previous constructions to ideal Saccheri quadrilaterals, Huang and Papadopoulos [32, Theorem 3.3 and Theorem 4.7] constructed, for any complete hyperbolic torus .T with one boundary component and any chain-recurrent geodesic lamination .λ in the convex hull of .T , a family of complete hyperbolic tori .{Tt }t≥1 and a family of Lipschitz homeomorphisms .ft : T → Tt such that • • • •

the convex hull of .Tt has the same boundary length as that of .T, ft is t-Lipschitz and multiplies the arclength of .λ by the factor t, .ft is an isometry outside a compact neighbourhood of the convex hull of .T, if the boundary length of the convex hull of .T is at most .4arcsinh(1), .ft maps the convex hull of .T onto the convex hull of .Tt . .

In particular, for any hyperbolic torus T with one geodesic boundary component of length at most .4arcsinh(1), the construction above gives a family of hyperbolic tori .{Tt }t≥1 , each of which has exactly one geodesic boundary component, and a family of Lipschitz homeomorphisms .ft : T → Tt such that • .Tt has the same boundary length as that of T , • .ft is t-Lipschitz, multiplies the arclength of .λ by the factor t, and maps the boundary of T linearly to the boundary of .Tt by the factor 1. Consider a hyperbolic surface X with punctures or geodesic boundary components. If X contains a hyperbolic torus T with one geodesic boundary component, whose boundary length is at most .4arcsinh(1), then for any chain-recurrent geodesic lamination in T the above construction gives a family of hyperbolic surfaces .Xt and a family of Lipschitz homeomorphisms .ft : X → Xt such that • .ft is t-Lipschitz, and multiplies the arclength of .λ; • .ft is an isometry on the closure of .X\T .

2.3.2 Deforming Crowned Surfaces via Harmonic Diffeomorphisms Using harmonic maps of high energy, Pan and Wolf constructed piecewise harmonic stretch lines and harmonic stretch lines [49]. Recall that a differentiable map .f : (M, σ |dz|2 ) → (N, ρ|dw|2 ) between Riemannian surfaces is said to be harmonic if it satisfies the Euler–Lagrange equation: fz¯z + (log ρ)w fz fz¯ = 0.

.

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If M and N are compact, then f is harmonic if and only if it is a critical point of the energy functional  E(f ) :=

e(f )σ |dz|2

.

M (z)) 2 2 where .e(f ) := ρ(f σ (z) (|fz | + |fz¯ | ) is the energy density of f . Notice that the energy depends on the conformal structure on M and the metric on N . The basic existence result of harmonic maps was established by Eells and Sampson in [20] and by Hamilton in [30], if the target manifold has nonpositive sectional curvature. The uniqueness was obtained by Al’ber [1] and Hartman [31] if the target manifold has negative sectional curvature and if the image is not contractible to a point or a geodesic. Moreover, Sampson [61] and Schoen-Yau [66] proved that any harmonic map between compact surfaces which is homotopic to a diffeomorphism is a diffeomorphism, provided that the target surface has nonpositive curvature. Suppose that f is harmonic. Consider the pullback of .ρ by f :

f ∗ (ρ) = ρfz fz¯ dz2 + e(f )σ dzd z¯ + ρfz fz¯ d z¯ 2 .

.

The .(2, 0)-part of .f ∗ (ρ) is called the Hopf differential of f . The harmonicity of f implies that the Hopf differential of f is holomorphic (see [36, 65]). If we choose the coordinate .z = x + iy such that the Hopf differential .Ф = dz2 and choose .σ to be the singular flat metric induced by .|Ф|, then .f ∗ (ρ) can be simply expressed as f ∗ ρ = (e + 2)dx 2 + (e − 2)dy 2 .

.

By [43, Lemma 3.2 and Lemma 3.3], the energy density satisfies .e(z) = 2+O(e−2r ) where r represents the distance from z to the zeros of the Hopf differential. In particular, for points far away from the zeros of the Hopf differential, the pullback metric .f ∗ ρ is nearly .4dx 2 . Roughly speaking, at regions far away from the zeros of the Hopf differential, the harmonic map f linearly maps horizontal leaves of the Hopf differential to hyperbolic geodesics by a factor of 2 while contracting the vertical leaves exponentially. This observation is the starting point of the construction of Pan and Wolf. Theorem 2.3.4 ([49], Theorem 1.7) Let .Y ∈ T (S) be any closed hyperbolic surface, and let .λ be any geodesic lamination. Then for any surjective harmonic diffeomorphism .f : X → Y \ λ from some (possibly disconnected) punctured surface X, there is a family of new hyperbolic surfaces Yt := stretch(Y, λ, f ; t) ∈ T (S)

.

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depending analytically on .{t > 0} such that (a) the identity map .ft : X → Yt \ λ is a surjective harmonic map .ft : X → Yt \ λ with Hopf differential .tHopf(f ); √ −1 (b) for any .0 < s < t, the identity map (.f√ t ◦fs ) is . t/s-Lipschitz with (pointwise) Lipschitz constant strictly less than √ . t/s in .S − λ, and exactly expands arc length of .λ by the constant factor . t/s. , f; t) constructed above is A family of hyperbolic structures .stretch(X, λ; Ф called a piecewise harmonic stretch line. It admits a canonical orientation coming from the orientation of the positive real ray .{t > 0}. With that orientation, a piecewise harmonic stretch line is a (reparametrized) geodesic in the Thurston metric. Whenever we say a piecewise harmonic stretch line, we mean a directed line.

2.3.3 The Constructions of Guéritaud-Kassel and Alessandrini-Disarlo Using a totally different method, Guéritaud and Kassel extended Thurston’s construction to a geometrically finite setting and to higher dimension. Let .𝚪0 be a discrete group. For a pair .(j, ρ) of representations of .𝚪0 into .PO(n, 1) = Isom(Hn ) with j geometrically finite. Guéritaud and Kassel investigate the set of .(j, ρ)equivariant Lipschitz maps from .Hn to itself having the minimal Lipschitz constants (such maps do exists if .ρ is reductive [28, Lemma 4.10]), establish the existence of maximally stretched locus .E(j, ρ) (which is a geodesic lamination if the minimal Lipschitz constant is at least one), and prove the existence of optimal Lipschitz maps whose stretched locus is exactly the maximally stretched locus .E(j, ρ) of the representations .(j, ρ) [28, Theorem 1.3]. This allows Guéritaud and Kassel to study proper actions of discrete subgroups of .PO(n, 1) × PO(n, 1) on .PO(n, 1) by left and right multiplication [28, Theorem 1.8 and Theorem 1.9]. One of the key ideas of Guéritaud and Kassel is to consider the “average of Lipchitz maps”. This idea is used by Alessandrini and Disarlo to construct extremal Lipschitz maps between hyperbolic surfaces with geodesic boundary components [2]. We now sketch the construction of Alessandrini and Disarlo [2]. Let X be a finitearea hyperbolic surface with geodesic boundary components. A geodesic lamination .λ on X is called maximal if any connected component of its complementary region .X\λ is either an ideal triangle, a right-angled quadrilateral with two consecutive ideal vertices, a right-angled pentagon with one ideal vertex, or a right-angled hexagon. These four types of polygons are called geometric pieces. For any maximal geodesic lamination on X, Alessandrini and Disarlo constructed a family of hyperbolic surface .Xλt homeomorphic to X and a family of generalized stretch maps .Ψ t : X → Xλt which realizes the minimal Lipschitz constant from X to .Xλt and maximally stretches along .λ. Their construction has three steps.

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Step 1: Generalized stretch map between geometric pieces. Combine Thurston’s stretch maps and Guéritaud-Kassel’s idea of averaging Lipschitz maps to construct implicit extremal maps between geometric pieces of the same type which stretch along certain sides. However, it is unclear whether the resulting maps between geometric pieces are homeomorphic or not. So one can’t simply “glue” these new pieces and the corresponding stretch maps. To overcome this issue, Alessandrini and Disarlo adapted a different idea in the remaining steps. Step 2: Decomposition of X. Let .B ⊂ X be the union of geometric pieces of .X\λ which are not ideal triangles. The surface B is a crowned hyperbolic surface. Let .C ⊂ B be the union of crown ends of B and set .BC = B\C ⊂ B. Finally, let .XC := X\BC ⊂ X. Step 3: Define .Xλt and the generalized stretch map .ψ t : X → Xλt . For each t t t .t ≥ 0, Alessandrini and Disarlo construct hyperbolic surfaces .B , B , X C C homeomorphic to .B, BC , XC respectively. The new surfaces come with preferred (locally) isometric embeddings .ξ t : C t → B t and .ht : C t → BCt . The new surface .Xλt is then defined to be Xλt := (B t ∪ BCt )/ ∼,

.

where .ξ t (z) ∼ ht (z) for all .z ∈ C t . The generalized stretch map .Ψ t : X → Xλt is defined by glueing together suitable generalized stretch maps .β t : B → B t and t t .ψ from an open dense subset of .BC to .X . (For the details of the construction, C we refer to [2, Section 6.2, 7.4, 8.1 and 8.2].) Remark 2.3.5 (1) It is unclear whether the generalized stretch maps are homeomorphisms or not. (2) Based on the generalized stretch maps and the generalized stretch lines, Alessandrini and Disarlo proved that the Teichmüller space of bordered hyperbolic surfaces with the arc metric is geodesic, and provided new geodesics of the Thurston metric in the Teichmüller space of closed surface.

2.3.4 The Construction of Daskalopoulos-Uhlenbeck In [16, 17], Daskalopoulos and Uhlenbeck constructed extremal Lipschitz maps between manifolds using infinity harmonic maps. Here we sketch the construction of [17], which works more generally than hyperbolic surfaces. Let .(M, g) be a compact Riemannian manifold with boundary .∂M (possibly empty) and .dim(M) = n. Let .(N, h) be a closed Riemannian manifold of nonpositive sectional curvature and let .W 1,p (M, N ) denote the Soblev space of maps p .f : M → N such that f and its weak derivative have a finite .L norm. For any map f : M → N ∈ W 1,p (M, N ) ∩ C 0 (M, N ),

.

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define the .Jp -functional of f as  Jp (f ) :=

Tr(Q(df )p ) ∗ 1,

.

M

where .1 ≤ p < ∞ and where .Q(df )2 = df df T is a non-negative symmetric linear map mapping the tangent space T N to itself. The integrand .Tr(Q(df )p ) is essentially the sum of the p-th powers of the singular eigenvalues of df . The EulerLagrange equations of the .Jp functional are D ∗ Q(df )q−2 (df )2 = 0,

.

where .D = Df is the pullback of the Levi-Civita connection on .f −1 T N . If the domain manifold .(M, g) has nonempty boundary, there are two types minimizing problems: the Dirichlet problem and the Neumann problem. For the Dirichlet problem we fix a continuous map .f : M → N and seek a minimizer in the homotopy class of f relative to the boundary values of f . For the Neumann problem we only fix a homotopy class and no boundary condition at all. Here is the existence theorem concerning the .Jp functional. Theorem 2.3.6 ([17], Theorem 1.1) Let .(M, g) be a compact Riemannian manifold with boundary .∂M (possibly empty) and .dim(M) = n. Let .(N, h) be a closed Riemannian manifold of non-positive sectional curvature. Then for each .p > N, there exists a minimizer f in .W 1,p (M, N ) of the functional .Jp with either the Dirichlet or the Neumann boundary conditions in a homotopy (relative homotopy) class. Furthermore, if .∂M /= ∅, the solution of Dirichlet problem is unique. The next result is about the construction of extremal Lipschitz maps using .Jp harmonic maps. Recall that a Riemannian manifold M with boundary is said to be convex if it can be isometrically embedded as a convex subset of a manifold of the same dimension without boundary. Theorem 2.3.7 ([17], Theorem 1.4) Let .M, N be Riemannian manifolds where M is convex with possibly non-empty boundary and N is closed and has non-positive curvature. Given a sequence .p → ∞, there exists a subsequence (denoted also by p) and a sequence of .Jp minimizers .fp : M → N in the same homotopy class (either in the absolute sense or relative to the boundary depending on the context) such that f = lim fp ∈ W 1,s (M, N ) for all s.

.

p→∞

Furthermore, .fp → f uniformly and f is an extremal Lipschitz map in the (relative) homotopy class. A critical point of the .Jp functional (with either the Dirichlet problem or the Neumann problem) is called a Schatten-von Neumann p-harmonic map or simply

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a .Jp harmonic map . The limit map obtained in Theorem 2.3.7 is called infinity harmonic map. About the regularity, Daskalopoulos and Uhlenbeck conjecture that for .p > 2, any .Jp harmonic map between hyperbolic surfaces is of .C 1,α [17, Conjecture 4.19]. We now return to the hyperbolic surface setting. Let .f : M → N be a Lipschitz map between hyperbolic surfaces M and N . Here M and N are not necessary homeomorphic. Suppose further that the minimal Lipschitz constant in the homotopy class of f is at least one.1 In this setting, the maximally stretched locus, i.e. the intersection of maximally stretched loci of all extremal Lipschitz maps from M to N in the given homotopy class, is a chain recurrent geodesic lamination. In particular, the maximally stretched loci of the infinity harmonic map obtained in Theorem 2.3.7 contains the maximally stretched lamination. For the inverse direction, they proved: Theorem 2.3.8 ([17], Theorem 1.7) If the maximally stretched lamination .λ for a homotopy class of maps .M → N consists of a finite number of simple closed geodesics, then the maximally stretched loci of the infinity harmonic map obtained in Theorem 2.3.7 is exactly .λ. Daskalopoulos and Uhlenbeck conjectured that Theorem 2.3.8 holds in general [17, Conjecture 8.12]: the maximally stretched loci of any infinity harmonic map, whose Lipschitz constant is at least one, is the maximally stretched lamination [17, Conjecture 8.12]. Furthermore, they conjectured that the limit infinity harmonic maps are not homeomorphisms [17, Conjecture 8.5].

2.3.5 Concatenations of Geodesic Segments In the previous subsections, we discussed the constructions of various geodesic rays. Thurston constructed another type of geodesics, that is, concatenations of Thurston stretch segments [74, Theorem 8.5]. This construction applies to other segments as well.

2.4 Harmonic Stretch Lines In Sect. 2.3, we discussed various constructions of geodesic rays under the Thurston metric. In this section, we will focus on a special type of piecewise harmonic stretch lines, namely, the harmonic stretch lines. Throughout this section, we assume that

1 If the minimal Lipschitz constant is less than one, then the maximally stretched loci may not be a geodesic lamination [28, Section 9.4]. In fact, Guéritaud and Kassel conjectured that in this case the maximally stretched loci is a “gramination” (contraction of “graph” and “lamination”) [28, Conjecture 1.4].

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the underlying topological surface S is a closed orientable surface with genus at least two.

2.4.1 Harmonic Stretch Lines Recall that the construction of piecewise harmonic stretch lines through .Y ∈ T (S) depends on the choice of a geodesic lamination .λ and a surjective harmonic diffeomorphism .f : X → Y \λ from some (possibly disconnected) punctured surface X. If X has several components, the harmonic map f can be prescribed independently on each component. If we impose some additional “compatibility conditions” on the harmonic maps from distinct components, we arrive at the harmonic stretch lines. Originally, the harmonic stretch lines are defined via harmonic map rays, of which we now recall the definition. Let .X ∈ T (S) and let .Ф be a holomorphic quadratic differential on X. The harmonic map ray determined by X and .Ф is the set of hyperbolic surfaces .{Yt }t>0 such that the Hopf differential of harmonic map from X to .Yt is .tФ [77]. A piecewise harmonic stretch line is said to be a harmonic stretch line if it arises as a limit of some sequence of harmonic map rays in .T (S). The basic theorem about those harmonic stretch lines is the following. Theorem 2.4.1 ([49], Theorem 1.8) For any two distinct hyperbolic surfaces Y, Z ∈ T (S), there exists a unique harmonic stretch line proceeding from Y to Z.

.

Actually, one can say more about the relationship between harmonic map rays and Thurston geodesic rays. To that end, we need another definition, the harmonic map dual rays. Let .X ∈ T (S) and let .λ be a measured foliation on X. The harmonic map dual ray determined by X and .λ is the of hyperbolic surfaces such that the horizontal foliation of the Hopf differential of the harmonic map from .Xt to X is .tλ (for the existence and uniqueness of such a .Xt , see [78, Theorem 3.1]). For any two hyperbolic surfaces .X' , X, we denote by .HR(X' , X) the harmonic map which starts at .X' and passes through X. Theorem 2.4.2 ([49], Theorem 1.1) Let .X ∈ T (S) be a hyperbolic surface and .λ a measured lamination. Then the family of harmonic map rays .HR(Xt , X) converge to a Thurston geodesic locally uniformly as .Xt diverges along the harmonic map dual ray determined by X and .λ. Remark 2.4.3 The conclusion still holds if we replace the harmonic map dual ray by a Teichmüller ray [49, Theorem 1.2]. Similar results also hold for harmonic map dual rays. We fix .X ∈ T (S) and a holomorphic quadratic differential .Ф on X. Suppose that .Yt diverges along the harmonic map ray determined by X and .Ф. Then the family of harmonic map dual rays, each of which starts at .Yt and passes through X, converges to the Teichmüller geodesic determined by X and .Ф locally uniformly [49, Theorem 1.4].

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If we remove the assumption that .Xt diverges along a harmonic map ray, then we have the following sub-convergence result: Theorem 2.4.4 ([49], Theorem 1.3) For any fixed .X ∈ T (S), let .Xn ∈ T (S) be an arbitrary divergent sequence with .Xn → ∞. Then the sequence of harmonic map rays .HR(Xn , X) contains a subsequence which converges to some Thurston geodesic locally uniformly. Next, we extend the existence/uniqueness theory of harmonic stretch lines to rays whose terminal point is a projective measured lamination, representing a point on the boundary of the Thurston compactification of the Teichmüller space .T (S). Theorem 2.4.5 ([49], Theorem 1.11) For any .Y ∈ T (S) and any .[η] ∈ PML(S), there exists a unique harmonic stretch ray starting at Y , which converges to .[η] ∈ PML(S) in the Thurston compactification. Moreover, these rays foliate .T (S) if we fix Y and let .[η] vary in .PML(S), or if we fix .[η] and let Y vary in .T (S). Finally, let us mention the following continuity property, where .HSL(X, X' ) represents the harmonic stretch line proceeding from .X ∈ T (S) through .X' ∈ T (S) ∪ PML(S) Proposition 2.4.6 ([49], Proposition 12.14 and Proposition 13.8) Let Y and Z be two distinct points in .T (S). Let .[η] ∈ PML(S). (i) Suppose that .Yn → Y and .Zn → Z in .T (S), then the sequence of harmonic stretch lines .HSL(Yn , Zn ) locally uniformly converges to the harmonic stretch line .HSL(Y, Z). (ii) Suppose that .Yn → Y in .T (S) and .[ηn ] → [η] in .PML(S), then the sequence of harmonic stretch lines .HSL(Yn , [η]n ) locally uniformly converges to the harmonic stretch line .HSL(Y, [η]). Remark 2.4.7 Using the identification between .ML(S) and .TY∗ T (S), Theorem 2.4.5 induces a bijection from .TY∗ T (S) to .T (S) which sends rays through the origin in .TY∗ T (S) to harmonic stretch rays in .T (S). (By Proposition 2.4.6 such a bijection is a homeomorphism.) A natural question is to improve Theorem 2.4.5 to an exponential map from .TY T (S) to .T (S). Since the Thurston norm is not strictly convex, this does not follow directly from Theorem 2.4.5. The existence part of the exponential map is clear, see [49, Remark 1.12]. However, the uniqueness part is unknown yet.

2.4.2 Two Versions of the Geodesic Flow for the Thurston metric Theorem 2.4.5 allows us to define the Thurston geodesic flow ψt : T (S) × PML(S) −→ T (S) × PML(S)

.

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such that the orbit through .(Y, [η]) ∈ T (S) × PML(S) is the harmonic stretch line determined by Y and .[η] via Theorem 2.4.5. Moreover, every harmonic stretch line appears as a (forward) orbit. There is another version of the Thurston geodesic flow φt : T (S) × PML(S) → T (S) × PML(S)

.

such that the orbit through .(Y, [η]) ∈ T (S) × PML(S) is the stretch line obtained from Theorem 2.4.2. For a further discussion about those flows, we refer to [49].

2.5 Thurston Boundary The Teichmüller space .T (S) parametrizes hyperbolic structures on the topological surface S. A sequence of hyperbolic surfaces .Xn ∈ T (S) which tends to infinity has the property that some closed curve .γ on S becomes infinitely long, that is, the hyperbolic length .𝓁γ (Xn ) → ∞. The length of long curves can be measured quantitatively by looking at their transverse measures with respect to measured foliations. This observation of Thurston [72] resulted in a natural compactification of .T (S), which is called the Thurston compactification. Recall that a measured foliation .(F, μ) is a foliation F on S equipped with an invariant transverse measure .μ. We require that the singularities of F are similar to the singularities of holomorphic quadratic differentials (or meromorphic quadratic differentials with at most simple pole at the punctures, when S is a punctured surface). For simplicity, we denote .(F, μ) by .μ. For any simple closed curve .γ , the intersection number .i(γ , μ) is defined by  i(γ , μ) = inf'

.

γ

γ'

dμ,

where .γ ' is taken over all simple closed curves homotopic to .γ . Two measured foliations .μ1 , μ2 are said to be equivalent if .i(γ , μ1 ) = i(γ , μ2 ) for all simple closed curves .γ on S. Let .MF be the set of equivalence classes of measured foliations on S. The set of projective equivalence classes of measured foliations is denoted by .PMF(S). As shown by Thurston (see [23]), .PMF(S) is topologically a sphere of dimension .6g − 7 + 2n. Moreover, .PMF(S) is the Thurston boundary of .T (S). By definition, a sequence of hyperbolic surfaces .Xn ∈ T (S) converges to the projective class of .μ if there is some .cn > 0 such that cn 𝓁γ (Xn ) → i(γ , μ)

.

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for all simple closed curves .γ on S. To glue .T (S) together with .PMF(S), we can embed both .T (S) and .PMF(S) into .PR∞ . There are many applications of Thurston’s theory of measured foliations. Below, we will limit ourselves to illustrating some of the results related to the Thurston metric. The following result, first proved by Papadopoulos [50, Theorem 5.1], implies there is direct relation between the Thurston metric and the Thurston boundary: Theorem 2.5.1 Let .λ be a maximal geodesic lamination on .X ∈ T (S) with .μ = Fλ (X) being the transverse horocycle foliation. Then the corresponding Thurston stretch ray converges to the projective class .[μ] ∈ PMF (S). Remark 2.5.2 The problem of the convergence of anti-stretch lines (stretch lines in backward direction) towards the Thurston boundary was studied by Théret [70]. If the anti-stretch line is directed by a complete geodesic lamination .λ which is the completion of a uniquely ergodic measured lamination .μ, then it converges to the projective class of .μ. For general anti-stretch lines, the problem of the convergence or non-convergence remains open. It is true that every geodesic ray of the Thurston metric has a unique limit in the Thurston boundary. This is a corollary of the following result by Walsh [76, Theorem 3.6]. Theorem 2.5.3 The horofunction boundary of .T (S), endowed with the Thurston metric, is homeomorphic to the Thurston boundary. The horofunction compactification and the corresponding boundary of a proper metric space are introduced by Gromov [27]. Remark 2.5.4 An asymmetric metric, called arc metric, is defined on the Teichmüller space of compact hyperbolic surfaces with boundary. It is an analogue of the Thurston metric defined on the Teichmüller space of complete finite area hyperbolic surfaces without boundary. Theorem 2.5.3 was generalized to the arc metric by [3]. Geodesics for the arc metric, which are analogues of Thurston’s stretch lines, were constructed by [2, 49]. By [49, Theorem 14.1], the arc metric coincides with the Lipschitz metric, which is defined using Lipschitz maps similarly as (2.1). Let us briefly mention the work of Bonahon [9] on geodesic currents. Let us endow the surface S with a hyperbolic metric. For simplicity, we assume that S is a closed hyperbolic surface. A geodesic current is a locally finite Radon (positive) measure on the space of geodesics on S. Alternatively, a geodesic current is a finite Radon measure on the unique tangent bundle of S that is invariant under the geodesic flow. Closed geodesics or weighted sums of closed geodesics on S are examples of geodesic currents. Denote the space of geodesic currents on S by .C(S), endowed with the weak.∗ topology. We can embed .MF(S) into .C(S), by representing each measured foliation as a geodesic lamination with an invariant transverse measure. On the other hand,

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every hyperbolic structure .X ∈ T (S) corresponds to a Liouville current. By taking projective classes, the Teichmüller space embeds into .PC(S), and Bonahon [9] proved that the boundary of the image is exactly .PMF(S).

2.6 Isometry Rigidity In this section, we will discuss the rigidity of the Thurston metric, especially the isometry rigidity.

2.6.1 Global Rigidity We start with the global rigidity due to Walsh [76]. Using an identification of the horofunction boundary of the Thurston metric with the Thurston boundary (Theorem 2.5.3), Walsh proved that Theorem 2.6.1 ([76], Theorem 7.8) If S is not a sphere with four or less punctures, nor a torus with two or fewer punctures, then every isometry of .(T (S), dT h ) is an element of the extended mapping class group. Theorem 2.6.1 is an analogue of Royden’s theorem for the Teichmüller metric. Note that the proof of Walsh is geometric. An important step consists of reducing the problem to showing that every isometry of the Thurston metric induces an automorphism of the curve complex. Furthermore, distinct surfaces give rise to distinct Teichmüller spaces: Theorem 2.6.2 ([76], Theorem 7.8) Let .Sg,n and .Sg ' ,n' be orientable surfaces of finite type with negative Euler characteristics. Assume that .{(g, n), (g ' , n' )} is not one of the three sets: {(1, 1), (0, 4)}, {(1, 2), (0, 5)}, {(2, 0), (0, 6)}.

.

If .(g, n) /= (g ' , n' ), then .(T (S), dT h ) and .(T (Sg ' ,n' ), dT h ) are not isometric.

2.6.2 Infinitesimal Rigidity The infinitesimal rigidity of the Thurston metric was first studied by Dumas– Lenzhen–Rafi–Tao [19] for once-punctured tori, then independently by Pan [47] and Huang-Ohshika-Papadopoulos [34] for surfaces of higher complexity.

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Theorem 2.6.3 ([19, 34, 47]) Let S be a surface of finite type with negative Euler characteristic. Let .X, Y ∈ T (S). Then there exists an isometry of normed vector spaces (TX T (S), ‖ • ‖Th ) → (TY T (S), ‖ • ‖Th )

.

if and only X and Y are in the same extended mapping class group orbit. We can also use the Thurston norm to distinguish the topology of the underlying hyperbolic surfaces. Theorem 2.6.4 ([47]) Let S and .S ' be two surfaces of finite type with negative Euler characteristics. Then there exists an isometry of normed vector spaces (TX T (S), ‖ • ‖Th ) → (TY T (S ' ), ‖ • ‖Th )

.

for some .X ∈ T (S) and .Y ∈ T (S ' ) if and only S and .S ' are homeomorphic. Remark 2.6.5 The original statement of Theorem 2.6.4 is slightly differently from the statement here. In [47, Theorem 1.6], the isometry is assumed to be .R-linear. As pointed out in [34, Remark 1.16], this assumption is unnecessary, as isometries of the Thurston norm are also isometries of its (additive) symmetrisation, and the Mazur–Ulam theorem ensures that any isometry of the Thurston norm must be affine, and hence there exists an isometric translate of the affine isometry which is a linear map. Using the infinitesimal rigidity, we have the local rigidity: Theorem 2.6.6 ([19, 34, 47]) Let S be a surface of finite type with negative Euler characteristic. Let .U ⊂ T (S) be a connected open subset, considered as a metric space with the restriction of the Thurston metric. Then any isometric embedding .(U, dT h ) → (T (S), dT h ) is the restriction to U of an element of the extended mapping class group. Intuitively, this says that the quotient of .T (S) by the mapping class group is “totally unsymmetric”: each ball fits into the space isometrically in only one place. As a direct consequence of Theorem 2.6.6, this reproves Theorem 2.6.1 and extends it to the exceptional surfaces: Theorem 2.6.7 ([19, 34, 47, 76]) Let S be a surface of finite type with negative Euler characteristic. Then every isometry of .T (S), dT h ) is induced by an element of the extended mapping class group. To pass from the infinitesimal rigidity to the local rigidity, one needs the following regularity result about the Thurston norm: Theorem 2.6.8 ([19], Theorem 6.1) Let S be a surface of finite type with negative Euler characteristic. Then the Thurston norm function .T T (S) → R is locally Lipschitz.

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2.7 Coarse Geometry In this section, we will discuss the coarse geometry of the Thurston metric.

2.7.1 Short Markings A pants decomposition on S is a collection of mutually disjoint curves which cut S into pairs of pants. A marking .μ on S is a pants decomposition P with additionally a set of transverse curves Q satisfying the following properties. We require each curve .α ∈ P to have a unique transverse curve .β ∈ Q that intersects .α minimally (once or twice depending on whether .α is separating or not) and is disjoint from all other curves in P . We will often say .α and .β are dual to each other, and write .α¯ = β or .β¯ = α. Given .X ∈ T (S), a short marking .μX on X is a marking where the pants decomposition is constructed using the algorithm that picks the shortest curve on X, then the second shortest disjoint from the first, and so on. Once the pants decomposition is complete, the transverse curves are then chosen to be as short as possible. Note that a short marking on X may not be unique, but all short markings on X form a bounded set in the curve complex. Thus, we will refer to .μX as the associated short marking on X.

2.7.2 Curve Graphs Given two curves .α and .β on S, we define their intersection number .i(α, β) to be the minimal number of intersections between any representatives of homotopy classes of .α and .β. Notice that the minimum of intersection numbers for any two distinct (homotopy classes of) simple closed curves is 1 for the once-punctured torus, 2 for the four-holed sphere, and 0 for all other surfaces. The curve graph .C(S) is defined as follows: the vertices are homotopy classes of nontrivial simple closed curves and two vertices are connected by an edge if the corresponding curves can be realized to have minimal possible intersection numbers. We equip .C(S) with a metric by assigning length one to every edge. If S is an annulus, the above definition does not give interesting objects since there is only one homotopy class of simple closed curves on any annulus. We need a different definition. To emphasize the difference, we denote by A the (topological) annulus. By an arc on A we always mean a homotopy class of a simple arc .ω connecting the two boundary components of A where the homotopy is taken relative to the endpoints of .ω. The intersection .i(ω, ω' ) of two arcs is the minimal number of intersections between any representatives of homotopy classes of .ω and .ω' . The

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vertices of the curve graph .C(A) are arcs on A and the edges are pairs of arcs with zero intersection. We also equip .C(A) with a metric as above.

2.7.3 Subsurface Projection For any subsurface Y of S, the subsurface projection πY : C(S) → P(C(Y ))

.

from .C(S) to the set of subsets of the vertices of .C(Y ) is defined as follows. Suppose first that Y is not an annulus. If .α is disjoint from Y , then .πY (α) = ∅. If .α is contained in Y , then .πY (α) = α. If .α intersects Y , then .α ∩ Y is a union of arcs. Let .ω be such an arc. Let .δ and .δ ' be the boundary components of Y intersecting .ω. Let .αω be the collection of boundary components of the .ϵ-regular neighbourhood of .δ ∪ ω ∪ δ ' for sufficiently small .ϵ. The projection .πY (α) is then defined as the collection of .αω for all arcs of .α ∩ Y . From the construction we see that the diameter of .πY (α) is at most two. We now consider the case where S is an annulus A with core curve .γ . The Gromov compactification of the annulus cover of S corresponding to .γ ∈ π1 (S) is well-defined and is independent of the choice of the hyperbolic metrics on S. For any simple closed curve .α ∈ C(S), the projection .πA (α) is defined to be the lifts of .α to the annulus cover that connects two boundary components of the annulus. Notice that any lift has exactly two well-defined endpoints in the Gromov compactification. The diameter of .πA (α) is also at most two. The projection of multicurves is defined as the union of the projection of each simple closed curve.

2.7.4 Combinatorial Model Recall that for any X and Y in .T (S), dT h (X, Y ) = log sup

.

α

𝓁α (Y ) , 𝓁α (X)

where .α ranges over all essential simple closed curves on S. This implies that there (Y ) is a simple closed curve .α such that .log 𝓁𝓁αα (X) is a good estimate of .dT h . Lenzhen, Rafi, and Tao proved that such a curve can be chosen within any short marking .μX of X: Theorem 2.7.1 ([37], Theorem E) There exists a constant C depending on the topology of S such that for any X and Y in .T (S),

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   𝓁α (Y )   . dT h (X, Y ) − log max ≤ C.  α∈μX 𝓁α (X)  Theorem 2.7.1 is a consequence of the following estimate of lengths of simple closed curves in terms of short markings. Proposition 2.7.2 ([37], Proposition 3.1) There exists a positive constant C depending on the topology of S, such that for any .X ∈ T (S), any simple closed curve .γ , and any short marking .μX of X, C −1



.

i(γ , α)𝓁α¯ (X) ≤ 𝓁α (X) ≤ C

α∈μX



i(γ , α)𝓁α¯ (X).

α∈μX

Remark 2.7.3 (1) A similar estimate also holds for the extremal length ([37], Proposition 3.1): C −1



.

i(γ , α)2 Extα¯ (X) ≤ Extα (X) ≤ C

α∈μX



i(γ , α)2 Extα¯ (X).

α∈μX

(2) For similar estimates using twists with respect to some pants decomposition, see [44].

2.7.5 Short Curves One way to understand the behavior of Thurston geodesics in the moduli space of Riemann surfaces is to characterize short curves along the corresponding family of hyperbolic surfaces. Such a characterization is known for Teichmüller geodesics (see [59, Theorem 1.1] or [37, Theorem 2.3]). However, the situation is more involved for Thurston geodesics. Definition 2.7.4 (1) Let K be a positive constant. Two points X and Y in .T (S) are said to have K-bounded combinatorics if dC(Σ) (πΣ (μX ), πΣ (μY )) ≤ K

.

for any subsurface .Σ ⊂ S, where .μX and .μY are respectively the short markings of X and Y . (2) A path in .T (S) is said to be .ϵ cobounded if it is contained in the .ϵ thick part of .T (S). Theorem 2.7.5 ([37], Theorem A) Let X and Y be two points in the .ϵ-thick part of .T (S). If X and Y have K-bounded combinatorics, then every Thurston geodesic

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from X to Y fellow travels the Teichmüller geodesic from X to Y , and hence is ϵ ' -cobounded, where .ϵ ' depends on .ϵ and K.

.

This reflects a negative-curvature phenomenon of the Thurston metric. More precisely, Theorem 2.7.6 ([37], Theorem C and Corollary D) Let .G be a Thurston geodesic in .T (S) whose endpoints have bounded combinatorics. Then the closed point projection to .G is strongly contracting. In particular, any quasi-geodesic with the same endpoints as .G fellow travels .G. Remark 2.7.7 From Theorem 2.7.5, we see that bounded combinatorics implies ϵ-cobounded. One may wonder whether the inverse still holds or not. For the Teichmüller metric, the converse also holds. In fact, two points X and Y in .T (S) have bounded combinatorics if and only if the Teichmüller geodesic connecting them is cobounded ([59, Theorem 1.1], see also [37, Theorem 2.2]). However, this is not the case for the Thurston metric. There exists .ϵ0 > 0 such that for any .ϵ > 0, there are points X and Y in .T (S) and a Thurston geodesic from X to Y that stays in the .ϵ0 thick part of .T (S), whereas the associated Teichmüller geodesic from X to Y does not stay in the .ϵ-thick part of .T (S) [38, Theorem 1.4].

.

Recall that the Thuston metric is not uniquely geodesic. If we remove the bounded combinatorics assumption, then the fellow traveling property fails. In fact, for any .D > 0, there are points .X, Y, Z in .T (S) and two Thurston geodesics .G1 , .G2 from X to Y with the following properties [38, Theorem 1.1]. • Geodesics .G1 and .G2 do not fellow travel each other; the point Z lies in the path .G1 but is at least D away from any point in .G2 ; • The geodesic .G1 parameterized in any way in the reverse direction is not a geodesic. In fact, the point Z is at least D away from any point in any geodesic from Y to X. On the other hand, if we consider the projection to the curve complex, which sends a point in .X ∈ T (S) to the set of shortest simple closed curves, then all Thurston geodesics with common endpoints nearly look the same: Theorem 2.7.8 ([38], Theorem 1.2) The shadow of a Thurston geodesic to the curve graph is a reparameterized quasi-geodesic. Remark 2.7.9 Since the curve graph is Gromov hyperbolic [42], quasi-geodesics with common endpoints fellow travel each other. Hence, for any two points X and Y in .T (S), the shadow to the curve graph of different Thurston geodesics form X to Y fellow travel each other. In [19], the authors provided another characterization of short curves along Thurston geodesics. Before stating the result, let us introduce the notion of absolute twisting .dα (X, Y ) of two hyperbolic surfaces X and Y relative to a simple closed curve .α. Let .ω be a geodesic arc on X which is orthogonal to the geodesic representative of .α on X. Let .ω' be a geodesic arc on Y which is orthogonal to

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 → X be the annulus cover of X the geodesic representative of .α on Y . Let .X  Let . α be the core curve of .X. ω and . ω' be respectively lifts corresponding to .α. Let . '  The absolute twisting .dα (X, Y ) of X and Y relative to .α of .ω and .ω intersecting .X. is defined to be: dα (X, Y ) = min i( ω,  ω' )

.

 has where the minimum ranges over all such .ω and .ω' and their lifts. (Notice that .X ω,  ω' ) is considered the minimum a well-defined Gromov compactification, and .i( intersection number in the given homotopy class relative to their endpoints.) Recall that for any pair of distinct points X and Y in .T (S), there is a well-defined lamination, the maximally stretched lamination .Λ(X, Y ) (see Sect. 2.2.3). A simple closed curve .α is said to interact with the maximally stretched lamination .Λ(X, Y ) if it belongs to the lamination or intersects it essentially. Theorem 2.7.10 ([19], Theorem 1.2) There exists a constant .ϵ0 such that the following statement holds. Let X and Y be two points in the .ϵ0 -thick part of .T (S) and let .α be a simple closed curve on S that interacts with .Λ(X, Y ). Then the minimum length .𝓁α of .α along any Thurston geodesic from X to Y satisfies: C −1 · dα (X, Y ) − C ' ≤

.

1 1 Log ≤ C · dα (X, Y ) + C ' 𝓁α 𝓁α

where C and .C ' are two constants that depend only on .ϵ0 , and where .Log(x) := max{1, log(x)}. For more interesting discussions about short curves, we refer to [19, Section 3]. Based on the above characterization of short curves, Telpukhovskiy proved that Masur’s criterion does not hold in the Thurston metric: Theorem 2.7.11 ([69]) There are Thurston stretch paths in the Teichmüller space T (S0,7 ) with minimal, filling, but not uniquely ergodic horocyclic foliation, that stay in the thick part for the whole time.

.

2.7.6 Length Spectrum Metric and the Product Theorem Inspired by Minsky’s product theorem for the Teichmüller metric, Choi and Rafi proved an analogue for the symmetrized metric .dL of the Thurston metric: dL (X, Y ) := max{dT h (X, Y ), dT h (Y, X)}.

.

The metric .dL is called the length spectrum metric.

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For any positive constant .ϵ, consider the .ϵ-thick part .Thickϵ (S) and the .ϵ-thin part .Thinϵ (S) defined as below: Thickϵ (S) := {X ∈ T (S) : 𝓁α (X) ≥ ϵ for every simple closed curve α}

.

Thinϵ (S) := {X ∈ T (S) : 𝓁α (X) < ϵ for some simple closed curve α}. For a multicurve .𝚪, we define the subset .Thinϵ (𝚪; S) of the .ϵ-thin part as the set of points .X ∈ T (S) satisfying .𝓁α (X) < ϵ if and only if .α ∈ 𝚪. Let .T (S\𝚪) be the Teichmüller space of the punctured surface homeomorphic to .S\𝚪, let .Ui := {(x, y) ∈ R : y > 1/ϵ} be the subset of the upper half plane. Then the Fenchel– Nielsen coordinates on .T (S) provide a natural homeomorphism: Π : Thinϵ (𝚪; S) → T (S\𝚪) × U1 × · · · × Uk

.

Let .dL(S\𝚪) be the Lipschitz metric on .T (S\𝚪). Let .dL(γi ) be a modified metric of the hyperbolic metric on .Ui . Let dL𝚪 := sup{dL(S\𝚪) , dL(γ1 ) , · · · , dL(γk ) }.

.

Choi and Rafi proved that Theorem 2.7.12 ([15], Theorem C) There exists a constant c depending on .ϵ and the topology of S such that for any multicurve .𝚪 and for any X and Y in .Thinϵ (𝚪; S), we have .

  dL (X, Y ) − dL (Π(X), Π(Y )) ≤ c. 𝚪

Before closing this subsection, let us mention the comparison between the length spectrum metric and the Teichmüller metric. In the thick part of .T (S), these two metrics are almost isometric [15, Theorem B], whereas in the whole Teichmüller space they are not comparable [15, 39]. On the other hand, they are almost isometric in the moduli space [40, 41].

2.8 Counting Lattice Points In this section, we will discuss the lattice counting problem about the Thurston metric. Before stating the counting result precisely, we need several definitions. Following Bonahon [9], we can embed the Teichmüller space .T (S) as a subset of the space .C(S) of geodesic currents in such a way that for every .Y ∈ T (S) and for every curve .γ we have i(γ , Y ) = 𝓁Y (γ ).

.

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H. Pan and W. Su

One can also embed .T (S) into the space of functions on .C(S) as follows. Let .X ∈ T (S). Define the function .DX : DX : C(S) −→ R

.

η I−→

i(η, λ) . λ∈ML(S) 𝓁X (λ) sup

It is clear that .DX : C(S) → R is positive, homogeneous and continuous. Let .μT h be the Thurston volume on .ML(S). For any positive, homogeneous and continuous function f on .C(S), define m(f ) := μT h ({λ ∈ ML(S) : i(α, λ) ≤ 1}).

.

In particular, for any .Y ∈ T (S), m(Y ) := μT h ({λ ∈ ML(S) : 𝓁Y (λ) ≤ 1}),

.

and m(DY ) := μT h ({λ ∈ ML(S) : DY (λ) ≤ 1}).

.

Finally, let  mS :=

.

M(S)

m(Y )dvolwp

where .M(S) is the moduli space and .dvolwp represents the Weil-Petersson volume. Theorem 2.8.1 ([60], Theorem 1.1) Let S be a closed orientable surface of genus at least two. Let X and Y be two points in .T (S). Then .

lim

R→∞

#{φ ∈ MCG(S) : dT h (X, φ(Y )) ≤ R} m(DX )m(Y ) = . mS e(6g−6)R

Remark 2.8.2 Theorem 2.8.1 is a consequence of a more general counting result due to [60], where the authors counted the mapping class group orbits of filling currents (for more information about this type of counting problems, we refer to [21, 22] and references therein). The analogous result under the Teichüller metric was first proved in [5] and improved with an error term in [4].

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2.9 Shearing Coordinates The shearing coordinates for .T (S) were introduced by Thurston [74]. The shearing operation is a generalization of the earthquake deformation. In Thurston’s preprint, the construction is called cataclysm coordinates. Let .Δ be an ideal triangle in the hyperbolic plane. There is a unique circle in .Δ tangent to each edge of .Δ. If .γ is an edge of .Δ, then the circle intersects .γ at some distinguished point. Consider two ideal triangles .Δa , Δb , with disjoint interiors and a common edge .γ . The shear parameter .sγ is defined to be the signed distance of the two distinguished points on .γ . The quantity .sγ measures the magnitude of shift (or twist) when we glue .Δa and .Δb . We refer to [8] for details. Fix a finite ideal triangulation .λ of S. For each hyperbolic surface .X ∈ T (S), .λ is represented as a geodesic lamination with finitely many leaves whose complementary components are ideal triangles. We can associate with each edge .γ of .λ a shear parameter .sγ (X). Then the map sλ : T (S) → R|λ|   X I→ sγ (X) γ ∈λ

.

is an embedding, which is called shearing coordinates of .T (S). Note that the image of .sλ is an open convex cone of a linear subspace of the dimension .6g + 2n − 6. For example, every pants decomposition of S can be completed to be a finite ideal triangulation. Then, the corresponding Fenchel–Nielsen parameters of .T (S) can be extended (up to linear transformation) to shearing coordinates. Remark 2.9.1 In Mirzakhani’s work, random hyperbolic surfaces are defined by gluing random pairs of pants. It is also natural to construct random hyperbolic surfaces by gluing ideal triangles randomly. The above construction works for general maximal geodesic laminations. Fix a maximal geodesic lamination .λ. Then each .X ∈ T (S) can be associated with a horocyclic measured foliation .Fλ (X) and a shearing cocycle. In brief, a shearing cocycle is similar to some transverse weights on a train track carrying .λ, and the weights measure the shifts of the shear map when one hyperbolic surface is deformed into another. Conversely, for any shearing cocycle, one can construct a measured foliation .Fλ in a neighborhood of .λ, whose leaves are hocycles orthogonal to .λ. It turns out that there is a unique hyperbolic structure X on S such that .Fλ extends to .Fλ (X).

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Let .MF(λ) be the set of measured foliations that is totally transverse to .λ (see [74, Proposition 9.4]). Then Theorem 2.9.2 (Thurston) Let .λ be a maximal geodesic lamination on S. Then the shear map Fλ : T (S) → MF(λ)

.

X I→ Fλ (X) is a real analytic parametrization of .T (S). The shearing coordinates can be also interpreted as transverse cycles, as done by Bonahon [10]. There are several nice properties and developments of this notion: (1) The shear map pulls back the Thurston form on .MF(S) to the Weil–Petersson symplectic form on .T (S) [52, 67]. This has been very useful when Mirzakhani [45] proved that the earthquake flow is measured isomorphic to the horocycle flow on moduli space. (2) The complexified version of transverse cocycles can be used to measure the bending of pleated surfaces, and thus is used to extend Thurston’s parametrization to the quasi-Fuchsian space [10]. Furthermore, Thurston’s parametrization can be extended to Hitchin representations into .PSLn (R) [11, 12]. (3) There are many related works on decorated Teichmüller spaces, quantum Teichmüller spaces and higher Teichmüller spaces, such as [24–26], etc. (4) There is a construction of shearing coordinates on the universal Teichmüller space, see [57, 63, 64] When .λ is not a maximal geodesic lamination, the horocyclic foliation is no longer well-defined. Such a difficultly was already encountered in the case of surfaces with boundaries, when Alessandrini–Disarlo [2] constructed geodesics for the arc metric. In general, Calderon–Farre [13] defined orthogeodesic foliations associated with .λ. The orthogeodesic foliations are constructed on complementary subsurfaces. They showed that the restriction of the orthogeodesic foliation to each component of .S \ λ completely determines the hyperbolic structure on that piece. Thus they generalized the shearing coordinates of .T (S). We conclude this section with the following result of Théret [71], which improves the result of [8, Theorem 1.3]. Theorem 2.9.3 ([71], Theorem 1.1) Let .μ be a measured geodesic lamination transverse to the complete geodesic lamination .λ on S. Then the length function .𝓁μ is a convex function of the shear coordinates defined over .T (S) associated with .λ, and .𝓁μ is strictly convex whenever .μ intersects all leaves of .λ. In particular, lengths functions are convex along stretch lines. An interesting corollary is that the length of the horocyclic foliation is strictly decreasing on a

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stretch line and converges to zero as the stretch line diverges to infinity in the forward direction.

2.10 Generalization of the Thurston Metric In this section, we shall discuss some generalizations of the Thurston metric. As we mentioned earlier, there is a generalization to the setting of hyperbolic surface with geodesic boundary components, the arc metric, see Remark 2.5.4. Guéritaud-Kassel [28] considered a pair .(j, ρ) of representations of a discrete group .𝚪0 into .G = Isom(Hn ) such that j is injective with .j (𝚪0 ) discrete and geometrically finite. They showed that, when the minimal Lipschitz constant is strictly greater than 1, there exists a geodesic lamination that is “maximally stretched” by any extremal .(j, ρ)-equivariant Lipschitz map .f : Hn → Hn . As an application, they generalized the two-dimensional results and constructions of Thurston and extend the Thurston metric on Teichmüller space to higher dimension. Let us denote the minimal Lipschitz constant of the .(j, ρ)-equivariant map by .L(j, ρ). A related constant is K(j, ρ) = log sup

.

γ

λ(ρ(γ )) , λ(j (γ ))

where .λ denotes the translation length of the element, and the supremum is taken over all hyperbolic elements in .γ ∈ 𝚪0 . It is obvious that K(j, ρ) ≤ L(j, ρ).

.

Let .M = Hn / 𝚪0 be a geometrically finite hyperbolic manifold. The Teichmüller space .T (M) is defined to be the set of conjugacy classes of geometrically finite representations of .𝚪0 . For .j, ρ ∈ T (M), denote the critical exponents of j and .ρ by .δ(j ) and .δ(ρ), respectively. Then Theorem 2.10.1 ([28]) The function  δ(ρ) dTh (j, ρ) = log L(j, ρ) δ(j )

.

defines an asymmetric metric on .T (M). We call the above metric a generalized Thurston metric. Another version is  δ(ρ) .d Th (j, ρ) = log K(j, ρ) . δ(j ) As pointed out by Guéritaud-Kassel, the two metrics differ in general.

(2.4)

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Question 1 Is the generalized Thurston metric .dTh Finsler? How to construct geodesics of .dTh ? Recently, Sapir [62] extended the Thurston metric to the space of projective filling currents; Carvajales-Dai-Pozzetti-Wienhard [14] generalized the Thurston metric and the associated Finsler norm to Anosov representations of surface groups, including the space of Hitchin representations. The distance formula is similar to (2.4). The Thurston metric has an analogue in Outer space, see [75] for a discussion. Just like the Thurston metric on .T (S), this metric is not symmetric. It was used by Bestvina [7] to give a Bers-like proof of the train track theorem for fully irreducible automorphisms of .Fn . Regarding the extremal Lipschitz maps, Thurston [74] suggested that I currently think that a characterization of minimal stretch maps should be possible in a considerably more general context (in particular, to include some version for all Riemannian surfaces). . .

For related results along this line, we refer to [16, 17, 49]. Finally, let us mention that it is impossible to deform hyperbolic metrics of finite area in .T (S) such that no length of any closed geodesic is increased. This is a typical example of the (marked) length spectrum rigidity problem, which is extensively studied in geometry and dynamics, see [18, 29, 46] for examples. Acknowledgments We would like to thank Athanase Papadopoulos for the opportunity of writing this article, for his patience during the preparation of the manuscript, and for suggesting several improvements on the exposition. The first author is supported by NSFC 12371073. The second author is supported by NSFC 12371076.

References 1. S.I. Al’ber, On n-dimensional problems in the calculus of variations in the large. Sov. Math. Dokl. 5, 26 (1964) 2. D. Alessandrini, V. Disarlo, Generalizing stretch lines for surfaces with boundary. Int. Math. Res. Not. 2022(23), 18919–18991 (2022) 3. D. Alessandrini, L. Liu, A. Papadopoulos, W. Su, The horofunction compactification of Teichmüller spaces of surfaces with boundary. Topol. Appl. 208, 160–191 (2016) 4. F. Arana-Herrera, Effective mapping class group dynamics, i: counting lattice points in Teichmüller space. Duke Math. J. 1(1), 1–93 (2023) 5. J. Athreya, A. Bufetov, A. Eskin, M. Mirzakhani, Lattice point asymptotics and volume growth on Teichmüller space. Duke Math. J. 161(6), 1055–1111 (2012) 6. A. Bar-Natan, Geodesic Envelopes in Teichmuller Space Equipped with the Thurston Metric. ProQuest LLC, Ann Arbor (2022). Thesis (Ph.D.)—University of Toronto (Canada) 7. M. Bestvina, A Bers-like proof of the existence of train tracks for free group automorphisms. Fund. Math. 214(1), 1–12 (2011) 8. M. Bestvina, K. Bromberg, K. Fujiwara, J. Souto, Shearing coordinates and convexity of length functions on Teichmüller space. Am. J. Math. 135(6), 1449–1476 (2013)

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35. J.H. Hubbard, Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, vol. 2. Matrix Editions (2016) 36. J. Jost, Harmonic Maps Between Surfaces, vol. 1062. Lecture Notes in Mathematics (Springer, Berlin, 1984) 37. A. Lenzhen, K. Rafi, J. Tao, Bounded combinatorics and the Lipschitz metric on Teichmüller space. Geom. Dedicata 159, 353–371 (2012) 38. A. Lenzhen, K. Rafi, J. Tao, The shadow of a Thurston geodesic to the curve graph. J. Topol. 8(4), 1085–1118 (2015) 39. Z. Li, Length spectrums of Riemann surfaces and the Teichmüller metric. Bull. Lond. Math. Soc. 35(2), 247–254 (2003) 40. L. Liu, W. Su, Almost-isometry between Teichmüller metric and length-spectrum metric on moduli space. Bull. Lond. Math. Soc. 43(6), 1181–1190 (2011) 41. L. Liu, H. Shiga, W. Su, Y. Zhong, Almost-isometry between the Teichmüller metric and the length-spectrum metric on reduced moduli space for surfaces with boundary. Trans. Am. Math. Soc. 369(9), 6429–6464 (2017) 42. H.A. Masur, Y.N. Minsky, Geometry of the complex of curves. I. Hyperbolicity. Invent. Math. 138(1), 103–149 (1999) 43. Y.N. Minsky, Harmonic maps, length, and energy in Teichmüller space. J. Differ. Geom. 35(1), 151–217 (1992) 44. Y.N. Minsky, Extremal length estimates and product regions in Teichmüller space. Duke Math. J. 83(2), 249–286 (1996) 45. M. Mirzakhani, Ergodic theory of the earthquake flow. Int. Math. Res. Not. 2008(9), 1073– 7928 (2008) 46. J.-P. Otal, Le spectre marqué des longueurs des surfaces à courbure négative. Ann. Math. 131(1), 151–162 (1990) 47. H. Pan, Local Rigidity of the Teichmüller space with the Thurston metric. Sci. China Math. 66, 1751–1766 (2023) 48. H. Pan, M. Wolf, Envelopes of the Thurston metric on Teichmüller space (in preparation) 49. H. Pan, M. Wolf, Ray structures on Teichmüller space (June 2022). arXiv:2206.01371 50. A. Papadopoulos, On Thurston’s boundary of Teichmüller space and the extension of earthquakes. Topol. Appl. 41(3), 147–177 (1991) 51. A. Papadopoulos (ed.), Handbook of Teichmüller Theory. Vol. III, vol. 17. IRMA Lectures in Mathematics and Theoretical Physics (European Mathematical Society, Zürich, 2012) 52. A. Papadopoulos, R.C. Penner, The Weil-Petersson symplectic structure at Thurston’s boundary. Trans. Am. Math. Soc. 335(2), 891–904 (1993) 53. A. Papadopoulos, W. Su, On the Finsler structure of Teichmüller’s metric and Thurston’s metric. Expo. Math. 33(1), 30–47 (2015) 54. A. Papadopoulos, G. Théret, On Teichmüller’s metric and Thurston’s asymmetric metric on Teichmüller space, in Handbook of Teichmüller Theory. Vol. I, vol. 11. IRMA Lect. Math. Theor. Phys. (European Mathematical Society, Zürich, 2007), pp. 111–204 55. A. Papadopoulos, G. Théret, Some Lipschitz maps between hyperbolic surfaces with applications to Teichmüller theory. Geom. Dedicata 161, 63–83 (2012) 56. A. Papadopoulos, S. Yamada, Deforming hyperbolic hexagons with applications to the arc and the Thurston metrics on Teichmüller spaces. Monatsh. Math. 182(4), 913–939 (2017) 57. R.C. Penner, Universal constructions in Teichmüller theory. Adv. Math. 98(2), 143–215 (1993) 58. R.C. Penner, J.L. Harer, Combinatorics of Train Tracks, vol. 125. Annals of Mathematics Studies (Princeton University Press, Princeton, 1992) 59. K. Rafi, A characterization of short curves of a Teichmüller geodesic. Geom. Topol. 9, 179– 202 (2005) 60. K. Rafi, J. Souto, Geodesic currents and counting problems. Geom. Funct. Anal. 29(3), 871– 889 (2019) 61. J.H. Sampson, Some properties and applications of harmonic mappings. Ann. Sci. École Norm. Sup. (4) 11(2), 211–228 (1978)

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62. J. Sapir, An extension of the Thurston metric to projective filling currents. arXiv preprint arXiv:2210.08130 (2022) 63. D. Šari´c, Circle homeomorphisms and shears. Geom. Topol. 14(4), 2405–2430 (2010) 64. D. Šari´c, Y. Wang, C. Wolfram, Circle homeomorphisms with square summable diamond shears (2022). arXiv preprint arXiv:2211.11497 65. R.M. Schoen, Analytic aspects of the harmonic map problem, in Seminar on Nonlinear Partial Differential Equations (Berkeley, Calif., 1983), vol. 2. Math. Sci. Res. Inst. Publ. (Springer, New York, 1984), pp. 321–358 66. R. Schoen, S.T. Yau, On univalent harmonic maps between surfaces. Invent. Math. 44(3), 265–278 (1978) 67. Y. Sözen, F. Bonahon, The Weil-Petersson and Thurston symplectic forms. Duke Math. J. 108(3), 581–597 (2001) 68. W. Su, Problems on the Thurston metric, in Handbook of Teichmüller Theory. Vol. V, vol. 26. IRMA Lect. Math. Theor. Phys. (Eur. Math. Soc., Zürich, 2016), pp. 55–72 69. I. Telpukhovskiy, Masur’s criterion does not hold in the Thurston metric. Groups Geom. Dyn. 17(4), 1435–1481 (2023) 70. G. Théret, On the negative convergence of Thurston’s stretch lines towards the boundary of Teichmüller space. Ann. Acad. Sci. Fenn. Math. 32(2), 381–408 (2007) 71. G. Théret, Convexity of length functions and Thurston’s shear coordinates (2014). arXiv:1408.5771 72. W.P. Thurston, Earthquakes in two-dimensional hyperbolic geometry, in Low-Dimensional Topology and Kleinian Groups (Coventry/Durham, 1984), vol. 112. London Math. Soc. Lecture Note Ser. (Cambridge University Press, Cambridge, 1986), pp. 91–112 73. W.P. Thurston, The Geometry and Topology of Three-Manifolds: With a Preface by Steven P. Kerckhoff, vol. 27 (American Mathematical Society, Providence, 2022) 74. W.P. Thurston, Minimal stretch maps between hyperbolic surfaces, in Collected Works of William P. Thurston with Commentary. Vol. I. Foliations, Surfaces and Differential Geometry (Amer. Math. Soc., Providence, 2022), pp. 533–585. 1986 preprint, 1998 eprint 75. K. Vogtmann, On the geometry of outer space. Bull. Am. Math. Soc. 52(1), 27–46 (2015) 76. C. Walsh, The horoboundary and isometry group of Thurston’s Lipschitz metric, in Handbook of Teichmüller Theory. Vol. IV, vol. 19. IRMA Lect. Math. Theor. Phys. (Eur. Math. Soc., Zürich, 2014), pp. 327–353 77. M. Wolf, The Teichmüller theory of harmonic maps. J. Differ. Geom. 29(2), 449–479 (1989) 78. M. Wolf, Measured foliations and harmonic maps of surfaces. J. Differ. Geom. 49(3), 437–467 (1998)

Chapter 3

Thurston’s Metric on the Teichmüller Space of Flat Tori Binbin Xu

Abstract We briefly review Thurston’s metric theory for complete hyperbolic surfaces with finite area, as well as some recent results about it. Then we review the generalization of Thurston’s metric theory for flat tori studied in Greenfield and Ji (Asian J Math 25(4):477–504, 2021) and Saˇglam (Int Electron J Geom 14(1):59– 65, 2021) [13]. In particular, we review the connection between Lipschitz-extremal homeomorphisms and affine maps between flat tori and show that Thurston’s metric on the Teichmüller space of flat tori coincides with the Teichmüller metric. Keywords Teichmüller space · Flat tori · Thurston’s metric · Lipschitz-extremal homeomorphisms · Affine maps · Maximally stretched foliations AMS Classification: 30F60, 32G15, 51F30, 53A15, 53C12, 57M50

3.1 Introduction Let .S = Sg,n be an oriented surface of genus g with n cusps such that .2 − 2g − n < 0. The Teichmüller space .T (S) of S is defined to be the moduli space of marked hyperbolic structures on S. The geometry of .T (S) is one main subject studied in classical Teichmüller theory. Hyperbolic structures on S are related to other structures on S such as conformal structures, complex structures, etc. which yields various definitions of .T (S). By studying the deformation of these structures, we obtain various ways of defining metrics on .T (S). For example, the Teichmüller metric on .T (S) arises from studying the deformation of conformal structures on S, while the Weil–Petersson metric on .T (S) comes from studying the deformation of Riemannian metrics on S. In [14],

B. Xu () School of Mathematical Sciences, Nankai University, Tianjin, China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 K. Ohshika, A. Papadopoulos (eds.), In the Tradition of Thurston III, https://doi.org/10.1007/978-3-031-43502-7_3

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Thurston introduced an asymmetric metric on .T (S) by studying hyperbolic metrics on S from the Lipschitz geometry point of view.

3.1.1 Thurston’s Original Work In [14], for any ordered pair of hyperbolic structures on S, Thurston studied all homeomorphisms of S isotopic to the identity and considered the infimum of their Lipschitz constants with respect to these two given hyperbolic structures. By taking the logarithm of this infimum for each ordered pair of hyperbolic structures on S, Thurston defined a map .

dT h : T (S) × T (S) → R≥0 ,

and proved that it satisfies almost all properties of a distance function except symmetry. Moreover, he showed that for any ordered pair of hyperbolic structures on S, this infimum actually can be realized by some homeomorphism which is called a Lipschitz-extremal homeomorphism for this pair of hyperbolic structures. The asymmetric metric on .T (S) given by the map .dT h is now called the Thurston’s metric. The main tools used in this study are maximally stretched lamination and stretch path. The maximally stretched lamination for a Lipschitz-extremal homeomorphism consists of lines on S on which the extremal Lipschitz constant is realized. More precisely, a curve is the image of a continuous map from .S 1 to S. A curve is simple if it has no self-intersection. It is essential if it is not homotopic to a point in S and is non-peripheral if it is not homotopic to a cusp of S. An essential curve is primitive if it is not homotopic to a curve on S going along an essential curve more than once. In the following, we will consider only essential non-peripheral primitive curves on S. Given any curve .α on S and any point .X ∈ T (S), there is a unique X-geodesic representative in the homotopy class .[α]. We denote its X-length by .lX ([α]). For any pair of points .X, Y ∈ T (S), we define the following quantity K(X, Y ) := sup

.

[α]

lY ([α]) , lX ([α])

where the supremum is taken over all homotopy classes of curves on S. Thurston [14, Proposition 3.5] showed that the supremum is the same if we consider only simple curves, instead of all curves. Consider the following ratio .

lY ([α]) lX ([α])

as a function over the space of homotopy classes of simple curves on S. It can be extended to a continuous function over .PML(S) the space of projective measured

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laminations on S. (See [3] for more details about .PML(S)). Since .PML(S) is a compact space, the supremum is actually a maximum and is realized by a projective measured lamination. Let us denote it by .μ. A geodesic lamination on X is a closed subset of X which is a union of a collection of pairwise disjoint simple complete geodesics on X. Each complete geodesic contained in this union is called a leaf of this geodesic lamination. By forgetting the measure and taking the geodesic representative, we can map the space .ML(S) of measured laminations on S into the space .GL(X) of geodesic laminations on X. We will still call a geodesic lamination a measured lamination if it is in the image of this map. Let .GLC (X) denote the subset of .GL(X) consisting of compact geodesic laminations on X. Then .GLC (X) contains all measured laminations, and can be equipped with the Hausdorff distance. A geodesic lamination in .GLC (X) is called a chain recurrent lamination, if it is the limit of a sequence of measured laminations with respect to the Hausdorff distance. The maximally stretched lines of a Lipschitz-extremal homeomorphism from X to Y is a geodesic lamination .Λ(X, Y ) containing .μ which is called the maximally stretched lamination for .(X, Y ). The lamination .Λ(X, Y ) is not always a measured lamination, however it is always a chain recurrent lamination. Chain recurrent laminations can be used to construct stretch paths in .T (S) which are geodesics with respect to .dT h . To be more precise, given any point .X ∈ T (S) and any chain recurrent lamination .λ on X, we consider its complement in X. Each connected component could be either an ideal polygon or a hyperbolic surface with non trivial fundamental group. When there is a connected component different from an ideal triangle, we may add certain simple complete geodesics to get a new geodesic lamination strictly containing .λ. We say that a geodesic lamination is complete if there is no other geodesic lamination strictly containing it. When .λ is not complete, the above discussion also shows us a way to complete .λ, as well as the fact that there are different ways to complete .λ. By adding some simple complete geodesics if necessary, we complete .λ into a geodesic lamination  .λ on X whose complement in X is a disjoint union of ideal triangles. By stretching each ideal triangle with a same factor .et ≥ 1 [14, Proposition 2.2], we obtain a hyperbolic surface Xt = (X,  λ, t).

.

Let t run over an interval .I ⊂ R≥0 starting from 0, we get a stretch path {Xt }t∈I ⊂ T (S),

.

starting from X. Thurston showed that a stretch path constructed in this way is a geodesic for .dT h [14, Corollary 4.2]. Using this result, Thurston proved that for any given point .X, Y ∈ T (S), there is a geodesic for .dT h connecting X to Y which is a concatenation of stretch paths, and the maximally stretched lamination for each

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of these stretch path contains .Λ(X, Y ). Moreover Thurston showed the following equality dT h (X, Y ) = log K(X, Y ).

.

Notice that non-uniqueness of completions of a chain recurrent lamination suggests that .T (S) is not uniquely geodesic with respect to Thurston’s metric.

3.1.2 Thurston’s Metric on the Teichmüller Space of the Once-Punctured Torus To have a detailed picture of geodesics for Thurston’s metric, in [2], Dumas, Lenzhen, Rafi and Tao studied the union of geodesics for Thurston’s metric from one point .X ∈ T (S) to another Y which is called the envelope .E(X, Y ) for .(X, Y ). In particular, they considered Thurston’s metric and its induced topology on .T (S), and showed that when S is a once-punctured torus (i) the envelope .E(X, Y ) is a compact set; (ii) .E(X, Y ) varies continuously in the Hausdorff topology as a function on X and Y; (iii) if .Λ(X, Y ) is not a simple closed geodesic, then .E(X, Y ) is a segment on a stretch path (which is the unique geodesic from X to Y ); (iv) if .Λ(X, Y ) is a simple closed geodesic .α, then .E(X, Y ) is a geodesic quadrilateral in .T (S) with X and Y as a pair of opposite vertices, and each edge of the quadrilateral is a stretch path stretching along a completion of a chain recurrent lamination properly containing .α. Notice that (iii) shows that there exists a pair of points in .T (S) connected by a unique geodesic, and that it is possible that different completions of a chain recurrent lamination yield a same stretch path, despite the fact that Thurston’s metric is not uniquely geodesic. In [2], the coarse geometry of Thurston’s metric is also studied. Similar as for studying the coarse geometry of the Teichmüller metric on .T (S), the curve complex of S plays an important role. The curve complex .C(S) of S is a simplicial complex, where for any .k ∈ N, a ksimplex of .C(S) is a .(k + 1)-tuples of distinct homotopy classes of simple curves on S which are pairwise disjoint. By setting the length of each dimension 1 simplex to be 1, the 1-skeleton .C1 (S) becomes a metric space which is Gromov hyperbolic (See [8] for details). When S is the once-punctured torus, .C(S) can be identified with the Farey triangulation of the hyperbolic plane. A geodesic in .C1 (S) passes a sequence of simple curves on S which is called a pivot sequence (See [9] for details).

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Let .Ω = {Xt }t∈I be a stretch path in .T (S) from X to Y . For any simple curve .α on S, we define lΩ ([α]) = inf lXt ([α]).

.

t∈I

Let .ϵ > 0 be a constant choose once for all. A homotopy class .[α] of simple curves is said to be short along .Ω if lΩ ([α]) ≤ ϵ.

.

When S is the once-punctured torus, by considering short curves along stretch paths, Dumas, Lenzhen, Rafi and Tao gave a combinatorial description of geodesics for Thurston’s metric in term of pivot sequences. Moreover they showed that for any pair X and Y in .T (S), the pivot sequence associated to any geodesic from X to Y for Thurston’s metric is the same and equal to the one for geodesics from Y to X, despite the fact that the actual Thurston’s metric is not uniquely geodesic and is asymmetric. Another part of the result in [2] is about the rigidity of Thurston’s metric. More precisely, for any surface S with negative Euler characteristic and possibly with cusps, we consider the Thurston’s norm induced by Thurston’s metric on the tangent bundle .TT (S). The main question concerned here is whether the restriction of the Thurston’s norm at .TX T (S) can determine the base point .X ∈ T (S). In [2], this question is answered affirmatively for the case when S is the once-punctured torus. For the general case, this question is answered affirmatively independently in [11] by Pan and in [6] by Huang, Ohshika and Papadopoulos. A consequence of this result is that the isometry group of Thurston’s metric on .T (S) is the extended mapping class group of S.

3.1.3 Generalizations of Thurston’s Metric Theory Thurston’s metric theory has various generalizations in different contexts. One way of generalizing Thurston’s metric theory is to consider hyperbolic surfaces with geodesic boundary. If X is a hyperbolic surface with geodesic boundary, it is possible to deform the hyperbolic structure on X such that all closed geodesic lengths increase. One construction is based on an idea original to Thurston, which is now called the strip deformation (See [12] for details). Therefore for a given Lipschitz-extremal homeomorphism from X, there could be geodesic arcs, which are geodesics intersecting the boundary of X at both ends, realizing the extremal Lipschitz constant, instead of some chain recurrent lamination. This is not difficult to understand intuitively. We double S along its boundary to get a closed surface DS. Consider an order 2 homeomorphism of DS preserving each boundary component of S and use it to extend a hyperbolic structure on S

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to one on DS. This gives us an embedding of .T (S) into .T (DS). When we apply Thurston’s work in [14], the maximally stretched lamination could be one on DS intersecting the boundary of S. This suggests that it is indeed possible that the extremal Lipschitz constant is realized by geodesic arcs in S. On the other hand, this also tells us how to generalize Thurston’s metric to the Teichüller space of hyperbolic surfaces with geodesic boundary (See [5, 7]). Another generalization of Thurston’s metric theory is given by considering flat structures on surfaces. For the moment, the most studied case is the torus case. Let S be a torus. Given any flat structure m on S, rescaling it by a factor .t ∈ R>0 does not change it in an essential way. Therefore in order to study the Teichmüller space of flat tori, a renormalization is needed. For any flat structure m on S and any homotopy class .[α] of a curve .α on S, the m-geodesic representatives in .[α] are not unique but have the same m-length. We denote this quantity by .lm ([α]). In [1], Belkhirat, Papadopoulos and Troyanov chose once and for all a homotopy 2 ([α]). class .[α] of curves on S, and renormalized each flat metric m on S by .1/ lm Hence for any homotopy class .[β] of curves on S, its renormalized m-length is given by .

lm ([β]) . lm ([α])

Notice that this quantity is invariant when m is rescaled by a constant factor. By taking this renormalization, following the idea of Thurston, in [1], the authors defined two weak metrics considering the Lipschitz constant and the ratio between geodesic lengths respectively. Here a weak metric .d on a set X is a map d : X × X → R,

.

which is positive, and satisfies the triangle inequality. Moreover for any .x ∈ X, we have .d(x, x) = 0. We do not require .d to be symmetric, and it is possible that there are distinct points x and y in X, such that .d(x, y) = 0. In [1], the authors showed that same as in Thurston’s work, these two weak metrics coincide with each other and are asymmetric. Moreover the symmetrizations of these weak metrics coincide with the Teichmüller metric on .T (S). In [10], Miyachi, Ohshika and Papadopoulos studied this metric and showed that this metric is weak Finsler. They gave moreover a geometric description of the unit tangent circle at each point in .T (S) with respect to this metric. Instead of the above renormalization, one may consider renormalizing a flat metric m on S by the inverse of its area, so that the renormalized metric has area 1. Thurston’s metric theory of .T (S) for this renormalization is studied independently in [13] by Saˇglam and in [4] by Greenfield and Ji. It is proved in their work that in this case Thurston’s metric coincides with the Teichmüller metric. Meanwhile, by identifying .T (S) with the upper half plane .H, it is known that the Teichmüller metric coincides with the hyperbolic metric on .H. Hence Thurston’s metric constructed here is a Riemannian metric which is a constant multiple of the hyperbolic metric.

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The method used in [13] and that used in [4] are different. In [13], following the work in [1], Saˇglam considered lifting flat tori to the Euclidean plane, and did an explicit computation to get the extremal Lipschitz constant using the coordinates of the Euclidean plane. In [4], for any integer .n ≥ 2, Greenfield and Ji consider flat n-tori and their Teichmüller space denoted by .T (n). They identified .T (n) with the symmetric space .SL(n, R)/SO(n, R) and studied Thurston’s metric, the Teichmüller metric and the Weil–Petersson metric on this space using tools in Lie theory. In particular, the result about Thurston’s metric on .T (S) is a special case of their result. The key point of work in [4] and [13], as well as in [1], on .T (S) is to relate the Lipschitz-extremal homeomorphisms to affine maps between flat tori. Then the property of Thurston’s metric comes from the study of affine maps between flat tori. This also build a connection between Thurston’s metric and the Teichmüller metric on .T (S), due to the connection between the quasiconformal geometry and the affine geometry. And more generally, these also suggests that when staying in the Euclidean world, the Lipschitz geometry is closely related to the affine geometry. This could be used to extended Thurston’s metric theory further in the Euclidean world. In this chapter, we will give an explicitly explanation on Thurston’s metric theory for flat tori of area 1. Organization In Sect. 3.2, we recall some necessary background on the Teichmüller space of flat tori. In Sect. 3.3, we discuss the affine maps between flat tori. In Sect. 3.4, we discuss geodesic foliations on flat tori and their relation to affine maps between flat tori. In Sects. 3.5 and 3.6, we follow Thurston’s idea and discuss the Lipschitz constant of a homeomorphism between flat tori, and the geodesic length ratio for two flat tori. In Sect. 3.7, we show that the construction of Thurston also works in this case, and yields a metric on the Teichmüller space of flat tori. We then prove that this metric coincides with the Teichmüller metric on the Teichmüller space. In Sect. 3.8, we end this chapter by some further discussions.

3.2 Teichmüller Space of Flat Tori Let S be the oriented closed surface of genus 1. A flat structure on S is a maximal atlas where each chart is sent to the Euclidean plane .R2 and transition maps are restrictions of orientation-preserving isometries of .R2 . When S is equipped with a flat structure m, we call it a flat torus, and denote it by .(S, m). As mentioned in the introduction, two flat structures can be different by a rescaling which is not essential. Hence in the rest of this chapter, we will take a renormalization and assume that all flat structures on S have area 1. Let m be a flat structure on S. A marked flat structure on S is a pair .(m, f ) where m is a flat structure on S marked by an orientation-preserving homeomorphism f of S. Two marked flat structures .(m1 , f1 ) and .(m2 , f2 ) on S are said to be equivalent

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if there exists an orientation-preserving isometry ϕ : (S, m1 ) → (S, m2 ),

.

such that .ϕ ◦ f1 and .f2 are homotopic to each other. The Teichmüller space .T (S) of S is then defined by: T (S) := {equivalence classes of marked flat structures on S}

.

In each equivalence class .[m, f ] of marked flat structures, there is one whose marking is the identity map, denoted by .id. By fixing marking to be .id, we may identify the Teichmüller space with the spaces of isotopy classes of flat structures on S: T (S) := {flat structures on S}/isotopy.

.

The universal cover . S of S is homeomorphic to .R2 . Given any flat structure m on S, it can be lifted to a flat structure .m  on . S equivariant with respect to the deck transformation of the fundamental group .π1 (S) which can be realized as isometries of .( S, m ). Since .( S, m ) is isometric to .R2 , we have a representation ρm : π1 (S) → Isom+ R2 ,

.

where .Isom+ R2 is the group of orientation-preserving isometries of .R2 . We denote by .𝚪m the image of .ρm . The group .𝚪m is a discrete subgroup of .Isom+ R2 isomorphic to .π1 (S), hence isomorphic to .Z2 . Notice that .ρm depends on the isometry from .( S, m ) to .R2 . If ' two choices of this isometry induce .ρm and .ρm respectively, then there exists an isometry .h ∈ Isom+ R2 , such that ' ρm = h ◦ ρm ◦ h−1 .

.

Conversely, given any discrete faithful representation ρ : π1 (S) → Isom+ R2 ,

.

we denote its image by .𝚪. Consider the .𝚪-action on .R2 . The quotient space .R2 / 𝚪 is a flat torus. We call the volume of .R2 / 𝚪 the covolume of .ρ and also the covolume of .𝚪. Notice that two conjugate representations induce two isometric flat tori. Hence flat structures on S one-to-one correspond to the conjugacy classes of discrete faithful representations of .π1 (S) into .Isom+ R2 . Up to isotopy, a homeomorphism f of S to itself can be determined by the f image of two curves on S, intersecting each other once and corresponding to a pair of generators of .π1 (S). This is because the complement of such a pair in S is topologically a disk. A homeomorphism of a disk is determined up to isotopy

3 Thurston’s Metric on the Teichmüller Space of Flat Tori

53

by its restriction to the boundary circle. Hence if we consider a flat structure on S as a conjugacy class of discrete faithful representation of .π1 (S) into .Isom+ R2 , the marking in a marked flat structure can be considered as choice of a pair of generators of .π1 (S). From this point of view, choose once and for all a pair of generators, the space .T (S) can also be defined as follows: T (S) := {discrete faithful representations ρ : π1 (S)

.

→ Isom+ R2 of covolume 1}/conjugacy. Given any vector .u ∈ R2 , we can define a translation isometry of .R2 by Tu (p) = p + u,

.

for any .p ∈ R2 . Let m be any flat structure on S. Assume that the associated representation .ρm sends the chosen generators of .π1 (S) to translations .Th and .Tv respectively, where .h = (a, b) and .v = (c, d) satisfying  a c = ad − bc = 1. . det bd 

Then the image .𝚪 of .ρ is generated by .Th and .Tv , and we call .h and .v the horizontal vector and the vertical vector for .(m, ρ). Let p be any point of .R2 . The parallelogram P determined by p, .p +h, .p +v and 2 .p + h + v is a fundamental domain for the .𝚪-action on .R . Conversely, the vectors .h and .v can be recovered by choosing p as a marked vertex of P and considering the orientation of P . Moreover, if we conjugate .𝚪 by an isometry .h ∈ Isom+ R2 , then the parallelogram .h(P ) is the fundamental domain for the .h ◦ 𝚪 ◦ h−1 -action on .R2 , and the marked vertex p of P is sent to the vertex .h(p) of .h(P ). Hence the Teichmüller space .T (S) can be identified with {area 1 parallelograms with a marked vertex in R2 }/Isom+ R2 .

.

3.3 Affine Maps between Flat Tori Given any pair of flat structures m and .m' on S, we consider two parallelogram .Pm and .Pm' associated with m and .m' respectively as discussed above. Then there is an affine transformation f on .R2 sending each point p to f (p) = Ap + u,

.

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where .A ∈ SL(2, R) and .u ∈ R2 , such that f sends .Pm to .Pm' . The differential of f is then given by Df = A.

.

Since we consider the parallelogram for each flat structure on S up to isometry of R2 , if we fix .Pm , and change .Pm' to

.

Qm' = h(Pm' ) = R(Pm' ) + v,

.

with .h ∈ Isom+ R2 defined by .R ∈ SO(2, R) and .v ∈ R2 , then the associated affine transformation .f1 sending .Pm to .Qm' is given by the formula f1 (p) = RAp + Ru + v.

.

Therefore the differential of .f1 is given by Df1 = RA.

.

To see the Lipschitz constant of an affine transformation f defined by L(f ) := sup

.

p,q∈Pm p/=q

d(f (p), f (q)) , d(p, q)

with .d the Euclidean metric on .R2 , we consider the singular value of Df which is the square root of the eigenvalues of AT A,

.

where .AT is the transpose of A. Notice that this value stays the same when we change the map f to .f1 , since (RA)T RA = AT R T RA = AT A.

.

Now assume that the Jordan form of .At A is   t 0 . 0 t −1 √ with .t ≥ 1. Then the Lipschitz constant of f , and that of .f1 as well, are both . t. Now we glue the parallelograms .Pm and .Pm' back to .(S, m) and .(S, m' ) respectively. The affine transformation f from .Pm to .Pm' induces an affine map from .(S, m) to .(S, m' ) still denoted by f . The previous discussion shows that f has

3 Thurston’s Metric on the Teichmüller Space of Flat Tori

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√ Lipschitz constant . t. Notice that when fixing the image of one point on S, this affine map f from .(S, m) to .(S, m' ) is unique.

3.4 Geodesic Foliation on a Flat Torus Although the notion of measured foliation is purely topological, it is easier to describe it by considering geodesics with respect to a flat structure m on S. Given any flat structure m on S, a geodesic foliation .λ is a decomposition of .(S, m) into union of pairwise disjoint complete geodesics. Each geodesic in .λ is called a leaf of .λ. Similarly, we can define geodesic foliations on the Euclidean plane .R2 . In each geodesic foliation of .R2 , the leaves are straight lines parallel to one given direction. We can identify the universal cover of .(S, m) with .R2 . Then there is a one-to-one correspondence between geodesic foliations on .(S, m) and geodesic foliations on .R2 . Let us consider the flat structure m on S whose associated parallelogram is the square determined by .(0, 0), .(1, 0), .(1, 1) and .(0, 1). The horizontal vector and vertical vector for .(m, ρ) are given by .h = (1, 0) and .v = (0, 1). 2 Given any non-zero vector .u ∈ R2 , there is a unique geodesic foliation  .λ on .R whose leaves are parallel to .u. Consider the directed angle .θ counted from .(1, 0) to .u or .−u taking value in .(−π/2, π/2], with counter-clock direction to be the positive direction. We call τ = tan θ

.

2 .λ, with the convention .tan(π/2) = ∞. The leaves of  .λ are lines in .R the slope of  parallel to .u. We say that the geodesic foliation .λ on .(S, m) is along the slope .τ if it 2 lifts to  .λ a geodesic foliation on .R with slope .τ . We denote by .GF(S, m) the space of geodesic foliations on .(S, m). By considering slopes of geodesic foliations, we can identify .GF(S, m) with .RP1 which induces a topology on .GF(S, m). Leaves on .(S, m) for .λ are obtained by projecting leaves of  .λ using a covering map. A leaf of .λ on .(S, m) is periodic if it is compact, or if when we walk along this leaf with unit speed, we will come back to the starting point. A geodesic foliation on .(S, m) is periodic if all its leaves are periodic.

Proposition 4.1 Geodesic foliations on .(S, m) can be classified into two kinds by considering the dynamical property of their leaves: (i) Geodesic foliations along rational slopes: Every leaf of a geodesic foliation on .(S, m) along a rational slope is periodic, which is a closed geodesic in .(S, m). (ii) Geodesic foliations along irrational slopes: Every leaf of a geodesic foliation on .(S, m) along an irrational slope is dense in .(S, m). Moreover, periodic geodesic foliations on .(S, m) form a dense subset of .GF(S, m).

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Remark 4.2 As a convention, the slope .∞ is also counted as a rational slope. Let .m' be any flat structure on S, and f be an affine transformation from .(S, m) to .(S, m' ). The map f sends a geodesic foliation .λ on .(S, m) to a geodesic foliation ' ' .f (λ) on .(S, m ). Similarly we denote by .GF(S, m ) the space of geodesic foliations ' on .(S, m ). The map f induces a bijective map: f : GF(S, m) → GF(S, m' ), .

λ I→ f (λ).

Notice that f sends leaves of .λ to leaves of .f (λ). Moreover the dilatation along each leave of .λ is the same. Therefore we can define the following quantity, Lm,f (λ) :=

.

dm' (f (pλ ), f (qλ )) , dm (pλ , qλ )

where .pλ and .qλ are points on a same leaf of .λ close enough so that the geodesic realizing their distance is mapped to a geodesic realizing the distance between .f (pλ ) and .f (qλ ). A geodesic foliation .λ on .(S, m) is periodic if and only if its image .f (λ) on ' .(S, m ) is also periodic. Assume that .λ is periodic. Let .α be a leave of .λ which is a closed geodesic on .(S, m). Its image .f (α) is then a closed geodesic on .(S, m' ). We have Lm,f (λ) =

.

lm' (f (α)) , lm (α)

where .lm (α) (resp. .lm' (f (α))) is the m-length (resp. .m' -length) of .α (resp. .f (α)). Proposition 4.3 There is a unique geodesic foliation .λmax ∈ GF(S, m) satisfying Lm,f (λmax ) = max{Lm,f (λ) | λ ∈ GF(S, m)}.

.

Proof Since for each leaf of any geodesic lamination .λ, the dilatation along this leaf is evenly distributed on it, it is enough to consider what happens in a small neighborhood of a point on this leaf. We consider a disk .D ⊂ S, such that for any pair of points in D, the geodesic connecting them realizing their distance is contained in D. Moreover we ask the radius of D to be small enough, so that for any pair of points in .f (D), the geodesic connecting them realizing their distance is contained in .f (D). Notice that .f (D) is bounded by an ellipse. Hence .λmax exists and is the one parallel to the long axis of .f (D). ⨆ ⨅

3 Thurston’s Metric on the Teichmüller Space of Flat Tori

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The discussion in the above proof also shows that the function .Lm,f is continuous. Since periodic geodesic foliations form a dense subset of .GF(S, m), we have the following corollary Corollary 4.4 For any affine map f from .(S, m) to .(S, m' ), we have Lm,f (λmax ) =

.

lm' (f (α)) . α closed geodesic lm (α) sup

in (S,m)

Similarly, for any pair .m1 and .m2 of flat structures on S, given any affine map from .(S, m1 ) to .(S, m2 ), the following quantity is well-defined Lm1 ,f (λmax ) = max{Lm1 ,f (λ) | λ ∈ GF(S, m1 )}.

.

and equal to .

lm2 (f (α)) . α closed geodesic lm1 (α) sup

in (S,m1 )

Assume that f is isotopic to the identity map of S. We call the geodesic foliation λmax the maximally stretched foliation for the pair of points .([m1 ], [m2 ]) in .T (S).

.

3.5 Lipschitz Constants of Homeomorphisms between Flat Tori In this section, we consider more general homeomorphisms between flat tori. Let m and .m' be a pair of flat structures on S. Given any orientation-preserving homeomorphism .ϕ of S, its Lipschitz constant with respect to .(m, m' ) is defined by Lm,m' (ϕ) := sup

.

p,q∈S p/=q

dm' (ϕ(p), ϕ(q)) , dm (p, q)

where .dm and .dm' stand for the distance functions for m and .m' respectively. Notice that this quantity may be infinite, however since every homeomorphism of S is isotopic to a diffeomorphism, the following quantity is always finite Lm,m' ([ϕ]) = 'inf Lm,m' (ϕ ' ),

.

where .[ϕ] is the isotopy class of .ϕ.

ϕ ∈[ϕ]

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Let .[m] and .[m' ] be two points in .T (S). From the above discussion, we have the following well-defined quantity associated with .([m], [m' ]) L([m], [m' ]) := Lm,m' ([id])

.

where .id is the identity map on S. Hence we have a map L : T (S) × T (S) → R, .

([m], [m' ]) I→ L([m], [m' ]),

Remark 5.1 Our notation is slightly different from the one used in [14]. For our convenience, we do not take the logarithm when defining L. Proposition 5.2 For any points .[m], [m' ] ∈ T (S), we have L([m], [m' ]) ≥ 1.

.

Proof We follow the proof of Thurston for the hyperbolic surface case in [14], and prove this proposition by contradiction. Assume that there exists a pair .([m], [m' ]) ∈ T (S) × T (S), such that L([m], [m' ]) < 1.

.

Then we have a constant .0 < ϵ < 1, such that L([m], [m' ]) ≤ 1 − ϵ.

.

Hence there exists a homeomorphism ϕ : S → S,

.

isotopic to .id, such that ϵ Lm,m' (ϕ) ≤ 1 − . 2

.

Therefore for any embedded disk D in the flat surface .(S, m), we have the following area comparison:  ϵ Area(ϕ(D)) ≤ 1 − Area(D). 2

.

This gives the comparison between the area of .(S, m) and that of .(S, m' ):   ϵ ϵ 1 = Area(S, m' ) ≤ 1 − Area(S, m) = 1 − < 1, 2 2

.

which is a contradiction.

⨆ ⨅

3 Thurston’s Metric on the Teichmüller Space of Flat Tori

59

3.6 Curve Length Ratios for Two Flat Structures on S To get more properties of L, we will follow the idea of Thurston in [14] and study the ratio between lengths of curves on torus with respect to two flat structures. Since the fundamental group of S is abelian, all curves on S are homotopic to simple curves. We use .C0 (S) to denote the collection of homotopy classes of curves in S. Let .[α] and .[β] be two homotopy class of curves on S. We call their intersection number the following quantity:     1 1 ' ' t) ∈ S i([α], [β]) := min × S | α (s) = β (t)} {(s, . '

.

α ∈[α] β ' ∈[β]

Two homotopy classes .[α] and .[β] corresponds to a pair of generators of .π1 (S) if and only if i([α], [β]) = 1.

.

A triple of homotopy classes .[α], .[β] and .[γ ] form a Farey triple if i([α], [β]) = i([α], [γ ]) = i([β], [γ ]) = 1.

.

Let .γ be a curve in S. We then have a well-defined map lγ : T (S) → R>0 , .

[m] I→ l[γ ] (m) := lm ([γ ]),

where .lm ([γ ]) is the same as defined in the introduction. We first recall the finite marked length spectrum rigidity for a flat structure on S. Proposition 6.1 Let .[α], [β], [γ ] ∈ C0 (S) be a Farey triple. Then every point .[m] ∈ T (S) is determined by the triple  .

l[α] (m), l[β] (m), l[γ ] (m) ∈ (R>0 )3 .

Proof In the previous section, we showed that .T (S) is identified with the space of parallelograms of area 1 in .R2 up to isometry. A parallelogram is determined up to isometry by its side lengths and the length of one of its diagonal. Given any flat torus .(S, m), we can cut S along one geodesic representative .α ' in ' .[α] and one geodesic representative .β in .[β]. Since i([α], [β]) = 1,

.

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the complement .P = S \ (α ' ∪ β ' ) is a Euclidean parallelogram. Since .[α], [β], [γ ] is a Farey triple, there exists a geodesic representative .γ ' ∈ [γ ] passing through the intersection point .α ' ∩ β ' . It is cut into one diagonal in P . By the above discussion, the geometry of P is determined by the lengths of .α ' , .β ' and .γ ' . ⨆ ⨅ For any pair of points .([m], [m' ]) in .T (S), we can consider the following quantity: K([m], [m' ]) :=

.

l[γ ] (m' ) . [γ ]∈C0 (S) l[γ ] (m) sup

Remark 6.2 Same as for the map L, our notation for K is slightly different from the one used in [14]. For our convenience, we do not take the logarithm when defining K. The above finite marked length spectrum rigidity has the following corollary. Corollary 6.3 If .[m] = / [m' ], then we have K([m], [m' ]) /= 1.

.

Proof Let .[α], [β], [γ ] ∈ C0 (S) be a Farey triple. By the previous proposition, we have .[m] = [m' ] if and only if  .

 l[α] (m), l[β] (m), l[γ ] (m) = l[α] (m' ), l[β] (m' ), l[γ ] (m' ) .

Therefore, if .K([m], [m' ]) = 1, the above identity holds, hence .[m] = [m' ].

⨆ ⨅

This can be completed as follows by considering the other direction which holds trivially by considering the definition of K. Corollary 6.4 For any .[m], [m' ] ∈ T (S), we have .K([m], [m' ]) = 1 if and only if ' .[m] = [m ]. We compare the maps L and K, and have the following theorem. Theorem 6.5 The two maps L and K coincide with each other. Proof Let .[m], [m' ] be two points in .T (S). We first show that L([m], [m' ]) ≥ K([m], [m' ]).

.

Let .ϕ be a homeomorphism of S isotopic to .id. For any .[γ ] ∈ C0 (S), we denote by γ ' a geodesic representative of .[γ ]. Then we have

.

.

l[γ ] (m' ) lm' (ϕ(γ ' )) ≤ ≤ Lm,m' (ϕ), l[γ ] (m) lm (γ ' )

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where .lm (γ ' ) (resp. .lm' (ϕ(γ ' ))) is the m-length (resp. .m' -length) of .γ ' (resp. .ϕ(γ ' )). Therefore, we have .

l[γ ] (m' ) ≤ L([m], [m' ]). l[γ ] (m)

By considering all curves on S, we have L([m], [m' ]) ≥ K([m], [m' ]).

.

On the other hand, in the previous section, we have shown that the quantity K([m], [m' ]) can be realized as a Lipschitz constant of a affine map from .(S, m) to .(S, m' ), therefore we have the inequality for the other direction

.

L([m], [m' ]) ≤ K([m], [m' ]).

.

As a conclusion, we have the equality L([m], [m' ]) = K([m], [m' ]).

.

⨆ ⨅ Combining the study on K and the above theorem, we have the following corollaries. Corollary 6.6 For any .[m], [m' ] ∈ T (S), we have .L([m], [m' ]) = 1 if and only if ' .[m] = [m ]. Corollary 6.7 For any .[m], [m' ] ∈ T (S), the extremal Lipschitz constant ' ' .L([m], [m ]) is realized by an affine map from .(S, m) to .(S, m ). Remark 6.8 Notice that by fixing the image of one point on S, the affine map from (S, m) to .(S, m' ) is unique. However, it is shown in [4, Proposition 4.6] that there exists a pair of flat tori for which there is a infinite family of homeomorphisms realizing the extremal Lipschitz constant, even under the condition of fixing the image of one point on S.

.

3.7 Thurston’s Metric on T (S) Using the extremal Lipschitz constant for a pair .([m], [m' ]) in .T (S), we define the following map dT h : T (S) × T (S) → R≥0 , .

([m], [m' ]) I→ log L([m], [m' ]).

The goal of this section is to show the following result.

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Theorem 7.1 The map .dT h is a distance function on .T (S). As a consequence of this theorem, the map .dT h gives a metric on .T (S), which is called the Thurston’s metric. Proof Previously, we showed that for any pair .[m] and .[m' ], we have • .L([m], [m' ]) ≥ 1; • .L([m], [m' ]) = 1 if and only if .[m] = [m' ]. Hence we have • .dT h ([m], [m' ]) ≥ 0; • .dT h ([m], [m' ]) = 0 if and only if .[m] = [m' ]. The symmetry of .dT h comes from the fact that the extremal Lipschitz constant can be realized by an affine map. For any pair .(m, m' ) of flat structures on S, consider the affine map f : (S, m) → (S, m' ),

.

which is isotopic to the identity. The inverse f −1 : (S, m' ) → (S, m),

.

is also isotopic to the identity. Moreover, consider their differential at each point we have D(f −1 ) = (Df )−1 .

.

Since Df is in .SL(2, R), the above two matrices have the same maximal singular value. Hence we have dT h ([m], [m' ]) = dT h ([m' ], [m]).

.

Now we prove the triangle inequality. Let m, .m' and .m'' be three flat structures on S. For any given homeomorphisms ϕ : (S, m) → (S, m' ),

.

ϕ ' : (S, m' ) → (S, m'' ),

their composition is a homeomorphism ψ = ϕ ' ◦ ϕ : (S, m) → (S, m'' ).

.

For any pair of points p and q on S, we have .

dm'' (ψ(p), ψ(q)) dm'' (ψ(p), ψ(q)) dm' (ϕ(p), ϕ(q)) = . dm (p, q) dm' (ϕ(p), ϕ(q)) dm (p, q)

3 Thurston’s Metric on the Teichmüller Space of Flat Tori

63

Therefore, we have dT h ([m], [m'' ]) ≤ dT h ([m' ], [m'' ]) + dT h ([m], [m' ])

.

As a conclusion, we prove that .dT h is a distance function on .T (S).

⨆ ⨅

As mentioned in the introduction, there are various metrics on .T (S). Of particular interests, we consider the Teichmüller metric which comes from the study of quasiconformal maps between Riemann surfaces. The following is one definition for this metric. Let m and .m' be two flat structures on S. Let f be any diffeomorphism from ' .(S, m) to .(S, m ). We define the dilatation of f as follows D(f ) := sup

.

p∈S

sup{|(Df )p (u)| | u ∈ Tp S, |u| = 1} , inf{|(Df )p (u)| | u ∈ Tp S, |u| = 1}

here .| · | stands for the norm of tangent vectors given by the associated flat structure. Then the Teichmüller distance on .T (S) between .[m] and .[m' ] is given by dT eich ([m], [m' ]) =

.

1 inf{log D(f ) | f ∈ Diff0 (S)}, 2

where .Diff0 (S) is the group of diffeomorphism of S isotopic to identity. In [1, Theorem 5], Belkhirat, Papadopoulos and Troyanov proved the following result. Theorem 7.2 The affine map from .(S, m) to .(S, m' ) is extremal for Teichmüller metric. Notice that if f is an affine map, let A denote its differential. Then .D(f ) is the ratio between two singular values of A. Hence we have the following corollary. Corollary 7.3 The distance functions .dT h and .dT eich coincide with each other. It is known that up to a constant multiple the Teichmüller metric coincides with the hyperbolic metric, when we identify .T (S) with the upper half plane .H. Hence as a corollary we have Corollary 7.4 Thurston’s metric on .T (S) is a Riemannian metric which is a constant multiple of the hyperbolic metric.

3.8 Further Discussions on Extremal Lipschitz Constants and Singular Values of Affine Maps If we review the study of the maps L and K defined above, the restriction on flat structures to have area 1 is not essential. We may repeat the whole study for any pair of flat structures on S, the only change that we have to make is to consider the group + .GL (2, R) of matrices with positive determinants instead of .SL(2, R).

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To be more precise, given any pair of flat structures on S, denoted by m and .m' respectively, we can define the quantities .L([m], [m' ]) and .K([m], [m' ]) in the exact same way as for area 1 flat structures. Hence the functions L and K can be extended to the space .F(S) of isotopy classes of flat structures on S, with no restriction on their areas. By a same proof as in the previous sections, we can show that the equality .L = K still holds in this case. Again with the same proof as in the previous sections, we can show that the extremal Lipschitz constant can be realized by the affine map f from .(S, m) to + ' .(S, m ), which is the maximal singular value of .Df ∈ GL (2, R). Notice that the ' value of .L([m], [m ]) could be smaller than 1. For example, when .m' = tm with .0 < t < 1, then we have L([m], [m' ]) =

.

√ t.

In general, we have the following equality. Proposition 8.1 For any flat structures m and .m' on S, and for any positive real numbers t and .t ' , we have

'

'

L([tm], [t m ]) =

.

t' L([m], [m' ]). t

We consider the renormalization on flat structures taken in [1]. Using the same notation, let .ϵ be the curve on S chosen once and for all. Given any flat structure m 2 ([ϵ]), and denote the resulting flat structure by .m . on S, we renormalize it by .1/ lm 0 For any pair of flat structures m and .m' on S, using the above relation, we have L([m0 ], [m'0 ]) = .

lm ([ϵ]) L([m], [m' ]) lm' ([ϵ])

=

L([m], [m' ]) lm' ([ϵ])/ lm ([ϵ])

≥

K([m], [m' ]) = 1. K([m], [m' ])

Notice that the equality holds if and only if K([m], [m' ]) =

.

lm' ([ϵ]) , lm ([ϵ])

and this is possible. Rescaling a flat structure gives a trivial example for this to be true. To see a less trivial example, we consider the previous correspondence between flat structures of area 1 on S and parallelograms of area 1 on .R2 up to isometry. Assume that .ϵ corresponds to one pair of opposite sides of the parallelogram, and m

3 Thurston’s Metric on the Teichmüller Space of Flat Tori

65

and .m' correspond to Q and .Q' respectively which are both rectangle. Assume that lm ([ϵ]) < lm' ([ϵ]),

.

then we may verify that K([m], [m' ]) =

.

lm' ([ϵ]) . lm ([ϵ])

Therefore, when considering the renormalization used in [1], the function L can only induces a weak metric instead of a metric, since there are distinct points having weak distance 0. Another way to see the above inequality is to use the fact that .L([m], [m' ]) equals to the maximal singular value of the differential of the affine map f which realizes the extremal Lipschitz constant. Under the renormalization in [1], the matrix is always conjugate to   1a ∈ GL+ (2, R). . 0b If m and .m' are such that a is 0 and b smaller than 1, then the maximal singular value of Df is one, hence the weak metric cannot distinguish m and .m' . At the same time, the following relation still holds D(f −1 ) = (Df )−1 .

.

Hence the weak metric constructed in [1] is asymmetric. This point of view also shows the reason why the symmetrization of the weak metrics considered in [1] is the Teichmüller metric. If we consider the differential of f , and assume its diagonalization is  .

t1 0 0 t2



with .t1 ≥ t2 > 0, then the Teichmüller distance between .[m] and .[m' ] is given by 1 .dT eich ([m], [m ]) = log 2 '

=



 √ t1 1 = log t1 + log t2−1 t2 2

log L([m], [m' ]) + log L([m' ], [m]) , 2

which is the equality proved in [1, Theorem 5]. Notice that since .ϵ always has length 1, we have t1 ≥ 1 ≥ t2 .

.

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In [4, Theorem 5.4], the same relation between the extremal Lipschitz constant and the singular value of the affine map between two flat tori is proved to be still true when considering n-tori. The differential of the Lipschitz extremal affine map f is then in .SL(n, R). The relation D(f −1 ) = (Df )−1 ,

.

still holds, but the maximal singular value of Df equals to that of .D(f −1 ) is not always true for volume 1 flat n-tori when .n > 2. Hence the Thurston metric in this case is asymmetric. Acknowledgments The author would like to thank Athanase Papadopoulos for carefully reading the early version of this chapter and for the valuable comments and suggestions. The author was supported by the Fundamental Research Funds for the Central Universities, Nankai University (Grant Number 63231055), Natural Science Foundation of Tianjin (Grant number 22JCYBJC00690) and LPMC at Nankai University.

References 1. A. Belkhirat, A. Papadopoulos, M. Troyanov, Thurston’s weak metric on the Teichmüller space of the torus. Trans. Amer. Math. Soc. 357(8), 3311–3324 (2005) 2. D. Dumas, A. Lenzhen, K. Rafi, J. Tao, Coarse and fine geometry of the Thurston metric. Forum Math. Sigma 8, paper no. e28, 58 pages (2020) 3. A. Fathi, F. Laudenbach, V. Poénaru, Travaux de Thurston sur les surfaces, Astérisque, vols. 66–67 (Société Mathématique de France, Paris, 1979) 4. M. Greenfield, L. Ji, Metrics and compactifications of Teichmüller spaces of flat tori. Asian J. Math. 25(4), 477–504 (2021) 5. Y. Huang, A. Papadopoulos, Optimal Lipschitz maps on one-holed tori and the Thurston metric theory of Teichmüller space. Geom. Dedicata 214, 465–488 (2021) 6. Y. Huang, K. Ohshika, A. Papadopoulos, The infinitisimal and global Thurston geometry of Teichmüller space (2021). arXiv:2111.13381 7. L. Liu, A. Papadopoulos, W. Su, G. Théret, On length spectrum metrics and weak metrics on Teichmüller spaces of surfaces with boundary. Ann. Acad. Sci. Fenn. Math. 35(1), 255–274 (2010) 8. H.A. Masur, Y.N. Minsky, Geometry of the complex of curves. I. Hyperbolicity. Invent. Math. 138(1), 103–149 (1999) 9. Y.N. Minsky, The classification of punctured-torus groups. Ann. of Math. 149(2), 559–626 (1999) 10. H. Miyachi, K. Ohshika, A. Papadopoulos, Tangent spaces of the Teichmüller space of the torus with Thurston’s weak metric. Ann. Fenn. Math. 47(1), 325–334 (2022) 11. H. Pan, Local rigidity of Teichmüller space with the Thurston metric. Sci. China Math. 66(8), 1751–1766 (2023) 12. A. Papadopoulos, G. Théret, Shortening all the simple closed geodesics on surfaces with boundary. Proc. Amer. Math. Soc. 138(5), 1775–1784 (2010) ˙ Saˇglam, On the moduli space of flat tori having unit area. Int. Electron. J. Geom. 14(1), 13. I. 59–65 (2021) 14. W.P. Thurston, Minimal stretch maps between hyperbolic surfaces, Collected works of William P. Thurston with commentary. Vol. I. Foliations, surfaces and differential geometry (2022), pp. 533–585. 1986 preprint, 1998 eprint

Chapter 4

The Anti-de Sitter Proof of Thurston’s Earthquake Theorem Farid Diaf and Andrea Seppi

Abstract Thurston’s earthquake theorem asserts that every orientation-preserving homeomorphism of the circle admits an extension to the hyperbolic plane which is a (left or right) earthquake. The purpose of these notes is to provide a proof of Thurston’s earthquake theorem, using the bi-invariant geometry of the Lie group .PSL(2, R), which is also called Anti-de Sitter three-space. The involved techniques are elementary, and no background knowledge is assumed apart from some twodimensional hyperbolic geometry. Keywords Hyperbolic geometry · Teichmüller theory · Anti-de Sitter geometry MSC 2020: 30F60, 53C50, 57M50

4.1 Introduction Since the 1980s, earthquake maps have played an important role in the study of hyperbolic geometry and Teichmüller theory. These are (possibly discontinuous) maps of the hyperbolic plane to itself that, roughly speaking, are isometric in the complement of a subset of the hyperbolic plane which is a disjoint union of geodesics, and they “slip” along the “faults” represented by these geodesics. In particular, they may have points of discontinuity there. In general, an earthquake map can be complicated, and it is an isometry only on the connected components of the complement of a measured geodesic lamination. To achieve the solution of the Nielsen realization problem [11], Steven Kerckhoff proved the so-called earthquake theorem for closed hyperbolic surfaces, that is, the existence of a left (right) earthquake map between any two closed hyperbolic surfaces of the same genus. In [20], William Thurston gave a generalization,

F. Diaf · A. Seppi () University of Grenoble Alpes, Grenoble, France e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 K. Ohshika, A. Papadopoulos (eds.), In the Tradition of Thurston III, https://doi.org/10.1007/978-3-031-43502-7_4

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proved by independent methods, to a universal setting, which is the statement that we consider in the present notes: he proved that every orientation-preserving homeomorphism of the circle admits an extension to the hyperbolic plane which is a (left or right) earthquake. Earthquake maps have been extensively studied later in various directions, see [9, 10, 13, 14, 16–18]

4.1.1 Mess’ Groundbreaking Work and Later Developments In his 1990 pioneering paper [12], Geoffrey Mess has first highlighted the deep connections between the Teichmüller theory of hyperbolic surfaces, and threedimensional Lorentzian geometries of constant sectional curvature. In particular, the so-called Anti-de Sitter geometry is the Lorentzian geometry of constant negative curvature—that is, the Lorentzian analogue of hyperbolic geometry. One of the models of Anti-de Sitter three-space is simply the Lie group .PSL(2, R), endowed with a Lorentzian metric which is induced by the (bi-invariant) Killing form on its Lie algebra. This is the model that we adopt in the present work. Mess has then observed that convex hulls in Anti-de Sitter space can be used, together with a Gauss map construction for spacelike surfaces, to prove earthquake theorems in hyperbolic geometry. In [12], Mess outlined the proof of the earthquake theorem between closed hyperbolic surfaces. His groundbreaking ideas have been improved and implemented by several authors, leading to many results of existence of earthquake maps in various settings [1, 2, 4, 15] and of other interesting types of extensions [3, 5, 6, 19]. See also the paper [7], which is a detailed introduction to Anti-de Sitter geometry, contains a general treatment of the Gauss map, but only sketches some of the ideas that appear in the proof of Mess. The literature seems to lack a complete proof of the earthquake theorem, in Thurston’s universal version, which relies on Anti-de Sitter geometry. In this note, we will provide a detailed proof of Thurston’s earthquake theorem (Theorem 4.1), and we will then recover (Corollary 4.6) the existence of earthquake maps for closed hyperbolic surfaces. While the proofs that appear in [12], and in several of the aforementioned subsequent works, make use of a computation of the holonomy, here we will simply work with the definition of earthquake map. In fact, the proof presented here, although going through several technical steps, entirely involves elementary tools. The only required knowledge for these notes is the hyperbolic plane geometry in the upper half-space model, and the very basic definitions of Lie groups theory and Lorentzian geometry.

4.1.2 A Quick Comparison of the Two Proofs It is also worth remarking that the proof presented here, and suggested by Mess, is not entirely different in spirit from Thurston’s proof in [20]. Indeed, the starting

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point of Thurston’s proof consists in considering, given an orientation-preserving homeomorphism f or the circle, those isometries .γ of the hyperbolic plane such that the composition .h := γ ◦ f is extreme left: that is, such that h has a lift .h˜ : R → R satisfying .h(x) ≤ x and whose fixed point set is non-empty. In Thurston’s words, “h moves points counterclockwise on the circle, except for those points that it fixes”. Then Thurston defines the earthquake map to be equal to .γ −1 on the convex hull of the fixed points of h. This has an interpretation in terms of Anti-de Sitter geometry. Spacelike planes in Anti-de Sitter space, which is simply the Lie group .PSL(2, R), are isometrically embedded copies of the hyperbolic plane, and are parameterized by elements of .PSL(2, R) itself, via a natural duality. For instance, the dual plane to the identity consists of all elliptic elements of order two, which is identified with the hyperbolic plane itself via the fixed point map. The “extreme left condition” as above is then exactly equivalent to the condition that the spacelike plane dual to .γ is a past support plane of the convex hull of the graph of f , which can be seen as a subset of the boundary at infinity of Anti-de Sitter space. The proof presented here then consists in considering the left and right projections, defined on the past boundary components of the convex hull, and to consider the composition E of one projection with the inverse of the other. It turns out that this composition map E is indeed equal to .γ −1 on the convex hull of the fixed points of .γ ◦ f , as in Thurston’s ansatz. Of course one can replace extreme left by extreme right, and past boundary with future boundary, to obtain right earthquakes instead of left earthquakes. We remark that the main statement proved by Thurston also includes a uniqueness part. In fact, the earthquake map is not quite unique, but it is up to a certain choice that has to be made at every geodesic where it is discontinuous. We will give an interpretation of this phenomenon in terms of a choice of support plane at the points of the boundary of the convex hull that admit several support planes, but we will not provide a proof of the uniqueness part here.

4.1.3 Main Elements of the Anti-de Sitter Proof Despite the above analogies with Thurston’s original proof of the existence of left and right earthquakes, developing the proof in the Anti-de Sitter setting then leads to remarkable differences with respect to Thurston’s proof. A large part of our proof is actually achieved by a reduction to the situation of an orientation-preserving homeomorphism of the circle which is equal to the restriction of an element .γi of .PSL(2, R) on an interval .Ii (.i = 1, 2), where .I1 ∪ I2 equals the circle. In this situation the earthquake extension is already well-known, and consists of a simple earthquake. However, understanding this example in detail from the perspective of Anti-de Sitter geometry—which corresponds to the situation where a boundary component of the convex envelope of f is the union of two totally geodesic half-planes meeting along a geodesic—then permits to prove easily some of the

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fundamental properties that one has to verify in order to show that the composition map E is an earthquake map. There are furthermore two main technical statements that we have to prove. The first is the fact that the left and right projections (although they can be discontinuous) are bijective—which is essential since the earthquake map is defined as the inverse of the left projection post-composed with the right projection, and implies that E itself is a bijection of the hyperbolic plane. While injectivity is easy using the aforementioned example of two totally geodesic planes meeting along a geodesic, surjectivity requires a more technical argument. The second statement is an extension lemma, which ensures that the left and right projections (although sometimes discontinuous) extend continuously to the boundary, and the extension is simply the projection from the graph of f onto the first and second factor. This ensures that the composition E of the right projection with the inverse of the left projection extends to f itself on the circle at infinity. Some of the above steps do of course involve a number of technical difficulties, but the language of Anti-de Sitter geometry is, in our opinion, extremely effective, and permits to stick to quite elementary techniques in the entire work.

4.2 Earthquake Maps Throughout this chapter, we will use the upper half-plane model of the hyperbolic plane .H2 , that is, .H2 is the half-space .Im(z) > 0 in .C endowed with the Riemannian metric .|dz|2 /Im(z)2 of constant curvature .−1. Its visual boundary .∂∞ H2 is therefore identified with .R ∪ {∞}, and .H2 = H2 ∪ ∂∞ H2 is endowed with the topology given by the one-point compactification of the closed half-plane .Im(z) ≥ 0. The isometry group of .H2 is identified with the group .PSL(2, R) acting by homographies, and its action naturally extends to .∂∞ H2 . Definition 4.1 A geodesic lamination .λ of .H2 is a collection of disjoint geodesics that foliate a closed subset .X ⊆ H2 . The closed set X is called the support of .λ. The geodesics in .λ are called leaves. The connected components of the complement 2 .H \ X are called gaps. The strata of .λ are the leaves and the gaps. Given a hyperbolic isometry .γ of .H2 , the axis of .γ is the geodesic .𝓁 of .H2 connecting the two fixed points of .γ in .∂∞ H2 . Therefore the axis .𝓁 is preserved by .γ , and when restricted to .𝓁, .γ |𝓁 : 𝓁 → 𝓁 acts as a translation with respect to any constant speed parameterization of .𝓁. Given two subsets .A, B of .H2 , we say that a geodesic .𝓁 weakly separates A and B if A and B are contained in the closure of different connected components of 2 .H \ 𝓁.

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Definition 4.2 A left (resp. right) earthquake of .H2 is a bijective map .E : H2 → H2 such that there exists a geodesic lamination .λ for which the restriction .E|S of E to any stratum S of .λ is equal to the restriction of an isometry of .H2 , and for any two strata S and .S ' of .λ, the comparison isometry Comp(S, S ' ) := (E|S )−1 ◦ E|S '

.

is the restriction of an isometry .γ of .H2 , such that: • .γ is different from the identity, unless possibly when one of the two strata S and ' .S is contained in the closure of the other; • when it is not the identity, .γ is a hyperbolic transformation whose axis .𝓁 weakly separates S and .S ' ; • moreover, .γ translates to the left (resp. right), seen from S to .S ' . Let us clarify the meaning of this last condition. Suppose .f : [0, 1] → H2 is smooth path such that .f (0) ∈ S, .f (1) ∈ S ' and the image of f intersects .𝓁 transversely and exactly at one point .z0 = f (t0 ) ∈ 𝓁. Let .v = f ' (t0 ) ∈ Tz0 H2 be the tangent vector at the intersection point. Let .w ∈ Tz0 H2 be a vector tangent to the geodesic .𝓁 pointing towards .γ (z0 ). Then we say that .γ translates to the left (resp. right) seen from S to .S ' if .v, w is a positive (resp. negative) basis of .Tz0 H2 , for the standard orientation of .H2 . It is important to observe that this condition is independent of the order in which we choose S and .S ' . That is, if .Comp(S, S ' ) translates to the left (resp. right) seen from S to .S ' , then .Comp(S ' , S) translates to the left (resp. right) seen from .S ' to S. We remark that an earthquake E is not required to be continuous. In fact, in some cases it will not be continuous, for instance when the lamination .λ is finite, meaning that .λ is a collection of a finite number of geodesics. This is best visualized in the following simple example. Example 4.1 The map E : H2 → H2

.

defined in the upper half-space model of .H2 by:

E(z) =

.

⎧ ⎪ ⎪ ⎨z

if Re(z) < 0

az ⎪ ⎪ ⎩bz

if Re(z) > 0

if Re(z) = 0

is a left earthquake if .1 < a < b, and a right earthquake if .0 < b < a < 1. The lamination .λ that satisfies Definition 4.2 is composed of a unique geodesic, namely the geodesic .𝓁 with endpoints 0 and .∞.

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It is clear that the earthquake map E from Example 4.1 is not continuous along .𝓁. Despite the lack of continuity, Thurston proved that any earthquake map extends continuously to an orientation-preserving homeomorphism of .∂∞ H2 , meaning that there exists a (unique) orientation-preserving homeomorphism .f : ∂∞ H2 → ∂∞ H2 such that the map E(z) =

.

 E(z) if z ∈ H2 f (z)

if z ∈ ∂∞ H2

is continuous at every point of .∂∞ H2 . Then Thurston provided a proof of the following theorem, that he called “geology is transitive”: Theorem 4.1 Given any orientation-preserving homeomorphism .f : ∂∞ H2 → ∂∞ H2 , there exists a left earthquake map of .H2 , and a right earthquake map, that extend continuously to f on .∂∞ H2 . We remark that the earthquake map is not unique, as shown by Example 4.1, which provides a family of left (resp. right) earthquake maps extending the homeomorphism

f (x) =

.

⎧ ⎪ ⎪x ⎨ bx ⎪ ⎪ ⎩∞

if x ≤ 0 if x ≥ 0

,

if x = ∞

parameterized by the choice of .a ∈ (1, b) (resp. .a ∈ (b, 1)). Thurston’s theorem is actually stronger than the statement of Theorem 4.1 above, since it characterizes the non-uniqueness as well. In short, the range of choices of the earthquake extension as in Example 4.1 is essentially the only indeterminacy that occurs, and it happens exactly on each leaf of the lamination where the earthquake is discontinuous. We will not deal with the uniqueness part as in Thurston’s work here. Nevertheless, in Sect. 4.6.4 we will show that our proof permits to recover the existence of earthquake maps between homeomorphic closed hyperbolic surfaces, not relying on the uniqueness property.

4.3 Anti-de Sitter Geometry In this section, we will introduce the fundamental notions in Anti-de Sitter geometry. For more details, the reader can consult [7, Section 3].

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4.3.1 First Definitions The three-dimensional Anti-de Sitter space .AdS3 is the Lie group .PSL(2, R), that is, the group of orientation-preserving isometries of .H2 , endowed with a bi-invariant metric of signature .(2, 1) (namely, a Lorentzian metric) which we now construct. Consider first the double cover .SL(2, R) of .PSL(2, R), which we realize as the subset of matrices of unit determinant in the four-dimensional vector space .M2 (R) of 2-by-2 matrices. Endow .M2 (R) with the quadratic form q: q(A) = − det(A) .

.

It can be checked that q has signature .(2, 2). The associated bilinear form is expressed by the formula: 1 〈A, B〉 = − tr(A · adj(B)) 2

.

(4.1)

for .A, B ∈ M2 (R), where .adj denotes the adjugate matrix, namely     d −b ab . = .adj −c a cd Then .SL(2, R) is realized as the subset of .M2 (R) defined by the condition q(A) = −1, and the restriction of .〈·, ·〉 to the tangent space of .SL(2, R) at every point defines a pseudo-Riemannian metric of signature .(2, 1). We will still denote by .〈·, ·〉 this metric on .SL(2, R), and by q the corresponding quadratic form. It can be shown that this metric has constant curvature .−1, and the restriction of .〈·, ·〉 to the Lie algebra .sl(2, R) coincides with .1/8 times the Killing form of .SL(2, R). Clearly both .SL(2, R) and q are invariant under multiplication by minus the identity matrix, hence the quotient .PSL(2, R) = SL(2, R)/{±1} is endowed with a Lorentzian metric of constant curvature .−1, and is what we call the (threedimensional) Anti-de Sitter space .AdS3 . It turns out that the group of orientationpreserving and time-preserving isometries of .AdS3 is the group .PSL(2, R) × PSL(2, R), acting by left and right multiplication on .PSL(2, R) ∼ = AdS3 :

.

(α, β) · γ := αγβ −1 .

.

Although orientation-preserving and time-preserving are notions that do not depend on a chosen orientation, we will fix here an orientation and a time-orientation of .AdS3 ∼ = PSL(2, R). To define an orientation on a Lie group, it actually suffices to define it in the Lie algebra, namely the tangent space at the identity .1. Hence

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we declare that the following is an oriented basis (which is actually orthonormal) of sl(2, R):

.

 V =

.

01 10



 W =

 1 0 0 −1

 U=

0 −1 1 0

 (4.2)

Observe that the vectors .V , W are spacelike (i.e. .q(V , V ) > 0 and .q(W, W ) > 0), while U is timelike (.q(U, U ) < 0). One can check that U is the tangent vector to the one-parameter group of elliptic isometries of .H2 fixing .i ∈ H2 , parameterized by the angle of clockwise rotation; V and W are vectors tangent to the one-parameter groups of hyperbolic isometries fixing the geodesics with endpoints .(−1, 1) and .(0, ∞) respectively. Analogously, to define a time-orientation it suffices to define it in the Lie algebra, and we declare that U is a future-pointing timelike vector.

4.3.2 Boundary at Infinity The boundary at infinity of .AdS3 is defined as the projectivization of the cone of rank one matrices in .M2 (R): ∂∞ AdS3 = P{A | q(A) = 0, A /= 0} .

.

We endow .AdS3 = AdS3 ∪ ∂∞ AdS3 with the topology induced by seeing both 3 3 .AdS and .∂∞ AdS as subsets of the (real) projective space .PM2 (R) over the vector space .M2 (R). Hence .AdS3 is the compactification of .AdS3 in .PM2 (R). It will be extremely useful to consider the homeomorphism between .∂∞ AdS3 and .RP1 ×RP1 , which is defined as follows: .

δ : ∂∞ AdS3 → RP1 × RP1 [X] I→ (Im(X), Ker(X))

(4.3)

where of course in the right-hand side we interpret .RP1 as the space of onedimensional subspaces of .R2 . Since we have .Im(AXB −1 ) = A · Im(X) and −1 ) = B · Ker(X), the map .δ is equivariant with respect to the action .Ker(AXB of the group .PSL(2, R) × PSL(2, R), acting on .∂∞ AdS3 as the natural extension of the group of isometries of .AdS3 , and on .RP1 × RP1 by the obvious product action. The following is a useful characterization of sequences in .AdS3 converging to a point in the boundary (see [7, Lemma 3.2.2]): for .(γn )n∈N a sequence of isometries of .H2 , we have: γn → δ −1 (x, y) ⇔ there exists z ∈ H2 such that γn (z) → x and γn−1 (z) → y .

⇔ for every z ∈ H2 , γn (z) → x and γn−1 (z) → y (4.4)

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where of course here we are using the standard identification between .RP1 and the visual boundary .R ∪ {∞} = ∂∞ H2 , mapping the line spanned by .(a, b) to .a/b. A fundamental step in the proof of the earthquake theorem is that to any map 2 2 3 .f : ∂∞ H → ∂∞ H we can associate a subset of .∂∞ AdS , namely (via the map .δ) the graph of f . By the equivariance of the map .δ introduced in (4.3), we see immediately that, for .(α, β) ∈ PSL(2, R) × PSL(2, R): (α, β) · graph(f ) = graph(βf α −1 ) .

.

(4.5)

In the rest of this paper, we will omit the map .δ, and we will simply identify ∂∞ AdS3 with .RP1 × RP1 .

.

4.3.3 Spacelike Planes We conclude the preliminaries by an analysis of totally geodesic planes in .AdS3 . They are all obtained as the intersection of .AdS3 with a projective subspace in the projective space .PM2 (R) over .M2 (R). Hence they are all of the following form: P[A] = {[X] ∈ PSL(2, R) | 〈X, A〉 = 0}

.

(4.6)

for some nonzero 2-by-2 matrix A. The notation .P[A] is justified by the observation that the plane .PA defined in the right-hand side of (4.6) depends only on the projective class of A. The totally geodesic plane .P[A] is spacelike (resp. timelike, lightlike) if and only if .q(A) = − det(A) is negative (resp. positive, null). It will be called the dual plane of .[A], since it can be seen as a particular case of the usual projective duality between points and planes in projective space. In particular, the dual plane .Pγ of an element .γ ∈ PSL(2, R) is a spacelike totally geodesic plane. Example 4.2 The first example, which is of fundamental importance for the following, is for .γ = 1 is the identity of .PSL(2, R). By (4.1), .P1 is the subset of .PSL(2, R) consisting of projective classes of unit matrices X with .tr(X) = 0. By the Cayley–Hamilton theorem, .X2 = −id, hence the elements of .P1 are order–two isometries of .H2 , that is, elliptic elements with rotation angle .π. Observe that .P1 is invariant under the action of .PSL(2, R) by conjugation, which corresponds to the diagonal in the isometry group .PSL(2, R) × PSL(2, R) of .AdS3 . Using (4.4), one immediately sees that the boundary of .P1 in .∂∞ AdS3 ∼ = RP1 × RP1 is the diagonal; more precisely: ∂∞ P1 = graph(1) ⊂ RP1 × RP1 .

.

(4.7)

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Given a point .z ∈ H2 , let us denote by .Rz the order–two elliptic isometry with fixed point z. We claim that the map ι : H2 → P1

.

ι(z) = Rz

is an isometry with respect to the hyperbolic metric of .H2 and the induced metric on .P1 ⊂ AdS3 . First, the inverse of .ι is simply the fixed-point map .Fix : P1 → H2 sending an elliptic isometry to its fixed point, which also shows that .ι is equivariant with respect to the action of .PSL(2, R) on .H2 by homographies and on .P1 by conjugation, since .Fix(αγ α −1 ) = α(Fix(γ )). That is, we have the relation ι(α · p) = α ◦ ι(p) ◦ α −1 .

.

(4.8)

This immediately implies that .ι is isometric, since the pull-back of the metric of .P1 is necessarily .PSL(2, R)-invariant and has constant curvature .−1, hence it coincides with the standard hyperbolic metric on the upper half-space. This example is actually the essential example to understand general spacelike totally geodesic planes. Indeed, every spacelike totally geodesic plane is of the form .Pγ for some .γ ∈ PSL(2, R). To see this, observe that the action of the isometry group of .AdS3 on spacelike totally geodesic planes is transitive, and that .Pγ = (γ , 1)P1 because the isometry .(γ , 1) maps .1 to .γ , and therefore maps the dual plane of .1 to the dual plane of .γ . By (4.5) and (4.7), we immediately conclude the following: Lemma 4.1 Every spacelike totally geodesic plane of .AdS3 is of the form .Pγ for some orientation-preserving isometry .γ of .H2 , and ∂∞ Pγ = graph(γ −1 ) ⊂ RP1 × RP1 .

.

4.3.4 Timelike Planes Let us now consider a matrix .A ∈ M2 (R) such that .det(A) = −1. Hence the plane defined by Eq. (4.6) is a timelike totally geodesic plane. Associated with 2 .[A] is an orientation-reversing isometry .η of .H . Indeed, the action of A by 1 1 homography on .CP preserves .RP and switches the two connected components of the complement, that is, the upper and the lower half-spaces. The matrix A thus induces an orientation-reversing isometry, up to identifying these two components via .z I→ z¯ . We will thus denote .P[A] by .Pη , by a small abuse of notation. The totally geodesic plane .Pη can be parameterized as follows. Consider the map I I→ Iη ,

.

(4.9)

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defined on the space of reflections .I along geodesics of .H2 , with values in .PSL(2, R) ∼ = AdS3 . Its image is precisely .Pη . Indeed, it is useful to remark that by the Cayley–Hamilton theorem, a matrix X with .det(X) = −1 is an involution if and only if and .tr(X) = 0. Now, because .det(A) = −1, .adj(A) = −A−1 , and therefore .〈XA, A〉 = 0 if and only if .tr(X) = 0, that is, if and only if X is an involution. This shows that the image of the map (4.9) is the entire plane .Pη . Similarly to the spacelike case, using the transitivity of the action of the group of isometries on timelike planes, every timelike plane is of the form above. Thanks to this description, we can show the following. Lemma 4.2 Every timelike totally geodesic plane of .AdS3 is of the form .Pη for some orientation-reversing isometry .η of .H2 , and ∂∞ Pη = graph(η−1 ) ⊂ RP1 × RP1 .

.

Proof It only remains to check the identity for .∂∞ Pη . For this, we will use the characterization (4.4) together with the parameterization (4.9) of .Pη . Suppose the sequence .In is such that .In η(z) → x ∈ ∂∞ H2 , for any .z ∈ H2 . Then, using that .In is an involution and the continuity of the action of .η on .H2 , .(In η)−1 (z) = −1 −1 η−1 I−1 ⨆ ⨅ n (z) = η In (z) → η (x). This concludes the proof. Remark 4.1 It is worth remarking that, since reflections of .H2 are uniquely determined by (unoriented) geodesics, we can consider the map (4.9) as a map from the space .G(H2 ) of unoriented geodesics of .H2 to .PSL(2, R). It turns out that this map is isometric with respect to a natural metric on .G(H2 ) which makes it identified with the two-dimensional Anti-de Sitter space .AdS2 , see [8, Example 6.1] for more details.

4.3.5 Lightlike Planes The only case left to consider consists of lightlike totally geodesic planes. Those are of the form .P[A] for a nonzero matrix A with .det(A) = 0, that is, for .rank(A) = 1. We describe their boundary in the following lemma. It is important to remark that, unlike spacelike and timelike planes considered above, the boundary will not be a graph in .RP1 × RP1 . Lemma 4.3 Every lightlike totally geodesic plane of .AdS3 is of the form .P[A] for some rank one matrix A, and

1 .∂∞ P[A] = Im(A) × RP ∪ RP1 × Ker(A) .

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In other words, .∂∞ P[A] is the union of two circles in .RP1 × RP1 , one horizontal and one vertical, which intersect exactly at the point in .RP1 × RP1 corresponding to .[A] ∈ ∂∞ AdS3 via the map .δ introduced in (4.3). Proof The points in .∂∞ P[A] are projective classes of rank one matrices X satisfying 〈X, A〉 = 0, that is, such that .tr(Xadj(A)) = 0. Since .Xadj(A) has vanishing determinant, by the Cayley-Hamilton theorem .Xadj(A) is traceless if and only if it is nilpotent, that is, if and only if .Xadj(A)Xadj(A) = 0. Since image and kernel of both X and .adj(A) are all one-dimensional, it is immediate to see that this happens if and only if

.

Im(adj(A)) = Ker(X)

.

or

Im(X) = Ker(adj(A)) .

(4.10)

Now, since .det(A) = 0 implies .adj(A)A = Aadj(A) = 0, the relations Ker(adj(A)) = Im(A) and .Im(adj(A)) = Ker(A) hold. Hence .X ∈ P[A] if and only if .Im(X) = Im(A) or .Ker(X) = Ker(A), which concludes the proof, by the definition of .δ. ⨆ ⨅

.

4.4 Convexity Notions In this section we develop the necessary tools to tackle the proof of Thurston’s earthquake theorem.

4.4.1 Affine Charts The starting point of the proof rests in considering the graph of an orientationpreserving homeomorphism .f : RP1 → RP1 as a subset of .∂∞ AdS3 , and taking its convex hull. However, the convex hull of a set in projective space can be defined in an affine chart, but .AdS3 is not contained in any affine chart. The following lemma serves to show that the convex hull of the graph of f is well-defined. Lemma 4.4 Let .f : RP1 → RP1 be an orientation-preserving homeomorphism. Then: 1. There exists a spacelike plane .Pγ in .AdS3 such that .∂∞ Pγ ∩ graph(f ) = ∅. 2. Moreover, given any point .(x0 , y0 ) ∈ / graph(f ), there exists a spacelike plane .Pγ such that .∂∞ Pγ ∩ graph(f ) = ∅ and .(x0 , y0 ) ∈ ∂∞ Pγ . Before providing the proof, let us discuss an important consequence of the first item. Given a (spacelike) plane .Pγ in .AdS3 , let .Pγ be the unique projective subspace in .PM2 (R) that contains .Pγ , which is defined by the Eq. (4.6) (where .γ = [A]). Let

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us denote by .Aγ the complement of .Pγ , which we will call a (spacelike) affine chart. The first item of Lemma 4.4 can be reformulated as follows: Corollary 4.1 Let .f : RP1 → RP1 be an orientation-preserving homeomorphism. There exists a spacelike affine chart .Aγ containing .graph(f ). The proof of Lemma 4.4 below is largely inspired by Bonsante et al. [1, Lemma 6.2, Lemma 6.3]. Proof of Lemma 4.4 Clearly the second item implies the first. However, we will first prove the first item, and then explain how to improve the proof to achieve the second item. Recall that .PSL(2, R) acts transitively on pairs of distinct points of .RP1 ∼ = R∪ {∞}—actually, it acts simply transitively on positively oriented triples. Hence for the first point we may assume, up to the action of the isometry group of .AdS3 by post-composition on f (recall (4.5)), that .f (0) = 0 and .f (∞) = ∞. Then f induces a monotone increasing homeomorphism from .R to .R. Since .f (0) = 0, f preserves the two intervals .(−∞, 0) and .(0, ∞). Let now .γ = Ri be the order–two elliptic isometry fixing i. Clearly .γ is an involution that maps 0 to .∞, and switches the two intervals .(−∞, 0) and .(0, ∞). Hence .f (x) /= γ (x) for all .x ∈ R ∪ {∞}, that is, .graph(f )∩graph(γ ) = ∅. By Lemma 4.1 and the fact that .γ is an involution, .graph(f ) ∩ ∂∞ Pγ = ∅. To prove the second item, we will make full use of the transitivity of the .PSL(2, R)-action on oriented triples, and we will apply both pre and postcomposition of an element of .PSL(2, R). As a preliminary step, let .(x0 , y0 ) ∈ / graph(f ), and observe that we can find points x and .x ' such that f maps the unoriented arc of .RP1 connecting x and .x ' containing .x0 to the unoriented arc connecting .f (x) and .f (x ' ) not containing .y0 . The proof is just a picture, see Fig. 4.1. Since f preserves the orientation of .RP1 , up to switching x and .x ' , we have that .(x, x0 , x ' ) is a positive triple in .RP1 , while .(f (x), y0 , f (x ' )) is a negative triple.

p

y0 f (x')

graph(f )

f (x) x

x0

x'

Fig. 4.1 The proof of a claim in Lemma 4.4, drawn in the torus .RP1 ×RP1 (identify opposite sides by a translation). Given .p ∈ / graph(f ), consider any orientation-reversing homeomorphism g of 1 ' .RP . Then .graph(f ) and .graph(g) (dashed) intersect in two points, and let .x, x the corresponding solutions of the equation .f = g. Then f maps an arc from x to .x ' containing .x0 , to an arc from ' .f(x) to .f(x ) not containing .y0

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Having made this observation, using simple transitivity on oriented triples we can assume .(x, x0 , x ' ) = (0, 1, ∞) and .(f (x), y0 , f (x ' )) = (0, −1, ∞). Then the choice .γ = Ri as in the first part of the proof satisfies the condition in the second item as well, since .γ (1) = −1. ⨆ ⨅

4.4.2 Convex Hulls Corollary 4.1 permits to consider the convex hull of .graph(f ), in any affine chart Aγ that contains .graph(f ).

.

Example 4.3 Given .σ ∈ PSL(2, R), the convex hull of .graph(σ ) is the closure of the totally geodesic spacelike plane .Pσ −1 in .AdS3 . Indeed by Lemma 4.1 the boundary at infinity of .Pσ −1 equals .graph(σ ), and moreover .Pσ −1 is convex, since spacelike geodesics of .AdS3 (which are the intersections of two transverse spacelike planes) are lines in an affine chart, and any two points in .∂∞ H2 are connected by a geodesic. Hence .Pσ −1 is clearly the smallest convex set containing .graph(σ ). This is the only case in which .graph(f ) is contained in a plane, and therefore its convex hull has empty interior. If f is not the restriction to .RP1 of an element of .PSL(2, R), then the convex hull of .graph(f ) is a convex body in the affine chart .Aγ . Let us study one more important property of the convex hull of .graph(f ). Proposition 4.1 Let .f : RP1 → RP1 be an orientation-preserving homeomorphism, let .Pγ in .AdS3 be a spacelike plane such that .∂∞ Pγ ∩ graph(f ) = ∅, and let K be the convex hull of .graph(f ) in the affine chart .Aγ . Then: • The interior of K is contained in .AdS3 . • The intersection of K with .∂∞ AdS3 equals .graph(f ). In particular, .K ⊂ AdS3 . Before proving Proposition 4.1, we give another technical lemma, which is proved by an argument in a similar spirit as the proof of Lemma 4.4. Lemma 4.5 Let .f : RP1 → RP1 be an orientation-preserving homeomorphism and let .Pγ in .AdS3 be a spacelike plane such that .∂∞ Pγ ∩ graph(f ) = ∅. Given any two distinct points .(x, f (x)) and .(x ' , f (x ' )) in .graph(f ), there exists a spacelike plane, disjoint from .Pγ , containing them in its boundary at infinity. Proof Applying the action of .PSL(2, R) × PSL(2, R) we can assume that .γ = 1. The hypothesis .∂∞ P1 ∩ graph(f ) = ∅ then tells us that f has no fixed point. We are looking for a .σ ∈ PSL(2, R) such that • .P1 ∩ Pσ −1 = ∅; • .(x, f (x)), (x ' , f (x ' )) ∈ ∂∞ Pσ −1 = graph(σ ).

4 The Anti-de Sitter Proof of Thurston’s Earthquake Theorem

𝓁'

x

f (x' ) f (x)

𝓁2

𝓁1

𝓁 x'

x

81

f (x)

x'

f (x' )

x f (x) = x'

f (x' )

Fig. 4.2 Several cases in the proof of Lemma 4.5

For the first condition to hold, it clearly suffices that the boundaries of .P1 and .Pσ −1 do not intersect, that is to say, .σ (y) /= y for all .y ∈ RP1 . This is equivalent to saying that .σ does not have fixed points on .RP1 , namely, .σ is an elliptic isometry. The second condition is equivalent to .σ (x) = f (x) and .σ (x ' ) = f (x ' ). Since f has no fixed points, .f (x) /= x and .f (x ' ) /= x ' . There are various cases to distinguish (see also Fig. 4.2). First, suppose .(x, f (x), x ' ) is a positive triple. Then either .(x, f (x ' ), f (x), x ' ) or .(x, f (x), x ' , f (x ' )) are in cyclic order, because the remaining possibility, namely that .(x, f (x), f (x ' ), x ' ) are in cyclic order, would imply that f has a fixed point. If .(x, f (x ' ), f (x), x ' ) are in cyclic order, then the hyperbolic geodesics .𝓁 connecting x to .f (x) and .𝓁' connecting .x ' to .f (x ' ) intersect, and the order two elliptic isometry .σ fixing .𝓁 ∩ 𝓁' maps x to .f (x) and .x ' to .f (x ' ). If ' ' ' .(x, f (x), x , f (x )) are in cyclic order, then the geodesics .𝓁1 connecting x to .x and ' .𝓁2 connecting .f (x) to .f (x ) intersect, and one can find an elliptic element .σ fixing ' ' ' .𝓁1 ∩ 𝓁2 sending x to .f (x) and .x to .f (x ). Second, if .(x, f (x), x ) is a negative triple, then the argument is completely analogous. Finally, there is the possibility that .f (x) = x ' . If .f (x ' ) /= x, the .σ we are looking for is for instance an order– three elliptic isometry with fixed point in the barycenter of the triangle with vertices ' ' ' .x, f (x) = x and .f (x ). If instead .f (x ) = x, then clearly we can pick any order– two elliptic isometry with fixed point on the geodesic .𝓁 from x to .x ' . ⨆ ⨅ In particular, Lemma 4.5 shows that given any spacelike affine chart .Aγ containing .graph(f ) and any two distinct points in .graph(f ), the line connecting them is contained in .AdS3 ∩ Aγ (except for its endpoints, which are in .∂∞ AdS3 ), and is a spacelike geodesic of .AdS3 . We are now ready to prove Proposition 4.1. Proof of Proposition 4.1 Given a point p in .∂∞ AdS3 \ graph(f ), by the second item of Lemma 4.4 there exists a spacelike plane .Pγ ' passing through p that does not intersect .graph(f ). This implies that .Pγ ' ∩K = ∅, and therefore .K ∩∂∞ AdS3 = graph(f ). Since K is connected, it is contained in the closure of one component of the complement of .∂∞ AdS3 in .Aγ . But K is connected and intersects .AdS3 \ Pγ because, by Lemma 4.5, the line segment connecting any two points of .graph(f ) in the affine chart .Aγ is contained in .AdS3 ∩ Aγ . Hence K is contained in .AdS3 and its interior is contained in .AdS3 . ⨆ ⨅

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By Corollary 4.1 and Proposition 4.1, we can now give the following definition: Definition 4.3 Given an orientation-preserving homeomorphism .f : RP1 → RP1 , we define .C(f ) to be the subset of .AdS3 which is obtained as the convex hull of .graph(f ) in any spacelike affine chart .Aγ such that .∂∞ Pγ ∩ graph(f ) = ∅. The definition is well posed—that is, it does not depend on the chosen affine chart .Aγ —because lines and planes are well defined in projective space, hence the change of coordinates from an affine chart to another preserves convex sets. When referring to convexity notions in the following, we will implicitly assume we have chosen a spacelike affine chart .Aγ containing .graph(f ).

4.4.3 Support Planes Let us recall a basic notion in convex analysis. Given a convex body K in an affine space of dimension three, a support plane of K is an affine plane Q such that K is contained in a closed half-space bounded by Q, and .∂K ∩ Q /= ∅. If .p ∈ ∂K ∩ Q, one says that Q is a support plane at the point p. As a consequence of the Hahn– Banach theorem, there exists a support plane at every point .p ∈ ∂K. We will adopt this terminology for the convex hulls .C(f ) in .AdS3 : we say that a totally geodesic plane P is a support plane of .C(f ) (at .p ∈ ∂C(f )) if .p ∈ C(f ) ∩ P ⊂ AdS3 and, in an affine chart containing .graph(f ), .C(f ) lies in a closed halfspace bounded by the affine plane that contains P . As usual, one easily sees that this definition does not depend on the affine chart as long as it contains .graph(f ). Remark 4.2 Equivalently, we can say that a totally geodesic plane P is a support plane for .C(f ) if there exists a continuous family .{Pt }t∈[0,ϵ) of totally geodesic planes, pairwise disjoint in .AdS3 , such that .P0 = P and .Pt ∩ C(f ) = ∅ for .t > 0. Also, recall that we have the following identity for convex hulls: if X is a set, C(X) its convex hull and Q an affine support plane for .C(X), then .Q ∩ C(X) = C(Q ∩ X). Applying this identity in our setting, we obtain for any totally geodesic support plane P :

.

P ∩ C(f ) = C(∂∞ P ∩ graph(f )) .

.

(4.11)

In the following proposition, we see that all support planes of .C(f ) are allowed to be spacelike, and lightlike only if they touch .C(f ) at a boundary point. Proposition 4.2 Let .f : RP1 → RP1 be an orientation-preserving homeomorphism, and let P be a support plane of .C(f ) at a point .p ∈ ∂C(f ). Then: • If .p ∈ AdS3 , then P is a spacelike plane. • If .p ∈ ∂∞ AdS3 , then P is either spacelike or lightlike.

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Proof The basic observation is that if P is a support plane, then .∂∞ P and graph(f ) = C(f ) ∩ ∂∞ AdS3 do not intersect transversely. To clarify this notion, we say that an intersection point .p ∈ ∂∞ P ∩ graph(f ) is transverse if, for a small neighbourhood U of p such that .(graph(f ) \ p) ∩ U has two connected components, these two connected components are contained in different connected components of .U \ ∂∞ P . From Lemma 4.2, if P is timelike, then .∂∞ P is the graph of an orientation-reversing homeomorphism of .RP1 , hence it intersects .graph(f ) transversely. From Lemma 4.3, if P is lightlike, then .∂∞ P is the union of the two circles .{x} × RP1 and .RP1 × {y}. So if .p ∈ ∂∞ P ∩ graph(f ) and p is not the point .p0 = (x, y), then .∂∞ P and .graph(f ) intersect transversely. So the sole possibility for P to be a lightlike support plane is to intersect .graph(f ) only at the point .p0 . It remains to show that .P ∩ C(f ) consists only of the point .p0 , that is, it does not contain any point of .AdS3 . By contradiction, if .q ∈ P ∩ C(f ) is different from .p0 , then by (4.11) .∂∞ P ∩ graph(f ) would contain another point different by .p0 as well, because the left-hand side must contain not only .p0 but also q. This would give a contradiction as above. ⨆ ⨅ .

Given a spacelike support plane P of .C(f ) at a point p, we say that P is a future (resp. past) support plane if in a small simply connected neighbourhood U of p in 3 .AdS , .C(f ) is contained in the closure of the connected component of .U \ P which is in the past (resp. future) of P . This means that there exist future-oriented (resp. past-oriented) timelike curves in U leaving .C(f ) ∩ U and reaching .P ∩ U . Clearly .C(f ) cannot have a future and past support plane at p at the same time, unless .C(f ) has empty interior, which is precisely the situation when f is an element of .PSL(2, R) as in Example 4.3. In the following we will always assume .int C(f ) /= ∅. As a consequence of the previous discussion, we have the following useful statement on the convergence of support planes. Lemma 4.6 Let .f : RP1 → RP1 be an orientation-preserving homeomorphism which is not in .PSL(2, R), .pn a sequence of points in .∂C(f ), and .Pγn a sequence of future (resp. past) spacelike support planes at .pn , for .γn ∈ PSL(2, R). Up to extracting a subsequence, we can assume .pn → p and .Pγn → P . Then: • If .p ∈ AdS3 , then .P = Pγ is a future (resp. past) support plane of .C(f ), for .γn → γ ∈ PSL(2, R). • If .p ∈ ∂∞ AdS3 , then either P is a lightlike plane whose boundary is the union of two circles meeting at p, or the conclusion of the previous point holds. Proof The proof is straightforward, having developed all the necessary elements above. It is clear that we can extract converging subsequences from .pn and .Pγn , by compactness of .C(f ) and of the space of planes in projective space. Also, the limit of the sequence of support planes .Pγn at .pn is a support plane P at p, since both conditions that .pn ∈ C(f ) and that .C(f ) is contained in a closed half-space bounded by .Pγn are closed conditions. By Proposition 4.2, if the limit p is in .AdS3 , then P is a spacelike support plane, which is of course future (resp. past) if all the .Pγn are future (resp. past). This situation can also occur analogously if .p ∈ ∂∞ AdS3 ; the other

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possibility being that P is lightlike, and in this case the proof of Proposition 4.2 shows that .P = P[A] if p is represented by the projective class of the rank–one matrix A. ⨆ ⨅ Corollary 4.2 Let .f : RP1 → RP1 be an orientation-preserving homeomorphism which is not in .PSL(2, R). Then .∂C(f ) is the disjoint union of .graph(f ) = C(f ) ∩ ∂∞ AdS3 and of two topological discs, of which one only admits future support plane, and the other only admits past support planes. Proof It is a basic fact in convex analysis that .∂C(f ) is homeomorphic to .S2 ; by Proposition 4.1, its intersection with .∂∞ AdS3 equals .graph(f ) and is therefore a simple closed curve. By the Jordan curve theorem, the complement of .graph(f ) is the disjoint union of two topological discs, each of which is contained in .AdS3 again by Proposition 4.1. By Lemma 4.6, the set of points .p ∈ ∂C(f ) admitting a future support plane is closed. But it is also open because its complement is the set of points admitting a past support plane, for which the same argument applies. Hence each connected component of the complement of .graph(f ) admits only future support planes, or only past support planes. Finally, .C(f ) necessarily admits both a past and a future support plane, otherwise it would not be compact in an affine chart. This concludes the proof. ⨆ ⨅ By virtue of Corollary 4.2, we will call the connected component of .∂C(f ) \ graph(f ) that only admits future support planes the future boundary component, and denote it by .∂+ C(f ); similarly, the connected component that only admits past support planes is the past boundary component, denoted by .∂− C(f ).

4.4.4 Left and Right Projections We are now ready to introduce the left and right projections, which will play a central role in the proof of the earthquake theorem. These are maps πl± : ∂± C(f ) → H2

.

πr± : ∂± C(f ) → H2

defined on the future or past components of .∂C(f ), constructed as follows. Given a point .p ∈ ∂± C(f ), let P be a support plane of .C(f ) at p. By Proposition 4.2, the support plane is necessarily spacelike, hence of the form .P = Pγ for some .γ ∈ PSL(2, R). Remark 4.3 It is important to remark here that .Pγ might not be unique, if .∂± C(f ) is not .C 1 at p. Hence we choose a support plane .Pγ at p. Moreover we require that the choice of support planes is made so that the support plane is constant on any connected component of the subset of .∂± C(f ) consisting of those points that admit more than one support plane. The definition of the projections then depends (although quite mildly, see Corollary 4.4 below) on the choice of .Pγ .

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Now, having chosen the support plane .Pγ at p, left or right multiplication by γ −1 maps .γ to .1, and therefore maps .Pγ to .P1 , which we recall from Example 4.2 is the space of order–two elliptic elements and is therefore naturally identified with 2 2 .H via the map .Fix : P1 → H . Denote by .Lγ −1 : PSL(2, R) → PSL(2, R) and .Rγ −1 : PSL(2, R) → PSL(2, R) the left and right multiplications by .γ −1 ; in other words, there are the actions of the elements .(γ , 1) and .(1, γ −1 ) of .PSL(2, R) × PSL(2, R). By what we said above, .Lγ −1 (p) and .Rγ −1 (p) are elements of .P1 , since .p ∈ Pγ , and .Lγ −1 (p) (resp. .Rγ −1 (p)) maps bijectively .Pγ to .P1 . We can finally define: .

πl± (p) = Fix(Rγ −1 (p))

.

πr± (p) = Fix(Lγ −1 (p)) .

(4.12)

It might seem counterintuitive to define the left projection using right multiplication, and vice versa. However, this is the most natural choice by virtue of the property of Lemma 4.7 below. Another reason to justify this choice is that these projections can be naturally seen as the left and right components of the Gauss map of spacelike surfaces in .AdS3 with values in the space of timelike geodesics of .AdS3 , which is naturally identified with .H2 × H2 , see [7, Section 6.3] for more details and for several other equivalent definitions. Lemma 4.7 Let .f : RP1 → RP1 be an orientation-preserving homeomorphism, and let .(α, β) ∈ PSL(2, R) × PSL(2, R). Let us denote .K = C(f ) and .Kˆ = (α, β) · C(f ) and let .πl± , πr± : ∂± K → H2 and .πˆ l± , πˆ r± : ∂± Kˆ → H2 be the left and right projections of K and .Kˆ respectively. Then .

πˆ l± ◦ (α, β) = α ◦ πl±

πˆ r± ◦ (α, β) = β ◦ πr± .

(4.13)

To clarify the statement, let us remark that the isometry .(α, β) maps a point ˆ and maps support planes at .p ∈ K to support planes p ∈ K to a point .pˆ ∈ K, at .p. ˆ Hence the relation (4.13) holds when we consider the projections .πˆ l± and .πˆ r± defined with the choice of support planes of .Kˆ given by the images .Pˆ of the support planes P chosen in the definitions of .πl± and .πr± .

.

ˆ and for Proof As remarked above, for any .p ∈ ∂ ± K, we have .pˆ := (α, β) · p ∈ K, a chosen support plane .P = Pγ for K at p, .(α, β) · P = Pγˆ is the chosen support plane for .Kˆ at .p. ˆ By the duality, .γˆ = (α, β) · γ = αγβ −1 . Hence we have: .



 πˆ l± (p) ˆ = Fix Rγˆ −1 (p) ˆ = Fix R(βγ −1 α −1 ) (αpβ −1 )

 = Fix R(γ −1 α −1 ) (αp) = Fix α ◦ Rγ −1 (p) ◦ α −1 = α(Fix(Rγ −1 (p))) = απl± (p) .

The computation is completely analogous for the right projection.

⨆ ⨅

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Example 4.4 The simplest example that we can consider is the situation where f = σ ∈ PSL(2, R), so that .C(f ) = Pσ −1 as in Example 4.3. This case is somehow degenerate, because .C(σ ) has empty interior, hence Corollary 4.2 does not hold and it does not quite make sense to talk about the future and past components of the boundary. However, we can still define the left and right projections. Since .Pσ −1 itself is the unique support plane at any of its points, from (4.12) we have the following simple expressions for the left and right projections .πl , πr : Pσ −1 → H2 .

.

πl (p) = Fix(p ◦ σ )

.

πr (p) = Fix(σ ◦ p) .

(4.14)

Observe that .πl and .πr extend to the boundary of .Pσ −1 : recalling that the boundary of .Pσ −1 is the graph of .σ (Lemma 4.1), we have πl (x, σ (x)) = x

.

πr (x, σ (x)) = σ (x) .

(4.15)

Equation (4.15) is indeed immediately checked when .σ = 1, because in that case πl and .πr coincide with the fixed point map .Fix : P1 → H2 , and we have already observed in Example 4.3, using (4.4), that .Fix extends to the map .(x, x) I→ x from 2 .∂∞ P1 to .∂∞ H . The general case of Eq. (4.15) then follows from Eqs. (4.5) and (4.13), that is, by observing that the isometry .(1, σ ) maps .graph(1) to .graph(σ ) and .P1 to .Pσ −1 . Finally, we can compute the map of .H2 obtained by composing the inverse of the left projection with the right projection. Indeed, this is induced by the map .P1 → P1 sending an order–two elliptic element .R = p ◦ σ ∈ P1 to .σ ◦ p = σ ◦ R ◦ σ −1 . Hence we have .

πr ◦ πl−1 = σ : H2 → H2 .

.

(4.16)

In conclusion, the composition .πr ◦πl−1 is an isometry and its extension to .∂∞ H2 is precisely the map .f = σ of which .∂∞ Pσ −1 is the graph. In the next sections we will see that this fact is extremely general, that is, for any orientation-preserving homeomorphism of the circle f , the compositions .πr± ◦ (πl± )−1 associated with .∂± C(f ) will be the left and right earthquake maps extending f .

4.5 The Case of Two Spacelike Planes Before moving to the proof of Thurston’s earthquake theorem, we will now consider another very concrete example, which is only slightly more complicated than Example 4.4. Nevertheless, we will see that this example represents a very general situation, and its comprehension is the essential step towards the proof of the full theorem.

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Fig. 4.3 Given two elements .γ1 , γ2 such that .γ2 ◦ γ1−1 is a hyperbolic isometry, there are two possible configurations. On the left, we see the future boundary component of .C(fγ±1 ,γ2 ), on the right the past boundary component of .C(fγ∓1 ,γ2 )

4.5.1 The Fundamental Example The idea here is to consider piecewise totally geodesic surfaces in .AdS3 , which are obtained as the union of two connected subsets, each contained in a totally geodesic spacelike plane, meeting along a common geodesic. See Fig. 4.3. To formalize this idea, we will consider the union of two half-planes, each contained in a totally geodesic spacelike plane .Pγ1 or .Pγ2 . The first important observation is the following. Lemma 4.8 Let .γ1 /= γ2 ∈ PSL(2, R). Then .Pγ1 and .Pγ2 intersect in .AdS3 if and only if .γ2 ◦ γ1−1 is a hyperbolic isometry. Proof Since .Pγi is the convex envelope of .∂∞ Pγi = graph(γi−1 ) (Example 4.3), the closures .P γ1 and .P γ2 intersect in .AdS3 if and only if .graph(γ1−1 ) ∩ graph(γ2−1 ) /= ∅. Moreover, by (4.11), .Pγ1 and .Pγ2 intersect in .AdS3 if and only .graph(γ1−1 ) ∩ graph(γ2−1 ) contains at least two different points. Now, .(x, y) ∈ RP1 × RP1 is in .graph(γ1−1 ) ∩ graph(γ2−1 ) if and only if .y = γ1−1 (x) = γ2−1 (x), which is equivalent to the condition that x is a fixed point of −1 −1 .γ2 ◦ γ 1 . But .γ2 ◦ γ1 is an element of .PSL(2, R), hence it has two fixed points in 1 .RP if and only if it is a hyperbolic isometry. ⨆ ⨅

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Now, let .I1 and .I2 be two closed intervals in .RP1 such that .RP1 = I1 ∪ I2 and −1 .I1 ∩ I2 consists precisely of the two fixed points of .γ2 ◦ γ 1 . Clearly there are two possibilities to produce a homeomorphism of .RP1 by combining the restrictions of −1 −1 .γ 1 and .γ2 to the intervals .Ij ’s, that is:  + .fγ ,γ (x) 1 2

=

γ1−1

γ2−1

if x ∈ I1 if x ∈ I2

and

fγ−1 ,γ2 (x)

=

 γ2−1 γ1−1

if x ∈ I1

. if x ∈ I2 (4.17)

One easily checks that .fγ±1 ,γ2 actually are orientation-preserving homeomorphisms, since .γ1−1 and .γ2−1 map homeomorphically the intervals .I1 and .I2 to the same intervals .J1 := γ1−1 (I1 ) = γ2−1 (I1 ) and .J2 := γ1−1 (I2 ) = γ2−1 (I2 ), which intersect only at their endpoints. Let us also denote by .Di the convex hull of .Ii in .H2 , and by .𝓁 = D1 ∩ D2 the axis of .γ2 ◦ γ1−1 . Proposition 4.3 Suppose that .γ2 ◦ γ1−1 is a hyperbolic isometry that translates along .𝓁 to the left, as seen from .D1 to .D2 . Then: • The future boundary component .∂+ C(fγ+1 ,γ2 ) coincides with the union of the convex envelope of .graph(γ1−1 |I1 ) and of the convex envelope of .graph(γ2−1 |I2 ). • The past boundary component .∂− C(fγ−1 ,γ2 ) is the union of the convex envelope of −1 −1 .graph(γ 1 |I2 ) and of the convex envelope of .graph(γ2 |I1 ). If instead .γ2 ◦ γ1−1 translates along .𝓁 to the right as seen from .D1 to .D2 , then: • The past boundary component .∂− C(fγ+1 ,γ2 ) coincides with the union of the convex envelope of .graph(γ1−1 |I1 ) and of the convex envelope of .graph(γ2−1 |I2 ). • The future boundary component .∂+ C(fγ−1 ,γ2 ) is the union of the convex envelope of .graph(γ1−1 |I2 ) and of the convex envelope of .graph(γ2−1 |I1 ). Proof Let us consider the case where .γ2 ◦ γ1−1 translates to the left along .𝓁, and let us prove the first item. Let .x, x ' be the fixed points of .γ2 ◦ γ1−1 , and let .y = γ1−1 (x) = γ2−1 (x) and .y ' = γ1−1 (x ' ) = γ2−1 (x ' ). Then the convex envelope of −1 .graph(γ i |Ii ) is a half-plane .Ai in .Pγi bounded by the geodesic .Pγ1 ∩ Pγ2 , which has endpoints .(x, y) and .(x ' , y ' ). Clearly both the convex envelope of .graph(γ1−1 |I1 ) and the convex envelope of .graph(γ2−1 |I2 ) are contained in .C(fγ+1 ,γ2 ). Nevertheless, we can be more precise. We claim that .Pγ1 and .Pγ2 are future support planes for .C(fγ+1 ,γ2 ). This claim implies that the union of .A1 and .A2 is contained in the future boundary component .∂+ C(fγ+1 ,γ2 ), because every point .p ∈ A1 ∪ A2 admits a future support plane through p, which is either .Pγ1 or + .Pγ2 . However .A1 ∪ A2 is a topological disc in .∂+ C(fγ ,γ ), whose boundary is 1 2

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precisely the curve .graph(fγ+1 ,γ2 ) by construction. Hence the claim will imply that + .A1 ∪ A2 = ∂+ C(fγ ,γ ). 1 2 We prove the claim for .Pγ1 , the proof for .Pγ2 being completely analogous. it is convenient to assume that .γ1 = 1 and .γ2 = γ is a hyperbolic isometry with fixed points x and .x ' , translating to the left seen from .D1 to .D2 . Indeed, we can apply the isometry .(1, γ1 ), which sends .Pγ1 to .P1 , .Pγ2 to .Pγ2 γ −1 , and (by (4.5)) .graph(fγ+1 ,γ2 ) 1 to .graph(f + −1 ). 1,γ2 γ1 Having made this assumption, consider a path .σt , for .t ∈ [0, ϵ) of elliptic elements fixing a given point .z0 ∈ H2 , that rotate clockwise by an angle t. As in the proof of Lemma 4.8, the planes .Pσt are pairwise disjoint in .AdS3 , because −1 .σt2 ◦ σt is still an elliptic element fixing .z0 for .t1 /= t2 , hence it has no fixed point 1 1 in .RP . Moreover, observe that .γ −1 is an isometry fixing .𝓁 and translates along .𝓁 on the right as seen from .D1 to .D2 . Since .f1+,γ equals the identity on .I1 and .γ −1 on .I2 , it fixes .I1 pointwise, and moves points of .I2 clockwise. In particular, the equation .f1+,γ (x) = σt−1 (x) has no solutions for .t > 0, because .σt−1 = σ−t moves all points counterclockwise if t is positive. This shows that .Pσt ∩ C(f1+,γ ) = ∅ for .t > 0, and thus .P1 is a support plane for .C(f1+,γ ) by Remark 4.2. Moreover it is a future support plane: indeed one can check (for instance using (4.1)) that .σt+π/2 = Rz0 ◦ σt ∈ Pσt , and the path .t I→ σt is future-directed because, from the discussion after (4.2), its tangent vector is future-directed, hence .C(f1+,γ ) is locally in the past of .P1 . This concludes the proof of the first point. The other cases are completely analogous. ⨆ ⨅

See also Fig. 4.3 to visualize the different configurations. The following is an important consequence of the proof of Proposition 4.3. Corollary 4.3 Suppose that .γ2 ◦ γ1−1 is a hyperbolic isometry that translates along −1 .𝓁 to the left (resp. right), as seen from .D1 to .D2 , and write .γ2 ◦ γ = exp(a) 1 for .a ∈ sl(2, R). Let p be a point in the future (resp. past) boundary component of + .C(fγ ,γ ). Then: 1 2 • If .p ∈ int(A1 ), then .Pγ1 is the unique support plane of .C(fγ+1 ,γ2 ) at p. • If .p ∈ int(A2 ), then .Pγ2 is the unique support plane of .C(fγ+1 ,γ2 ) at p. • If .p ∈ A1 ∩ A2 = Pγ1 ∩ Pγ2 , then the support planes of .C(fγ+1 ,γ2 ) at p are precisely those of the form .Pσ γ1 where .σ = exp(ta) for .t ∈ [0, 1]. Recall the notation from the proof of Proposition 4.3: .Ai ⊂ Pγi is the convex envelope of .graph(γi−1 |Ii ), which is a half-plane bounded by the geodesic .Pγ1 ∩Pγ2 . Of course we could provide an analogous statement for .C(fγ−1 ,γ2 ), but we restrict to .fγ+1 ,γ2 for simplicity.

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Proof From Proposition 4.3, the pleated surface which is obtained as the union of A1 ⊂ Pγ1 and .A2 ⊂ Pγ2 coincides with .∂+ C(fγ+1 ,γ2 ) if .γ2 ◦ γ1−1 is a hyperbolic isometry that translates along .𝓁 to the left, and with .∂− C(fγ+1 ,γ2 ) if it translates to the right, by Proposition 4.3. The first two items are then obvious, since .Pγi are smooth surfaces, hence .Ai is smooth at any interior point, and therefore has a unique support plane there. The last item can be proved in the same spirit as Proposition 4.3. First, we can assume .γ1 = 1 and .γ2 = γ is a hyperbolic isometry translating on the left (resp. right) along .𝓁. By (4.11), if .Pσ is a support plane at p, then p is in the convex hull of the pairs .(y, σ −1 (y)) where y satisfies .σ −1 (y) = f1±,γ (y). The only possibility is then .

that p lies in the geodesic connecting the points .(x, x) and .(x ' , x ' ) in .RP1 × RP1 , where x and .x ' are the fixed points of .γ . Hence .σ must have the same fixed points of .γ . That is, .σ is a hyperbolic isometry with axis .𝓁 (or the identity). Moreover, by an analogous argument as in Proposition 4.3, .Pσ is in the future (resp. past) of + .C(fγ ,γ ) if and only if .σ translates on the left (resp. right), and its translation length 1 2 is less than that of .γ . Hence .σ is of the form .exp(ta) for .t ∈ [0, 1]. ⨆ ⨅

4.5.2 Simple Earthquake We can now conclude the study of orientation-preserving homeomorphisms obtained by combining two elements of .PSL(2, R). The following proposition shows that in that situation, the composition of the projections .πl± and .πr± provide the earthquake map as in Example 4.1. This is not interesting in its own, since we recover a simple earthquake map which we had already defined explicitly. However, the following proposition will be an important tool to complete the proof of the earthquake theorem in Sect. 4.6. Proposition 4.4 Let .γ1 , γ2 ∈ PSL(2, R) be such that .γ2 ◦ γ1−1 is a hyperbolic isometry, and let .πl± , πr± be the projections associated with the convex envelope of + .fγ ,γ . Then: 1 2 1. .πl± , πr± : ∂± C(fγ+1 ,γ2 ) → H2 are bijections; 2. Assume .γ2 ◦γ1−1 translates along .𝓁 to the right (resp. left), as seen from .D1 to .D2 . Then the composition .πr− ◦ (πl− )−1 : H2 → H2 (resp. .πr+ ◦ (πl+ )−1 : H2 → H2 ) is a left (resp. right) earthquake map extending .fγ+1 ,γ2 . Again, we considered the case of .fγ+1 ,γ2 for the sake of simplicity, but one could give an analogous statement for .fγ−1 ,γ2 . Moreover, we remark that Proposition 4.4 holds for any choice of support planes that is needed to define the projections. Proof For the first point, recall that .Ai ⊂ Pγi , and that the union .A1 ∪ A2 is the past (resp. future) boundary component of .C(fγ+1 ,γ2 ) if .γ2 ◦ γ1−1 translates along .𝓁 to the right (resp. left).

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Hence .(πl± )int(Ai ) and .(πr± )|int(Ai ) are the restrictions of the projections associated with the totally geodesic plane .Pγi , which are described in Example 4.4. In particular, .(πl± )int(Ai ) and .(πr± )|int(Ai ) are the restrictions to .int(Ai ) of global isometries of .AdS3 (defined by multiplication on the left or on the right by .γi−1 ) sending .Pγi to .P1 , post-composed with the usual isometry .Fix : P1 → H2 . As a consequence, .(πl± )int(Ai ) and .(πr± )|int(Ai ) map geodesics of .Pγi to geodesics of ± −1 2 .H . Moreover, by Eq. (4.15), .π l maps .int(∂∞ (Ai )) = graph(γi |int(Ii ) ) to .int(Ii ). Hence .πl± (int(Ai )) = int(Di ). Analogously, .πr± (int(Ai )) = γ1−1 (int(Di )) = γ2−1 (int(Di )). To see that .πl± and .πr± are bijective, it remains to show that the image of the geodesic .A1 ∩ A2 = Pγ1 ∩ Pγ2 via .πl± is the geodesic .𝓁 = D1 ∩ D2 , while the image via .πr± is the geodesic .γ1−1 (𝓁) = γ2−1 (𝓁). The definition of .πl± and .πr± on .A1 ∩ A2 actually depends on the choice of a support plane. Recall that we must choose the same support plane at any point .p ∈ A1 ∩ A2 . From Corollary 4.3, the possible choices of support planes at p are of the form .Pσ γ1 , where .σ has the same fixed points as .γ2 ◦ γ1−1 , which are precisely the common endpoints of .I1 and .I2 . Using the notation from Lemma 4.8, we thus see that the endpoints at infinity of −1 ' ' ' .A1 ∩ A2 are the points .(x, y) and .(x , y ) where .x, x are the fixed points of .γ2 ◦ γ 1 (and of .σ ). Hence from Eq. (4.15) we have (for any choice of .σ as in the third item of Corollary 4.3) .πl± (x, y) = x and .πl± (x ' , y ' ) = x ' . Since .πl± is, as before, the restriction of an isometry between .Pσ γ1 and .H2 , it maps geodesics to geodesics, hence .πl± (A1 ∩ A2 ) = 𝓁. Analogously, .πr± (x, y) = y and .πr± (x ' , y ' ) = y ' , from which it follows that .πl± (A1 ∩ A2 ) = γ1−1 (𝓁) = γ2−1 (𝓁). This concludes the proof of the first item. For the second item, define .E := πr− ◦ (πl− )−1 , which is a bijection of .H2 . Consider the geodesic lamination of .H2 which is composed by the sole geodesic .𝓁. Hence the strata of .𝓁 are: .int(D1 ), int(D2 ) and .𝓁. We will show that the comparison isometries .Comp(S, S ' ) := (E|S )−1 ◦ E|S ' all translate to the right or to the left seen from one stratum to another, according to as .γ2 ◦ γ1−1 translates to the left or to the right seen from .D1 to .D2 . Let us first consider .S = int(D1 ) and .S ' = int(D2 ). Then by Example 4.4 (see in particular Eq. (4.16)) E equals .γi−1 on .int(Di ), because .(πl± )−1 (int(Di )) = int(Ai ) ⊂ Pγ −1 . Hence the comparison isometry .Comp(int(D1 ), int(D2 )) equals i

γ1 ◦ γ2−1 , and it translates to the left (resp. right) seen from .int(D1 ) to .int(D2 ) exactly when .γ2 ◦ γ1−1 , which is its inverse, translates to the right (resp. left). The proof when one of the two strata S or .S ' is .𝓁 is completely analogous, by using the third item of Corollary 4.3. Indeed (recalling Remark 4.3), by any choice of .σ of the form .σ = exp(ta) with .t ∈ (0, 1), .Comp(𝓁, int(D2 )) = σ ◦ γ2−1 translates to the left (resp. right) seen from .𝓁 to .int(D2 ), and .Comp(int(D1 ), 𝓁) = γ1 ◦ σ −1 translates to the left (resp. right) seen from .int(D1 ) to .𝓁. If instead .σ = exp(ta) with .t ∈ {0, 1}, then .σ coincides either with .γ1 or with .γ2 , which means that one of the comparison isometries .Comp(int(D1 ), 𝓁) and .Comp(𝓁, int(D2 )) translates to the left, and the

.

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other is the identity, which is still allowed in the definition of earthquake because .𝓁 is in the boundary of .int(Di ). ⨆ ⨅

4.5.3 The Example Is Prototypical The case of simple earthquakes that we have considered above may appear as very special. However, it turns out that it is the prototypical example, that will serve to treat the general case in the proof of the earthquake theorem. The following lemma shows that the situation of two intersecting planes occurs quite often. Lemma 4.9 Let .f : RP1 → RP1 be an orientation-preserving homeomorphism which is not in .PSL(2, R). Then: 1. Any two future support planes of .C(f ) at points of .∂+ C(f ) intersect in .AdS3 . Analogously, any two past support planes of .C(f ) at points of .∂− C(f ) intersect in .AdS3 . 2. Given a point .p ∈ ∂± C(f ), if there exist two support planes at p, then their intersection (which is a spacelike geodesic) is contained in .∂± C(f ). As a consequence, any other support plane at p contains this spacelike geodesic. Proof Let us consider future support planes, the other case being analogous. For the first item, let P and Q be support planes intersecting .∂+ C(f ), which are spacelike by Proposition 4.2, and suppose by contradiction P and Q that they are disjoint. We can slightly move them in the future to get spacelike planes .P ' and .Q' such that P , Q, .P ' and .Q' are mutually disjoint and .P ' ∩ ∂+ C(f ) = Q' ∩ ∂+ C(f ) = ∅. (For instance, if .P = Pγ1 and .Q = Pγ2 , then we can use Lemma 4.8 and consider ' ' .P = Pσ γ1 and .Q = Pσ γ1 for .σ an elliptic element of small clockwise angle of rotation.) Now, observe that .AdS3 \ (P ' ∪ Q' ) is the disjoint union of two cylinders and P and Q lie in different connected components of this complement. See Fig. 4.4. However, .∂+ C(f ) is connected and has empty intersection with P and Q, leading to a contradiction. For the second item, let .P = Pγ1 and .Q = Pγ2 be support planes such that −1 .p ∈ ∂+ C(f ) ∩ P ∩ Q. By Lemma 4.8, .γ2 ◦ γ 1 is hyperbolic. Up to switching the −1 roles of .γ1 and .γ2 , we can assume that .γ2 ◦ γ1 translates to the left seen from .D1 to .D2 , where as usual .Di is the convex envelope of the interval .Ii , and the common endpoints .x, x ' of .I1 and .I2 are the fixed points of .γ2 ◦γ1−1 . Hence .∂∞ Pγ1 ∩∂∞ Pγ2 = {(x, y), (x ' , y ' )} where .y = γ1−1 (x) = γ2−1 (x) and .y ' = γ1−1 (x ' ) = γ2−1 (x ' ). Now, by (4.11), .Pγi ∩ graph(f ) consists of at least two points for .i = 1, 2. We claim that .graph(f ) ∩ Pγi contains at least .(x, y) and .(x ' , y ' ). Indeed, since .Pγ2 is a support plane, .C(f ) ∩ Pγ1 is contained in the half-plane .A1 ⊂ Pγ1 . If ' ' .graph(f ) ∩ Pγ1 had not contained .(x, y) and .(x , y ), then .C(f ) ∩ Pγ1 would not contain the boundary geodesic .A1 ∩ A2 , and thus would not contain p. The same

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Fig. 4.4 The setting of the proof of Lemma 4.9

argument applies for .Pγ2 . This shows that both .(x, y) and .(x ' , y ' ) are in .graph(f ), and therefore the spacelike geodesic .Pγ1 ∩ Pγ2 is in .∂± C(f ). ⨆ ⨅ Remark 4.4 In the first item of Lemma 4.9, the hypothesis that P and Q are support planes at points of .∂ ± C(f ) (hence not at points of .graph(f ) ⊂ ∂∞ AdS3 ) is necessary. Recall that by Proposition 4.2 support planes of .C(f ) are either spacelike or lightlike, and they are necessarily spacelike if they intersect .C(f ) at points of .∂± C(f ). Now, if one of the two planes P or Q is a support plane at a point of .graph(f ), then the proof only shows that P and Q must intersect in .AdS3 , but not necessarily in the interior. It can perfectly happen that two future (or two past) support planes (one of which possibly lightlike) at a point .(x, f (x)) of .graph(f ) intersect at 3 .(x, f (x)) but not in the interior of .AdS . Lemma 4.9 has an important consequence. Recall that the definition of the projections .πl± , πr± : ∂± C(f ) → H2 depends on the choice of a support plane at all points p that admit more than one support plane. Moreover, we require that this support plane is chosen to be constant on any connected component of the subset of .∂± C(f ) consisting of those points that admit more than one support plane (Remark 4.3). We will now see that, roughly speaking, their image does not depend on this choice of support plane. Corollary 4.4 Let .f : RP1 → RP1 be an orientation-preserving homeomorphism which is not in .PSL(2, R), and suppose .p ∈ ∂± C(f ) has at least two support planes. Then there exist .γ1 , γ2 ∈ PSL(2, R) with .γ2 ◦ γ1−1 = exp(a) a hyperbolic element, such that all support planes at p are precisely those of the form .Pσ γ1 where .σ = exp(ta) for .t ∈ [0, 1]. The same conclusion holds for all other points .p' ∈ Pγ1 ∩Pγ2 . In particular, the image of the spacelike geodesic .Pγ1 ∩ Pγ2 under the projections ± ± 2 2 .π , πr : ∂± C(f ) → H is a geodesic in .H that does not depend on the choice of l ± the support plane as in the definition of .πl and .πr± .

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Proof Suppose .Pγˆ1 and .Pγˆ2 are (say, future) distinct support planes at p. Write γˆ2 ◦ γˆ1−1 = exp(ˆa), which is a hyperbolic element by Lemma 4.8 and the first item of Lemma 4.9. By the second item of Lemma 4.9, any other support plane at p must be of the form .Pσ γˆ1 for .σ an element having the same fixed points as .γˆ2 ◦ γˆ1−1 . That is, .σ is of the form .exp(s aˆ ) for some .s ∈ R. We claim that the set

.

I = {s ∈ R | exp(s aˆ ) is a support plane of C(f ) at p}

.

is a compact interval. This will conclude the proof, up to applying an affine change of variable mapping the interval .I = [s1 , s2 ] to .[0, 1], and defining .γi = exp(si aˆ ). To prove the claim, suppose that .s, s ' ∈ I . Then .C(f ) is contained in the past of a pleated surface obtained as the union of two half-spaces, one contained in .Pexp(s a) ˆ γˆ1 and the other in .Pexp(s ' a) ˆ γˆ1 , meeting along the spacelike geodesic .Pγˆ1 ∩ Pγˆ2 . Then every support plane for this pleated surface is a support plane for .C(f ) as well. That is, by the last item of Corollary 4.3, .[s, s ' ] ⊂ I . This shows that I is an interval. It is compact by Lemma 4.6, applied to the constant sequence .pn = p and to .γn = exp(sn aˆ )γˆ1 , showing that .sn must be converging (up to subsequences) and its limit is in I . This concludes the proof. ⨆ ⨅

4.6 Proof of the Earthquake Theorem We are now ready to enter into the details of the proof of the earthquake theorem. The outline of the proof is now clear: given an orientation-preserving homeomorphism .f : RP1 → RP1 (which we can assume is not in .PSL(2, R)), we consider the projections .πl± , πr± : ∂± C(f ) → H2 , and we want to show that the composition ± −1 ± .πr ◦ (π ) is well-defined and is a (left or right) earthquake map extending f . l We will prove this in several steps: the proof of Theorem 4.1 will follow from Proposition 4.6, Corollary 4.5 and Proposition 4.7 below.

4.6.1 Extension to the Boundary The first property we study is the extension of the projections .πl± and .πr± to the boundary. Proposition 4.5 The projections .πl± , πr± : ∂± C(f ) → H2 extend to .graph(f ). More precisely, if .pn ∈ ∂± C(f ) → (x, y) ∈ graph(f ), then .πl± (pn ) → x and ± .πr (pn ) → y. Observe that the conclusion of Proposition 4.5 holds for any choice of the projections .πl± and .πr± , regardless of the chosen support planes when several choices

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are possible, as in Remark 4.3. The proof involves two well-known properties of isometries in plane hyperbolic geometry; for the sake of completeness, we provide elementary, self-contained proofs in the Appendix. Proof Let .pn ∈ ∂± C(f ) be a sequence converging to .(x, y) ∈ graph(f ), and let Pγn be a sequence of support planes of .C(f ) at .pn , which are necessarily spacelike by Proposition 4.2. By Lemma 4.6, up to extracting a subsequence, there are two possibilities: either .γn → γ and .Pγn converges to the spacelike support plane .Pγ , or .γn diverges in .PSL(2, R) and .Pγn converges to the lightlike plane whose boundary is .({x} × RP1 ) ∪ (RP1 ∪ {y}). We will treat these two situations separately, and we will always use the characterization of the convergence to the boundary given in (4.4). Consider the former case, namely when .γn → γ . We have by hypothesis that .

pn (z0 ) → x

and

.

pn−1 (z0 ) → y ,

(4.18)

for any point .z0 ∈ H2 . Observe moreover that, from the definition of the projections, πl± (pn ) = Fix(pn γn−1 )

.

πr± (pn ) = Fix(γn−1 pn ) .

and

(4.19)

Recalling (see (4.7)) that the boundary of .P1 is identified with .RP1 via the map .(x, x) I→ x, we thus have to show (choosing for instance the point .z0 = i) that: −1 −1 −1 .pn γn (i) → x and .γn pn (i) → y. However, since .γn → γ , .pn γn (i) is at −1 bounded distance from .pn γ (i). Applying the hypothesis (4.18) to .z0 = γ −1 (i), we have .pn γ −1 (i) → x and therefore .pn γn−1 (i) → x. The argument is analogous to show that .γn−1 pn (i) → y, except that it is useful to observe that .γn−1 pn = pn−1 γn since it is an order–two isometry. Now .pn−1 γn (i) is at bounded distance from −1 −1 .pn γ (i), which converges to y by hypothesis. Hence .pn γn (i) → y and the proof is complete for this case. Let us move on to the latter case, that is, .γn diverges in .PSL(2, R). Here we must use not only the previous assumption (4.18), but also the following: γn (z0 ) → x

and

.

γn−1 (z0 ) → y ,

(4.20)

for any .z0 ∈ H2 . The condition (4.20) holds because .γn converges to the projective class of a rank one matrix A, such that .P[A] is a lightlike support plane; we have already observed that the boundary at infinity of .P[A] must be equal to .({x}×RP1 )∪ (RP1 ∪ {y}). Combining (4.3), (4.4) and Lemma 4.3, we deduce that .γn (z0 ) → x and .γn−1 (z0 ) → y as claimed. Having made this preliminary observation, now we can rewrite (4.19) as the identities: pn = Rπ ± (pn ) ◦ γn

.

l

and

pn−1 = Rπr± (pn ) ◦ γn−1 ,

(4.21)

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where we recall that .Rw denotes the order two elliptic isometry with fixed point w ∈ H2 . Up to extracting a subsequence, we can assume that .πl± (pn ) → xˆ± and ± .πr (pn ) → y ˆ± , for some points .xˆ± , yˆ± ∈ H2 ∪∂∞ H2 . We need to show that .xˆ± = x and .yˆ± = y. For this purpose, suppose by contradiction .xˆ± /= x. Suppose first that .xˆ± ∈ H2 . We will use the fact (Lemma 4.10 in the Appendix) that if .wn → w ∈ H2 , then 2 2 .Rwn converges to .Rw uniformly on .H ∪ ∂∞ H . From (4.21), and the fact that, from (4.18) and (4.20), both .pn (z0 ) and .γn (z0 ) converge to x, we would then have .

x = lim pn (z0 ) = lim(Rπ ± (pn ) (γn (z0 ))) = Rxˆ± (x) /= x

.

n

n

l

since .Rxˆ± does not have fixed points on .∂∞ H2 , thus giving a contradiction. If .yˆ± ∈ H2 , we get a contradiction by an analogous argument. Finally, if .xˆ± ∈ ∂∞ H2 , we can find a neighbourhood U of .xˆ± not containing x, such that for n large .Rπ ± (pn ) maps the complement of U inside U (see Lemma 4.11 l in the Appendix). This gives a contradiction with (4.21) because .pn (z0 ) and .γn (z0 ) are in the complement of U for n large, but at the same time .Rπ ± (pn ) (γn (z0 )) should l ⨆ ⨅ be in U for n large. The argument for .yˆ± is completely analogous. Remark 4.5 We remark that the proof of Proposition 4.5 does not use the full hypothesis that the surface on which the projections are defined is a boundary component of .C(f ), but only the property that whenever a sequence .Pγn of spacelike support planes converges to a lightlike plane, then this limit is a support plane too, which is true for any convex surface.

4.6.2 Invertibility of the Projections The next step in the proof is to show that the projections .πl± and .πr± are bijective. Proposition 4.6 The projections .πl± , πr± : ∂± C(f ) → H2 are bijective. Proof We give the proof for .πl± , the proof for .πr± being completely identical. Let us first show that .πl± and .πr± are injective. Given .p1 , p2 ∈ ∂± C(f ), let .Pγ1 and .Pγ2 be the support planes at .p1 and .p2 respectively. (If there are several support planes, we choose one, as in the definition of .πl± and .πr± —see Remark 4.3.) By Lemma 4.8 and Lemma 4.9, .γ2 ◦ γ1−1 is a hyperbolic isometry; let .D1 and .D2 be the convex envelopes in .H2 of the two intervals .I1 and .I2 with endpoints the fixed points of .γ2 ◦γ1−1 . Up to switching .γ1 and .γ2 , we can moreover assume that .γ2 ◦γ1−1 translates to the left seen from .D1 to .D2 . Now, we will use the example studied in Sect. 4.5. Let .fγ+1 ,γ2 be defined as in (4.17). By Corollary 4.3, .Pγi is the support plane of .C(fγ+1 ,γ2 ) at the point ± + .pi ∈ ∂± C(fγ ,γ ), for .i = 1, 2. Hence .π (pi ) = π ˆ l± (pi ), where .πˆ l± is the left l 1 2

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projection associated with .C(fγ+1 ,γ2 ). Since .πˆ l± (pi ) is bijective by Proposition 4.4, ± ± .π (p1 ) /= π (p2 ). This shows the injectivity. l l To prove the surjectivity, we first show that the image is closed. Suppose .zn = πl± (pn ) is a sequence in the image, with .lim zn = z ∈ H2 . Up to extracting a subsequence, we can assume .pn → p ∈ ∂± C(f ) ∪ graph(f ). From Proposition 4.5, we have that .p ∈ ∂± C(f ), because if .p = (x, y) ∈ graph(f ), then .πl± (pn ) → x ∈ ∂∞ H2 , thus contradicting the hypothesis .zn → z ∈ H2 . Now, let .Pγn be a support plane at .pn , which is spacelike by Proposition 4.2. By Lemma 4.6, up to extracting a subsequence, .γn → γ ∈ PSL(2, R) and .Pγ is a spacelike support plane at p. It is important to remark that .∂± C(f ) might admit several support planes at p, and .Pγ might not be the support plane that has been chosen in the definition of ± .π ; however, by Corollary 4.4 the image does not depend on this choice. Hence we l can assume that .Pγ is the support plane chosen at p. That is, from (4.12), .πl± (p) = Fix(p ◦ γ −1 ). We can now conclude that z is in the image of .πl± : on the one hand ± −1 .zn = π (pn ) = Fix(pn ◦ γn ) converges to z by hypothesis, and on the other l ± it converges to .πl (p) = Fix(p ◦ γ −1 ) because .pn → p, .γn → γ and .Fix is continuous. This shows that .z ∈ πl± (∂± C(f )), and therefore the image is closed. We now proceed to show that .πl± is surjective. Suppose by contradiction that there is a point .w ∈ H2 which is not in the image of .πl± . Let .r0 = inf{r | B(w, r) ∩ πl± (∂± C(f )) /= ∅}, where .B(w, r) is the open ball centered at w of radius r with respect to the hyperbolic metric of .H2 . Since the image of .πl± is closed, we have that .r0 > 0, .B(w, r0 ) is disjoint from the image of .πl± , and there exists a point ± ± .z ∈ ∂B(w, r0 ) which is in the image of .π . Say that .z = π (p). We will obtain a l l contradiction by finding points close to p which are mapped by .πl± inside .B(w, r0 ). Let .Pγ be a support plane of .C(f ) at p. By (4.11), .Pγ ∩ C(f ) is the convex hull of .∂∞ Pγ ∩ graph(f ), which contains at least two points. If p is in the interior of .Pγ ∩ C(f ) (which is non-empty if and only if .∂∞ Pγ ∩ graph(f ) contains at least three points), then the restriction of .πl± to the interior of .Pγ ∩ C(f ) is an isometry onto its image in .H2 , because .Pγ is the unique support plane at interior points .p' , and .πl± (p' ) = Fix(p ' ◦ γ −1 ). Hence .πl± maps a small neighbourhood of p to a neighbourhood of z, which intersects .B(w, r0 ), giving a contradiction. We are only left with the case where p is not in the interior of .Pγ ∩ C(f ). In this case, there is a geodesic L contained in .Pγ ∩ C(f ) such that .p ∈ L. (The geodesic L might be equal to .Pγ ∩ C(f ) or not.) As before, the image of L is a geodesic .𝓁 in ± 2 .H because .(π )|L is an isometry onto its image, and .z ∈ 𝓁. We claim that in the l image of .πl± there are two sequences of geodesics .𝓁n ⊂ Im(πl± ) converging to .𝓁 (in other words, such that the endpoints of .𝓁n converge to the endpoint of .𝓁); moreover the two sequences are contained in different connected components of .H2 \ 𝓁. This will give a contradiction, because for one of these two sequences, .𝓁n must intersect .B(w, r0 ) for n large. To show the claim, and thus conclude the proof, observe that L disconnects .∂± C(f ) in two connected components, and let .pn be a sequence converging to p contained in one connected component of .∂± C(f ) \ L. Let .Pγn be the support plane

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for .C(f ) at .pn which has been chosen to define .πl± . By Lemma 4.6, .Pγn converges to a support plane .Pγ at p, which as before we can assume is the support plane that defined .πl± at p, since the image does not depend on this choice by Corollary 4.4. Also, we can assume that each .pn is contained in a geodesic .Ln in .Pγn ∩ ∂± C(f ): indeed, it suffices to replace .pn by the point in .Pγn ∩ ∂± C(f ) which is closest to p (where closest is with respect to the induced metric on .∂± C(f ), or to any auxiliary Riemannian metric). If .Pγn ∩∂± C(f ) is not already a geodesic, with this assumption .pn now belongs to a boundary component which is the geodesic .Ln . As observed before, .πl± maps the geodesic .Ln to a geodesic .𝓁n = πl± (Ln ) in .H2 , and (as in the argument that showed that .Im(πl± ) is closed), the limit of .πl± (pn ) is a point in ± .𝓁 = π (L). l Moreover .𝓁n ∩ 𝓁 = ∅, and the .𝓁n are all contained in the same connected component of .H2 \𝓁: this follows from observing again (compare with the injectivity at the beginning of this proof) that .(πl± )|Ln ∪L equals the left projection associated with the surface .∂± C(fγ+n ,γ ) studied in Sect. 4.5, where .fγ+1 ,γ2 is defined in (4.17), and thus maps .∂± C(f ) ∩ Pγn (which in particular contains .Ln ) to a subset (containing .𝓁n ) disjoint from .𝓁 and included in a connected component of .H2 \ 𝓁 which does not depend on n. This implies that .𝓁n converges to .𝓁 as .n → +∞. Clearly if we had chosen .pn in the other connected component of .∂± C(f ) \ L, then the .𝓁n would be contained in the other connected component of .H2 \ 𝓁. This concludes the claim and thus the proof. ⨆ ⨅ As a consequence, the composition .πr± ◦(πl± )−1 is well-defined and is a bijection of .H2 to itself. Combining with Proposition 4.5, we get: Corollary 4.5 The composition .πr± ◦(πl± )−1 extends to a bijection from .H2 ∪∂∞ H2 to itself, which equals f on .∂∞ H2 and is continuous at any point of .∂∞ H2 . Proof Since .πl± and .πr± are bijective and extend to the bijections from .graph(f ) to ± −1 2 ± .∂∞ H sending .(x, y) to x and .y = f (x) respectively, the composition .πr ◦ (π ) l 2 2 extends to a bijection of .H ∪ ∂∞ H to itself sending x to .f (x). We need to check that this extension is continuous at any point of .∂∞ H2 . Proposition 4.5 shows that the extensions of .πl± and .πr± to .∂± C(f ) ∪ graph(f ) are continuous at any point of .graph(f ). Hence it remains to check that .(πl± )−1 is continuous at any point of .∂∞ H2 . This follows from a standard argument: let .zn be a sequence in .H2 ∪ ∂∞ H2 converging to .x ∈ ∂∞ H2 , and let .pn = (πl± )−1 (zn ). Up to extracting a subsequence, ± .pn → p. The limit p must be in .graph(f ), because if .p ∈ ∂± C(f ), although .π l might not be continuous there, we have already seen in Proposition 4.6 (see the proof that the image of .πl± is closed) that .limn πl± (pn ) = limn zn is a point of .H2 , thus giving a contradiction with .limn zn = x ∈ ∂∞ H2 . If .p ∈ graph(f ), then we can use the continuity and injectivity of .πl± on .graph(f ) to infer that .p = (πl± )−1 (x). This concludes the proof. ⨆ ⨅

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4.6.3 Earthquake Properties The last step which is left to prove is the verification that .πr± ◦ (πl± )−1 satisfies the properties defining earthquake maps. Proposition 4.7 The composition .πr− ◦ (πl− )−1 : H2 → H2 is a left earthquake map. Analogously, .πr+ ◦ (πl+ )−1 : H2 → H2 is a right earthquake map. Proof First, let us define a geodesic lamination .λ. Let us consider all the support planes .Pγ of .C(f ) at points of .∂± C(f ) (which are necessarily spacelike by Proposition 4.2). Define .L to be the collection of all the connected components of .(Pγ ∩∂± C(f ))\int(Pγ ∩∂± C(f )), as .Pγ varies over all support planes. As observed before, by (4.11) .Pγ ∩ ∂± C(f ) is the convex hull in .Pγ of .∂∞ Pγ ∩ graph(f ), which consists of at least two points. If it consists of exactly two points, then .Pγ ∩ ∂± C(f ) is a spacelike geodesic L; otherwise .Pγ ∩ ∂± C(f ) has nonempty interior and each connected component of its boundary is a spacelike geodesic. Now, ± .π l is an isometry onto its image when restricted to any .L ∈ L (which might depend on the choice of a support plane if there are several support planes at points of L, but the image does not depend on this choice by Corollary 4.4). Hence we define .λ to be the collection of all the .πl± (L) as L varies in .L. To show that .λ is a geodesic lamination, we first observe that the geodesics .𝓁 ∈ λ are pairwise disjoint, because the spacelike geodesics L in .L are pairwise disjoint and .πl± is injective. Then it remains to show that their union is a closed subset of .H2 . This follows immediately from the proof of Proposition 4.6. Indeed, suppose that ± ± ± .𝓁n = π (Ln ) converges to .𝓁 = π (L), and let .zn = π (pn ) ∈ 𝓁n be a sequence l l l converging to .z ∈ 𝓁. Since .Im(πl± ) is closed, .z ∈ Im(πl± ), and since .πl± is injective, ± .z = π (p) for some .p ∈ L. Then in the last part of the proof of Proposition 4.6 we l have shown that in this situation .𝓁n converges to .𝓁. Having shown that .λ is a geodesic lamination, we are ready to check that .πr− ◦ − −1 (πl ) is an earthquake map. Observe that the gaps of .λ are precisely the images under .πl± of the interior of the sets .Pγ ∩ ∂± C(f ) (when this intersection is not reduced to a geodesic), as .Pγ varies among all support planes. Let .S1 and .S2 be two strata of .λ, and let .Σi = (πl± )−1 (Si ). Hence .Σi ⊂ Pγi ∩ ∂± C(f ), where .Pγi is a support plane. As usual, there might be several support planes at points of .Σi , and this can occur only if .Σi is reduced to a geodesic by Lemma 4.9. Recalling from Remark 4.3 that the chosen support plane is assumed to be constant along .Σi , we can suppose that .Pγi is the support plane chosen in the definition of .πl± and .πr± . Now we proceed as in the proof of injectivity in Proposition 4.6. Consider first the case that .γ1 /= γ2 . By Lemmas 4.8 and 4.9, .γ2 ◦ γ1−1 is a hyperbolic isometry; let .D1 and .D2 be the convex envelopes in .H2 of the two intervals .I1 and .I2 with endpoints the fixed points of .γ2 ◦ γ1−1 . Up to switching .γ1 and .γ2 , we assume that −1 ± .γ2 ◦ γ ˆ l± )|Σi and 1 translates to the left seen from .D1 to .D2 . Then .(πl )|Σi = (π ± ± ± ± .(πr )|Σi = (π ˆ r )|Σi , where .πˆ l and .πˆ r are the left and right projections associated

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with .C(fγ+1 ,γ2 ), and moreover .Si ⊂ Di . By the second part of Proposition 4.4, the  1 , D2 ) of the map .πˆ r± ◦ (πˆ ± )−1 translates to the left comparison isometry .Comp(D l

(for .πr− and .πl− ) or right (for .πr+ and .πl+ ) seen from .D1 to .D2 . Then .Comp(S1 , S2 ),  1 , D2 ), translates to the left (or right) seen from which is indeed equal to .Comp(D .S1 to .S2 . Finally, we instead consider the case .γ1 = γ2 , which can only happen either if .Σ1 = Σ2 (hence .S1 = S2 ) or if .Σ1 has nonempty interior and .Σ2 is one of its boundary components (or vice versa). In this case we clearly have .Comp(S1 , S2 ) = id. But the comparison isometry is indeed allowed in Definition 4.2 to be the identity, when one of the two strata is contained in the closure of the other. This concludes the proof. ⨆ ⨅ The proof of Thurston’s earthquake theorem (Theorem 4.1) is thus complete.

4.6.4 Recovering Earthquakes of Closed Surfaces In this final section, we recover (Corollary 4.6) the existence of earthquake maps between two homeomorphic closed hyperbolic surfaces. Given a group G and two representations .ρ, ϱ : G → PSL(2, R), we say that a map F from .H2 (or .∂∞ H2 ) to itself is .(ρ, ϱ)-equivariant if it satisfies F ◦ ρ(g) = ϱ(g) ◦ F

.

for every .g ∈ G. Corollary 4.6 Let S be a closed oriented surface and let .ρ, ϱ : π1 (S) → PSL(2, R) be two Fuchsian representations. Then there exists a .(ρ, ϱ)-equivariant left earthquake map of .H2 , and a .(ρ, ϱ)-equivariant right earthquake map. Proof Let .f : ∂∞ H2 → ∂∞ H2 be the unique .(ρ, ϱ)-equivariant orientationpreserving homeomorphism. We claim that there exists a left (resp. right) earthquake as in Theorem 4.1, which is itself .(ρ, ϱ)-equivariant. For this purpose, observe that for any .g ∈ π1 (S), the pair .(ρ(g), ϱ(g)) ∈ PSL(2, R) × PSL(2, R) acts on .∂∞ AdS3 preserving .graph(f ), since by (4.5) and the definition of .(ρ, ϱ)-equivariant, (ρ(g), ϱ(g)) · graph(f ) = graph(ϱ(g) ◦ f ◦ ρ −1 (g)) = graph(f ) .

.

Hence the convex hull .C(f ) is preserved by the action of .(ρ(g), ϱ(g)) for all .g ∈ π1 (S). To conclude the proof, we need to show that we can choose support planes at every point of both boundary components of .C(f ) \ graph(f ) in such a way that this choice of support planes is also preserved by the action of .(ρ(g), ϱ(g)) for all .g ∈ π1 (S). (Clearly it suffices to consider the situation at points that admit more

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than one support plane, because if .p ∈ ∂± C(f ) has a unique support plane P , then (ρ(g), ϱ(g)) · P is the unique support plane at .(ρ(g), ϱ(g)) · p.) When we have shown this, we will take the left and right projections .πl± , πr± defined via this invariant choice of support planes. By Lemma 4.7, we will then deduce that the left projection .πl± : ∂± C(f ) → H2 is equivariant with respect to the action of .(ρ(g), ϱ(g)) on .∂± C(f ) and the action of .ρ(g) on .H2 ; analogously the right projection .πr± : ∂± C(f ) → H2 is equivariant with respect to the action of .(ρ(g), ϱ(g)) on .∂± C(f ) and the action of .ϱ(g) on .H2 . Following the proof of Theorem 4.1, the left and right earthquake maps obtained as the composition ∓ −1 ◦ π ∓ will be .(ρ, ϱ)-equivariant, and the proof will be concluded. .(πr ) l First, we need to prove an intermediate claim. Suppose .p ∈ ∂± C(f ) admits several support planes. By Lemma 4.9, there is a spacelike geodesic .L ⊂ ∂± C(f ) containing p. Let .g ∈ π1 (S) be such that .(ρ(g), ϱ(g)) · L = L. Then we claim that .(ρ(g), ϱ(g)) maps every support plane at p to itself. To prove this claim, we use Corollary 4.4 and suppose up to an isometry (so that, in the notation of Corollary 4.4, .γ1 = 1) that all the support planes at p are of the form .Pexp(ta) with .t ∈ [0, 1], where .γ := exp(a) is a hyperbolic element. Clearly .(ρ(g), ϱ(g)) must preserve the pair of “extreme” support planes .P1 and .Pγ . Hence there are two possibilities: either .(ρ(g), ϱ(g)) maps .1 to .1 and .γ to .γ , or it switches .1 and .γ . However, the latter possibility cannot be realized, since the identities .ρ(g)ϱ(g)−1 = γ and .ρ(g)γ ϱ(g)−1 = 1 would imply that .γ has order two, and this is not possible for a hyperbolic element. We thus have .(ρ(g), ϱ(g)) · 1 = 1 and .(ρ(g), ϱ(g)) · γ = γ . This implies first that .ρ(g) = ϱ(g). Moreover .ρ(g)γρ(g)−1 = γ , which shows that .ρ(g) = ϱ(g) = exp(sa) for some −1 = exp(ta) for all t, that is .(ρ(g), ϱ(g)) = .s ∈ R. Therefore .ρ(g) exp(ta)ρ(g) (ρ(g), ρ(g)) maps every support plane .Pexp(ta) to itself. Having shown the claim, we can conclude as follows. Observe that the set of points .p ∈ ∂± C(f ) that admit several support planes form a disjoint union of spacelike geodesics in .∂± C(f ), and that this set (say X) is invariant under the action of .(ρ(g), ϱ(g)) for all .g ∈ π1 (S). Pick a subset .{Li }i∈I of this family of geodesics such that its .π1 (S)-orbit is X, and that the orbits of .Li and .Lj are disjoint if .i /= j . Pick a support plane .Pi at .p ∈ Li , and then we declare that .(ρ(g0 ), ϱ(g0 )) · Pi is the chosen support plane at every point of .(ρ(g0 ), ϱ(g0 ))·Li . This choice is well-defined by the above claim, which showed that if .(ρ(g), ϱ(g)) leaves .Li invariant, then it also leaves every support plane at .Li invariant. Moreover this choice of support planes is invariant by the action of .π1 (S) by construction. This concludes the proof. ⨆ ⨅ .

Appendix: Two Lemmas in the Hyperbolic Plane We provide here the proofs of two properties on the action on .H2 ∪ ∂∞ H2 of sequences of elements in .PSL(2, R). We prove them by elementary arguments in the specific case of sequences of order–two elliptic isometries.

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The first elementary property that we prove here is the uniform convergence of the action of elliptic isometries on the compactification of .H2 . Lemma 4.10 Let .wn be a sequence in .H2 converging to .w ∈ H2 . Then .Rwn converges to .Rw uniformly on .H2 ∪ ∂∞ H2 . Proof Up to conjugation, we may assume .w = i. Writing .wn = |wn |eiηn , it is easy to check that Rwn (z) =

.

cos(ηn )z − |wn | . |wn |−1 z − cos(ηn )

Let us conjugate .Rwn by the map .ψ(z) = (iz + 1)/(z + i), which maps .H2 to the disc, and show that it converges to .z I→ −z uniformly on the closed disc. For 2 2 .z ∈ H ∪ ∂∞ H we have ψ ◦ Rwn ◦ ψ −1 (z) + z

.

(|wn |−1 − |wn | − 2i cos(ηn ))z2 + (|wn |−1 − |wn | + 2i cos(ηn )) (|wn |−1 − |wn | − 2i cos(ηn ))z + i(|wn | + |wn |−1 )

= Hence

|ψ ◦ Rwn ◦ ψ −1 (z) + z| ≤

.

2|αn | |αn z + βn |

where .αn = ||wn |−1 − |wn | − 2i cos(ηn )| and .βn = i(|wn | + |wn |−1 ). Thus |ψ ◦ Rwn ◦ ψ −1 (z) + z| ≤

.

2 |z +

βn αn |

≤

2 || αβnn | − |z||

Since .|βn | ≥ 2, .|αn | → 0 and .|z| ≤ 1, there exists .n0 ∈ N such that the right-hand side is smaller than .ϵ for all z in the closed disc. This completes the proof. ⨆ ⨅ The second property is a special case of the so-called North-South dynamics. Lemma 4.11 Let .wn be a sequence in .H2 converging to .w ∈ ∂∞ H2 . Then, for every neighbourhood U of w, there exists .n0 such that .Rwn ((H2 ∪ ∂∞ H2 ) \ U ) ⊂ U for .n ≥ n0 . Proof We adopt the same notation as in the proof of Lemma 4.10. Up to conjugation, we may assume that .w = ∞. It is sufficient to consider neighbourhoods U of the form .Ur = {|z| > r} ⊂ H2 ∪ ∂∞ H2 . By a direct computation, |Rwn (z)| =

.

|wn | − | cos(ηn )||z| | cos(ηn )z − |wn || ≥ . ||wn |−1 z − cos(ηn )| |wn |−1 |z| + | cos ηn |

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Since .wn converges to .∞, for all r we have .|wn | ≥ r ≥ |z| ≥ | cos ηn ||z| if n is sufficiently large and z is in the complement of .Ur . Then |Rwn (z)| ≥

.

|wn

|wn | − r −1 | r + | cos η

n|

−→ +∞.

It follows that .|Rwn (z)| > r for .n ≥ n0 , that is, .Rwn maps the complement of .Ur to .Ur . ⨅ ⨆ Acknowledgments We would like to thank Pierre Will for a remark on the description of timelike planes via composition of orientation-reversing isometries, that is used in Sect. 4.3.4. We are grateful to Filippo Mazzoli and Athanase Papadopoulos for useful suggestions that helped improving the exposition.

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16. D. Šari´c, Real and complex earthquakes. Trans. Am. Math. Soc. 358(1), 233–249 (2006) 17. D. Šari´c, Bounded earthquakes. Proc. Am. Math. Soc. 136(3), 889–897 (2008) 18. D. Šari´c, Some remarks on bounded earthquakes. Proc. Am. Math. Soc. 138(3), 871–879 (2010) 19. A. Seppi, Maximal surfaces in Anti-de Sitter space, width of convex hulls and quasiconformal extensions of quasisymmetric homeomorphisms. J. Eur. Math. Soc. 21(6), 1855–1913 (2019) 20. W.P. Thurston, Earthquakes in two-dimensional hyperbolic geometry, in Low Dimensional Topology and Kleinian Groups, Symposium Warwick and Durham 1984. London Mathematical Society Lecture Note Series, vol. 112 (1986), pp. 91–112

Chapter 5

Homeomorphism Groups of Self-Similar 2-Manifolds Nicholas G. Vlamis

Abstract The class of self-similar 2-manifolds consists of manifolds exhibiting a type of homogeneity akin to the 2-sphere and the Cantor set, and includes both the 2-sphere and the 2-sphere with a Cantor set removed. This chapter aims to provide a narrative thread between recent results on the structure of homeomorphism groups/mapping class groups of self-similar 2-manifolds, and also connections to classical structural results on the homeomorphism group of the 2-sphere and the Cantor set. In order to do this, we provide a survey of recent results, an exposition on classical results about homeomorphism groups, provide a treatment of the structure of stable sets, and prove extensions/strengthenings of the recent results surveyed. Of particular note, we establish the following theorems: (1) A characterization of homeomorphisms of (orientable) perfectly self-similar 2-manifolds that normally generate the group of (orientation-preserving) homeomorphisms—a strengthening of a result of Malestein–Tao. (2) The homeomorphism group of a perfectly self-similar 2-manifold is strongly distorted—an extension of a result of Calegari–Freedman for spheres. (3) The homeomorphism group of a perfectly tame 2-manifold is Steinhaus, and hence has the automatic continuity property— an extension of a result of Mann in dimension two—providing the first examples of homeomorphism groups of infinite-genus 2-manifolds with the Steinhaus property. Keywords Big mapping class group · Infinite-type surface · Self-similar 2-manifold · Strong distortion · Automatic continuity · Rokhlin property · Coarsely bounded groups Mathematics Subject Classification 2022 57K20, 57S05, 54H11, 20F65

N. G. Vlamis () Department of Mathematics, CUNY Graduate Center, New York, NY, USA Department of Mathematics, CUNY Queens College, Flushing, NY, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 K. Ohshika, A. Papadopoulos (eds.), In the Tradition of Thurston III, https://doi.org/10.1007/978-3-031-43502-7_5

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5.1 Introduction Over the last decade, there has been growing interest in the study of mapping class groups of infinite-type 2-manifolds, often referred to as big mapping class groups. Recently, Mann and Rafi’s work [55] has provided a framework for partitioning infinite-type 2-manifolds into several natural classes, which has led to a flurry of activity. We focus here on a particular subclass of 2-manifolds that exhibit a high degree of homogeneity at infinity, which leads to their homeomorphism groups and mapping class groups having “low complexity” from the viewpoint of their geometry and subgroup structure (as compared to other 2-manifolds). The recent progress in this subclass connects back to the work of Anderson, which has been adapted to this setting. In 1958, Anderson [3] introduced a simple yet ingenious technique to establish the algebraic simplicity of a class of transformation groups, including the group of orientation-preserving self-homeomorphisms of the 2-sphere and the group of selfhomeomorphisms of the Cantor set. In 1947, Ulam and von Neumann announced this same result for the 2-sphere, but it was never published. It is worth noting that Anderson’s method fails in the context of diffeomorphism groups, and the analogous theorem in this setting is due to Thurston [70]. Thurston did not publish his proof, but the details are presented in [9]. A short proof has been given by Mann (see [53] and the set of notes [51, Section 2]).

5.1.1 The Main Object of Study: Self-Similar 2-Manifolds The subclass of 2-manifolds we restrict ourselves to are known as self-similar 2manifolds, and they exhibit a high-degree of homogeneity at infinity, which is meant to be captured in the following definition. Definition 5.1.1 (Self-Similar 2-Manifold) A subset .K of a manifold .M is displaceable if there exists a homeomorphism .f : M → M such that .f (K) ∩ K = ∅. A 2-manifold .M is self-similar if every proper compact subset is displaceable and, for any separating simple closed curve .c in .M, there is a component of .M \ c that is homeomorphic to .M with a point removed. We further partition the class of selfsimilar 2-manifolds into two subclasses: A self-similar 2-manifold .M is perfectly self-similar if .M#M is homeomorphic to .M; otherwise, it is uniquely self-similar. It follows from the Jordan–Schoenflies theorem that the 2-sphere and the plane are self-similar; moreover, the 2-sphere is perfectly self-similar, but the plane is not. These two examples are the only finite-type self-similar 2-manifolds; all other cases must necessarily have infinitely many ends or infinite genus. The simplest infinite-type examples of self-similar 2-manifolds are the 2-sphere with a Cantor set removed, the plane with an infinite discrete set removed, and the orientable oneended infinite-genus 2-manifold. Among these examples, only the 2-sphere with the

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Cantor set removed is perfectly self-similar. We encourage the reader to keep these examples in mind throughout the chapter, as they capture the essence of self-similar 2-manifolds. More generally, a self-similar 2-manifold can be obtained by removing a compact countable subset of Cantor–Bendixson degree one from the 2-sphere; this procedure results in an uncountably family of pairwise non-homeomorphic self-similar 2manifolds (all of which are uniquely self-similar). If one imagines blowing up the highest rank accumulation point in the previous examples to a Cantor set, an uncountable collection of perfectly self-similar 2-manifolds is obtained. The original definition of a self-similar 2-manifold was given by Malestein and Tao in [50] and was framed in terms of the topological ends of the manifold, which in turn used the terminology introduced by Mann–Rafi in [55]. The advantage of the definition above is that we do not rely on the notion of an end of a manifold, and hence the definition clearly includes the 2-sphere, which allows us to see how the recent developments in the theory of homeomorphism groups of (perfectly) selfsimilar 2-manifolds are a natural extension of results regarding the homeomorphism group of the 2-sphere.

5.1.2 Goals and Outline Many of the recent articles investigating the structure of mapping class groups of self-similar 2-manifolds have been released in a brief period of time, and hence, this chapter benefits by cross-pollinating ideas between the articles. For instance, we are able to give a stronger version of a result of Malestein and Tao [50] using ideas from Calegari and Chen [16] (see Theorem 5.8.1). This is an example of our first goal: strengthen and/or extend the current results in the literature. Our second goal is to generalize recent results about mapping class groups to the setting of homeomorphism groups (e.g., Theorem 5.11.1), and in the process, provide the reader with a background in the topology of homeomorphism groups. This provides a narrative thread between recent progress in the study of big mapping class groups with the history of studying homeomorphism groups. The next goal, realized through various corollaries, is to exhibit a clear connection between the homeomorphism groups of 2-manifolds and the homeomorphism groups of second-countable Stone spaces, which factors through the classification of surfaces. In particular, many results about the homeomorphism group of the 2-sphere generalize naturally to homeomorphisms groups of perfectly self-similar 2-manifolds, including the 2-sphere with a Cantor set removed. We can therefore see how a single proof can simultaneously establish properties of the homeomorphisms group of the 2-sphere and the Cantor set. Our final goal is to collect, in one place, useful tools and results for understanding and working with end spaces of 2-manifolds and their stable sets. These tools are used throughout the chapter in various arguments.

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With these goals in mind, the chapter has been written in a nearly self-contained fashion relying in most places only on a minimal set of prerequisites, including point-set topology, algebraic topology (e.g., [37, Chapters 0, 1, and 2]), and the basics of topological group theory (e.g., [58, Chapter 1]). However, this is not to imply that the chapter makes for easy reading, as many ideas are introduced quickly and treated concisely. In the places where providing all the required details would send us too far afield, we provide appropriate references for the reader. We also point the reader to [5], a chapter in an earlier volume of this series written by Aramayona and the author giving an introduction to big mapping class groups. The outline of the paper is as follows: • Section 5.2 provides a survey of recent results on homeomorphism groups and mapping class groups of self-similar 2-manifolds. • Section 5.3 gives an overview of the key topological properties of 2-manifolds, including their classification up to homeomorphism. • Section 5.4 introduces tools for studying end spaces of 2-manifolds, and in particular, develops the structure theory of stable sets. • Section 5.5 introduces the notion of Freudenthal subsurfaces and introduces a version of Anderson’s method that is well suited to our purposes. • Section 5.6 is an exposition on fundamental properties of homeomorphism groups of 2-manifolds. • Section 5.7 gives several equivalent notions of self-similarity. • Sections 5.7–5.12 provide proofs of the various theorems surveyed in Sect. 5.2.

5.2 Overview of Results Let us begin by setting basic notation. In what follows, a 2-manifold (resp., surface) refers to a connected second-countable Hausdorff topological space in which every point admits an open neighborhood homeomorphic to the plane .R2 (resp., a closed half-plane, that is, .{(x, y) ∈ R2 : y ≥ 0}). In particular, a 2-manifold is a surface whose boundary is empty. Note that every 2-manifold is metrizable. A 2-manifold is of finite type if it can be realized as the interior of a compact surface; otherwise, it is of infinite type. A simple closed curve on a surface .S is the image of a topological embedding of the circle in .S; it is separating if its complement is disconnected. Given a 2-manifold .M, let .Homeo(M) be the homeomorphism group of .M, that is, the group of homeomorphisms .M → M. Equipped with the compact-open topology, .Homeo(M) is a Polish group, that is, a separable and completely metrizable topological group (see Sect. 5.6 for more details). If .M is orientable, .Homeo+ (M) denotes the subgroup of .Homeo(M) consisting of orientation-preserving homeomorphisms, which, as a closed subgroup, is also Polish. Notation Let .M be a 2-manifold. If .M is orientable, we set .H (M) = Homeo+ (M); otherwise, we set .H (M) = Homeo(M).

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The mapping class group of a 2-manifold .M, denoted .MCG(M), is the quotient group .H (M)/H0 (M), where .H0 (M) is the connected component of the identity in .H (M), or equivalently, .MCG(M) is the group of isotopy classes of elements in .H (M); this equivalence is established in Sect. 5.6. Since .H0 (M) is a closed subgroup of .H (M), it follows that .MCG(M) is a Polish topological group when equipped with the quotient topology (see [42, Theorem  8.19]). We write .2N to represent the countable product . n∈N {0, 1} equipped with the product topology, where .{0, 1} is given the discrete topology. With this topology, .2N is homeomorphic to the Cantor set. In what follows, we survey and motivate a number of recent results. In the following sections, we provide proofs of all the numbered theorems and corollaries presented in this section. For each of the numbered theorems, we are able to give some extension of the original result in the literature, such as extending to a larger classes of surfaces and including non-orientable surfaces, or generalizing from mapping class groups to homeomorphism groups. Despite these extensions, the proofs generally follow the existing proofs in the literature with appropriate adaptation to our setting.

5.2.1 Normal Generation and Purity As noted in the introduction, Anderson [3] proved that .H (S2 ) and .Homeo(2N ) are simple groups (i.e., they only have two normal subgroups). The first theorem we introduce is a natural generalization of these results to the setting of perfectly selfsimilar 2-manifolds (and their end spaces) and strengthens a result of Malestein and Tao [50, Theorem A]. The theorem relies on what we call Anderson’s method (see Sect. 5.5 and, more specifically, Proposition 5.5.7). The version of Anderson’s method presented is suited to our purposes and is a straightforward generalization of Calegari and Chen’s [16, Lemma 1], which itself is a variation of Anderson’s original technique presented in [3]. The commutator of two elements .g and .h in a group is the group element .[g, h] = ghg −1 h−1 . A group .G is perfect if it is equal to its commutator subgroup .[G, G]— the subgroup generated by the commutators in .G. A group .G is uniformly perfect if there exits .p ∈ N such that every element of .G can be written as a product of .p commutators; the minimal such .p is called the commutator width of .G. An element .g of a group normally generates the group if every element can be expressed as a product of conjugates of .g. To state the theorem, we need the notion of a half-space: a closed subset .D of a self-similar 2-manifold .M is a half-space if .D is a subsurface of .M with connected compact boundary and the closure of .M \ D is homeomorphic to .D (see Definition 5.7.3). For example, in the 2-sphere, every embedded closed disk is a half-space.

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Theorem 5.8.1 Let .M be a perfectly self-similar 2-manifold, and let .G denote either .H (M) or .MCG(M). (1) .G is uniformly perfect and its commutator width is at most two. (2) If .g ∈ G displaces a half-space of .M, then every element of .G can be expressed as a product of at most eight conjugates of .g and .g −1 . In particular, .G is normally generated by .g. (3) If .n ∈ N such that .n ≥ 2, then every element of .G can be expressed as the product of at most eight elements of order .n. In the case of the 2-sphere minus a Cantor set, Theorem 5.8.1(1) was established in a blog post of Calegari, and Theorem 5.8.1(2) was established by Calegari and Chen [16]. Under the additional assumption of orientability, Malestein and Tao [50] established Theorem 5.8.1(1) (with commutator width three instead of two) and Theorem 5.8.1(3) restricted to .n = 2. As already noted, the 2-sphere is perfectly self-similar, and moreover, every nontrivial homeomorphism must displace a disk (i.e., a half-space). Therefore, by Theorem 5.8.1, every nontrivial element of .H (S2 ) normally generates the group, yielding: Corollary 5.2.1 (Anderson [3]) .H (S2 ) is a simple group.

⨆ ⨅

A quasimorphism of a group .G is a function .q : G → R such that there exists d ∈ R+ satisfying .|q(gh) − q(g) − q(h)| < d for all .g, h ∈ G (see [15, Chapter 2] for an introduction to quasimorphisms and their properties, and [44] for an informal introduction to quasimorphisms and additional references). By Theorem 5.8.1, if .M is a perfectly self-similar 2-manifold, then every element of .H (M) is the product of at most two commutators; this implies that every quasimorphism of .H (M) is bounded.

.

Corollary 5.2.2 If .M is a perfectly self-similar 2-manifold, then every quasimorphism of .H (M) (resp., .MCG(M)) is bounded. ⨆ ⨅ Informally, Corollary 5.2.2 implies that .H (M) admits no “interesting” geometric actions; we will not be more explicit here, as we will prove a stronger geometric restriction in the following subsection. Next, let us explain how Theorem 5.8.1 is also a generalization of Anderson’s theorem stating that .Homeo(2N ) is simple. An end of a perfectly self-similar 2manifold is maximally stable if it corresponds to a nested sequence of half-spaces (see Definition 5.7.3 for the general definition of a maximally stable end and see Sect. 5.3 for an introduction to ends). Recall that a 2-manifold is planar if it is homeomorphic to an open subset of .R2 . Let .E be the end space of a planar perfectly self-similar 2-manifold .M. From Richards’s work on the classification of surfaces [61], we obtain the following two facts (see Sect. 5.3 for more details): (1) The end space of a planar 2-manifold is a second-countable Stone space,1 and conversely, every second-countable Stone space can be realized as the end space of a planar

1A

Stone space is a compact zero-dimensional Hausdorff topological space.

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2-manifold. (2) The action of .H (M) on .E induces an epimorphism of .H (M) onto Homeo(E). We say that a second-countable Stone space .E is (perfectly/uniquely) self-similar if it is realized as the end space of a planar (perfectly/uniquely) self-similar 2manifold .M and that .e ∈ E is maximally stable if it is a maximally stable end of .M (intrinsic definitions can be given with the language introduced in Sect. 5.4). Together with Theorem 5.8.1, the above facts yield the following corollary:

.

Corollary 5.2.3 Let .E be a perfectly self-similar second-countable Stone space. Every element of .Homeo(E) that permutes the maximally stable points nontrivially normally generates .Homeo(E). In particular, .Homeo(2N ) is simple. ⨆ ⨅ As another corollary, we give an extension of the Purity Theorem of Calegari and Chen [16]. The Purity Theorem below completely characterizes the normal generators of .H (M) for a non-compact perfectly self-similar 2-manifold, and therefore, is the natural generalization of Anderson’s theorem, Corollary 5.2.1, to the non-compact setting. Corollary 5.2.4 (Purity Theorem) Let .M be a non-compact perfectly self-similar 2-manifold, and let .G denote either .H (M) or .MCG(M). Then, every proper normal subgroup of .G is contained in the kernel of the action of .G on the set of maximally stable ends of .M. In particular, an element of .H (M) normally generates .H (M) if and only if it induces a nontrivial permutation of the maximally stable ends of .M. Moreover, .G does not contain a proper normal subgroup of countable index. We give a proof of Corollary 5.2.4 in Sect. 5.8. Let .K ⊂ R2 ⊂ S2 be a Cantor space (that is, homeomorphic to .2N ). The Purity Theorem was first proved by Calegari and Chen [16] in the special case of .S2 \ K (they also give a statement for finite-type surfaces with a Cantor space removed). To foreshadow a subsection below, we note that in this special case, the author [72] had previously established the weaker result that neither .H (S2 \ K) nor .H (R2 \ K) contain a proper normal subgroup of countable index. The proof given in [72] is an example application of the automatic continuity of these homeomorphism groups (established by Mann [54]), which we explore in a later subsection. In a separable topological group, any open subgroup must have countable index. Therefore, Corollary 5.2.1 (in the case of the 2-sphere) and Corollary 5.2.4 (for all other perfectly self-similar 2-manifolds) yield the following corollary, which we will later generalize to the setting of all self-similar 2-manifolds. Corollary 5.2.5 If .M is a perfectly self-similar 2-manifold, then neither .H (M) nor MCG(M) contain a proper open normal subgroup. ⨆ ⨅

.

As a last corollary to Theorem 5.8.1, we give an extension of [16, Lemma 2.5]. Corollary 5.2.6 Let .M be a planar perfectly self-similar 2-manifold. Every proper normal subgroup of .H (M) and .MCG(M) is torsion free. In particular, every torsion element of .H (M) (resp., .MCG(M)) is a normal generator.

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The proof of Corollary 5.2.6 is given in Sect. 5.8. The planarity assumption in Corollary 5.2.6 is necessary: by the work of Aougab, Patel, and the author (see [4, Lemmas 3.5 & 3.6]), the pure mapping class group of an orientable infinite-genus self-similar 2-manifold with no planar ends contains an isomorphic copy of every countable group; in particular, it contains torsion elements that do not normally generate. Here, the pure mapping class group refers to the kernel of the action of the mapping class group on the end space.

5.2.2 Strong Distortion In the previous section, we showed the extent to which the algebraic simplicity of .H (S2 ) can be generalized to the setting of perfectly self-similar 2-manifolds. Here, we will give a strong notion of geometric simplicity that is exhibited by 2 .H (S ), which can also be seen in homeomorphism groups of perfectly self-similar 2-manifolds. A group .G is strongly distorted if there exists .m ∈ N and a sequence .{wn }n∈N ⊂ N such that, given any sequence .{gn }n∈N in .G, there exists a set .S ⊂ G of cardinality at most .m such that .gn ∈ S wn for every .n ∈ N. One immediate consequence is that every non-identity element of .G is distorted in the standard sense, that is, if .g ∈ G is not the identity, then there exists a finitely generated subgroup .𝚪 of .G such that the word length of .g n in .𝚪 grows sublinearly in .n (equivalently, the homomorphism from .Z to .𝚪 given by .n I→ g n fails to be a quasi-isometric embedding). Quasi-morphisms are often used to detect undistorted group elements, and so Corollary 5.2.2 suggests that the elements of .H (M) are distorted, and indeed: Theorem 5.9.2 If .M is a perfectly self-similar 2-manifold, then .H (M) and MCG(M) are strongly distorted.

.

The proof of Theorem 5.9.2 is an adaptation of Calegari–Freedman’s proof of [17, Theorem C], and setting .M = S2 , we recover Calegari–Freedman’s result in dimension two. Corollary 5.2.7 (Calegari–Freedman [17]) .H (S2 ) is strongly distorted. Strong distortion is inherited by quotients, and so arguing as in the preceding subsection, we deduce the following corollary for second-countable Stone spaces. Corollary 5.2.8 The homeomorphism group of a perfectly self-similar secondcountable Stone space is strongly distorted; in particular, .Homeo(2N ) is strongly distorted. ⨆ ⨅ To the best of the author’s knowledge, this result does not appear in the literature; however, it is known that .Homeo(2N ) is strongly bounded [45].

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There are several other consequences of being strongly distorted; for instance, if G is strongly distorted, then:

.

• .G is strongly bounded (i.e., every isometric action of .G on a metric space has bounded orbits, and in particular, every left-invariant metric on .G is bounded), • .G has uncountable cofinality (i.e., .G is not the union of a countable strictly increasing sequence of subgroups), and • .G has the Schreier property (i.e., every countable subset of .G is contained in a finitely generated subgroup of .G). A more in-depth discussion of these various properties, and how they are related, can be found in the introduction of [67].

5.2.3 Coarse Boundedness Thus far, we have only discussed perfectly self-similar 2-manifolds, and we have motivated the results by comparing to the structure of .H (S2 ). We will now broaden our discussion to include uniquely self-similar 2-manifolds. Ideally, we would like to compare the structure of homeomorphism groups of uniquely selfsimilar 2-manifolds with that of .H (R2 ). Let us quickly summarize the analogs of Theorems 5.8.1 and 5.9.2 for .H (R2 ): every element of .H (R2 ) has commutator length at most three (see Corollary 5.6.13), and .H (R2 ) is strongly distorted [67, Theorem 1.6]. Unfortunately, we cannot generalize these statements to the entire class of uniquely self-similar 2-manifolds. Let .L denote the orientable one-ended infinitegenus 2-manifold, often referred to as the Loch Ness monster surface. Domat and Dickmann [21] showed that .H (L) admits an epimorphism to .R, which implies .H (L) fails to be perfect, strongly distorted, strongly bounded, and fails to have uncountable cofinality and the Schreier property. Moreover, realizing .R as a direct sum of a continuum worth of copies of .Q, we see that .H (L) has many countableindex subgroups. Despite this, we can obtain some positive comparisons to .H (R2 ) if we take into account the topology of .H (M). Rosendal has laid the foundation for extending the tools of geometric group theory to the setting of non-locally compact topological groups (see [65]), and in particular, understanding when such a group has a canonical metric (up to quasi-isometry). Mann and Rafi [55] have classified the mapping class groups that admit such a metric. Here, we will focus on the portion of Mann– Rafi’s work dedicated to understanding which 2-manifolds’ homeomorphism groups behave—geometrically—like finite (or compact) groups in Rosendal’s theory. A topological group .G is coarsely bounded if every .G-orbit is bounded whenever 2 .G acts continuously on a metric space by isometries. In particular, every continuous

2 This

also appears in the literature as the topological Bergman property or property (OB).

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left-invariant (pseudo-)metric on a coarsely bounded topological group has finite diameter, and in fact, this is equivalent to being coarsely bounded. In the setting of Polish groups, there are numerous equivalent conditions (see [65, Proposition 2.7]). Note that if .G is strongly bounded, then it is coarsely bounded. A historical remark: in the article introducing the notion of a coarsely bounded group [64], Rosendal proved that the homeomorphism group of a sphere is coarsely bounded. The preprint appeared within a day of the preprint of Calegari and Freedman [17], establishing strong distortion for these homeomorphism groups. In a later version of [17], an appendix, written by Corulier, was added showing that strong distortion implies strongly bounded, and hence strengthening the result of Rosendal. In [55, Proposition 3.1], Mann–Rafi establish that the mapping class group of every self-similar 2-manifold is coarsely bounded. We will see that their proof readily adapts to the case of homeomorphism groups and non-orientable 2manifolds. Theorem 5.10.1 Let .M be a self-similar 2-manifold. Then, .H (M) and .MCG(M) are coarsely bounded. Being coarsely bounded is inherited by quotients by closed subgroups, and so we have the following. Corollary 5.2.9 The homeomorphism group of a uniquely self-similar secondcountable Stone space is coarsely bounded. ⨆ ⨅ There are many natural examples of uniquely self-similar second-countable Stone spaces, such as every compact countable Hausdorff space of Cantor– Bendixson degree one. Using the classification of compact countable Hausdorff spaces [56], every such space of Cantor–Bendixson degree one is homeomorphic to an ordinal space3 of the form .ωα + 1, where .ω is the first countable ordinal and .α is a countable ordinal. When .α = 1, we have from Corollary 5.2.9 that .Sym(N), the symmetric group on .N, is coarsely bounded. This also follows from the work of Bergman [10].

5.2.4 Rokhlin Property In Theorem 5.10.1, by taking into account the topological structure of homeomorphism groups, we were able to establish a weaker version of Theorem 5.9.2 in the case of uniquely self-similar 2-manifolds. The next result is motivated by asking to what extent can the topological group structure be used to recover a weaker version of Theorem 5.8.1 for uniquely self-similar 2-manifolds. We offer one answer, which

3 Given an ordinal .β, the ordinal space associated to .β, also denoted .β, is the space .{η : η < β} equipped with the order topology.

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is to show that Corollary 5.2.5, a corollary of Theorem 5.8.1, can be extended to include uniquely self-similar 2-manifolds. A topological group has the Rokhlin property if it contains a dense conjugacy class. Note that the existence of a dense conjugacy class precludes a topological group from having any proper open normal subgroups (since the complement of an open normal subgroup is an open conjugation-invariant subset). A classification of mapping class groups of orientable 2-manifolds with the Rokhlin property was given independently by Lanier and the author [46] and by Hernández et al. [40]; this result was extended to non-orientable 2-manifolds by Estrada [28]. We give the relevant part of this classification here, extended to homeomorphism groups: Theorem 5.11.1 If .M is a uniquely self-similar 2-manifold, then .H (M) and MCG(M) have the Rokhlin property.

.

The Rokhlin property was introduced in [34], and it was shown in [35] that H (S2d ) has the Rokhlin property, where .d ∈ N and .S2d is the .(2d)-dimensional sphere (see Theorem 5.11.4). It follows from this fact, Theorem 5.11.1, the results of [40, 46], and forthcoming work of Lanier and the author [47] that if .M is a 2manifold, then .H (M) has the Rokhlin property if and only if .M ∼ = S2 or .M is uniquely self-similar. Given the discussion above, Theorem 5.11.1 shows that .H (M) has no proper open normal subgroup whenever .M is uniquely self-similar, which allows us to extend Corollary 5.2.5 to all self-similar 2-manifolds. Before stating the corollary, we also note that, in a Polish group .G, any closed countable-index subgroup .N of .G is open. To see this, observe that the left cosets of .N cover .G, and hence .N must have nonempty interior, as a Polish space cannot be expressed as a countable union of nowhere dense subsets (a consequence of the Baire Category Theorem). To finish, observe that any subgroup of a topological group with nonempty interior is open, since the subgroup is a union of the translates of any open subset contained in the subgroup. .

Corollary 5.2.10 If .M is a self-similar 2-manifold, then neither .H (M) nor MCG(M) contain a proper open normal subgroup. Moreover, any proper closed normal subgroup of .H (M) or .MCG(M) has uncountable index. ⨆ ⨅

.

We are also able to establish Corollary 5.2.2 in the topological category for all self-similar 2-manifolds. As quasimorphisms are bounded on conjugacy classes (see [15, Section 2.2]), any continuous quasimorphism of a group with the Rokhlin property must be bounded. Using this fact for uniquely self-similar 2-manifolds together with Theorem 5.11.1 and Corollary 5.2.2 in the perfectly self-similar case, we have the following corollary. Corollary 5.2.11 If .M is a self-similar 2-manifold, then every continuous quasimorphism of .H (M) (resp., .MCG(M)) is bounded. There is a fascinating body of work establishing automatic continuity properties for homomorphisms between classes of groups. When coupled with the Rokhlin property, the topological condition of having a group element with a dense

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conjugacy class has algebraic consequences; in particular, there are a plethora of countable groups that we know cannot appear as the quotient of a Polish group with a unique open normal subgroup. A group .H is cm-slender if the kernel of any abstract homomorphism from a completely metrizable topological group to .H is open. We readily see that any homomorphism from a Polish group with the Rokhlin property to a cmslender group is trivial. Examples of cm-slender groups are abundant, and include free abelian groups [22]; torsion-free word-hyperbolic groups, Baumslag–Solitar groups, and Thompson’s group .F (see [18]); non-exceptional spherical Artin groups (e.g., braid groups) [19]; and any torsion-free subgroup of a mapping class group of a finite-type 2-manifold [11]. In fact, Conner has made the following conjecture: Conjecture 5.2.12 (Conner’s Conjecture) A countable group is cm-slender if and only if it is torsion free and does not contain an isomorphic copy of .Q. We capture this discussion in the following corollary: Corollary 5.2.13 Let .M be a self-similar 2-manifold. Every homomorphism from either .H (M) or .MCG(M) to a cm-slender group is trivial. ⨆ ⨅ The Rokhlin property is inherited by quotients by closed subgroups, yielding the following corollary. Corollary 5.2.14 The homeomorphism group of a uniquely self-similar secondcountable Stone space has the Rokhlin property. ⨆ ⨅ This corollary tells us that .Sym(N) has the Rokhlin property. A much stronger property is known for the symmetric group, namely .Sym(N) has ample generics (see [43]).

5.2.5 Automatic Continuity A topological group .G has the automatic continuity property if every abstract homomorphism from .G to a separable topological group is continuous. This is a strong property indicating a deep connection between the algebra and topology of a group. Automatic continuity has a relatively long history of being studied, and we refer the reader to Rosendal’s excellent survey [63] for an introduction. We need a few brief definitions to begin. A subset .W of a group .G is countably syndetic if .G is the union of countably many left translates of .W . A topological group .G is Steinhaus if there exists .m ∈ N, depending only on .G, such that .W m contains an open neighborhood of the identity for any symmetric countably syndetic set .W ⊂ G. The notion of a Steinhaus topological group was introduced by Rosendal–Solecki in order to provide a general strategy for establishing automatic continuity; in particular, every Steinhaus topological group has the automatic continuity property [66, Proposition 2].

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Rosendal [62] showed that .H (M) is Steinhaus for every compact 2-manifold (and Mann [52] subsequently showed that .H (M) is Steinhaus for every compact manifold). It is therefore natural to ask: which non-compact manifolds have Steinhaus homeomorphisms groups? In [54, Question 2.4], Mann asks this question in the context of mapping class groups of 2-manifolds. For starters, we know that there exist 2-manifolds whose mapping class groups (and hence homeomorphism groups) fail to have the automatic continuity property. Mann gives such examples in [54]. Additionally, for the Loch Ness monster surface .L, the epimorphism .MCG(L) → R of Domat and Dickmann [21] mentioned previously fails to be continuous, and hence .MCG(L) does not have the automatic continuity property. The discontinuity of this homomorphism can be deduced in several ways, for instance, it follows from any one of the following facts: (1) .MCG(L) is coarsely bounded (Theorem 5.10.1), (2) .MCG(L) has the Rokhlin property (Theorem 5.11.1), and (3) .MCG(L) has a dense subgroup that is perfect (a fact we have not discussed, see [6] for a discussion). This example was bootstrapped by Malestein and Tao [50] to construct additional examples of surfaces whose mapping class group fails to have the automatic continuity property, including the plane with an infinite discrete set removed (i.e. the flute surface). Now, let .M be a 2-manifold obtained by removing the union of a totally disconnected perfect set and a finite set from a compact 2-manifold (e.g., removing a Cantor set from the 2-sphere). Then, Mann [54] showed that .H (M) and .MCG(M) are Steinhaus. The next theorem makes progress on [54, Question 2.4] mentioned above by extending Mann’s work to the setting of perfectly tame 2-manifolds. Definition 5.2.15 A 2-manifold .M is perfectly tame4 if it is homeomorphic to the connected sum of a finite-type 2-manifold and finitely many perfectly self-similar 2-manifolds, each of whose space of ends can be written as .P ∪ D, where .P is perfect5 , .D is a discrete set consisting of planar ends, and .D = P. The definition of a perfectly tame 2-manifold is engineered so that the arguments given by Mann in [54] can be adapted to establish: Theorem 5.12.2 The homeomorphism group of a perfectly tame 2-manifold is Steinhaus. In the proof of Theorem 5.12.2, we will see that the Steinhaus constant can be taken to be 60. In forthcoming work, a stronger version of Theorem 5.12.2 has been independently obtained by Bestvina–Domat–Rafi. If .G is a Polish Steinhaus group and .G' is a Polish group such that there exists an epimorphism .G → G' , then .G' is Steinhaus [66, Corollary 3], which yields the following corollary. Corollary 5.2.16 The mapping class group of a perfectly tame 2-manifold is Steinhaus. ⨅ ⨆ 4 The naming is motivated by the fact that a perfectly tame 2-manifold is tame in the sense of Mann and Rafi [55]. 5 Recall that the empty set is a totally disconnected, perfect, and discrete subset of every space.

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A noteworthy example of a perfectly tame 2-manifold is the blooming Cantor tree surface (i.e., the orientable 2-manifold whose end space is a Cantor set of non-planar ends), which is infinite genus; hence, Theorem 5.12.2 gives the first example of an infinite-genus 2-manifold with Steinhaus homeomorphism group. Dickmann [20] recently classified the surfaces whose pure mapping class groups have automatic continuity, and also includes examples of an infinite-genus surfaces with non-empty non-compact boundary whose mapping class group has the Steinhaus property. As already noted, the Steinhaus property implies automatic continuity, yielding: Corollary 5.2.17 If .M is a perfectly tame 2-manifold, then .H (M) and .MCG(M) have the automatic continuity property. ⨆ ⨅ As .2N can be realized as the end space of a planar perfectly tame 2-manifold, namely the 2-sphere with a Cantor set removed, we can appeal to the fact that quotients of Steinhaus groups by closed subgroups are Steinhaus to obtain the following corollary, which is a result of Rosendal and Solecki [66]: Corollary 5.2.18 .Homeo(2N ) is Steinhaus and hence has the automatic continuity property. ⨆ ⨅ One application of automatic continuity is establishing the uniqueness of a Polish group structure. Let .G be a group, and let .τ, τ ' be two topologies on .G such that .G equipped with each of the topologies is a Polish group. Assume .(G, τ ) has the automatic continuity property, so that the identity homomorphism .(G, τ ) → (G, τ ' ) is continuous. Let .V ∈ τ , and let us consider .V as a subset of .(G, τ ' ). Then .V , being the continuous image of an open set of a Polish space, is analytic in ' .(G, τ ). Similarly, .G \ V , being the continuous image of a closed subset of a Polish space, is analytic in .(G, τ ' ). A theorem of Suslin implies that .V is Borel in .(G, τ ' ) as both .V and its complement are analytic in .(G, τ ' ). In other words, the identity homomorphism .(G, τ ' ) → (G, τ ) is Baire measurable. Now, every Baire measurable homomorphism between Polish groups is continuous (see [63, Section 2]), and hence, the identity homomorphism is a topological isomorphism. Corollary 5.2.19 If .M is a perfectly tame 2-manifold, then .H (M) and .MCG(M) have unique Polish group topologies. ⨆ ⨅ Corollary 5.2.19 is most interesting in the case of mapping class groups. In the case of homeomorphism groups, Corollary 5.2.19 is much weaker than what is already known: Kallman [41] has given very general criteria for a closed subgroup of a homeomorphism group to have a unique Polish group topology. In particular, the homeomorphism group of any manifold has a unique Polish group topology, namely the compact-open topology.

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5.2.6 Commutator Subgroups Given a group .G, the group .G/[G, G] is called the abelianization of .G and is denoted .Gab ; it has the property that every homomorphism from .G to an abelian group factors through .Gab . In particular, a group is perfect if and only if its abelianization is trivial. Pure mapping class groups of finite-type 2-manifolds of genus at least three are perfect (see [29, Section 5.1.2] for a short proof and an overview of the history). In contrast, combining the work of Domat and Dickmann [21] and Aramayona, Patel, and the author [6], the abelianization of the pure mapping class group of an orientable infinite-type 2-manifold is infinite. On the other hand, we saw in Theorem 5.8.1 that .H (M) and .MCG(M) are uniformly perfect whenever .M is perfectly self-similar. Abelian groups always admit countable quotients, and hence the abelianization is useful in trying to understand the structure of countable-index subgroups. Therefore, there is a natural interest in computing abelianizations: Problem 5.2.20 Given a 2-manifold .M, compute the abelianization of .H (M) and MCG(M).

.

With this problem in mind, we give a slight generalization of a result of Field et al. [31] to the setting of homeomorphism groups and to include non-orientable 2-manifolds. Theorem 5.13.1 Let .M be a 2-manifold obtained by taking the connected sum of a finite-type 2-manifold and finitely many perfectly self-similar 2-manifolds. Then, the commutator subgroup of .H (M) is open. Given a subgroup .H of a topological group .G, the canonical map .G → G/H is open when .G/H is equipped with the quotient topology. This follows from the fact that the saturation of an open subset .U of .G is simply .U H . Using this basic fact about topological groups, together with the fact that the image of the commutator subgroup in the domain of an epimorphism is the commutator subgroup in the codomain, yields the following two corollaries: Corollary 5.2.21 Let .M be a 2-manifold obtained by taking the connected sum of a finite-type 2-manifold and finitely many perfectly self-similar 2-manifolds. Then, .[MCG(M), MCG(M)] is an open subgroup of .MCG(M). ⨆ ⨅ Corollary 5.2.22 Let .M be a 2-manifold obtained by taking the connected sum of a finite-type 2-manifold and finitely many perfectly self-similar 2-manifolds. The abelianization of .H (M) and .MCG(M) are isomorphic. Moreover, the abelianization of .H (M) and .MCG(M) are countable. Proof Let .q : H (M) → H (M)ab be the canonical homomorphism. Equip .H (M)ab with the quotient topology, so that .q is open and continuous. Moreover, since ab is discrete. As the continuous image .[H (M), H (M)] is open and .q is open, .H (M) ab of a separable space, .H (M) is separable, and hence countable (as it is separable

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and discrete). Now since .q is a continuous homomorphism to a discrete space, the connected component of the identity is in the kernel of .q; in particular, .q factors through .MCG(M). ⨆ ⨅

5.3 Topology of Surfaces This section has been written with the goal of providing enough background so that the chapter is accessible to non-experts, especially graduate students. In particular, we provide a thorough overview of the necessary background in the topology of surfaces, while assuming the reader is familiar with the basics of algebraic topology, namely the concepts of connected sums, orientation, and Euler characteristic. In the background of the following discussion is the fact that every 2-manifold admits a triangulation (see [69]), which allows one to appeal to the theory of piecewise-linear manifolds, and in particular the structure of regular neighborhoods (see [14] for a survey of the relevant theory). We will not explicitly appeal to these facts, but they are required if one wants to prove the statements given below. Most readers are likely familiar with the classification of compact 2-manifolds, but there is a classification for all 2-manifold in terms of certain 0-dimensional data. The majority of this section is dedicated to stating and understanding this classification, which is an essential ingredient in the proof of every theorem highlighted in the previous section. We will often work with subsurfaces, and so we will focus on presenting the classification of surfaces with compact boundary. The standard reference is [61]. The classification theory relies on the notion of a topological end. For the sake of variation, we give a slightly different (and more concise) definition than we did in an earlier volume of this series [5]. There are many equivalent definitions one can give, all of which have (dis)advantages. We state the definition in terms of surfaces, but it can be adapted to a much broader class of spaces (see [30] for a concise and detailed discussion of end spaces and the associated Freudenthal compactification). Given two compact subsets .K and .K ' of a surface .S such that .K ⊂ K ' there exists a projection .fK,K ' : π0 (S \ K ' ) → π0 (S \ K) mapping a component .U of ' .S \ K to the component of .S \ K containing .U . Definition 5.3.1 (Space of Ends) Let .S be a surface, and let .K be the directed partially ordered set of compact subsets of .S (ordered by inclusion). The space of ends, or end space, of .S, denoted .E(S), is the space whose underlying set is the inverse limit of the inverse system .({π0 (S \ K)}K∈K , {fK,K ' }K⊆K ' ∈K ), and whose topology is the limit topology, that is, the coarsest topology for which the natural projection .E(S) → π0 (S \ K) is continuous for each .K ∈ K. An end of .S is an element in .E(S). This definition is hard to digest (we will clarify shortly), but it is useful for establishing formal properties. For instance, we readily see that the space of ends is a topological invariant, and in particular, every homeomorphism .f : S → S induces

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a homeomorphism .f : E(S) → E(S). A subsurface of .S is a closed subset of .S that is a surface when equipped with the subspace topology. Every surface admits an exhaustion by compact subsurfaces, that is, if .S is a surface with compact boundary, there exists a sequence .{Σn }n∈N of compact subsurfaces such that .Σn is contained  in the interior of .Σn+1 and such that .S = n∈N Σn . As the sequence .{Σn }n∈N is cofinal in the poset .K, the inverse limit taken over this compact exhaustion— instead of all of .K—is still .E(S).  Appealing to a standard fact of inverse limits, this says that .E(S) embeds into . n∈N π0 (S \ Σn ) as a closed subset. Moreover, as  . π (S \ K) is a countable product of finite discrete spaces, it is Hausdorff, 0 n∈N second countable, zero-dimensional, and compact (by Tychonoff’s theorem). All of these properties are inherited by closed subsets, establishing the following: Proposition 5.3.2 The space of ends of a surface is Hausdorff, second countable, zero-dimensional, and compact.6 In particular, it is homeomorphic to a closed subset of the Cantor set .2N . ⨆ ⨅ Let us consider some basic examples: • • • • •

The end space of a compact 2-manifold is empty. The end space of .R2 is a singleton. 1 .S × R has two ends. The end space of .C \ N is the one-point compactification of .N. The end space of the manifold obtained by removing a Cantor space from .S2 is homeomorphic to the Cantor space .2N .

Let us now be more concrete and give a description of the clopen subsets of the space of ends as we will represent them throughout the chapter. From the definition, given a surface .S and any open subset .Ω of .S with compact boundary, there is a  of .E(S), which is given as follows: .Ω defines a point corresponding clopen subset .Ω  in .π0 (S \ ∂Ω), and .Ω is the preimage of this point under the natural projection  for some .E(S) → π0 (S \ ∂Ω). In fact, every clopen subset of .E(S) is of the form .Ω open subset .Ω with compact boundary. To simplify language, we say that .Ω is a . neighborhood of an end .e if .e ∈ Ω Definition 5.3.3 A Freudenthal subsurface of a 2-manifold is a subsurface whose boundary is connected and compact.  to be equal If .D is a Freudenthal subsurface with interior .Ω, then we define .D  to .Ω . A sequence of Freudenthal subsurfaces .{Dn }n ∈ N satisfying .Dn+1 ⊂ Dn and . Dn = ∅ defines an end, and in fact, the space of ends can be defined as equivalence classes of such sequences (c.f., [5, 61]). Given the discussion thus far, and using the fact that connected surfaces are path connected, we record the following lemma, which we will use throughout the chapter.

6 In

other words, it is a second countable Stone space.

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Lemma 5.3.4 Let .S be a surface with compact boundary. Every clopen subset  for some open Freudenthal subsurface .D. Moreover, of .E(S) is of the form .D given a collection .{Ui }i∈I of pairwise-disjoint clopen subsets of .E(S), there exists a collection .{Di }i∈I of pairwise-disjoint Freudenthal subsurfaces of .S such that i = Ui for each .i ∈ I . .D ⨆ ⨅ An end .e of .S is planar (resp., orientable) if there exists an open subset .Ω of .S , and .Ω is homeomorphic to an open subset of .R2 such that .∂Ω is compact, .e ∈ Ω (resp., is an orientable manifold); otherwise, it is non-planar (resp., non-orientable). The set of non-orientable ends of .S is denoted .Eno (S) and the set of non-planar ends is denoted .Enp (S). We then have that .Enp (S) and .Eno (S) are closed subsets of .E(S) and .Eno (S) ⊂ Enp (S). Notation Given a surface .S with compact boundary, the space of ends of .S will refer to the triple .(E(S), Enp (S), Eno (S)). Moreover, a homeomorphism between the space of ends of two surfaces .S and .R will refer to a homeomorphism .E(S) → E(R) mapping .Enp (S) onto .Enp (R) and .Eno (S) onto .Eno (R). Continuing towards the classification, we need the notion of genus. The genus of a compact surface with .n boundary components (possibly .n = 0) is defined to be the quantity .g = (2 − χ − n)/2, where .χ is the Euler characteristic of the surface. (In the non-orientable case, this is sometimes referred to as the reduced genus.) A surface .S is of finite genus if there exists a compact subsurface .Σ such that every compact subsurface of .S containing .Σ has the same genus as .Σ; we then define the genus of .S to be equal to the genus of .Σ. If .S is not of finite genus, then we say it is of infinite genus. Observe that if .S is of infinite genus, then .Enp /= ∅. If .S is not orientable, then it is of even (resp., odd) non-orientability if .S contains a compact subsurface .Σ such that each component of .S \ Σ is orientable and .Σ has integral (resp., half-integral) genus. If .S is not orientable, but of neither even nor odd orientability, then we say it is infinitely non-orientable; note that .S is infinitely non-orientable if and only if .Eno /= ∅. With these definitions, we partition the class of surfaces into four orientability classes: the orientable surfaces, the infinitely nonorientable surfaces, the surfaces of even non-orientablility, and the surfaces of odd non-orientability. Theorem 5.3.5 (The classification of surfaces with compact boundary) Let .S and .R be two surfaces with compact boundary such that they have the same number of boundary components, they are of the same orientability class, and they are of the same genus. Then, .S and .R are homeomorphic if and only if there is a homeomorphism .E(S) → E(R) mapping .Enp (S) onto .Enp (R) and .Eno (S) onto .Eno (R). ⨆ ⨅ There is a corresponding classification of surfaces that allows for non-compact boundary [13], but we will not require it here. As a consequence of Richards’s proof of the classification of surfaces, we obtain the following theorem:

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Theorem 5.3.6 Let .S be a surface with compact boundary. If .ϕ : E(S) → E(S) is a homeomorphism mapping .Enp (S) onto itself and mapping .Eno (S) onto itself, then ⨆ ⨅ there exists a homeomorphism .f : S → S such that .fˆ = ϕ. An important application of the classification of surfaces is the change of coordinates principle. Before, stating this corollary, we need to understand more about 1-submanifolds of 2-manifolds. The open annulus is the manifold .S1 × R and the open Möbius band is the manifold obtained from .[−1, 1] × (0, 1) by identifying 1 .(−1, y) and .(1, −y) for all .y ∈ (0, 1). In the former case, we call .S × {0} the core curve, and in the latter case, we call the image of .[−1, 1] × {0} in the quotient space the core curve. The standard reference for the following discussion regarding closed curves is [25]. A simple closed curve in a 2-manifold is a compact 1-submanifold, or equivalently, the image of an embedding of the circle .S1 . Every simple closed curve .a admits an open neighborhood .A such that there exists a homeomorphism mapping .A to either an open annulus or an open Möbius band and that sends .a to the core curve in either case; in the former case, the simple closed curve is said to be two sided and in the latter is said to be one sided. An important consequence of the above fact is that the closure of any complementary component of a subsurface is a subsurface.7 Corollary 5.3.7 (Change of Coordinates Principle) Let .S be a surface with compact boundary, and let .Σ be a subsurface of .S with compact boundary. If .Σ ' is a subsurface of .S homeomorphic to .Σ such that the closures of .S \ Σ and .S \ Σ ' are homeomorphic, then there exists a homeomorphism .S → S mapping .Σ onto ' .Σ . ⨆ ⨅ Let us finish this section with describing how one constructs a 2-manifold with prescribed orientability, genus, and end space (the construction is due to Richards [61]). First, let us recall some basic spaces and the basics of compact surfaces. Let 2 1 1 .S denote the 2-sphere. The torus is the space .S × S and the projective plane is the space obtained by identifying antipodal points on the 2-sphere. Note that the genus of the 2-sphere is 0, the genus of the torus is one, and the genus of the projective plane is one-half. Every compact orientable (resp., non-orientable) surface is homeomorphic to a surface obtained by taking the connected sum of the 2-sphere with finitely many tori (resp., projective planes) and removing the interiors of finitely many pairwisedisjoint closed 2-disks. In particular, a compact surface not homeomorphic to .S2 is of genus zero if and only if it is planar, that is, homeomorphic to a compact subsurface of .R2 . Using the Jordan-Schoenflies theorem (see [69]) and the fact that every subsurface admits an exhaustion by compact subsurfaces, we can deduce that

7 This is one of the ways in which 2-manifolds are special. The analogous statement in higher dimensions is false, as can be seen by the existence of the Alexander horned sphere in .R3 .

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for a surface .S with compact boundary, .Enp (S) is empty if and only if .S has finite genus. Now, let .E'' ⊂ E' ⊂ E be a nested triple of closed subsets of the Cantor set. Embed .E into .S2 , and identify .E with this embedding. From the above discussion, the prescribed genus can be finite if and only if .E' = ∅. So, if the prescribed genus is finite, call it .g, then we let .M be the 2-manifold obtained by taking the connected sum of .S2 \ E with .g tori in the orientable case or .2g projective planes otherwise. Now suppose the prescribed genus is infinite. In this case, we have that .E' is nonempty. First suppose that .E'' = ∅. Choose a sequence .{Dn' }n∈N of pairwisedisjoint closed disks in the complement of .E such that the .Dn' accumulate onto .E' . Let .M be the 2-manifold obtained as follows: for each .n ∈ N, perform a connected sum of .S2 \ E with a torus using the disk .Dn' , and then connect sum with a finite number of projective planes to have the desired non-orientable genus. Finally, if .E'' is not empty, choose a sequence .{Dn' }n∈N of pairwise-disjoint closed disks in the complement of .E such that the .Dn' accumulate onto .E' and a sequence .{Dn'' }n∈N of pairwise-disjoint closed disks contained in the complement of the union of .E with the .Dn' such that the .Dn'' accumulate onto .E'' . Let .M be the 2-manifold obtained as follows: for each .n ∈ N, perform a connected sum of .S2 \ E with a torus using the disk .Dn' , and with a projective plane using the disk .Dn'' . Then, .M is the desired 2-manifold.

5.4 Stable Sets The goal of this section is to introduce the notion of stable subsets of end spaces as defined by Mann and Rafi [55, Definition 4.14] and to establish a sequence of lemmas and propositions that have appeared in various forms in the literature. The main idea behind a stable set is to capture key characteristics of the Cantor space .2N in a more general setting. Recall that Brouwer’s theorem says that .2N is the unique—up to homeomorphism—nonempty second-countable zerodimensional perfect compact Hausdorff topological space (see [42, Theorem 7.4] for a reference). As a consequence, once we have definitions in place, this will imply that every clopen subset of .2N is stable. Remark Though we work in the setting of 2-manifolds in this section, we will not use the topology of the 2-manifold itself, and hence the results in this section are results about second-countable Stone spaces.

5.4.1 Definitions, Notations, and Conventions Given a 2-manifold .M, let .E = E(M) denote its space of ends, let .Enp denote the subset of .E consisting of non-planar ends, and let .Eno denote the subset of .E

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consisting of non-orientable ends (see Sect. 5.3 for more details and an introduction the space of ends). Let .HM (E) denote the group consisting of homeomorphisms .E → E mapping .Enp onto .Enp and .Eno onto .Eno (see Theorem 5.3.6). It follows from Richards’s proof of the classification of surfaces [61] that the canonical homomorphism .H (M) → HM (E) is surjective. To not overburden the reader with notation, if we say two subset .U and .U' of .E are homeomorphic, we mean there there is a homeomorphism .U → U' that sends ' ' .U ∩ Enp onto .U ∩ Enp and .U ∩ Eno onto .U ∩ Eno . Definition 5.4.1 (Stability) Let .M be a 2-manifold. An end .e of .M is stable if it admits a neighborhood basis in .E consisting of pairwise-homeomorphic clopen sets, and such a basis is called a stable basis of .e. A clopen subset .U of .E(M) is a stable neighborhood of a stable end if the end admits a stable neighborhood basis containing .U. A clopen subset of .E(M) is stable if there exists an end for which it is a stable neighborhood. Two stable ends are of the same type if they admit homeomorphic stable neighborhoods. Given a stable clopen subset .U of .E, let .S(U) denote the elements of .U for which U is stable neighborhood. Note that any two elements of .S(U) are necessarily of the same type. Now that we have the definition of a stable set, we reiterate that as an application of Brouwer’s theorem, every clopen subset of .2N is stable. In particular, every point in .2N is stable, and given any clopen subset .U ⊂ 2N , .S(U) = U. For another ¯ = N ∪ {∞}, the one-point compactification of .N and the end example, consider .N space of .C \ N. Every clopen neighborhood of .∞ is stable. More generally, given a compact Hausdorff space, the stable sets are exactly the clopen subsets of Cantor– Bendixson degree one. This is an application of the classification of countable compact Hausdorff spaces [56]: the Cantor–Bendixson degree and rank of a such space determine the space up to homeomorphism. More concretely, if .X is compact countable Hausdorff space, then it is homeomorphic to the ordinal space of the form α .ω · n + 1, where .ω is the first countable ordinal, .α + 1 is the Cantor–Bendixson rank of .X, and .n is the Cantor–Bendixson degree of .X. .

5.4.2 Structure of Stable Sets We proceed to establish the basic topological structure of stable sets. As mentioned earlier, the following lemmas, or versions of them, have appeared in several places. We will therefore avoid giving attributions, but we make the exception of observing that they all stem from the original introduction of stable sets by Mann and Rafi [55, Section 4] and their original contributions to their structure. We begin with a lemma that explains the extent to which stable ends of the same type are in fact the same. The proof is an example of a standard type of argument referred to as a back-and-forth argument.

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Proposition 5.4.2 Let .M be a 2-manifold and let .μ1 and .μ2 be a stable ends of M of the same type. If, for .i ∈ {1, 2}, .Ui is a clopen neighborhood of .μi in .E(M) contained in a stable neighborhood of .μi , then there exists a homeomorphism .U1 → U2 sending .μ1 to .μ2 .

.

Proof Let .V and .W be stable clopen neighborhoods of .μ1 and .μ2 , respectively, such that .U1 ⊂ V and .U2 ⊂ W. Let .{Vn }n∈N and .{Wn }n∈N be neighborhood bases of .μ1 and .μ2 , respectively, satisfying .V1 = U1 , .W1 = U2 , and, for .n ∈ N, .Vn ⊂ V and .Wn ⊂ W. Since .μ1 and .μ2 are of the same type, every element of .S(W) admits a stable neighborhood basis consisting of sets homeomorphic to .V; in particular, there is a homeomorphic copy of .Vm contained in .Wn for all .n, m ∈ N. Similarly, every element of .S(V) admits a stable neighborhood basis consisting of sets homeomorphic to .W, and there is a homeomorphic copy of .Wm contained in .Vn for all .n, m ∈ N. We claim there exists a homeomorphism .f1 from .V1 \ V2 onto an open subset of .W1 \ {μ2 }. To see this, let .W'1 be a clopen stable neighborhood of .μ2 contained in .W1 and homeomorphic to .W. First suppose that .S(W'1 ) contains an element ' '' .μ distinct from .μ2 . Then, we can choose a clopen stable neighborhood .W of 2 1 '' .μ2 homeomorphic to .W and that does not contain .μ2 . Then, .W contains an open 1 subset homeomorphic to .V1 , and we can choose a homeomorphism .f1 mapping .V1 onto an open subset of .W''1 . Then, the restriction of .f1 to .V1 \ V2 yields the desired map. Now, if .S(W'1 ) = {μ2 }, then let .f1 be any homeomorphism of .V1 onto an open subset of .W'1 . Then, we must have that .f1 (μ1 ) = μ2 , and hence restricting .f1 to .V1 \ V2 yields the desired map. Similarly, there exists a homeomorphism .g1 from .(W1 \W2 )\image(f1 ) onto an open subset of .V2 \ {μ1 }. Let .V'1 = (V1 \ V2 ) ∪ image(g1 ) and .W'1 = (W1 \ W2 ) ∪ image(f1 ), then .h1 : V'1 → W'1 defined by .h1 = f1 ⨆ g1−1 (that is, .h1 (x) = f1 (x) if .x ∈ V1 and .h1 (x) = g1−1 (x) if .x ∈ image(g1 )) is a homeomorphism. Following the same process as above, we recursively construct the homeomorphism .fn from .(Vn \ Vn+1 ) \ (V'1 ∪ · · · ∪ V'n−1 ) onto an open subset of ' ' .(Wn \ Wn+1 ) \ ({μ2 } ∪ W ∪ · · · ∪ W 1 n−1 ) and then choose a homeomorphism ' ' .gn of .(Wn \ Wn+1 ) \ (W ∪ · · · ∪ W 1 n−1 ∪ image(fn )) onto an open subset of ' ' .Vn+1 \ ({μ1 } ∪ V ∪ · · · ∪ V ). Let 1 n−1 V'n = ((Vn \ Vn+1 ) \ (V'1 ∪ · · · ∪ V'n−1 )) ∪ image(gn )

.

and W'n = ((Wn \ Wn+1 ) \ (W'1 ∪ · · · ∪ W'n−1 )) ∪ image(fn ).

.

Then, .hn : V'n →  W'n defined by .hn = fn ⨆ gn−1 is a homeomorphism. Observe ' that .U1 \ {μ1 } = n∈N Vn and .U2 \ {μ2 } = n∈N W'n , which allows us to define .h : U1 → U2 by .h|V' = hn and .h(μ1 ) = μ2 . It is readily checked that .h is the n desired homeomorphism. ⨆ ⨅

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Setting .μ2 = μ1 in Proposition 5.4.2, we obtain the following corollary. Corollary 5.4.3 Let .M be a 2-manifold. Any two stable neighborhoods of a stable end are homeomorphic. ⨆ ⨅ An important property—established in the next proposition—of a stable set is that its corresponding subset of stable ends is either a singleton or perfect, and in ¯ despite being special particular, it is compact. This also tells us that .2N and .N, examples, nonetheless are the right examples for considering the structure of stable sets in general. Proposition 5.4.4 Let .M be a 2-manifold and let .U be a stable clopen subset of E(M). Then, .S(U) is either a singleton or a perfect set.

.

Proof Suppose .S(U) contains at least two ends. We first show that no element of .S(U) is isolated. Let .μ ∈ S(U). Fix a stable neighborhood basis .{Un }n∈N of .μ. Then, as .Un is homeomorphic to .U by Corollary 5.4.3, .S(Un ) contains an end distinct from from .μ and hence .μ is not isolated in .S(U). We now argue that .S(U) is closed. Let .e be in the closure of .S(U). As .U is clopen, it must be that .U is a clopen neighborhood of .e. Let .V be any clopen neighborhood of .e contained in .U. Then, there exists .μ ∈ S(U) ∩ V, which implies, via Proposition 5.4.2, that .V is homeomorphic to .U. Therefore, .e has a stable neighborhood basis consisting of sets homeomorphic to and contained in .U; hence, .e ∈ S(U) and .S(U) is closed. ⨆ ⨅ The next sequence of lemmas and propositions is meant to capture properties of ¯ that hold more generally for stable sets. We begin by assuming .S(U) is 2N and .N perfect, and so .2N is our motivating example in this case.

.

Lemma 5.4.5 Let .M be a 2-manifold and let .U ⊂ E(M) be a clopen stable set such that .S(U) is perfect. Given any .e ∈ S(U), there exists a sequence  .{Un }n∈N of pairwise-disjoint open sets homeomorphic to .U such that .U \ {e} = n∈N Un . Proof Fix a stable neighborhood basis .{Vn }n∈N of .e contained in .U such that Vn+1 ⊂ Vn and such that .S(Vn ) \ Vn+1 /= ∅. Then, .Vn \ Vn+1 is a clopen neighborhood of an element of .S(U) and is contained in .U; hence, .Vn \ Vn+1 is homeomorphic to .U by Corollary 5.4.3. Then, .Un = Vn \ Vn+1 are the desired sets. ⨆ ⨅

.

Proposition 5.4.6 Let .M be a 2-manifold, let .U ⊂ E(M) be a clopen stable set such that .S(U) is perfect, and let .X be a countable discrete space. If .X is finite, then .U × X is homeomorphic to .U; otherwise, when .X is infinite, the one-point compactification of .U × X is homeomorphic to .U. Proof First assume .X is finite. Let .n = |X| and let .e1 , . . . , en ∈ S(U) be distinct points. Choose pairwise-disjoint clopen stable neighborhoods .U1 , . . . , Un ⊂ U of .e1 , . . . , en , respectively, such that each .Ui is homeomorphic to .U. By Proposition 5.4.2, .U' = U1 ∪ · · · ∪ Un is homeomorphic to .U. The result follows by observing that .U' is homeomorphic to .U × X.

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Now, suppose .X is infinite. Fix .e ∈ S(U). By Lemma 5.4.5, there exists a sequence .{Un }n∈N of  pairwise-disjoint open sets, each homeomorphic to .U, and such that .U \ {e} = n∈N Un . Enumerate the elements of .X, so .X = {xn }n∈N . Let .U∗ = (U × X) ∪ {∞} denote the one-point compactification of .U × X. Choose ∗ → U such that .ϕ(∞) = e and such that .ϕ restricted to .U × {x } is a .ϕ : U n homeomorphism from .U × {xn } to .Un . Then, .ϕ is a homeomorphism. ⨆ ⨅ Let us now turn back to stable sets more generally. The next lemma and proposition provide additional structure for stable sets that will be used to build homeomorphisms of 2-manifolds in the following sections. Lemma 5.4.7 Let .M be a 2-manifold and let .U ⊂ E(M) be a clopen stable set. If V is a clopen subset of .U such that .S(U) \ V /= ∅, then there exists .e ∈ S(U) and a stable neighborhood basis .{Un }n∈N of .e such that .Un \ Un+1 contains an open subset .Vn homeomorphic to .V for each .n ∈ N.

.

Proof Fix a metric .d on .E(M). Let .e1 ∈ S(U) \ V, let .U1 = U, and let .V1 = V. As .V1 is closed, we can find an open stable neighborhood .U2 of .e1 homeomorphic to .U that is contained in .U1 and disjoint from .V1 . By possibly shrinking .U2 , we may assume that .V1 ⊂ U1 \ U2 and that the diameter of .U2 is less than .1/2. Now, .U2 , being homeomorphic to .U, contains an open set .V2 homeomorphic to .V such that there exists .e2 ∈ S(U2 ) \ V2 . As before, we can now find an open stable neighborhood .U3 of .e2 of diameter less than .1/3 that is homeomorphic to .U and such that .V2 ⊂ U2 \ U3 . Continuing in this fashion, we build a sequence of pairwise-homeomorphic nested clopen stable sets .{Un }n∈N with diameters limiting to zero and a sequence of pairwise-disjoint pairwise-homeomorphic open sets .{Vn }n∈N such that .Vn ⊂ Un \ Un+1 and .Vn is homeomorphic to .V. Since .{S(U n )}n∈N is a nested sequence of closed subsets of a compact space, it must be that . S(Un ) /= ∅; moreover, as  the diameter of .Un tends to zero, there exists an end .e such that .{e} = S(Un ); hence, .{Un }n∈N is a stable neighborhood basis for .e. ⨆ ⨅ Proposition 5.4.8 Let .M be a 2-manifold and let .U ⊂ E(M) be a clopen stable set. If .V is a clopen set of .E(M) that is homeomorphic to an open subset of .U \ {e} for some .e ∈ S(U), then .U ∪ V is homeomoprhic to .U. Proof If .V ⊂ U, then the statement is trivial, so we assume that .V is not contained in .U. Moreover, since .U ∪ V = U ∪ (V \ U), we may assume that .U ∩ V = ∅. By Lemma 5.4.7, there exists .e' ∈ S(U), a stable neighborhood basis .{Un }n∈N of .e' , and a sequence .{Vn }n∈N of pairwise-disjoint clopen subsets such that .Vn ⊂ Un \{e} and .Vn is homeomorphic to .V. We can now construct a homeomorphism .ψ : U ∪ V → U as follows: fix a homeomorphism .ψ0 : V → V1 and, for .n ∈ N, fix a homeomorphism .ψn : Vn → Vn+1 . Define .ψ : U ∪ V→  .ψ|V = ψ0 , .ψ|Vn = ψn , and .ψ(x) = x for all .x in  U by the complement of .V ∪ Vn . It is readily verified that .ψ is a homeomorphism. ⨆ ⨅

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We finish this section with providing an equivalent definition of stability, which is referred to as self-similarity by Mann and Rafi [55]. A clopen subset .U of .E(M) is self-similar if given open subsets .U1 , . . . , Uk ⊂ E(M) such that .U = U1 ∪ · · · ∪ Uk , then there exists .i ∈ {1, . . . , k} such that .Ui contains an open subset homeomorphic to .U. Proposition 5.4.9 Let .M be a 2-manifold. Suppose .U is a clopen subset of .E(M). Then, the following are equivalent: (1) .U is stable. (2) .U is self-similar. (3) If .V, W ⊂ E(M) are open sets such that .U = V ∪ W, then .V or .W contains an open subset homeomorphic to .U. Proof Let us first show the equivalence of the latter two statements. Clearly (2) implies (3) by setting .k = 2. Conversely, suppose .U = (U1 ∪ · · · Uk−1 ) ∪ Uk with .U1 , . . . , Uk open subsets of .E(M). Then, by assumption, .Uk or .U1 ∪ · · · ∪ Uk−1 must contain an open subset homeomorphic to .U. If it is .Uk , then we are finished. Otherwise, .U1 ∪· · ·∪Uk−1 contains an open set .V homeomorphic to .U, and we repeat the above argument with .V = (V1 ∪ · · · Vk−2 ) ∪ Vk−1 , where .Vi = V ∩ Ui . Clearly, this process has finitely many steps and ends with an open subset homeomorphic to .U and contained in one of the .Ui . We now establish the equivalence of the first two statements. Clearly (1) implies (2). Let us finish by proving the converse. Fix a metric on .E(M). Cover .E(M) with finitely many balls of radius 1/2 (of course this can be done as .E(M) is compact). By assumption, taking the intersection of these balls with .U, one of these balls contains an open subset .U1 contained in and homeomorphic to .U; moreover, the diameter of .U1 is at most one. Now cover .E(M) by balls of radius .1/4, and again, by taking intersections with .U1 and using the assumption, we obtain an open subset .U2 contained in .U1 , homeomorphic to .U, and of diameter of at most 1/2. Continuing in this fashion, we build a sequence of nested open sets .{Un }n∈N such that .Un is homeomorphic to .U and of diameter at most .1/n. As .U is compact, it follows  that . n∈N Un is nonempty and, by construction, its diameter is 0; hence, there is a unique element .μ in the intersection, and .{Un }n∈N is a stable neighborhood basis of .μ. Therefore, .U is a stable neighborhood of .μ, and hence stable. ⨆ ⨅

5.5 Freudenthal Subsurfaces and Anderson’s Method The goal of this section is to adapt and generalize the technique used by Anderson in [3] to a general tool for 2-manifolds, which we call Anderson’s Method (see Proposition 5.5.7). It is worth noting that Anderson’s original presentation is itself in a general setting, which he then applied to several transformation groups (including 2 N .H0 (S ) and .Homeo(2 )). In the literature, there are several other instances of Anderson’s ideas being generalized to various settings (for instance, see [26]). That

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is to say, the notion of Anderson’s method is broader than the presentation given here, which is suited to our purposes.

5.5.1 Definitions, Notations, Conventions Given a subset .X of a topological space, we let .X denote its closure, .Xo denote its interior, and .∂X = X \ Xo denote its boundary. A family of subsets .{Xn }n∈N of a topological space converges to a point .x if for every open neighborhood .U of .x there exists .N ∈ N such that .Xn ⊂ U for all .n > N; it is locally finite if given any compact subset .K the set .{n ∈ N : Xn ∩ K /= ∅} is finite; it is convergent if it is either locally finite or converges to a point. The support of a homeomorphism .f : X → X, denoted .supp(f ), is the closure of the set .{x ∈ X : f (x) /= x}. Given a subset .A of .X, we say a homeomorphism .f : X → X is supported on .A if .supp(f ) ⊂ A. If .{fn }n∈N is a sequence of selfhomeomorphisms of a topological space such that .{supp(fn )} n∈N either converges to a point or is locally finite, then the infinite product .f = n∈N fn exists and is a homeomorphism. As function composition reads right to left, we read the infinite product right to left as well, e.g., given a point .x and .k = max{n ∈ N : x ∈ supp(fn )}, then .f (x) = fk ◦ fk−1 ◦ · · · ◦ f1 (x). Given a subset .S of a group .G, the normal closure of .S is the subgroup of .G generated by all the conjugates of the elements of .S. An element .g of .G normally generates .G or is a normal generator of .G if the normal closure of .{g} is .G.  to Given an open subset .Ω of a 2-manifold .M with compact boundary, define .Ω be the subset of .E(M) consisting of ends for which .Ω is a neighborhood in .M (see Sect. 5.3 for more details). Given a closed subset .Σ of .M with compact boundary,  = Σ o . Recall that surfaces, and hence subsurfaces, are required to be we set .Σ connected. Definition 5.5.1 (Types of Freudenthal Subsurfaces) Let .M be a 2-manifold. Recall (Definition 5.3.3) that a Freudenthal subsurface of .M is a subsurface with connected compact boundary. A Freudenthal subsurface .Δ is trim if  ∩ Enp (M) = ∅, and (i) .Δ is planar whenever .Δ  ∩ Eno (M) = ∅. (ii) .Δ is orientable whenever .Δ  is A Freudenthal subsurface .Δ is stable if it is trim and either .Δ is compact or .Δ stable (note that every compact stable Freudenthal subsurface is homeomorphic to the closed 2-disk). For a stable Freudenthal subsurface .Δ, we define

S(Δ) =

.

Δo if Δ is compact  S(Δ) otherwise

A Freudenthal subsurface .Δ is dividing if it stable, .S(Δ) is infinite, and .M \ Δ contains a Freudenthal subsurface homeomorphic to .Δ.

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 for We will often use the fact that every clopen subset of .E(M) is of the form .Δ some trim Freudenthal subsurface .Δ (see Lemma 5.3.4). This is equivalent to the fact that the interiors of the Freudenthal subsurfaces form a basis for the topology of the Freudenthal compactification of .M, a viewpoint not discussed in this chapter, but motivates our naming convention. On several occasions, we will need a locally supported homeomorphism of a non-orientable 2-manifold that allows us to reverse the orientation of a two-sided curve (in particular, the boundary of a Freudenthal subsurface); we introduce the required homeomorphism here. Let .B be a closed Möbius band; more specifically, let .B = [−1, 1]2 / ∼, where .∼ is the equivalence relation generated by .(1, y) ∼ (−1, −y). Let .D ⊂ B be the closed disk centered at the origin of radius 1/2. Then sliding .D along the central curve of .B until it comes back to itself yields an isotopy .H : B × [0, 1] → B. In fact, this isotopy can be taken relative to .∂B, and hence the homeomorphism .H1 of .B onto itself, given by .H1 (x) = H (x, 1), restricts to a homeomorphism of .B \ D o onto itself that reverses the orientation of .∂D while fixing .∂B pointwise. Moreover, the restriction of .H1 to .∂D is simply the reflection o → B \ D o is called a .(x, y) I→ (x, −y). The homeomorphism .H1 : B \ D boundary slide. The boundary slide is a standard tool in the study of mapping class groups of non-orientable surfaces (see [59]); it is based on the notion of a crosscap slide, or Y-homeomorphism, introduced by Lickorish [48]. Let .M be a non-orientable 2-manifold, and let .b be a separating simple closed curve in .M such that both components of .M \ b are non-orientable and such that neither are homeomorphic to a Möbius band. Label the closures of the components of .M \ b by .Mb+ and .Mb− . Viewing .Mb± as a surface, let .b± denote the boundary of .Mb± . Let .c be a two-sided simple closed curve in .M such that .|c ∩ b| = 2 and such that .c± = c ∩ Mb± has a neighborhood .B± in .Mb± admitting a homeomorphism o .p± : B \ D → B± sending .∂D to .b± (see Fig. 5.1). The boundary slide .p+ ◦ H1 ◦ −1 p+ of .b in .Mb+ along .c+ in .B+ does not extend to a homeomorphism of .M since it switches orientation on one side of .b but not the other; however, if we also perform −1 the boundary slide .p− ◦ H1 ◦ p− of .b in .Mb− along .c− in .P− , then we will have switched the orientation on both sides, which allows us to define a homeomorphism −1 .sb,c : M → M by setting .sb,c |B± = p± ◦ H1 ◦ p± and setting .sb,c to be the identity on the complement of .B+ ∪ B− . Definition 5.5.2 The homeomorphism .sb,c is called a double slide.

5.5.2 Anderson’s Method We begin with a baby version of (what we will refer to as) Anderson’s Method. It is now a common technique used to express local transformations as commutators and can be used in a fairly general setting.

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Fig. 5.1 The setup for one side of the double slide .sb,c , and hence the setup for a boundary slide. The disk with the “X” represents a crosscap, meaning the open disk is removed and the antipodal points of the boundary circle are identified

c+ B+ b+

Definition 5.5.3 Let .M be a 2-manifold and let .Σ ⊂ M be a subsurface. A homeomorphism .ϕ : M → M is a .Σ-translation if .ϕ n (Σ) ∩ ϕ m (Σ) = ∅ for any distinct integers .n and .m. If, in addition, .{ϕ n (Σ)}n∈N is convergent, then we say that .ϕ is a convergent .Σ-translation. Recall that the commutator of two elements .g and .h in a group is defined to be [g, h] = ghg −1 h−1 . The following proposition is motivated by Anderson’s proof of [3, Lemma 1].

.

Proposition 5.5.4 Let .M be a 2-manifold. Suppose .Σ and .Δ are closed subsets of .M such that .Σ ⊂ Δ and such that there exists a convergent .Σ-translation .ϕ supported in .Δ. If .h ∈ H (M) such that .supp(h) ⊂ Σ, then .h can be expressed as a commutator of two elements of .H (M), each of which is supported in .Δ. Proof By the convergence condition of .ϕ, we can define the homeomorphism σ =

∞ 

.

ϕ n hϕ −n .

n=0

Consider the commutator .[σ, ϕ] = σ (ϕσ −1 ϕ −1 ). Let us give an intuitive description of this commutator: .σ is performing .h on .ϕ n (Σ) for .n ≥ 0, and similarly, one views −1 ϕ −1 as performing .h−1 on .ϕ n (Σ) for .n > 0, so that .ϕσ −1 ϕ −1 restricts to .ϕσ the identity on .Σ. Leaving the details to the reader, we then see that .σ (ϕσ −1 ϕ −1 ) restricts to the identity outside of .Σ and restricts to .h on .Σ; in particular, .[σ, ϕ] = h, which is the desired result. ⨆ ⨅ Before getting to Anderson’s Method, we will need two lemmas, both of which involve using ambient homeomorphisms to displace Freudenthal subsurfaces. It readily follows from the definition of stability and the classification of surfaces that any two non-compact stable Freudenthal subsurfaces .Δ1 and .Δ2 of a 2-manifold are abstractly homeomorphic if and only if .S(Δ1 ) and .S(Δ2 ) contain a stable end of the same type. Lemma 5.5.5 below provides a sufficient condition for when this abstract homeomorphism can be promoted to an ambient homeomorphism.

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Lemma 5.5.5 Let .M be 2-manifold and let .Δ1 , Δ2 ⊂ M be abstractly homeomorphic stable Freudenthal subsurfaces. If .Σ is a subsurface of .M such that .∂Σ is compact, .Δ1 ∪ Δ2 is contained in the interior of .Σ and, for each .i ∈ {1, 2}, the interior of .Σ contains a Freudenthal subsurface .Δ'i disjoint from and homeomorphic to .Δi , then there exists .f ∈ H (M) with support in .Σ such that .f (Δ1 ) = Δ2 . Moreover, if .Δ1 ∩ Δ2 = ∅, then .f can be chosen to also satisfy .f (Δ2 ) = Δ1 . Proof If .Δ1 and .Δ2 are disjoint, then the statement immediately follows from the change of coordinates principle (Corollary 5.3.7). Now we suppose .Δ1 ∩ Δ2 /= ∅. Fix .μ ∈ S(Δ2 ). If .μ ∈ S(Δ1 ), then there exists a stable Freudenthal subsurface .Δ such that .Δ ⊂ Δ1 ∩Δ2 and .μ ∈ S(Δ). It follows that .Δ∩(Δ'1 ∪Δ'2 ) = ∅. Appealing to the disjoint case at the beginning of the proof, we see there exist homeomorphisms ' ' .f1 , f2 , f3 , f4 ∈ H (M) supported in .Σ such that .f1 (Δ1 ) = Δ , .f2 (Δ ) = Δ, 1 1 ' ' .f3 (Δ) = Δ , and .f4 (Δ ) = Δ2 , and hence, .f = f4 ◦ f3 ◦ f2 ◦ f1 is supported in 2 2 .Σ and maps .Δ1 onto .Δ2 . Otherwise, if .μ ∈ / S(Δ1 ), then, arguing similarly, we can find a Freudenthal subsurface .Δ abstractly homeomorphic to .Δ1 such that .Δ ⊂ Δ2 and such that .Δ ∩ Δ1 = ∅. Hence, we can find .f1 , f2 , f3 ∈ H (M) supported in .Σ such that .f1 (Δ1 ) = Δ, .f2 (Δ) = Δ'2 , and .f3 (Δ'2 ) = Δ2 , and hence, .f = f3 ◦f2 ◦f1 is supported in .Σ and maps .Δ1 onto .Δ2 . ⨆ ⨅ Lemma 5.5.6 Let .M be a 2-manifold, and let .D be a Freudenthal subsurface of M contained in the interior of a stable Freudenthal subsurface .Δ. If either .D is  /= ∅, then there exists a convergent .D-translation with compact or .S(Δ) \ D support contained in the interior of .Δ.

.

Proof Let us consider the case where .D—and hence .Δ—is not compact (the case  to obtain .e ∈ S(Δ) and  and .D with .D compact is similar). Apply Lemma 5.4.7 to .Δ a stable neighborhood basis .{Un }n∈Z of .e such that .Un \{e} contains a clopen subset  for each .n ∈ Z. Moreover, we can assume that .U0 = Δ  and .Vn homeomorphic to .D  .V0 = D. For each .n ∈ Z, fix a homeomorphism .ψn : V n → Vn+1 . Define .Ψ : E(M) → E(M) by setting .Ψ to be the identity outside of . Vn and to satsify .Ψ|Vn = ψn ;  . Let .{Dn }n∈Z be a sequence of then, .Ψ is a homeomorphism supported on .Δ pairwise-disjoint pairwise-homeomorphic Freudenthal subsurfaces such that .Dn ⊂ n = Vn , and .D0 = D. The existence of the .Dn is guaranteed by Lemma 5.3.4. Δ, .D It is now an exercise, using the classification of surfaces (in the manner of an infinite change of coordinates, see Theorem 5.3.5 and Corollary 5.3.7) and the fact that the canonical homomorphism .H (M) → HM (E(M)) is surjective (Theorem 5.3.6), to : M → M supported in .Δ such that .Ψ (Dn ) = Dn+1 construct a homeomorphism .Ψ is a lift of .Ψ. In particular, .Ψ is a convergent .D-translation. (In and such that .Ψ fact, the .D-translation can be chosen to be a shift map as defined in [1, Section 2.4], which is shown in Fig. 5.2). ⨆ ⨅ The second part of the following proposition is a version of Anderson’s ingenious technique—introduced in [3, Lemma 1]—suited to our needs. The version we give, and its proof, is a direct generalization of Calegari–Chen’s version presented in [16, Lemma 2.10], which deals with the sphere minus a Cantor set.

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Fig. 5.2 The support of a sending .Dn to .Dn+1 for all .n ∈ Z, as given by Lemma 5.5.6

D−3

.D-translation

D−2 D−1

e

D0 D1 D2

D3

Proposition 5.5.7 (Anderson’s Method) Let .M be a 2-manifold, and let .Σ and .Δ be Freudenthal subsurfaces of .M such that .Σ ⊂ Δo , .Δ is stable, and either .Σ is  /= ∅. Suppose .h ∈ H (M) such that .supp(h) ⊂ Σ. Then, compact or .S(Δ) \ Σ (1) .h can be expressed as a commutator of two elements of .H (M), each of which is supported in the interior of .Δ, and (2) if .f ∈ H (M) such that .f (Δ) ∩ Δ = ∅, then .h can be factored as a product of four conjugates of .f and .f −1 . Proof By Lemma 5.5.6, we can choose a convergent .Σ-translation .ϕ supported in Δ. For (1), we simply apply Proposition 5.5.4. Now let us consider (2). Let .Δ' = f (Δ). By assumption, .Δ ∩ Δ' = ∅. If .Δ is not orientable, then using the fact that .Δ is stable and by possibly shrinking .Δ, we may assume that the complement of ' n ' .Δ ∪ Δ is not orientable as well. For .n ∈ Z, let .Σn = ϕ (Σ) and .Σn = f (Σn ). By Lemma 5.5.5, there exists a homeomorphism .ψ ∈ H (M) such that .ψ(Δ) = Δ' and .ψ(Δ' ) = Δ. By possibly pre-composing .ψ with a double slide with support disjoint from .Δ' , we may assume that .f −1 ◦ ψ fixes .∂Δ pointwise. Similarly, we may assume that .f ◦ ψ fixes .∂Δ' pointwise. In particular, by further pre-composing ' .ψ with homeomorphisms supported in .Δ and .Δ , we may assume that .ψ|Δ = f |Δ −1 and .ψ|Δ' =  ϕ ◦ f |Δ' . n −n and let .τ = [σ, f ] = σf σ −1 f −1 . We claim that Let .σ = ∞ n=0 ϕ hϕ .

h = τ ψτ ψ −1 = (σf σ −1 )f −1 (ψσf σ −1 ψ −1 )(ψf −1 ψ −1 ),

.

and hence .h is a product of four conjugates of .f and .f −1 . We will provide the intuition and leave the details to the reader. We have already noted in the proof of Proposition 5.5.4 that we should view .σ as performing .h on .Δn for .n ≥ 0, and so we view .f σf −1 as performing .h−1 on .Δ'n for .n ≥ 0, and hence .τ = σ (f σ −1 f −1 ) is performing both .h on .Δn and .h−1 on .Δ'n for .n ≥ 0. Now, .ψτ ψ −1 is performing

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h−1 on .Δn for .n > 0 (and in particular is the identity on .Δ0 ) and is performing .h on .Δ'n for .n ≥ 0. And hence, .τ (ψτ ψ −1 ) = h. ⨆ ⨅

.

Our main application of Anderson’s method will involve dividing Freudenthal subsurfaces. Note that by Proposition 5.4.4, if .Δ is a dividing Freudenthal subsurface, then either .Δ is compact or .S(Δ) is perfect. Given a Freudenthal subsurface .Δ of a 2-manifold .M, let .𝚪Δ denote the normal closure of the subgroup of .H (M) consisting of homeomorphisms with support in .Δ. The following corollary will be our main use of Anderson’s method. To simplify the statement, we abuse notation by letting .f ∈ H (M) denote both itself as well as the element of .H (E(M)) induced by .f . Corollary 5.5.8 Let .M be a 2-manifold, and let .Δ ⊂ M be a dividing Freudenthal subsurface. Suppose .f ∈ H (M) and .x ∈ S(Δ) such that .f (x) /= x. Then, every element of .H (M) supported in .Δ is a product of four conjugates of .f and .f −1 ; in particular, .𝚪Δ is contained in the group normally generated by .f . Proof By continuity, there exists a stable Freudenthal subsurface .Δ' contained in .Δ such that .x ∈ S(Δ' ) and .f (Δ' ) ∩ Δ' = ∅. Moreover, by possibly shrinking .Δ' , we may assume that .M \ (Δ' ∪ f (Δ' )) contains a stable Freudenthal subsurface homeomorphic to .Δ. Choose any stable Freudehntal subsurface .Δ'' that is is homeomorphic to .Δ' , contained in the interior of .Δ' , and satisfies ' '' .S(Δ )\S(Δ )/=∅. By Lemma 5.5.5, any element supported in .Δ can be conjugated to be supported in .Δ'' . The result now follows from Anderson’s Method. ⨆ ⨅ Corollary 5.5.9 Let .M be a 2-manifold and let .Δ ⊂ M be a dividing Freudenthal subsurface. Then, every homeomorphism of .M with support in .Δ can be expressed as the commutator of two elements in .𝚪Δ ; in particular, .𝚪Δ is uniformly perfect with commutator width one. Proof Fix a dividing Freudenthal subsurface .Δ' such that .Δ ⊂ Δ' and such that ' .S(Δ) is a proper subset of .S(Δ ). Indeed, this is readily seen to be possible if .Δ is compact, and if .Δ is not compact, then it can be deduced from Proposition 5.4.6  with a homeomorphic copy of .Δ  in its complement. by taking the union of .Δ Then, by Anderson’s Method, for every .g ∈ H (M) with support in .Δ there exists ' .h, f ∈H (M) with support in .Δ such that .g = [h, f ]. It follows from Lemma 5.5.5 that .h, f ∈ 𝚪Δ , and as the conjugate of a commutator is a commutator, we can conclude that .𝚪Δ is uniformly perfect with commutator width one. ⨆ ⨅

5.6 Topology of Homeomorphism Groups In this section, we recall basic facts about the compact-open topology and establish classical results about homeomorphism groups of 2-manifolds. The aim is not to give the state of the art or the strongest statements but to prove what is necessary for the later sections and to do so in a way that highlights the use of Anderson’s

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method. Note also that statements will only be made for 2-manifolds; however, much of what is said can be generalized to higher dimensions utilizing the resolution of the annulus conjecture and the stable homeomorphism conjecture (see [23] for the relevant history), as well as the fragmentation lemma of Edwards–Kirby discussed below. We end the section by establishing the equivalence of the two standard definitions of the mapping class group seen in the literature.

5.6.1 The Compact-Open Topology Let us begin with considering the basics of the topology of homeomorphism groups before moving to manifolds. Let .X be a locally compact Hausdorff topological space. We equip .Homeo(X) with the compact-open topology, that is, the topology generated by sets of the form .U (K, W ) = {f ∈ Homeo(X) : f (K) ⊂ W }, where .K ⊂ X is compact and .W ⊂ X is open. One readily observes that if we require .W to be a precompact open subset, then this does not change the topology. Given a compact Hausdorff topological space .X with a metric .d, define ρd , μd : Homeo(X) × Homeo(X) → R

.

by .ρd (f, g) = max{d(f (x), g(x)) : x ∈ X} and .μd (f, g) = ρd (f, g) + ρd (f −1 , g −1 ). Observe that .ρd is invariant with respect to right multiplication. We leave the following lemma as an exercise (one may also refer to [8]): Lemma 5.6.1 Let .X be a compact Hausdorff topological space. Then, .Homeo(X), equipped with the compact-open topology, is a topological group. Moreover, if .(X, d) is a compact metric space, then both .ρd and .μd are metrics on .Homeo(X) inducing the compact-open topology, and .μd is complete. ⨆ ⨅ The next two propositions are due to Arens [7], and we leave the interested reader to either read Arens’s proofs or to treat them as exercises. Proposition 5.6.2 Let .X be a locally compact locally connected Hausdorff space. Then, .Homeo(X), equipped with the compact-open topology, is a topological group. ⨆ ⨅ A compactification of a non-compact locally compact locally connected Hausdorff space .X is a compact Hausdorff space .C such that .X can be topologically embedded in .C as a dense open subset. Every such .X admits a compactification (e.g., the one-point compactification and the Freudenthal compactification). In fact, Arens [7] proves that for a locally compact locally connected Hausdorff space, the compact-open topology agrees with the closed-open topology (i.e., the topology defined analogously to the compact-open topology but allowing .K to be closed instead of compact). This yields the following:

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Proposition 5.6.3 Let .X be a non-compact locally compact locally connected Hausdorff space, let .C be a compactification of .X, and let .H (C, X) = {f ∈ Homeo(C) : f (X) = X}. If every homeomorphism of .X extends continuously to a homeomorphism of .C, then .Homeo(X) and .Homeo(C, X) are isomorphic as topological groups, where .Homeo(X) and .Homeo(C) are equipped with their respective compact-open topologies, and .Homeo(C, X) is equipped with the corresponding subspace topology. ⨆ ⨅ Even though we will not tend to work with metrics on homeomorphism groups, it will arise on several occasions, and so we record an important fact that comes out of the above discussion. Corollary 5.6.4 Let .X be a locally compact locally connected Hausdorff space, and let .C be a compactification of .X such that every homeomorphism of .X extends continuously to a homeomorphism of .C. Suppose .C supports a metric .d, and define .ρˆd , μˆ d : Homeo(X)2 → R given by .ρˆd (f, g) = ρd (fˆ, g) ˆ and .μˆ d (f, g) = μd (fˆ, g), ˆ where .fˆ and .gˆ are the extensions of .f and .g, respectively, to .C. Then, the metric topologies on .Homeo(X) given by .ρˆd and .μˆ d agree and are equal to the compact-open topology, and moreover, .ρˆd is right-invariant and .μˆ d is complete. ⨅ ⨆ The key takeaway of the above corollary is the following: a complete metric on a homeomorphism group giving rise to the compact-open topology comes from a metric on a compactification of the underlying space, rather than from the space itself.

To finish our quick recap of the basics, recall that a Polish group is a completely metrizable separable topological group. It is an exercise to see that if .X is a separable locally compact locally connected Hausdorff space, then .Homeo(X) equipped with the compact-open topology is also separable, yielding: Proposition 5.6.5 The homeomorphism group of a separable locally compact locally connected metrizable space is a Polish group when equipped with the compact-open topology. ⨆ ⨅ For the remainder of the chapter, we will always consider homeomorphism groups (resp., their subgroups) equipped with the compact-open topology (resp., the corresponding subgroup topology).

5.6.2 Homeomorphism Groups of 2-Manifolds In this subsection, we establish various topological and algebraic properties of homeomorphism groups. The majority of the results are classical and highlight the usage of Anderson’s method. In addition, we will establish several factorization results that are essential for arguments throughout the chapter. In what follows, we will regularly apply isotopies, and so we record two basic facts.

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Theorem 5.6.6 (Alexander’s Trick) Let .Dn = {x ∈ Rn : | x | ≤ 1} be the standard .n-disk. Any homeomorphism .f : Dn → Dn restricting to the identity on n n .∂D is isotopic to the identity via an isotopy that fixes .∂D pointwise at each stage. Moreover, if .f fixes the origin, then the isotopy can be chosen to fix the origin at each stage. Proof The function .H : Dn × [0, 1] → Dn given by

H (x, t) =

.

is the desired isotopy.

tf (x/t) | x | < t x |x | ≥ t ⨆ ⨅

We forgo the proof of the next proposition, but the interested reader should see [25, Lemmas 2.4 & 2.5]. Proposition 5.6.7 Let .ι : S1 → S1 × R be the inclusion map .ι(x) = (x, 0). If 1 1 .f : S → S × R is homotopic to .ι, then .f is isotopic to .ι via an ambient isotopy of 1 .S × R that is fixed outside of a compact subset. ⨆ ⨅ Motivated by Proposition 5.6.7, we introduce a class of open subsets of homeomorphism groups of 2-manifolds. Let .M be a 2-manifold, let .γ be a 2-sided simple closed curve in .M, and let .A be an open annular neighborhood of .γ . Fix an embedding .ι : S1 → A such that the image of .ι is .γ . Let .U + (γ , A) denote the subset of .H (M) consisting of elements .f ∈ U (γ , A) such that .f ◦ ι is isotopic to + .ι via an isotopy fixed outside of .A, and note that .U (γ , A) does not depend on the choice of .ι. Lemma 5.6.8 Let .M be a 2-manifold, let .γ be a 2-sided simple closed curve in .M, and let .A be an open annular neighborhood of .γ . Then, .U + (γ , A) is open in .H (M). Sketch of Proof Let .C be a compactification of .M, with .C = M if .M is compact. Let .d be a metric on .C and let .ρd be the corresponding metric on .H (M), as defined in Corollary 5.6.4. Fix an orientation on .γ , and fix .f ∈ U + (γ , A). Given a positive real number .ϵ, let .Uϵ be the .ϵ-ball (with respect to .ρd ) in .H (M) centered at .f . We can readily choose .ϵ small enough to guarantee that .g(γ ) does not bound a disk in .A for any .g ∈ Uϵ . And, by choosing .ϵ much smaller than the .d-diameter of .γ , we can guarantee that .g(γ ) is homotopic to .γ in .A. Indeed, if .g(γ ) was in the .ϵ-neighborhood of .γ and had the opposite orientation, then .g would have to move some point of .γ a distance more than .ϵ (that is to say, there is not a enough room to turn .γ around). Hence, .Uϵ is an open neighborhood of .f contained in .U + (γ , A), and therefore, .U + (γ , A) is open. ⨆ ⨅ An important tool used by Mann [52, 54] in her arguments establishing the automatic continuity of various homeomorphism groups is the Fragmentation Lemma [52, Proposition 2.3], which is deduced from the work of Edwards–Kirby (namely the proof of [24, Corollary 1.3]). The Fragmentation Lemma holds in all dimensions, and its proof is nontrivial. Let us state the Fragmentation Lemma,

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though we only use it once, as we present and sketch a proof of a much weaker result (Proposition 5.6.10) that is enough for our purposes in most cases. Theorem 5.6.9 (Fragmentation Lemma) Let .M be a compact manifold. Given any finite open cover .{V1 , . . . , Vm } of .M, there exists a neighborhood .U of the identity in .H0 (M) such that each .g ∈ U can be factored as a composition .g = g1 g2 . . . gm with .supp(gi ) ⊂ Vi and .gi ∈ H0 (M) for .i ∈ {1, . . . , m}. ⨆ ⨅ Let us now give a weakened version of the fragmentation lemma in dimension two. Proposition 5.6.10 Let .M be a compact 2-manifold, and let .X ⊂ M be finite. Then, there exists a finite collection of closed embedded 2-disks .Δ1 , . . . , Δk in .M and an open neighborhood .U of the identity in .H (M) such that • .{Δo1 , . . . , Δok } is an open cover of .M, • .|X ∩ Δi | ≤ 1 for each .i ∈ {1, . . . , k}, and • every .g ∈ U can be factored as a composition .g = g1 g2 . . . gk with .supp(gi ) ⊂ Δoi for .i ∈ {1, . . . , k}. Sketch of Proof Let us assume that .X is empty. The idea is as follows: cover the 2-manifold with finitely many closed 2-disks .Δ1 , . . . , Δk with simple intersection patterns (i.e., the intersection of any two disks is either empty or a bigon). Slightly shrink each disk so that the resulting disks, labelled .Δ'1 , . . . , Δ'k , continue to cover the surface. Then, letting .Ai ⊂ Δoi be an open annular neighborhood of .∂Δ'i , define k ' + .U = i=1 U (∂Δi , Ai ). Given .g ∈ U , there is an isotopy supported in .A1 sending .g(γ1 ) back to .γ1 . We can then apply Alexander’s trick to get a homeomorphism .g1 supported on .Δ1 such that .g1−1 ◦ g|Δ'1 is the identity. Proceed recursively. If .X is nonempty, then the same argument holds with the extra care of choosing the .Δi to satisfy the additional hypotheses. ⨆ ⨅ We can now prove a theorem of Fisher [32] regarding compact 2-manifolds, which generalizes the work of Anderson [3] on the 2-sphere. Also, we will require one additional fact: two self-homeomorphisms of a manifold .M are isotopic if and only if there is a path between them in .Homeo(M) (see Fox [33]). Given a closed 2-disk .Δ in a 2-manifold .M, recall that .𝚪Δ denotes the subgroup of .H (M) normally generated by homeomorphisms supported in .Δ. Theorem 5.6.11 Let .M be a 2-manifold, and let .Δ ⊂ M an embedded copy of the closed 2-disk. Then, .𝚪Δ is contained in every nontrivial normal subgroup of .H (M) and is a simple group. Moreover, if .M is of finite type, then (1) .H0 (M) is open and path connected, and (2) every element of .H0 (M) can be factored as a composition of homeomorphisms each of which is supported in a stable Freudenthal subsurface. In particular, if .M is compact, then .𝚪Δ = H0 (M) and .H0 (M) is a simple group.

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Proof Every nontrivial element of .H (M) must move some point of .M, and hence, some conjugate must move some point in .Δo = S(Δ); in particular, by Anderson’s Method (Proposition 5.5.7), we see that .𝚪Δ is contained in the normal subgroup generated by every nontrivial element of .H (M), and hence .𝚪Δ is contained in every nontrivial normal subgroup of .H (M). In particular, every nontrivial element of .𝚪Δ normally generates .𝚪Δ , and hence .𝚪Δ is a simple group. For the remainder of the proof, assume that .M is of finite type. Then, there exists a compact 2-manifold .N and a finite subset .X ⊂ N such that .M is homeomorphic to .N \ X (if .M is compact, then .N = M and .X = ∅). Then, by Corollary 5.6.4, the set .H (N, X) = {f ∈ H (N) : f (X) = X}, equipped with the subspace topology, is isomorphic to .H (M) as a topological group. Proposition 5.6.10 yields closed 2-disks .Δ1 , . . . , Δk embedded in .N and an open neighborhood of the identity .U in .H (N) such that every .f ∈ U can be factored as ' ' .f = f1 ◦ · · · ◦ fk with .supp(fj ) ⊂ Δj . Let .U = U ∩ H (N, X) and let .f ∈ U . Then, .f = f1 ◦ · · · ◦ fk with .supp(fj ) ⊂ Δj . For each .j ∈ {1, . . . , k}, .Δj contains at most one element of .X, and hence it must be that .fj ∈ H (N, X). We can apply Alexander’s trick to see that each .fj is isotopic to the identity relative to .X, and hence that .f is isotopic to the identity relative to .X. It follows that .U ' is contained the path component of the identity in .H (N, X); hence, the path component of the identity is open in .H (N, X) (since it is a union of translates of .U ' ). Every open subgroup of a topological group is also closed, so the path component of the identity is a clopen subset of .H0 (N, X), and hence equal to .H0 (N, X). Therefore, .H0 (M) is open and path connected, establishing (1). Now, observe that .Δj \X is a stable Freudenthal subsurface of .N \X. Since .U ' is an open neighborhood of the identity in the connected topological group .H0 (N, X), it must generate .H0 (N, X) (a standard exercise); hence, every element of .H0 (M) is a product of homeomorphisms each of which is supported in a stable Freudenthal subsurface, establishing (2). In particular, if .M is compact, then every Freudenthal subsurface is a disk, and hence .H0 (M) = 𝚪Δ . ⨆ ⨅ Before continuing, we note that, for a compact 2-manifold .M, Hamstrom and Dyer [36] appear to be the first to prove that .H0 (M) is path connected; in fact, they prove the stronger fact that .H0 (M) is locally contractible. For a compact 2-manifold .M, it follows from the simplicity of .H0 (M) (Theorem 5.6.11) that .H0 (M) is perfect. For a non-compact finite-type 2-manifold, simplicity of .H0 (M) does not hold since .𝚪Δ is a proper normal subgroup; however, we will need to know that .H0 (M) is perfect for any finite-type 2-manifold. This is a result of McDuff [57, Corollary 1.3], and we will prove it below. We note that McDuff’s work relies on a preprint of Ling that appears to never have been published, but the relevant result appears in Ling’s thesis [49]. The proof of Proposition 5.6.12 below is motivated by an argument in [49]. We will use the term standard open (resp., closed) annulus to refer to a subset of 1 1 .S × R of the form .S × I for some precompact open (resp., closed) interval .I ⊂ R. Proposition 5.6.12 If .f ∈ H0 (S1 ×R), then there exist .g1 , g2 , h1 , h2 ∈ H0 (S1 ×R) such that .f = [g1 , h1 ][g2 , h2 ]. Moreover, if .f restricts to the identity on a set of the

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form .S1 × [t, ∞) for some .t ∈ R, then there exists .t ' > t such that the .gi and .hi can be chosen to restrict to the identity on .S1 × [t ' , ∞). Proof The second statement of the proposition will be a consequence of the constructions in the proof, and we will not mention it explicitly. Let us begin by supposing that there exists a locally finite family .{A n }n∈Z of pairwise-disjoint standard closed annuli such that .f issupported on . n∈Z An . Let .f+ be the homeomorphism that agrees with .f on . n≥0 An and is the identity outside of this union. Choose .τ+ ∈ H0 (S1 × R) such that .τ+ (An ) = An+1 for .n ≥ 0 and such that .τ+ |An is the identity for .n < 0 (for instance, .τ+ can be chosen to be an appropriate conjugate of the homeomorphism given by .(z, t) I→ (z, 2t − 1) for .t ≥ 0 and that restricts to the identity on the negative reals). Using the fact that, for each .x ∈ S1 × R, the set .{k∈N : (τ+k ◦ f+ ◦ τ+−k )(x)/=x} is finite, together with the fact that .{An }n∈N is locally finite, we can define the homeomorphism F+ =

∞  

.

 τ+n ◦ f ◦ τ+−n .

n=0

A similar argument as in the proof of Proposition 5.5.4 shows that .f+ = [F+ , τ+ ]. Defining .f− , .τ− and .F− analogously, we can write .f− =[F− , τ− ] and .f =f− ◦ f+ . Moreover, since the support of .τ± and the support of .F± are disjoint from the supports of .F∓ and .τ∓ , we can write f = [F− , τ− ] ◦ [F+ , τ+ ] = [F− ◦ F+ , τ− ◦ τ+ ].

.

Now, suppose .f is arbitrary. We will decompose .f into a product of two homeomorphisms, each of which is supported in a locally finite union of pairwisedisjoint standard closed annuli (the proof follows the proof of Le Roux and Mann’s [67, Lemma 4.4]). Let .c0 = S1 × {0} and choose a standard open annulus .A0 in .S1 × R such that .c0 ∪ f (c0 ) ⊂ A0 . We can then find .n1 ∈ N such that 1 .c±1 = S × {±n1 } satisfies .(c±1 ∪ f (c±1 )) ∩ A0 = ∅ and such that .n1 ≥ 1. Choose standard open annuli .A±1 such that .A±1 ∩A0 = ∅ and .c±1 ∪f (c±1 ) ⊂ A±1 . Continuing in this fashion, we construct a locally finite sequence of pairwise-disjoint curves .{cn }n∈Z and a locally finite family of standard open annuli .{An }n∈Z with pairwise-disjoint closures such that .cn ∪ f (cn ) ⊂ An . For each .n ∈ Z, there is an ambient isotopy .Фn of .S1 × R fixed on the complement of .An such that .Фn (f (cn ), 0) = f (cn ), .Фn (f (cn ), 1) = cn , and letting .hn (x) = Фn (x, 1), we have .hn ◦ f restricts to the identity on a standard closed  −1 annular neighborhood .Cn ⊂ An of .cn . Let .f1 = n∈Z hn . Then, .f1 (as well as .f1 )  is supported on . n∈Z An . Let .f2 = f1 ◦f . Then, .f2 is supported on the complement  of . n∈Z Cn , and hence, both .f1 and .f2 are supported on the union of a locally finite family of pairwise-disjoint standard closed annuli. Therefore, .f = f1−1 ◦ f2 can be written as the product of two commutators. ⨆ ⨅

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Corollary 5.6.13 .H0 (S2 ), .H0 (R2 ), and .H0 (S1 × R) are uniformly perfect. Proof The case of .H0 (S1 × R) is immediate from Proposition 5.6.12. Suppose .f ∈ H0 (R2 ). Then, there exists an embedded closed 2-disk .Δ containing .0 and .f (0). Fix 2 o .g ∈ H0 (R ) such that .g(f (0)) = 0 and .supp(g) ⊂ Δ . By Corollary 5.5.9, .g can be expressed as a commutator. Now we can view .g ◦ f as a self-homeomorphism of .R2 \ {0}, which is of course homeomorphic to .S1 × R. Hence, we may express .g ◦ f as a product of two commutators, which allows us to express .f as a product of three commutators. Therefore, .H0 (R2 ) is uniformly perfect. A similar argument works for .H0 (S2 ). ⨆ ⨅ In fact, Tsuboi [71] has shown that the commutator width of .H0 (Sn ) is one. In the next theorem, we show that .H0 (M) is perfect when .M is finite-type. We note that Bowden et al. [12] have recently shown that if .M is a compact manifold of genus at least one, then .H0 (M) is not uniformly perfect, which answered a long-standing question. Theorem 5.6.14 Let .M be a finite-type 2-manifold. Then, .H0 (M) is perfect. Proof Let .f ∈ H0 (M). By Theorem 5.6.11, there exists .f1 , . . . , fk ∈ H0 (M) such that each .fi is supported in the interior of a stable Freudenthal subsurface .Δi and o 1 .f = f1 · · · fk . In particular, .Δ is homeomorphic to either the open 2-disk or .S ×R. i By Anderson’s Method in the former case or Proposition 5.6.12 in the latter, we have that .fi is a product of commutators of homeomorphisms with support in .Δoi . Hence, .f can be expressed as a product of commutators and .H0 (M) is perfect. ⨆ ⨅ From the above results regarding finite-type 2-manifolds, we record the following lemma: Lemma 5.6.15 Let .M be a 2-manifold and let .N be an open finite-type submanifold of .M. Then, there exists an open neighborhood .UN of the identity in .H (M) such that every element of .UN that is supported in .N o is contained in .[H (M), H (M)]. Proof As .N is finite-type, by Theorem 5.6.11, .H0 (N ) is open in .H (N); in particular, there is a finite collection of compact sets .K1 , . . . , K m of .N admitting precompact open neighborhoods .W1 , . . . , Wm in .N such that . ki=1 U (Ki , Wi ) is contained in .H0 (N ). Now, viewing the sets .U (Ki , Wi ) in .H (M), we let .UN = k o i=1 U (Ki , Wi ) ⊂ H (M). If .f ∈ UN such that .supp(f ) ⊂ N , then .f restricts to the identity on an open neighborhood of .∂N; therefore, viewing .f as a homeomorphism of .N and applying Theorem 5.6.14, there exists .g, h ∈ H0 (N ) such that .[g, h] = f . Now, Proposition 5.6.12 (and the construction in Theorem 5.6.14) guarantees that .g and .h can be chose such that .supp(g), supp(h) ⊂ N o , and can therefore be extended by the identity to .M, implying .f ∈ [H (M), H (M)]. ⨆ ⨅ To finish the subsection, we record a factorization result for non-compact 2manifolds.

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Proposition 5.6.16 Let .M be a non-compact 2-manifold. Suppose .N, Ω1 , . . . , Ωk is an open cover of .M consisting of connected sets satisfying the following conditions: (i) (ii) (iii) (iv)

the closure of .N is a finite-type subsurface of .M, Ωi is a Freudenthal subsurface for .i ∈ {1, . . . , k}, .Ωi ∩ Ωj = ∅ for distinct .i, j ∈ {1, . . . , k}, and for each .i ∈ {1, . . . , k}, there exists a simple closed curve .bi contained in .N such that .Ωi ∩ N is an open annular neighborhood of .bi . .

Then, there exists an open neighborhood .U of the identity in .H (M) such that every  element of .U can be factored as . ki=0 gi , where .supp(gi ) ⊂ Ωi for .i ∈ {1, . . . , k}, .supp(g0 ) ⊂ N, and .g0 ∈ [H (M), H (M)]. Proof Let .UN be the neighborhood of the identity in .H (M) given by Lemma 5.6.15. By shrinking .UN , we may assume that there exist compact subsets .K1 , . . . , Km of .M contained in .N admitting relatively compact open neighborhoods .W1 , . . . , Wm ,  also contained in .N, such that .UN = m i=1 U (Ki , Wi ). For each .i ∈ {1, . . . , k}, let .Ri be an open annular neighborhood of .bi whose closure is contained in .Ωi ∩ N and is disjoint from .W i for .i ∈ {1, . . . , m}. Let

k   + .U = UN ∩ U (bi , Ri ) . i=1

Note that .U is nonempty as it contains the identity. Fix .f ∈ U . By the definition of .U , we have .f (bi ) ⊂ Ri and .f (bi ) is isotopic to .bi for each .i ∈ {1, . . . , k} via an isotopy supported in .Ri . In particular, there exists .g ∈ H (M) supported in .R1 ∪· · ·∪Rk such that .g◦f restricts to the identity on an open annular neighborhood of .bi for all .i ∈ {1, . . . , k}. Therefore, we can write .g ◦ f = g0' g1 g2 · · · gk , where ' .gi is supported on .Ωi for .i ∈ {1, . . . , k} and .g is supported on .N . 0 Since .g is supported in .R1 ∪· · ·∪Rk , it follows that .g0 = g −1 ◦g0' is supported in .N and restricts to the identity on an open neighborhood of .∂N. Moreover, .g0 |Ki = f |Ki , which implies .g0 ∈ UN and hence .g0 ∈ [H (M), H (M)]. Therefore, .f = k ⨆ ⨅ i=0 gi is the desired decomposition.

5.6.3 Defining the Mapping Class Group Given a 2-manifold .M, we define the mappping class group8 of .M, denoted .MCG(M), to be the group .H (M)/H0 (M). In the literature, the mapping class group is often defined to be the group of isotopy classes of self-homeomorphisms of .M,

8 The

mapping class group is sometimes referred to as the homeotopy group in the literature.

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which agrees with the group .π0 (H (M)). We finish this section by proving that these definitions agree. We have already seen in Theorem 5.6.11 that if .M is of finite type, then .H0 (M) is path connected, which implies that .MCG(M) = π0 (H (M)). It is therefore left to check the infinite-type case, for which we require the following proposition, established by Hernández et al.[39] in the orientable case and Hernández and Hidber [38] in the non-orientable case (also see the work of Shapiro [68] for a more general result). Proposition 5.6.17 If a self-homeomorphism of an infinite-type 2-manifold fixes the isotopy class of every simple closed curve, then it is isotopic to the identity. ⨆ ⨅ The idea of the proof is as follows: fix a collection of simple closed curves that fill9 the 2-manifold and whose intersection pattern has “simple combinatorics”. Then, given some homeomorphism fixing the isotopy class of every simple closed curve, recursively perform isotopies—with increasing support—mapping the image of an increasing finite subset of curves in the collection back to itself. One does this in a such a way that the direct limit of the isotopies can be taken to obtain an isotopy returning the image of each curve to its original position simultaneously. It is then possible to apply the Alexander Trick simultaneously to each disk in the complement of the collection of curves. This is referred to as the Alexander Method (see [29, Section 2.3] for a discussion in the finite-type case). It is worth noting that Proposition 5.6.17 is false for a small number of finitetype 2-manifolds: for example, the hyperelliptic involution of the compact orientable genus two 2-manifold fixes the isotopy class of every simple closed curve. Theorem 5.6.18 Let .M be a 2-manifold. Then, .H0 (M) is path connected. Proof The finite-type case is handled in Theorem 5.6.11, so we may assume that M is of infinite-type. Let .c be a simple closed curve on .M. If .c is two-sided, let .Ac be an annular neighborhood of .c; otherwise, let .Ac be a Möbius band embedded in .M with .c as its core curve. Let .Uc = U (c, Ac ) ∩ H0 (M). Note that .Uc contains the identity and that every element of .Uc fixes the isotopy class of .c. Fix .f ∈ H0 (M). Then, as every open neighborhood of the identity in a connected topological group generates the group, we can write .f = f1 f2 · · · fk with each .fj ∈ Uc . Therefore, .f must fix the isotopy class of .c. As .c was arbitrary, it follows that every homeomorphism in .H0 (M) preserves the isotopy class of every simple closed curve in .M; hence, by Proposition 5.6.17, .f is isotopic to the identity, and the result follows. ⨆ ⨅ .

Corollary 5.6.19 Let .M be a 2-manifold. Then, .MCG(M) = π0 (H (M)).

⨆ ⨅

9 A set of simple closed curves on a 2-manifold fill if the complementary components of their union

are either disks or once-punctured disks.

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5.7 Equivalent Notions of Self-Similarity As noted in the introduction, the definition of a self-similar 2-manifold given differs from the original definition in [50]. The main purpose of this section is to establish the equivalence of the two definitions for non-compact 2-manifolds. Along the way, we will introduce the definition of maximally stable ends, half-spaces, and uniquely self-similar 2-manifolds. Let us recall the definition of self-similarity given previously and show the equivalence with the definition given by Malestein–Tao. First recall that a subset .Y of a topological space .X is displaceable if there exists a homeomorphism .f : X → X such that .f (Y )∩Y = ∅. A 2-manifold .M is self-similar if every proper compact subset of .M is displaceable and every separating simple closed curve has a complementary component homeomorphic to .M with a point removed. Definition 5.7.1 An end .μ of a 2-manifold .M is maximally stable if .E(M) is a stable neighborhood of .μ. Note that the existence of a maximally stable end is equivalent to .E(M) being stable. The following proposition establishes the equivalence of the definition of self-similar given here and the one given in [50] for non-compact 2-manifolds. Proposition 5.7.2 Let .M be a non-compact 2-manifold. Then, .M is self-similar if and only if every compact subset of .M is displaceable and .M has a maximally stable end. Proof First suppose that .M has a maximally stable end .μ and every compact subset of .M is displaceable. Let .c be a separating simple closed curve in .M. Then, there is a  . Then, by Proposition 5.4.2, component of .M \ c whose closure .Δ satisfies .μ ∈ Δ  is homeomorphic to .E(M), and .Δ is a stable Freudenthal subsurface. By the .Δ classification of surfaces, .Δ is homeomorphic to .M with an open disk removed, and hence .M is self-similar. Conversely, suppose that .M is self-similar. Let .U1 and .U2 be clopen subsets of .E(M) such that .E(M) = U1 ∪U2 . We can then find a separating simple closed curve i , where .Ω1 and .Ω2 are the complement components of .M \ c. .c such that .Ui = Ω The self-similarity of .M guarantees that at least one of .U1 and .U2 is homeomorphic to .E(M). In particular, by Proposition 5.4.9, .E(M) is stable, and hence contains a maximally stable end .μ. ⨆ ⨅ Definition 5.7.3 A half-space of a 2-manifold .M is a dividing Freudenthal subsurface .H such that the closure of .M \ H is homeomorphic to .H . Recall a self-similar 2-manifold is perfectly self-similar if .M#M is homeomorphic to .M. Proposition 5.7.4 A 2-manifold is perfectly self-similar if and only if it contains a half-space.

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Proof Let .M be a 2-manifold. If .M is compact, then .M contains a half-space if and only if .M is homeomorphic to the 2-sphere. So we may now assume that .M is not compact. First, let us assume that .M contains a half-space .H . Let .H ' be the closure of '  and .H ' are disjoint and .M \ H ; by assumption, .H is a half-space. In particular, .H '   . By homeomorphic. By Proposition 5.4.6, .E(M) = H ∪ H is homeomorphic to .H ' classification of surfaces, we have that .H and .H are both homeomorphic to .M with an open disk removed, and hence, .M is homeomorphic to .M#M. Conversely, suppose that .M is perfectly self-similar. Then, as .M is homeomorphic to .M#M, there exists a subsurface .H in .M that is homeomorphic to .M with an open disk removed and such that the closure of .M \H is homeomorphic to .H . Note that .H is a trim Freudenthal subsurface. Let us argue that .H is stable. Let .U1 and .U2 be a clopen subsets of .E(M) such that .E(M) = U1 ∪ U2 . Let .Δ be a Freudenthal  = U1 . Then, .K = ∂Δ ∪ ∂H is a compact subset of .M, and subsurface such that .Δ hence, by assumption, there exists .f ∈ H (M) such that .f (K) ∩ K = ∅. It follows that either .f (H ) ⊂ Δ, .f (H ) ⊂ M \ Δ, .Δ ⊂ f (H ), or .M \ Δ ⊂ f (H ); in any  and hence homeomorphic case, .U1 or .U2 contains an open set homeomorphic to .H  to .E(M). By Proposition 5.4.9, .E(M) is stable, and as .H is homeomorphic to .E(M), we see that .H is a stable Freudenthal subsurface. To finish, observe that .E(M), | > 1, and so by , is homeomorphic .H  × {1, 2}. This shows that .|H and hence .H Proposition 5.4.4, .S(H ) is infinite and .H is dividing. In particular, .H is a halfspace. ⨆ ⨅ Let us now exhibit a basic dichotomy of self-similar 2-manifolds. Lemma 5.7.5 A self-similar 2-manifold is perfectly self-similar if and only the set of maximally stable ends is perfect. Proof Let .M be a 2-manifold. If .M is non-compact and perfectly self-similar, then M has a maximally stable end by Proposition 5.7.2. Then, as .M is homeomorphic to .M#M, it must have at least two maximally stable ends, and hence a perfect set worth by Proposition 5.4.4. Conversely, using Proposition 5.4.2 in the non-compact case, we see that .M admits a half-space and hence is perfectly self-similar by Proposition 5.7.4. ⨆ ⨅

.

By Proposition 5.4.4, we know that, for a self-similar 2-manifold .M, the set of maximally stable ends is either a singleton or a perfect subset of .E(M), which motivates the following definition. Definition 5.7.6 A 2-manifold is uniquely self-similar if it is self-similar and has a unique maximally stable end. With this, we end with the following consequence of Lemma 5.7.5, which justifies the definition given in the introduction. Proposition 5.7.7 Every self-similar 2-manifold is either uniquely or perfectly selfsimilar. ⨆ ⨅

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5.8 Normal Generation and Purity The goal of this section is to prove a strengthened version of a theorem of Malestein and Tao [50, Theorem A]. Theorem 5.8.1 Let .M be a perfectly self-similar 2-manifold, and let .H = H (M). (1) .H is uniformly perfect and its commutator width is at most two. (2) If .g ∈ H displaces a half-space of .M, then every element of .H can be expressed as a product of at most eight conjugates of .g and .g −1 . In particular, .H is normally generated by .g. (3) If .n ∈ N such that .n ≥ 2, then every element of .H can be expressed as the product of at most eight elements of order .n. Before we prove Theorem 5.8.1, we need an additional result. Recall that, for a half-space .Δ of .M, we define .𝚪Δ to be the subgroup of .H (M) generated by the conjugates of homeomorphisms of .M supported in .Δ. By Lemma 5.5.5, if .Δ' is another half space of .M, then .𝚪Δ = 𝚪Δ' . The following theorem is a version of [50, Theorem 3.11], which allows for non-orientable 2-manifolds. Theorem 5.8.2 Let .M be a perfectly self-similar 2-manifold. Then, for every .h ∈ H (M) there exist .h1 , h2 ∈ H (M) and half-spaces .Δ1 and .Δ2 such that .h = h1 ◦h2 . Moreover, given any half-space .Δ, every element of .H (M) can be factored as the product of two homeomorphisms, each of which is conjugate to a homeomorphism with support in .Δ; in particular, .H (M) = 𝚪Δ . Proof Let .h ∈ H (M). Then, by taking a shrinking nested sequence of half-spaces, we are guaranteed to find a half-space .D1 such that the complement of .h(D1 ) ∪ D1 contains a half-space. In particular, we can find half-spaces .D2 and .D3 such that o .D2 ∩ (D1 ∪ h(D1 )) = ∅ and .D1 , h(D1 ), D2 ⊂ D . Therefore, by Lemma 5.5.5, 3 there exists .f1 ∈ H (M) with support in .D3 such that .f1 (h(D1 )) = D1 . If .M is orientable, then so is .f1 , and hence we can assume that .f1 ◦ h restricts to the identity on .∂D1 . However, if .M is non-orientable, this is not guaranteed by Lemma 5.5.5 as .f1 may be orientation-reversing in an annular neighborhood of .∂D1 . However, if this is case, then as both .D1 and .D2 are not orientable, we may find a simple closed curve .c such that the double slide .s∂D1 ,c is supported in .D3 . Then, replacing .f1 with .s∂D1 ,c ◦ f1 , we may assume that .f1 ◦ h restricts to the identity on .∂D1 in both the orientable and non-orientable cases. Let .f2 ∈ H (M) such that .f2 agrees with .(f1 ◦ h)−1 on .D1 and is the identity on the complement of .D1 . Then, .f2 is supported in .D1 ⊂ D3 and .f2 ◦ f1 ◦ h restricts to the identity on .D1 . Let .Δ1 be the half-space obtained by taking the closure of −1 .M \ D1 . Then, .h1 = f2 ◦ f1 ◦ h is supported in .Δ1 . Now, note that .h2 = (f2 ◦ f1 ) o is supported in .D3 . Set .Δ2 = D3 . Then, by construction, .∂Δ1 ⊂ Δ2 , .Δ1 ∩ Δ2 contains a half-space (namely .D2 ), .hi is supported in .Δi , and .h = h1 h2 . To finish, let .Δ be a half-space. Then, as .hi is supported in a half-space, it is conjugate to a homeomorphism supported in .Δ by Lemma 5.5.5, and hence .h ∈ 𝚪Δ . ⨆ ⨅

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Proof of Theorem 5.8.1 Let .f ∈ H and let .D be a half-space of .M such that g(D) ∩ D = ∅. Then, by Theorem 5.8.2, .f = h1 h2 , where .h1 , h2 ∈ 𝚪D . By Corollary 5.5.9, each of .h1 and .h2 can be expressed as a commutator of two elements in .𝚪D . By Proposition 5.5.7, each of .h1 and .h2 can be expressed as the product of four conjugates of .g. For the third statement, let . C be C denote the Riemann sphere and let .Σ0 ⊂  obtained by removing .n pairwise-disjoint open disks from . C such that each of the disks has the same radii and such that the center of each disk is an .nth root of unity. Then the rotation .z I→ e2π i/n z restricts to an order .n element .ϕ of .H (Σ0 ). For .i ∈ {1, . . . , n}, let .Σi be a copy of a half-space in .M. We can then construct the surface .M ' = (Σ0 ⨆ Σ1 ⨆ · · · ⨆ Σn )/ ∼, where .∼ identifies .∂Σi with a component of .∂Σ0 via a homeomorphism of the circle in such a way that .ϕ extends to an order .n homeomorphism of .M ' . Observe that .E(M ' ) is a disjoint union of .n copies of .E(M), and so by Proposition 5.4.6, .E(M ' ) is homeomorphic to .E(M), and moreover, .Σi , viewed as a Freudenthanl subsurface of .M ' , is a half-space of .M ' . Therefore, by the classification of surfaces, there exists a homeomorphism .h : M → M ' . To finish, −1 ◦ ϕ ◦ h is an order .n element of .H (M) displacing a half-space of .M, and the .h result follows from (2). ⨆ ⨅ .

As a corollary, we obtain a classical result for which the author is not certain how to attribute. We refer the interested reader to the history discussion in the introduction of [32]. We note that the corollary could have just as easily appeared in Sect. 5.6 as a consequence of Theorem 5.6.11 and Alexander’s trick. Corollary 5.8.3 .H (S2 ) is path connected. Proof By Theorem 5.8.1, .H (S2 ) is simple. But, by Theorem 5.6.11, .H0 (S2 ) is a normal subgroup of .H (S2 ), and hence, they must be equal. Also, by Theorem 5.8.1, 2 2 .H0 (S ) is path connected, and hence so is .H (S ). ⨆ ⨅ We finish the section by proving the corollaries mentioned in Sect. 5.2: Proof of the Purity Theorem, Corollary 5.2.4 Let .N be a proper normal subgroup of .G and let .𝚪 be the stabilizer of the maximally stable ends of .M. By continuity, if .g ∈ G \ 𝚪, then there exists a half-plane .D for which .g(D) ∩ D = ∅ (this follows from the fact that a maximally stable end has a neighborhood basis consisting of half-planes), and hence, by Theorem 5.8.1, every element of .G is a product of conjugates of .g. In particular, as .N is a proper normal subgroup, we must have that N .N < 𝚪. Moreover, the set of maximally stable ends of .M is homeomorphic to .2 (Proposition 5.4.4), and the action of .G on this set is transitive (Proposition 5.4.2). This action induces an isomorphism between .G/ 𝚪 and .Homeo(2N ). It follows that .G/ 𝚪, and hence .G/N , is uncountable. ⨆ ⨅ Proof of Corollary 5.2.6 The proof is identical to the one given in [16, Lemma 2.5]. Every torsion element of .MCG(M) has a finite-order representative in .H (M) by Afton et al. [2, Theorem 2], so it is enough to consider .H (M). Every homeomorphism of .M extends to a homeomorphism of the 2-sphere. This follows from basic properties of end spaces and the associated Freudenthal compactification (we have

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not discussed the Freudenthal compactification in detail: for a self-contained and succinct treatment, see [30, Section 2.1]). It is well known that every finite-order orientation-preserving homeomorphism of the 2-sphere is conjugate to a rotation (for instance, this can be deduced from [27, Theorem 2.8]). Since a nontrivial rotation has exactly two fixed points, we can conclude that any finite-order element of .H (M) does not act trivially on the (infinite) set of maximally stable ends of .M, and hence, by Theorem 5.8.1, must normally generate all of .H (M). ⨆ ⨅

5.9 Strong Distortion Recall that a group .G is strongly distorted if there exists .m ∈ N and a sequence {wn }n∈N ⊂ N such that, given any sequence .{gn }n∈N in .G, there exists a set .S ⊂ G of cardinality at most .m with .gn ∈ S wn for every .n ∈ N. Here, we establish the strong distortion property for homeomorphism groups of perfectly self-similar 2manifolds. We use the proof of Calegari and Freedman [17] for spheres as the outline and adapt where necessary. The strategy can be adapted to a fairly general setting (for instance, see [67, Construction 2.3]) and goes back to the work of Fisher [32].

.

Lemma 5.9.1 Let .M be a perfectly self-similar 2-manifold. If .D1 and .D2 are halfo spaces such that .D1 ∩ D2 contains a half-space and .∂Di ⊂ D3−i for .i ∈ {1, 2}, then every element of .H (M) can be factored as a product of six homeomorphisms, each of which is supported in either .D1 or .D2 . Proof Let .h ∈ H (M). By Theorem 5.8.2, there exists .h1 , h2 ∈ H (M) supported in half-planes .Δ1 , Δ2 ⊂ M, respectively, such that .h = h1 ◦ h2 . Observe that, by assumption, .M = D1o ∪ D2o . Therefore, up to relabelling, we may assume that o ' ' .D \ Δ1 contains a half-space .D ; note that .h1 restricts to the identity on .D . If 1 ' .D \ D2 does not contain a half-space, then there exists a half-space .D contained in o ' ' .D \ (D ∪ D2 ); otherwise, we can choose a half-space .D contained in .D \ D2 . In 1 the latter case, set .f1 to be the identity, and in the former case, using Lemma 5.5.5, choose .f1 ∈ H (M) supported in .D1 such that .f1 (D ' ) = D. Then, in either case, −1 .f1 ◦ h1 ◦ f 1 restricts to the identity on .D. Choose a separating simple closed curve .c such that .c is contained in the interior of .D1 ∩D2 , separates .∂D1 from .∂D2 , and the component of .(D1 ∩D2 )\c containing .∂D1 contains a half-space; let the closure of this component be labelled .T1 . Let .T2 denote the closure of .(M \ D) ∩ D1 , and let .Δ denote the closure of component of .M \ c containing .D. Then, by Proposition 5.4.8 and the classification of surfaces, we have that .D is homeomorphic to .Δ and .T1 is homeomorphic to .T2 (see Fig. 5.3 for a schematic). Therefore, by the change of coordinates principle, there exists an ambient homeomorphism .f2 of .M supported in .D1 that maps .D onto .Δ and .T2 onto .T1 ; in particular, .f2 (M \ D) ⊂ D2 . Let .g1 = f2 ◦ f1 , and note that .supp(g1 ) ⊂ D1 . Then, as .supp(g1 ◦ h1 ◦ g1−1 ) = f2 (supp(f1 ◦ h1 ◦ f1−1 )), .supp(f1 ◦ h1 ◦ f1−1 ) ⊂ M \ D o ,

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∂D2

c

T1

∂D1

D2

D T2

Fig. 5.3 A schematic of the subsurfaces in the proof of Lemma 5.9.1

and .f2 (M \ D) ⊂ D2 , we have .supp(g1 ◦ h1 ◦ g1−1 ) ⊂ D2 . Writing h1 = g1−1 ◦ (g1 ◦ h1 ◦ g1−1 ) ◦ g1 ,

.

factors .h1 as a product of three homeomorphisms, each of which is supported in .D1 or .D2 . Proceeding in a similar fashion for .h2 , we can factor .h as desired. ⨆ ⨅ Theorem 5.9.2 Let .M be a perfectly self-similar 2-manifold. Then, .H (M) is strongly distorted. Proof Let .{hn }n∈N be a sequence in .H (M). Fix half-spaces .D1 and .D2 in .M satisfying the hypotheses of Lemma 5.9.1. Then, applying Lemma 5.9.1, for each .n ∈ N, we can write hn = hn,1 ◦ hn,2 ◦ · · · ◦ hn,6

.

where each .hn,𝓁 is supported in .D1 or .D2 . Let .Δ be a half-space in .M. Consider the following 2-manifold .M ' : Begin with 2 2 .R . Then, for each .i, j ∈ Z, remove the open ball of radius .1/4 in .R centered at .(i, j ). Now for each boundary component of the resulting surface, glue in a copy of .Δ in such a way that the standard .Z2 action on .R2 extends to an action on the resulting manifold, which we call .M ' . Label the copy of .Δ glued in place of the disk centered at .(i, j ) by .Δij . The end space of .M ' is homeomorphic to the one-point compactification of .E(M) × Z2 , where .Z2 is given the discrete topology. Therefore, by Proposition 5.4.6 and the classification of surfaces, .M ' is homeomorphic to .M. Let us identify .M with .M ' . We will now proceed with a modified version of Anderson’s method. Let .ϕ, ψ ∈ H (M) be homeomorphisms such that • .ϕ(Δi,j ) = Δi+1,j for all .i, j ∈ Z, • .ψ(Δ0,j ) = Δ0,j +1 for all .j ∈ Z, and • .ψ restricts to the identity on .Δi,j for all .j ∈ Z and for all .i ∈ Z \ {0}.

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For .k ∈ {1, 2}, choose .ηk ∈ H (M) such that .ηk (Dk ) = Δ0,0 . For .n ∈ Z and 𝓁 ∈ {1, . . . , 6}, set .ηn,𝓁 = η1 if .hn,𝓁 is supported in .D1 ; otherwise set .ηn,𝓁 = η2 . For .𝓁 ∈ {1, . . . , 6}, define .f𝓁 ∈ H (M) by

.

f𝓁 =

.

∞  ∞    (ϕ i ◦ ψ j ◦ ηi,𝓁 ) ◦ hi,𝓁 ◦ (ϕ i ◦ ψ j ◦ ηi,𝓁 )−1 . i=0 j =0

Observe that for .i, j ∈ N ∪ {0}, the product .ϕ i ◦ ψ j ◦ ηi,𝓁 maps one of .D1 or .D2 (depending on .ηi,𝓁 ) onto .Δi,j . Therefore, intuitively, for .i ∈ N∪{0}, . f𝓁 is performing .hi,𝓁 on .Δi,j for all .j ∈ N ∪ {0}. In particular, the support of .f𝓁 is . i,j ∈N∪{0} Δi,j . Now, for .n ∈ N, consider gn,𝓁 = ϕ −n ◦ f𝓁 ◦ ϕ n ,

.

which intuitively translates the support of .f𝓁 so that .gn,𝓁 is performing .hn,𝓁 on .Δ0,j for all .j ∈ N ∪ {0}. Observe that .ψ ◦ gn,𝓁 ◦ ψ −1 is the identity on .D0,0 but otherwise agrees with .gn,𝓁 . Therefore, the same computation as in Propositions 5.5.4 yields −1 hn,𝓁 = ηn,𝓁 ◦ [gn,𝓁 , ψ] ◦ ηn,𝓁 .

.

Let .S ⊂ H (M) be the set consisting of the elements .f1 , . . . , f6 , ϕ, ψ, η1 , η2 , and their inverses. Then, we have shown that for each .n ∈ N and .𝓁 ∈ {1, . . . , 6}, the element .hn,𝓁 can be written as a word in .S of length .4n + 6, and hence .hn can be written as a word of length .24n + 36. In particular, as the sequence .{hn }n∈N was arbitrary, we have that .H (M) is strongly distorted, with .m = 20 and .wn = 24n+36. ⨆ ⨅

5.10 Coarse Boundedness Up to this point, we have mainly considered perfectly self-similar 2-manifolds, but here (and in the next section) we will consider the larger class of self-similar 2manifolds. Recall that a topological group is coarsely bounded if, whenever the group acts continuously by isometries on a metric space, every orbit is bounded. Mann and Rafi [55] proved that the mapping class group of a self-similar 2-manifold is coarsely bounded. With only slight modifications to Mann–Rafi’s proof, we prove this theorem in the setting of homeomorphism groups. Theorem 5.10.1 If .M is a self-similar 2-manifold, then .H (M) and .MCG(M) are coarsely bounded.

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The coarse boundedness of .H (M) implies the coarse boundedness of .MCG(M), so we will only work with .H (M). Before getting to the proof, we require two lemmas and a definition. Lemma 5.10.2 Let .G be a topological group. If for any open neighborhood .V of the identity in .G there exists a finite set .F ⊂ G and .k ∈ N such that .G = (F V )k , then .G is coarsely bounded. ⨆ ⨅ Proof Let .X be a metric space and suppose .G acts on .X continuously by isometries. Fix .x ∈ X and let .ϕ : G → X be the orbit map .g I→ g ·x. Let .B be the ball of radius 1 centered at .x. Then, .V = ϕ −1 (B) is an open neighborhood of the identity in .X. By assumption, there exists a finite set .F ⊂ G and .k ∈ N such that .G = (F V )k , and hence .G · x is bounded. ⨆ ⨅ In the setting of Polish groups, the converse of Lemma 5.10.2 is true as well (see [65, Proposition 2.7]), but we will not require this fact. Definition 5.10.3 A Freudenthal subsurface .Δ of a 2-manifold .M is maximal if .Δ is homeomorphic to .M with an open disk removed. Note that there are no half-spaces in a uniquely self-similar 2-manifold. The above definition is meant to be a weaker condition on a Freudenthal subsurface that is meant to capture some key features of half-spaces. In particular, every half-space is maximal. Also note that every maximal Freudenthal subsurface in a self-similar 2-manifold is stable. Lemma 5.10.4 Let .M be a self-similar 2-manifold. Suppose .Δ ⊂ M is a maximal Freudenthal subsurface and .D ⊂ M \ Δo is a Freudenthal subsurface. Then, there exists .f ∈ H (M) and a maximal Freudenthal subsurface .Δ' contained in the interior of .Δ satisfying: (i) .f (D) ⊂ Δ, (ii) .D ⊂ f (Δ), (iii) .supp(f ) ⊂ M \ Δ' , and Moreover, if .Z ⊂ M \ (D ∪ Δ) is a Freudenthal subsurface, then .f can be chosen to restrict to the identity on .Z. Proof As .Δ is homeomorphic to .M with an open disk removed, there exists .D ' ⊂ Δ homeomorphic to .D. Let .Δ' be a maximal Freudenthal subsurface contained in .Δ \ D ' . Then we can find a simple path .γ in .M \ Δ' connecting .∂D ' and .∂D (if .Z ⊂ M \ (D ∪ Δ) is a Freudenthal subsurface, then .γ can be chosen to be disjoint from .Z). Taking the closure of a small neighborhood of .D ∪ γ ∪ D ' yields a Freudenthal subsurface .N. By the change of coordinates principle, there exists a homeomorphism .f : M → M supported in .N such that .f (D) = D ' and ' .f (D ) = D. It is readily verified that .f is the desired homeomorphism. ⨆ ⨅ Proof of Theorem 5.10.1 Let .H = H (M). Let .V be an open neighborhood of the identity in .H . By possibly shrinking .V , we are guaranteed to find a maximal Freudenthal set .Δ such that .M \Δ is non-orientable if .M is non-orientable and such

5 Homeomorphism Groups of Self-Similar 2-Manifolds

153

that every homeomorphism supported in .Δ is contained in .V . Let .Σ = M \ Δo . Let .f ∈ H be the homeomorphism obtained by inputting .Δ into Lemma 5.10.4 with −1 }. We claim that .H = (F V )5 , which implies that .H is .D = Σ, and set .F = {f, f coarsely bounded by Lemma 5.10.2. Fix .g ∈ H . Note .M = Δ ∪ f (Δ), and so .g(Δ) ∩ Δ or .g(Δ) ∩ f (Δ) must contain a maximal Freudenthal subsurface .Δ1 . By shrinking .Δ1 , we may assume that if .Δ1 intersects .Σ (resp., .f (Σ)), then .Δ1 is contained in .Σ (resp., .f (Σ)). Let .ϕ be a .f (Σ)-translation supported in .Δ, as guaranteed by Lemma 5.5.6, and note that .ϕ ∈ V . Then, up to post-composing .g with .ϕ ◦ f if .Δ1 ⊂ Σ, or post-composing .g with .ϕ if .Δ1 ⊂ f (Σ), we may assume that .Δ1 is disjoint from .Σ ∪ f (Σ); in particular, Δ1 ⊂ g(Δ) ∩ Δ ∩ f (Δ).

.

(5.1)

Under this additional assumption, it is enough to show that .g ∈ (F V )4 as .(ϕ ◦ f )−1 ∈ F V . Apply Lemma 5.10.4 to .Δ1 and .f (Σ) ⊂ M \ Δo1 with .Z = Σ to obtain a homeomorphism .h ∈ H that restricts to the identity on .Σ and satisfies .h(f (Σ)) ⊂ Δ1 ⊂ g(Δ). Note that .h ∈ V as .supp(h) ⊂ Δ. Apply the complement to the containment .h(f (Σ)) ⊂ g(Δ) to see that .g(Σ) ⊂ h(f (Δ)). Moreover, .Σ ⊂ f (Δ) and .Σ ⊂ h(f (Δ)), as .Σ = h(Σ). Hence, Σ ∪ g(Σ) ⊂ h(f (Δ)).

.

(5.2)

As a consequence of .h being given by Lemma 5.10.4, there is a maximal Freudenthal subsurface .Δ2 contained in .Δ1 and fixed by .h. Using that .h(f (Σ)) is disjoint from .Σ, we again apply Lemma 5.10.4, this time to .Δ2 and .Σ ⊂ M \ Δo2 with .Z = h(f (Σ)), to obtain a homeomorphism .σ : M → M such that .σ (Σ) ⊂ Δ2 and such that .σ restricts to the identity on .h(f (Σ)). From this, we know that .h(f (Σ)) is disjoint from .Σ, so together with (5.1), we have σ (Σ) ∩ (h(f (Σ)) ∪ Σ ∪ g(Σ)) = ∅.

.

(5.3)

A schematic for all these sets and their intersection relations can be seen in Fig. 5.4. Applying the change of coordinates principle twice (first sending .g(Σ) to .σ (Σ) and then .σ (Σ) to .Σ), we find .v ∈ H such that .v(g(Σ)) = Σ. Moreover, from (5.2) and (5.3), we have that .h(f (Σ)) is disjoint from .Σ ∪ g(Σ) ∪ σ (Σ), and hence we may assume that .v restricts to the identity on .h(f (Σ)). Additionally, after possibly post composing .v with a double slide (to reverse the orientation of a neighborhood of .∂Σ) and then post composing with a homeomorphism supported in .Σ, we may assume that .v ◦ g restricts to the identity on .Σ, and hence .v ◦ g ∈ V . Observe that as .v restricts to the identity on .h(f (Σ)), we have .(hf )−1 v(hf ) restricts to the identity .Σ, and hence .(hf )−1 v(hf ) ∈ V . It follows that .v ∈ (F V )3 and 4 .g ∈ (F V ) , as desired. ⨆ ⨅

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M − Δ2

h(f (Σ))

M − Δ1 f (Σ)

σ(Σ)

g(Σ) Σ

Fig. 5.4 A schematic of the sets in the Proof of Theorem 5.10.1 with .M = R2

5.11 Rokhlin Property A topological group has the Rokhlin property if it contains an element whose conjugacy class is a dense set in the group. In this section, we will prove the following, which generalizes [46, Theorem 4.1] and the reverse direction of [40, Theorem 0.2] (these are identical statements proved independently) from mapping class groups to homeomorphism groups, and also accounts for non-orientable 2manifolds: Theorem 5.11.1 Let .M be a self-similar 2-manifold. If either .M is the 2-sphere or M is uniquely self-similar, then .H (M) has the Rokhlin property.

.

We note again that the converse of Theorem 5.11.1 is true, that is, the 2sphere is the only perfectly self-similar 2-manifold whose (orientation-preserving) homeomorphism group has the Rokhlin property; we direct the interested reader to [46] and [40]. Before beginning to prove Theorem 5.11.1, we require the following characterization of the Rokhlin property. A topological group .G has the joint embedding property, or JEP, if given any two nonempty open sets .U and .V in .G, there exists g /= ∅, where .V g = {gvg −1 : v ∈ V }. In other words, the .g ∈ G such that .U ∩ V action of .G on itself by conjugation is topologically transitive. We will also use the notation .v g = gvg −1 for .v, g ∈ G. The following lemma is a standard result; for completeness, we include a short proof, which is given in [43, Theorem 2.1] and its following remark. Lemma 5.11.2 A Polish group has the JEP if and only if it has the Rokhlin property.

5 Homeomorphism Groups of Self-Similar 2-Manifolds

155

Proof Let .G be a Polish group and  let .B be a countable basis for its topology. Then, for each .U ∈ B, define .DU = g∈G U g . If .G has the JEP, then .DU is necessarily adense open subset of .G. As an application of the Baire Category Theorem, .D = U ∈B DU is dense, as it is the intersection of countably many dense open subsets. Then, the conjugacy class of an any element of .D must be dense in .G: Indeed, let .h ∈ D. If .V is an open subset of .G, then there exists .U ∈ B such that .U ⊂ V , and, in turn, there exists .g ∈ G such .h ∈ U g ; hence, there is a conjugate of .h in .V . Conversely, if .h ∈ G such that the conjugacy class of .h is dense, then, for any open subsets .U and .V of .G, there exists .g1 , g2 ∈ G such that .hg1 ∈ U and .hg2 ∈ V . In particular, .hg1 = (hg2 )g3 ∈ U ∩ V g3 , where .g3 = g1 g2−1 , and .G has the JEP. ⨅ ⨆ We now proceed towards the proof of Theorem 5.11.1. The proofs that the (orientation-preserving) homeomorphism groups of the 2-sphere and of uniquely self-similar 2-manifolds have the Rokhlin property are similar in nature, but despite the general theme of giving unified proofs in this note, it is more natural in this case to separate out the 2-sphere. Lemma 5.11.3 If .U is an open subset of .H (S2 ), then there exists a closed disk .Δ in .S2 and a homeomorphism .g ∈ U such that such that .supp(g) ⊂ S2 \ Δo and such that .h ◦ g ∈ U for all .h ∈ H (S2 ) supported in .Δ. Proof Fix .f ∈ U . Let .d be the standard metric on .S2 , and let .ρd be the metric on .H (S2 ) defined by .ρd (f1 , f2 ) = max{d(f1 (x), f2 (x)) : x ∈ S2 } (see Corollary 5.6.4). Fix .ϵ ∈ R such that the .(2ϵ)-ball (with respect to .ρd ) in .H (S2 ) centered at .f is contained in .U . As a corollary to the Hairy Ball Theorem, there exists .z ∈ S2 such that .f (z) = z. For .r ∈ R, let .Br denote the closed ball of radius 2 o .r in .S centered at .z. Fix .δ ∈ R such that .δ < ϵ and .f (Bδ ) ⊂ Bϵ . Then, there exists 2 .f1 ∈ H (S ) supported in .Bϵ such that .f1 ◦ f restricts to the identity on .Bδ . Set o 2 .g = f1 ◦ f so .supp(g) ⊂ S \ B . δ In particular, as .f ◦ g −1 = f1−1 is supported in .Bϵ , we have that .ρd (f, g) < 2ϵ and .g ∈ U . Observe that if .h ∈ H (S2 ) is supported in .Bδ , then .f ◦ (h ◦ g)−1 = f1−1 ◦ h−1 is supported in .Bϵ ; hence, .ρd (f, h ◦ g) < 2ϵ and .h ◦ g ∈ U . To finish, set .Δ = Bδ . ⨆ ⨅ Theorem 5.11.4 .H (S2 ) has the Rokhlin property. Proof Let .U1 and .U2 be open subsets of .H (S2 ). For .i ∈ {1, 2}, let .gi ∈ Ui and 2 2 .Δi ⊂ S be obtained by applying Lemma 5.11.3 to .Ui . Fix .σ ∈ H (S ) such that o o σ σ 2 .σ (S \ Δ ) ⊂ Δ . Let .g = g1 ◦ g . Note that .g1 and .g commute as their supports 2 1 2 2 have disjoint interiors, so .g = g2σ ◦ g1 . It follows that .g = g2σ ◦ g1 ∈ U1 as σ 2 −1 ◦ h ◦ .supp(g ) ⊂ Δ1 . Now, if .h ∈ H (S ) is supported in .σ (Δ2 ), then .supp(σ 2 σ ) ⊂ Δ2 and .(σ −1 ◦ h ◦ σ ) ◦ g2 ∈ U2 ; in particular, .h ◦ g2σ ∈ U2σ . Hence, as o σ σ σ 2 .supp(g1 ) ⊂ S \Δ ⊂ σ (Δ2 ), we have .g = g1 ◦g ∈ U . Therefore, .U1 ∩U 1 2 2 2 /= ∅ 2 (as it contains .g), implying .H (S ) has the JEP, and hence the Rokhlin property by Lemma 5.11.2. ⨆ ⨅

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Theorem 5.11.4 was first proved by Glasner and Weiss [35, Theorem 3.4]. Their proof works for all even-dimensional spheres, and we simply note that the proof given above also works in all even dimensions (since the Harry Ball Theorem is true for even-dimensional spheres). The issue in odd dimensions is that the antipodal map is orientation preserving and admits an open neighborhood in which every element is a fixed point free homeomorphism. In particular, the group of orientationpreserving homeomorphisms of an odd-dimensional sphere contains an open set consisting of fixed-point free homeomorphisms and an open set in which every element fixes a point, and since both of these properties are conjugacy invariants, the group cannot have a dense conjugacy class. Let us move to the uniquely self-similar case. We first show that the (normal) subgroup of .H (M) consisting of homeomorphisms supported on the complement of a maximal Freudenthal subsurface in .M is dense. This will allow us to readily establish the JEP property. Lemma 5.11.5 Let .M be a uniquely self-similar 2-manifold. Let .𝚪 be the subgroup of .H (M) consisting of homeomorphisms that restrict to the identity on a maximal Freudenthal subsurface. Then, .𝚪 is dense in .H (M). Proof Let .H = H (M). It is enough to show that each set in a basis for .H intersects 𝚪 nontrivially. So, fix

.

U=

n 

.

U (Ki , Wi ),

i=1

where .K1 , . . . , Kn ⊂ M are compact and .W1 , . . . , Wn ⊂ M are precompact and open. Choose a maximal Freudenthal subsurface .Δ1 disjoint from .Ki and the closure of o .Wi for each .i ∈ {1, . . . , n}. Let .Σ = M \Δ , and let .f ∈ U . If .M is non-orientable, 1 then by possibly enlarging .Σ (and shrinking .Δ1 ), we may assume that .Σ is not orientable. Let .Δ2 be a maximal Freudenthal subsurface disjoint from .Σ ∪ f (Σ). By shrinking .Δ2 , we may assume that the surface co-bounded by .∂Σ and .∂Δ2 is not orientable if .M is not orientable. By Lemma 5.10.4, there exists .f1 ∈ 𝚪 supported in the complement of a maximal Freudenthal subsurface .Δ3 such that .f1 (Σ) ⊂ Δ2 . Then, using the fact that .f1 (Σ) ∩ f (Σ) = ∅, the change of coordinates principle yields a homeomorphism .f2 ∈ 𝚪 supported in the complement of .Δ3 such that −1 ◦ f −1 ◦ f (Σ) = Σ (See Fig. 5.5 for a .f2 (f (Σ)) = f1 (Σ). In particular, .f 1 2 schematic of the sets and their intersection relations). If .M is orientable, we may assume that .f −1 ◦ f2−1 ◦ f1 restricts to the identity on .∂Σ since it preserves orientation; however, if .M is non-orientable, it is possible that .f −1 ◦ f2−1 ◦ f1 reverses the orientation of .∂Σ. If this is the case, then by construction, there exists a double slide .s supported in .M \ Δ3 preserving .Σ and reversing the orientation of .∂Σ so that .f −1 ◦ f2−1 ◦ f1 ◦ s(Σ) restricts to the identity on .∂Σ. Up to replacing .f1 with .f1 ◦ s, we can assume that .f −1 ◦ f2−1 ◦ f1 restricts to the identity on .∂Σ. Therefore, there exists .f3 , f4 ∈ H such that .f3 is supported

5 Homeomorphism Groups of Self-Similar 2-Manifolds

f (Σ)

157

f1 (Σ)

Σ M − Δ2 M − Δ3

Fig. 5.5 A schematic of the subsurfaces in the proof of Lemma 5.11.5 with .M = R2

in .Σ, .f4 is supported in .M \ Σ o , and .f −1 ◦ f2−1 ◦ f1 = f4 ◦ f3 . It follows that −1 −1 .f = f ◦ f4 , and as .f4 restricts to the identity on .Σ, .f ◦ f4 (Ki ) = 2 ◦ f1 ◦ f3 f (Ki ) ⊂ Wi for all .i ∈ {1, . . . , n}; in particular, .f1−1 ◦ f2 ◦ f3−1 = f ◦ f4 ∈ U . The result follows as .f1 , f2 , f3 ∈ 𝚪. ⨆ ⨅ Observe that in the case .M is one-ended (i.e., .M is homeomorphic to either the plane, the Loch Ness monster, or the non-orientable Loch Ness monster), then Lemma 5.11.5 says that the subgroup of compactly supported homeomorphisms is dense in .H (M). The case of the Loch Ness monster was established by Patel and the author in [60] and was a key motivating example for the results in [6] and [46]. Theorem 5.11.6 If .M is a uniquely self-similar 2-manifold, then .H (M) has the Rokhlin property. Proof Let .H = H (M). As before, let .𝚪 be the subgroup of .H consisting of homeomorphisms that restrict to the identity on a maximal Freudenthal subsurface. Let .U1 and .U2 be nonempty open subsets of .H . We must find .g ∈ H such that g .U1 ∩ U /= ∅. Since both .U1 and .U2 must contain basic open sets, we may assume 2 that for .i ∈ {1, 2} Ui =

ni 

.

U (Kji , Wji ),

j =1

where each .Kji is a compact subset of .M and each .Wji is a precompact open subset of .M. Moreover, by Lemma 5.11.5, there exists .fi ∈ 𝚪 such that .fi ∈ Ui . Choose a maximal Freudenthal subsurface .Δ disjoint from the set ni  2   .

i=1 j =1

 i supp(fi ) ∪ Kji ∪ W j .

158

N. G. Vlamis

Let .Σ = M \ Δo . By Lemma 5.10.4, there exists .g ∈ 𝚪 such that .g(Σ) ⊂ Δ. Now, g .f1 is supported in .Σ and .f 2 is supported in .g(Σ); in particular, .f1 restricts to the g 2 identity on .g(Wk ) and .f2 restricts to the identity on .Kj1 for all .k ∈ {1, . . . , n2 } and .j ∈ {1, . . . , n1 }. g g Let .f = f1 ◦ f2 . We claim that .f ∈ U1 ∩ U2 . First observe that g

f (Kj1 ) = (f1 ◦ f2 )(Kj1 ) = f1 (Kj1 ) ⊂ Wj1

.

for all .j ∈ {1, . . . , n1 }, and hence .f ∈ U1 . Second, observe that g

f (g(Kj2 )) = (f1 ◦ f2 )(g(Kj2 )) = f1 (g(f2 (Kj2 ))) ⊂ f1 (g(Wj2 )) = g(Wj2 )

.

g

g

for .j ∈ {1, . . . , n2 }, and hence .f ∈ U2 . Thus, .f ∈ U1 ∩ U2 and .H has the JEP. The result follows from Lemma 5.11.2. ⨆ ⨅

5.12 Automatic Continuity Recall, from the introduction (Definition 5.2.15), that a 2-manifold .M is perfectly tame if it is homeomorphic to the connected sum of a finite-type 2-manifold and finitely many perfectly self-similar 2-manifolds, each of whose space of ends can be written as .P ∪ D, where .P is perfect, .D is a discrete set consisting entirely of planar ends, and .D = P. Now that we have the notions of a stable set and a Freudenthal subsurface at hand, we note that the definition of a perfectly tame 2-manifold is made exactly so that the following lemma holds: Lemma 5.12.1 If .M is a perfectly tame 2-manifold, then (i) .E(M) has a basis consisting of stable sets, and (ii) if .U is a stable clopen subset of .E(M), then either .S(U) is perfect or .U is a singleton consisting of a single planar end. The converse of Lemma 5.12.1 is true, but we will not require this fact. Recall that (1) a subset of a group is countably syndetic if the group is the union of countably many left translates of the set, and (2) a topological group .G is Steinhaus if there exists .m ∈ N such that .W m contains an open neighborhood of the identity in .G for any symmetric countably syndetic set .W in .G. As noted in the introduction, the definition of a perfectly tame 2-manifold is designed so that Mann’s proof in [54] can be extended to the class of perfectly tame 2-manifolds; in particular, we will establish the following theorem: Theorem 5.12.2 The homeomorphism group of a perfectly tame 2-manifold is Steinhaus.

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159

Before beginning we require several lemmas from [54]. The first is a standard result about Polish groups: Lemma 5.12.3 ([54, Lemma 3.1]) Let .G be a Polish group and .W ⊂ G a symmetric set. If .W is countably syndetic, then there exists a neighborhood .U of the identity in .G such that .W 2 is dense in .U . Proof A left translate of .W is meager if and only if .W is meager, and hence, as .G cannot be a countable union of meager sets, we must have that .W is not meager. Therefore, .W is dense is some open subset of .G. It follows that .W 2 is dense in an open subset .U of .G, and as .W is symmetric, .U is a neighborhood of the identity. ⨅ ⨆ We will split Mann’s [54, Lemma 3.2] into two separate lemmas. We say a family of subsets .{Xn }n∈N in a topological space is piece-wise convergent if there exists n n n .k ∈ N such that for each .n ∈ N there exist pairwise-disjoint subsets .Y , Y , . . . , Y k 1 2 n n n n such that .Xn = Y1 ∪ Y2 ∪ · · · ∪ Yk and .{Yi }n∈N is convergent for every .i ∈ {1, . . . , k}. Observe that given a sequence .{fn }n∈N of self-homeomorphisms of a topological space such that .{supp(fn )}n∈N is piece-wise convergent, the infinite  product . n∈N fn exists and is a homeomorphism. Lemma 5.12.4 Let .M be a manifold, and let .{Δi }i∈N be a piece-wise convergent family of pairwise-disjoint closed subsets of .M. Suppose .W is a symmetric  countably syndetic subset of .H (M) and .{gi }i∈N ⊂ H (M) such that .H (M) = i∈N gi W . Then, there exists .k ∈ N such that for each .f ∈ H (M) with .supp(f ) ⊂ Δk there exists an element of .W 2 agreeing with .f on .Δk and supported in the closure of  . i∈N Δi .  Proof Let .Δ denote the closure of . i∈N Δi . We claim that there exists .k ∈ N such that for each .f ∈ H (M) with .supp(f ) ⊂ Δk there exists .wf ∈ gk W supported in .Δ such that .wf |Δk = f |Δk . If not, then, for each .i ∈ N, there exists .fi ∈ H (M) with .supp(fi ) ⊂ Δi such that no element of .gi W supported in .Δ agrees with .f on .Δi . Let .w = i∈N fi , and note that .w is supported in .Δ. Then, by the assumption on .W , there exists .k ∈ N such that .w ∈ gk W . But, .w agrees with .fk on .Δk , a contradiction. Now, let .k be the natural number given by the previous paragraph, let .f ∈ H (M) with .supp(f ) ⊂ Δk , and let .wid and .wf be the homeomorphisms obtained −1 ◦ wf , we see that .w agrees with .f on .Δk and above. Then, letting .w = wid −1 2 .w ∈ (Wg )(g W ) = W . ⨆ ⨅ k k Lemma 5.12.5 Let .M be a manifold, and let .W ⊂ H (M) be a symmetric countably syndetic set. Suppose .A is a family of closed subsets of .M with the following property: there exists a piecewise-convergent collection .{Δi }i∈N of pairwise-disjoint closed sets such that, for each .i ∈ N, there is a piece-wise convergent collection of pairwise-disjoint sets .{Aj }j ∈N such that .Aj ∈ A and .Aj ⊂ Δi for each .j ∈ N. Then, there exists .A ∈ A such that .[a, b] ∈ W 8 whenever .a, b ∈ H (M) with .supp(a), supp(b) ⊂ A.

160

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Proof Let .k ∈ N be obtained by applying Lemma 5.12.4 to the sequence .{Δi }i∈N , and let .Δ = Δk . Let .{Aj }j ∈N be a convergent collection of pairwise-disjoint sets in .A, each of which is contained in .Δ. Apply Lemma 5.12.4 to .{Aj }j ∈N to obtain .m ∈ N such that for each .a ∈ H (M) with .supp(a) ⊂ Am there exists an element of 2 .W agreeing with .a on .Am . Let .A = Am . Let .a, b ∈ H (M) be supported in .A. As .A ⊂ Δ, .a is supported in .Δ and hence there exists .wa ∈ W 2 supported in the closure of . i∈N Δi and whose restriction 2 to .Δ (and hence to .A) agrees with .a. Similarly, there exists .wb ∈ W supported in the closure of . j ∈N Aj and whose restriction to .A agrees with .b. As .supp(wa ) ∩ supp(wb ) ⊂ A, one readily verifies that .[a, b] = [wa , wb ] and hence .[a, b] ∈ W 8 . ⨆ ⨅ The final lemma we require is a tailored version of [54, Proposition 3.3], as [54, Proposition 3.3] does not hold in the infinite-genus setting. We will only offer a sketch of the proof of this lemma as the full details would take us too far afield. Moreover, after a combinatorial setup, the proof will be identical to a part of the proof of Theorem 5.12.2, and hence we postpone the presentation of the sketch of the proof. Lemma 5.12.6 Let .M be a 2-manifold, and let .W be a symmetric countably syndetic subset of .H (M). If .Σ is a finite-type subsurface of .M, then there exists an open subset .U of the identity in .H (M) such that any element of .U supported in 36 . .Σ is an element of .W For the proof of Theorem 5.12.2, we will follow the proof presented by Mann [54] to the extent possible. The main tool we lose is the Fragmentation Lemma as presented in [52, Proposition 2.3] (see Theorem 5.6.9); however, Lemma 5.6.16 will suffice. Another superficial difference to note is that we will dispense with using an auxillary metric on .M and .H (M) and instead work directly with the compact-open topology. Proof of Theorem 5.12.2 If .M is of finite type, then the theorem follows directly from Lemma 5.12.6 by setting .Σ = M, so we may assume that .M is of infinite type. Let .W be a countably syndetic subset of .H (M). By Lemma 5.12.3, .W 2 is dense in some open neighborhood .U0 of the identity in .H (M). As a neighborhood of the identity, .U0 contains a set of the form .U (K1 , V1 ) ∩ · · · ∩ U (Kr , Vr ), where .Ki ⊂ Vi , .Ki is compact, and .Vi is open and precompact for .i ∈ {1, . . . , r}. Fix a finite-type subsurface .F of .M such that the closure of .V1 ∪ · · · ∪ Vr is contained in the interior of .F . Using Lemma 5.12.1, we can enlarge .F so that each component of .M \ F o is a non-compact dividing Freudenthal subsurface. Note that every homeomorphism with support in the complement of .F fixes each of the .Ki and hence is in .U0 . Let .B1 , . . . , Bk denote the components of .M \ F o . For each .i ∈ {1, . . . , k}, choose two disjoint stable Freudenthal subsurfaces .Δi , Δ'i ⊂ Bio each of which is i . For each .i ∈ {1, . . . , k}, let .Ri ' = B i ∪ Δ homeomorphic to .Bi and such that .Δ i ' and .Ri be open annular neighborhoods of .∂Δi and .∂Δ'i , respectively, such that .R i

5 Homeomorphism Groups of Self-Similar 2-Manifolds

B1

161

B2 Δ'1

Δ1

Δ'2

Δ2

Fig. 5.6 A schematic of the subsurface .N, in the proof of Theorem 5.12.2, realized as the complement of the shaded squares in the manifold .M = R2 '

and .R i are disjoint subsurfaces of .M, each of which is also disjoint from .∂Bi . Let

N =F ∪

k 

.



⎡

(Bi \ (Δi ∪ Δ'i )) ∪ ⎣

k 

⎤ (Rj ∪ Rj' )⎦

j =1

i=1

(see Fig. 5.6). Now, let .U1 be the open neighborhood of the identity in .H (M) obtained by setting .Σ = N in Lemma 5.12.6, and let .U2 be the open neighborhood of the identity obtained by applying Lemma 5.6.16 with the open cover o o ' o ' o .{N, Δ , . . . , Δ , (Δ ) , . . . , (Δ ) }. Set .U = U1 ∩ U2 . We now claim that k k 1 1 50 .U ⊂ W . Fix .f ∈ U . Then, since .f ∈ U2 , we have f = g0

k 

.

(gi gi' ),

i=1

where .supp(g0 ) ⊂ N, supp(gi ) ⊂ Δoi , and .supp(gi' ) ⊂ (Δ'i )o . It follows that 36 . .(g0 )|Kj = f |Kj for all .j ∈ {1, . . . , r}, and hence .g0 ∈ U1 . Therefore, .g0 ∈ W     Δi and .Δ' = Δ'i , and let .g = ki=1 gi and .g ' = ki=1 gi' , so Let .Δ = o ' that .supp(g) ⊂ Δ and .supp(g ) ⊂ (Δ' )o . Then, .f = g0 gg ' , and we claim that ' 12 , and hence .U ⊂ W 50 . We will prove the claim for .g (the case for .g ' is .g, g ∈ W identical). Let .A be defined as follows: .A ∈ A if and only if .A = A1 ∪· · ·∪Ak with .Ai ⊂ Δoi a stable Freudenthal subsurface satisfying .S(Ai )∩S(Δi ) /= ∅. Next, choose a stable Freudenthal subsurface .Di contained in .Δoi such that .S(Di ) is a proper subset of .S(Δi ). By Lemma 5.5.6, there exists a convergent .Di -translation .ϕi supported in .Δi . Again using Lemma 5.5.6 and the fact that .S(Di ) is perfect, we have that, for  each .n ∈ N, the set . ki=1 ϕin (Di ) contains a piece-wise convergent collection of pairwise-disjoint sets .{Aj }j ∈N contained in .A. Therefore, .A satisfies the conditions

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Fig. 5.7 The containment relations of the nested collection of stable Freudenthal subsurfaces Ci

Di

Ai

Δi

of Lemma 5.12.5, and hence there exists .A ∈ A such that the commutator of any two homeomorphisms with support in .A is contained in .W 8 . For each .i ∈ {1, . . . , k}, let .Ai = A ∩ Δi , so that .Ai ⊂ Di . Now, let .Ci ⊂ Aoi be a stable Freudenthal subsurface such that .S(Ci ) is a proper subset of .S(Ai ) (See Fig. 5.7 for the containment relations of these sets.) Using that .S(Ci ) is perfect and Lemma 5.5.5, there exists .σi ∈ H (M) supported in .Bi such that .σi (Δi ) ⊂ Ci and 2 .S(Ci ) \ σi (Δi ) /= ∅. Let .σ = σ1 ◦ · · · ◦ σk . Observe that .σ ∈ U0 , and as .W is 2 dense in .U0 , we may assume .σ ∈ W . Since .σ ◦ gi ◦ σ −1 is supported in .Ci , we can apply Anderson’s Method (Proposition 5.5.7), to express .gi as .[αi , βi ] with .supp(αi ), supp(βi ) ⊂ Ai . Now since the .Ai are pairwise disjoint, we have that σ gσ

.

−1

=

k   i=1

σ gi σ

−1



k  k k    [αi , βi ] = αi , βi = [α, β], = i=1

i=1

i=1

  where .α = ki=1 αi and .β = ki=1 βi are supported in .A. Hence, .σ ◦ g ◦ σ −1 ∈ W 8 , and therefore .g ∈ W 12 . ⨆ ⨅ Sketch of Proof of Lemma 5.12.6 The sketch presented here is an adaptation of [52, Section 3] to work in an infinite-type setting. By Lemma 5.12.3, there exists an open neighborhood .U0 of the identity in .H (M) such that .W 2 is dense in .U0 . By shrinking .U0 , we may assume that .U0 = U (K10 , V10 ) ∩ · · · U (Kq0 , Vq0 ) with 0 0 0 0 .K ⊂ V , .K compact, and .V open and precompact. Choose an open subset .F of i i i i .M such that .F is a finite-type subsurface of .M, each component of .∂F is separating, 0 .Σ ⊂ F , and .V i ⊂ F for each .i ∈ {1, . . . , q}. We can realize .F as .N \ X for some compact 2-manifold .N and some finite subset .X of .N . Moreover, by Proposition 5.6.3, .H (N, X) = {f ∈H (N) : f (X)=X} is topologically isomorphic to .H (F ). Now, triangulate .N such that each element of 0 .X is a vertex of some triangle and such that if a triangle intersects .K nontrivially i then it is contained in .Vi0 . Cover .N by a finite collection of open disks .D1 , . . . , Dn such that if .Dj intersects .Ki0 nontrivially then .Dj ⊂ Vi0 and such that the dual graph of the cover (i.e., the graph whose vertices correspond to each open set in the cover and where an edge denotes nontrivial intersection) is 3-colorable. Such

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Fig. 5.8 A 3-colored open cover of a triangle by disks

a cover can be found by first covering the vertices of the triangulation by disjoint disks, then covering the edges in an alternating fashion, and finishing with a single disk for each 2-cell, see Fig. 5.8. Moreover, we may assume that if .x ∈ X is disjoint from the closure of .Σ in .N, then the disk containing .x is disjoint from .Σ. Now, choose open disks .B1 , . . . , Bn such that .B i ⊂ Di and .{B1 , . . . , Bn } remains an open cover of .M. Note that this new cover is also 3-colorable. By the Fragmentation Lemma, Theorem 5.6.9, or arguing as in Proposition 5.6.10, there exists an open neighborhood .U1' of the identity in .H (N) such that every element of ' .U can be factored as .g1 ◦ g2 ◦ · · · ◦ gn with .supp(gi ) ⊂ Bi . Since .|gi ∩ X| ≤ 1, we 1 can conclude that if .g ∈ U1' ∩ H (N, X), then .g = g1 ◦ · · · gn with .supp(gi ) ⊂ Bi and .gi ∈ H (N, X). Let .U1'' be the open neighborhood of the identity in .H (F ) corresponding to the open set .U1' ∩ H (N, X) in .H (N, X). By shrinking .U1'' , we may write .U1'' = U (K1 , V1 )∩· · · U (Kr , Vr ) with .Ki ⊂ Vi , .Ki compact, and .Vi open and precompact. Viewing .F ⊂ M, each .Ki remains compact in .M and each .Vi remains open and precompact in .M. We can then view .U1 = U (K1 , V1 ) ∩ · · · U (Kr , Vr ) as an open neighborhood of the identity in .H (M). Set .U = U0 ∩ U1 . Let .g ∈ U such that .supp(g) ⊂ Σ ⊂ F . Replacing .Di and .Bi with .Di \ X and .Bi \ X, respectively, and viewing these sets in .M, we may write .g = g1 ◦ · · · ◦ gn with .supp(gi ) ⊂ Bi . Note that by the choice of cover, we can ignore that fact that the closure of .F has boundary, as .g restricts to the identity near .∂F . There are two cases to consider: (1) .Bi is an open disk and (2) .Bi is an annulus. For the remainder of the argument, we will assume that only the first case occurs and direct the interested reader to [54, Section 4] for the details of the second case. Using the 3-colorability of the cover, we can write .g = f1 ◦ f2 ◦ f3 where each .fk is a product of homeomorphisms supported on a disjoint union of the .Bi ’s. Proceeding identically as in the last three paragraphs of the proof of Theorem 5.12.2 and using

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the fact that every homeomorphism of .M supported in .Di is an element of .U0 , we see that .fk ∈ W 12 . Therefore, .g ∈ W 36 as desired. ⨆ ⨅

5.13 Commutator Subgroups In this section we prove a slight generalization of a result in [31]. We note that the class that appears in the statement of the theorem appears differently than in [31], but they are in fact equivalent as shown in [31, Lemma A.1]. The proof uses similar constructions that appears in the proof of Theorem 5.12.2. Theorem 5.13.1 Let .M be a 2-manifold obtained by taking the connected sum of a finite-type 2-manifold and finitely many perfectly self-similar 2-manifolds. Then, .[H (M), H (M)] is open in .H (M). Proof Let us write .M = F #M1 # · · · #Mk , where .F is a finite-type 2-manifold and Mj is a perfectly self-similar 2-manifold for .j ∈ {1, . . . , k}. We can then find a finite-type subsurface .Σ of .M such that .Σ is homeomorphic to .F with .k pairwisedisjoint open disks removed and such that .M \Σ has .k components whose closures, labelled .B1 , . . . , Bk , are pairwise disjoint and such that .Bj is homeomorphic to .Mj with an open disk removed. We can then find disjoint dividing Freudenthal subsurfaces .Δj and .Δ'j contained in .Bjo such that the complement of .(Δj ∪ Δ'j )o in .Bj is compact. For each .j ∈ {1, . . . , k}, fix an open annular neighborhood .Rj of .∂Δj and .Rj' of .∂Δ'j , each of

.

'

which is disjoint from .∂Bj , and such that .R j and .R j are subsurfaces. Let

N =Σ∪

k 

.

i=1



⎡

(Bi \ (Δi ∪ Δ'i )) ∪ ⎣

k 

⎤ (Rj ∪ Rj' )⎦ .

j =1

Let .U be the open neighborhood of the identity in .H (M) obtained by applying o o ' o ' o Lemma 5.6.16 with the open  cover .{N,  Δ1 , . . . , Δk , (Δ1 ) , . . . , (Δk ) }. Given .g ∈ k o ' U , we can write .g = g0 i=1 gi gi with .g0 ∈ [H (M), H (M)], .supp(gi ) ⊂ Δi , and .supp(gi' ) ⊂ (Δ'i )o for all .i ∈ {1, . . . , k}. By Proposition 5.5.7, .gi and .gi' can be expressed as commutators of homeomorphisms supported in .Bi for each .i ∈ {1, . . . , k}. Therefore, .U ⊂ [H (M), H (M)], and hence .[H (M), H (M)] is open as it can be written as union of translates of .U . ⨆ ⨅

Acknowledgments The proof of Theorem 5.6.18 is due to Mladen Bestvina, as communicated to the author by Jing Tao. The author thanks George Domat for pointing out that automatic continuity implies the existence of a unique Polish topology; Kathryn Mann for many helpful conversations; Diana Hubbard and Chaitanya Tappu for comments on an earlier draft; and the anonymous readers for their comments. The author is supported by NSF grant DMS–2212922 and PSC-CUNY Award # 64129-00 52. The author was also partially supported by the Faculty Fellowship Publication Program and thanks his cohort for reading a portion of a draft of this chapter.

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Chapter 6

Weil–Petersson Teichmüller Theory of Surfaces of Infinite Conformal Type Eric Schippers and Wolfgang Staubach

Abstract Over the past two decades the theory of the Weil–Petersson metric has been extended to general Teichmüller spaces of infinite type, including for example the universal Teichmüller space. In this chapter we give a survey of the main results in the Weil–Petersson geometry of infinite-dimensional Teichmüller spaces. This includes the rigorous definition of complex Hilbert manifold structures, Kähler geometry and global analysis, and generalizations of the period mapping. We also discuss the motivations of the theory in representation theory and physics beginning in the 1980s. Some examples of the appearance of Weil–Petersson Teichmüller space in other fields such as fluid mechanics and two-dimensional conformal field theory are also provided. Keywords Teichmüller space · Quasisymmetric maps · Quasiconformal maps · Weil-Petersson · Period embedding · Complex structure · Conformal field theory · Bers embedding · Kähler potential · Kähler metric · Universal Teichmüller space 2020 Mathematics Subject Classification Primary: 30F60, 32G15 Secondary: 30C62, 30F30, 30F35, 58J52, 81T40

E. Schippers (O) Department of Mathematics, University of Manitoba, Winnipeg, MB, Canada e-mail: [email protected] W. Staubach Department of Mathematics, Uppsala University, Uppsala, Sweden e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 K. Ohshika, A. Papadopoulos (eds.), In the Tradition of Thurston III, https://doi.org/10.1007/978-3-031-43502-7_6

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6.1 Introduction 6.1.1 History and Motivation We start by recalling the historical origins of the Weil–Petersson scalar product and the Weil–Petersson metric. In analytic number theory and in relation to the theory of automorphic forms, Petersson [64] defined a scalar product on the space of cusp forms .Sk of weight k, as follows. First let .r be a congruence subgroup of .SL(2, Z) and .F any fundamental domain for .r. Let .r¯ = r/{±I}. For the cusp forms .f, g ∈ Sk (r) (i.e. modular forms of weight k with zero constant coefficient in their Fourier series expansions), one defines the Petersson scalar product as 1 ] .(f, g) = [ SL2 (Z) : r¯

ff (Im(z))k f (z)g(z) F

dAz . (Im(z))2

(6.1.1)

dAz k Since the hyperbolic area element . (Im(z)) 2 and .(Im(z)) f (z)g(z) are both .rinvariant, the integral is well-defined and independent of the choice of fundamental domain. This scalar product converges since both f and g are cusp forms and ) ( −2π Im(z) as .Im(z) → ∞. Moreover, only one of the forms .|f (z)| = |g(z)| = O e f or g needs to be a cusp form for the Petersson product to converge. Using Petersson’s scalar product, and in the context of Kodaira–Spencer’s deformation theory of compact complex manifolds, Weil [131, 132], defined an inner product on the space of quadratic differentials on Riemann surfaces as follows. If a point of Teichmüller space is represented by a Riemann surface .R, then the cotangent space at that point can be identified with the space of quadratic differentials at .R. Since the Riemann surface has a natural hyperbolic metric (say in the case that .R has negative Euler characteristic), then one can define a Hermitian inner product on the space of quadratic differentials by integrating over the Riemann surface. This induces a Hermitian inner product on the tangent space to each point of Teichmüller space, and hence a Riemannian metric. Explicitly, let .T (R) be the Teichmüller space of compact Riemann surface .R of genus g, and let .Rt ∈ T (R) be a smooth path through .R := R0 ∈ T (R). The tangent vector .R˙ 0 = .dRt / dt|t=0 can be represented uniquely by a harmonic Beltrami differential

μ = ρ −1 ϕ¯

(6.1.2)

.

where .ρ is the hyperbolic metric and .ϕ ∈ Q(R) is a holomorphic quadratic differential. The Weil–Petersson metric (or WP metric for short) on .T (R) is given by || ||2 ˙ 0 || . ||R

WP

f =

||μ||2WP

=

f |μ| ρ = 2

R

R

|ϕ|2 ρ −1 ,

(6.1.3)

which could be compared with .(f, f ) using the Petersson scalar product (6.1.1).

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Now let us very briefly sketch Thurston’s involvement in Teichmüller theory and the theory of Weil–Petersson metric. For a much more complete account of the Thurston’s work in this as well as in other contexts, the reader is encouraged to read the essay by Ohshika and Papadopoulos [60] or consult the book by Fathi et al. [28]. William Thurston’s interest in Teichmüller theory stemmed from his interest in the theory of surfaces and deformation theory of Riemannian metrics on surfaces. His motivation was to develop a Teichmüller–space theory which was purely based on hyperbolic geometry and did not rely on Teichmüller–Ahlfors–Bers’ quasiconformal approach and the theory of quadratic differentials,1 see e.g. [60]. In accordance to his deformation theory-approach, given a surface R with Euler characteristic χ (R) < 0, one can always endow R with Riemannian metrics with curvature −1. Then one says that two Riemannian metrics g1 and g2 are isotopic if g1 stems from g2 via a diffeomorphism isotopic to the identity. Now if R is compact without boundary, the Teichmüller space T (R) can be defined as the set of isotopy classes of all Riemannian metrics of curvature −1. In connection with his utilization of Teichmüller theory in the theory of surfaces, one of Thurston’s remarkable achievements was the proof of the so-called Nielsen– Thurston classification theorem [123]. This result states that every homeomorphism f from a compact surface S to itself is isotopic to a homeomorphism F with at least one of the following properties: . F has finite order (i.e. some power of F is the identity), . F preserves some finite union of disjoint simple closed curves on R, . F is pseudo-Anosov (a dynamical term introduced by Thurston, see e.g. [123]). Let us see how Teichmüller theory enters the picture. Thurston considered a closed, connected, orientable surface R of genus g ≥ 2 and he realized that T(R), which is by a result of Fricke the space of hyperbolic metrics on R up to isotopy (and is an open ball of dimension 6g − 6), has a compactification that is homeomorphic to a closed ball. Furthermore the boundary of this ball is PMF(R), i.e. the space of projective classes of measured foliations on R. Moreover, the mapping class group of R, denoted by Mod(R), acts on this closed ball. Then, applying the Brouwer fixed point theorem, Thurston concluded that each element of Mod(R) fixes some point of T(R) ∪ PMF(R). Finally the analysis of the various cases for the fixed point, yields the classification theorem. An analytic proof for the Nielsen–Thurston classification theorem, based on Teichmüller theory of extremal quasiconformal mappings was given by Bers in [12]. Thurston also had his own approach to the Weil–Petersson metric, which has resulted in a clear understanding of the geometry on the Teichmüller space of compact surfaces. In this context, and once again motivated by hyperbolic geometry,

1 Indeed since every Teichmüller equivalence class contains infinitely many quasiconformal maps, there is much redundant information in the quasiconformal formulation. This motivates the consideration of extremal quasiconformal maps, though as is well known uniqueness fails in general.

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he introduced a Riemannian metric on Teichmüller space where the scalar product of two tangent vectors at a point represented by a hyperbolic surface is the second derivative of the length of a uniformly distributed sequence of closed geodesics on the surface (or the Hessian of length of a random geodesic). Moreover he introduced in [122] the concept of earthquakes that generalize the Fenchel— Nielsen deformation operation of cutting a hyperbolic surface along a simple closed geodesic and gluing it back with a twist. In [135], Wolpert showed that Thurston’s metric is a multiple of the Weil– Petersson metric. This added yet another component to the understanding of the picture for identification of the geometry on Teichmüller space coming from the hyperbolic surface geometry, for compact surfaces. For a comprehensive overview of many aspects of Teichmüller theory (including the Weil–Petersson geometry) in the compact finite dimensional setting, see Schumacher’s survey [102].

6.1.2 Weil–Petersson Geometry of Surfaces of Infinite Conformal Type This chapter is mainly about the Weil–Petersson geometry of the infinitedimensional quasiconformal Teichmüller spaces. Here, we briefly explain its geometric motivations from a bird’s-eye perspective. A more full picture of recent advances and their historical context is given in the remainder of the chapter. We hope to give an idea of what has compelled many researchers—often looking into Teichmüller theory from the outside—to stir the embers of the Ahlfors–Bers theory. The story begins with an obstacle. On surfaces of infinite type such as the disk, the Weil–Petersson pairing does not converge for arbitrary directions in the Teichmüller space. So to investigate the Riemannian geometry of the Teichmüller spaces of such surfaces, one must remove this obstacle. This requires developing a refinement of Teichmüller space whose tangent spaces are Hilbert spaces with respect to the Weil–Petersson product. Since this is not easy, it is worth first reflecting: why should there be any interest in the infinite-dimensional Teichmüller spaces in the first place (let alone their Weil–Petersson refinements)? After the Ahlfors–Bers period, it is probably safe to say that the focus moved away from quasiconformal Teichmüller theory, perhaps mainly because of the overwhelming force of the insights of Thurston. His largely hyperbolic geometric point of view of Riemann surfaces is in some sense Kleinian. That is, the universal cover is a homogeneous space: the quotient of all isometries of the disk by the stabilizer of a point. The hyperbolic geometry of Riemann surfaces is rigid. For example, a local isometry is determined by its value and its Jacobian matrix at one point. Also, if .f : X → Y is a holomorphic map of hyperbolic Riemann surfaces, then if it is a hyperbolic isometry at a single point it must lift to an isometry of the universal cover

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(see Hubbard [44, Proposition 3.3.4]). Thus hyperbolic geometry of a Riemann surface is essentially finite-dimensional. In contrast, locally conformal transformations are much less rigid—it requires a power series to specify a local biholomorphism fixing a point. Thus the local symmetry group of two-dimensional conformal geometry (complex analysis) is infinitedimensional. One therefore sees that the deformation theory of two-dimensional conformal geometry should also be infinite-dimensional. This is indeed the case: for example, the Teichmüller space of the disk is a Banach manifold.2 The local symmetry group also defies easy description as a Lie group. However, although the local symmetry group is not a Lie group according to the modern definition, it does fit Lie’s original conception of Lie groups as locally defined symmetry groups of PDEs (see the introductions to Pommaret [65] and Vinogradov [126]). In the case at hand, conformal transformations are the orientation-preserving symmetries of Laplace’s equation. It is then no surprise that the outside interest in quasiconformal Teichmüller theory, especially the universal Teichmüller space, has come from areas in which this local point of view cannot be avoided or postponed. These areas include fluid mechanics, infinite-dimensional groups, geometric PDEs, and areas relating to physics, such as two-dimensional conformal field theory, vertex operator algebras, string theory, and geometric quantization. In these fields the Ahlfors–Bers Teichmüller theory makes an appearance. We cannot resist making the following comparison. The imposing genius of Gould has had such overwhelming effect on listeners, that for a long time some of his interpretations of Bach, e.g. of the Goldberg variations and The art of the fugue, have been considered the ultimate renditions of these pieces. Indeed, since the 1950s, Gould’s way of rendering Bach has been viewed as a paradigm–shift in the way of playing Bach’s keyboard music. However one need neither dismiss Gould in order to see the beauty of entirely different paradigms—nor abandon the other paradigms to appreciate his insights. In the above analogy, in place of Gould one could substitute Klein, Ahlfors and Bers, Grothendieck, or Thurston, and in place of Bach one could substitute Riemann. In hindsight the Weil–Petersson theory of infinite type surfaces has a simplicity which fits into many paradigms. Analytic problems are a barrier to fitting infinite-dimensional phenomena into the standard geometric paradigms. For example, infinite-dimensional groups might not be continuous (e.g. left multiplication is not continuous in the universal Teichmüller space). Similarly, representations necessary for creating vector bundles and connections require solution of problems in functional analysis. In the case of Teichmüller theory, analytic advances have now removed this barrier. Returning to the two examples above, by a theorem of Takhtajan and Teo, the Weil–Petersson universal Teichmüller space is a topological group. Also, thanks to a theorem of Vodopyanov and Nag and Sullivan, one obtains a faithful representation of the

2 In fact, by Bers’ trick, Teichmüller spaces—the deformation spaces—are identified with spaces of conformal symmetries.

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universal Teichmüller space as bounded symplectomorphisms of the Sobolev .H 1/2 space of the unit circle. We have attempted in this chapter to give a survey of some of these advances, and provide a road map of the new Weil–Petersson analytic theory. Since it is still somewhat new, we have emphasized giving a literature review over giving the quickest possible introduction to the results. Throughout the chapter, we emphasize how new geometric and algebraic theorems motivate the analytic theory.3 Most importantly, the Weil–Petersson theory makes the Kähler geometry of the moduli spaces of infinite conformal type possible, and thereby also enriches our understanding of geometry, analysis, representation theory, and physics. For Kähler geometry one needs a complex manifold, and so much of the development of the Weil–Petersson Teichmüller theory has concerned the proof of the existence and compatibility of various complex structures. In particular we outline results on three complex structures: analogues of two classical complex structures obtained from the Bers embedding and harmonic Beltrami differentials, and a third complex structure. This third complex structure is in some sense a modification of Bers’ idea, and is motivated by two-dimensional conformal field theory. A fourth complex structure arises from generalizations of the classical period map. The study of complex structures in Teichmüller theory of course has a long history. It is striking how natural the results seem, no matter what angle they are viewed from. Indeed one feels that many of the results, e.g. the generalized period maps and their geometry, could have been motivated to mathematicians a century ago.

6.1.3 Outline We have tried to give a faithful account of the development of the infinitedimensional Weil–Petersson theory. As far as we know no such account has yet been given. The subject was initiated in the 1980s, but after some foundational results were established in the 2000s, the subject began to lift off. Now results are coming in quickly, and no doubt by the time this appears there will be new developments. Whenever possible we of course attempted to use familiar notation, but unfortunately the notation varies wildly in textbooks and papers. To maintain comprehensibility we were forced to be consistent, and it was therefore impossible to adhere to various authors’ choices. Needless to say, all theorems are invariant under change of notation. In Sect. 6.2 we establish notation and basic definitions. Since a large portion of the chapter involves the existence and equivalence of complex structures on the Weil–Petersson Teichmüller space, to set the stage we review three models of the 3 But

this also goes the other way—the analysis has informed the geometry and algebra.

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usual Teichmüller space and give their associated constructions of the complex structure in Sects. 6.3 and 6.4. Two of these models are classical ones, given by the Bers embedding and by harmonic Beltrami differentials. The third is a fiber model of Radnell and Schippers [73] which is motivated by conformal field theory. This model is used in some of the constructions and relates to some of the applications of Weil–Petersson Teichmüller theory. The section concludes with an overview of the differences between the classical .L∞ Teichmüller theory and the .L2 Weil–Petersson Teichmüller theory. In Sect. 6.5 we outline the main results regarding the Weil–Petersson universal Teichmüller space of Cui and Takhtajan–Teo. A description of polarizations and generalizations of the period map into the Siegel disk is given in Sect. 6.6. Because of their importance in the motivation of the Weil–Petersson theory, we include a historical outline. Section 6.7 briefly outlines some other refinements of Teichmüller spaces for the sake of completeness. Section 6.8 outlines the Weil–Petersson theory of general surfaces, due to Yanagishita and Radnell–Schippers–Staubach. Section 6.9 give results on the Kähler theory and global analysis of the Weil– Petersson metric. Section 6.10 gives a brief account of some applications to physics, fluid mechanics, and stochastic Loewner evolution.

6.2 Preliminaries 6.2.1 Surfaces, Borders, and Lifts ¯ denote the complex plane and the Riemann sphere respectively. Let Let .C and .C D+ = {z : |z| < 1} and .D− = {z : |z| > 1} ∪ {∞} denote the unit disk centered at 0 and .∞ respectively. We will denote the boundary of the disk by .S1 , which we assume to be oriented positively with respect to .D+ . In this chapter we consider mostly Riemann surfaces4 .E whose universal cover is the disk, that is, .E = D− /G where G is a Fuchsian group (here the quotient map is a covering map so we do not allow our Fuchsian groups to have fixed points). The limit set of a Fuchsian group G acting on .D+ or .D− is either the entire boundary .S1 or a nowhere dense subset of .S1 , in which case it is called of first or second kind. We will be concerned in this chapter with surfaces generated by Fuchsian groups of the second kind. These are bordered Riemann surfaces (see [6, 50]). If .E ∼ = D− /G for a Fuchsian group G of the second kind, and if B is the complement of the limit set of G in .S1 , then .D− /G ∪ B/G is a bordered Riemann surface, with .D− /G ∼ = E and .B/G the border. Thus every time we mention bordered Riemann surfaces, we are referring to surfaces that are obtained exactly in this way.

.

4 We

use .E to denote a generic bordered Riemann surface. For compact Riemann surfaces we use the notation .R .

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∼ C/G ¯ Such a surface also has a double .E d = so that .D+ /G is identified with the conjugate surface .E ∗ . This has an anti-holomorphic involution i : Ed → Ed

.

which takes .E to .E ∗ and vice versa, and which preserves the border. Surfaces of genus g whose border has n connected components, each of which is homeomorphic to .S1 , will be referred to as type .(g, n) bordered surfaces. We will always assume .n ≥ 1 when using this terminology. For such surfaces, the double is a compact Riemann surface of genus .2g + n − 1, and the border .B/G can be naturally viewed as an analytic curve .∂E = ∂E ∗ in the double.

6.2.2 Differentials Given a Riemann surface .E, an .(r, s)-differential on .E, .(r, s) ∈ Z × Z, is a quantity given in local coordinates by h(z) dzr O d z¯ s , which transforms under a holomorphic change of coordinates z = g(w) via ˜ h(z) dzr O d z¯ s = h(w) dwr O d w¯ s ,

.

s ˜ h(w) = h(g(w))g ' (w)r g ' (w) .

Here O denotes the symmetric tensor product. We shall denote the space of measurable (r, s)-differentials on E by Dr,s (E). The regularity will be imposed when the need arises. As is customary we will write the local coordinates form as h(z)dzr d z¯ s .

.

For example, quadratic differentials are (2, 0)-differentials, and Beltrami differentials are (−1, 1) differentials. It is easily seen that products of differentials are well-defined, and that the product of an (r1 , s1 )- and an (r2 , s2 )-differential is an (r1 + r2 , s1 + s2 )-differential. The hyperbolic metric on E can be viewed as a (1, 1)-differential, which we will denote by ρE , often written locally as ρE = h(z)|dz|2

.

where h(z) is a strictly positive function. Here it must be remembered that |dz|2 stands for dz O d z¯ . For example, the hyperbolic metric on D+ and D− are ρD+ =

.

d z¯ O dz d z¯ O dz , ρD− = (1 − |z|)2 (|z|2 − 1)2

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respectively. The metric induces a hyperbolic area measure given locally by dAhyp = h(z)

.

d z¯ ∧ dz 2i

2 Some sources also write this √ as h(z)|dz| . The line element λE = h(z)|dz| is a convenient object √ which transforms under √ change of coordinates z = g(w) as a density h(z)|dz| = h(g(w))||g ' (w)||dw|. The density can be used to construct an invariant L2 and L∞ norm. For α ∈ Dr,s (E) locally given by g(z) dzr d z¯ s , it is easily checked that |α/λr+s | is a well-defined global function which agrees with the local expressions √ E −r−s h(z) |g(z)|. Denote its essential supremum by ||α||∞ . We then set

L∞ r,s (E) = {α ∈ Dr,s (E) : ||α||∞ < ∞}

.

Similarly we obtain the L2 norm ff ||α||22 =

.

E

|α|2 λ−2r−2s dAhyp E

and the associated Hermitian inner product on L2 differentials, and set L2r,s (E) = {α ∈ Dr,s (E) : ||α||2 < ∞}.

.

For example 2 .L−1,1 (D− )

{ ff := μ ∈ D−1,1 (D− ) :

D−

} |μ(z)|2 dAz < ∞ (|z|2 − 1)2

where dAz denotes Euclidean area measure. If E = D± /G where G is a Fuchsian group, then any (r, s)-differential α has a unique lift αˆ = a(z)dzr d z¯ s to D± , satisfying the invariance a ◦ g · (g ' )r · (g ' )s = a for all g ∈ G.

.

(6.2.1)

2 Denote by L∞ r,s (D± , G) and Lr,s (D± , G) the spaces of differentials which are invariant in the sense of (6.2.1) which are L∞ and L2 respectively on a fundamental domain D ⊂ D± of G. Note that an element of L∞ r,s (D± ) is essentially bounded on D± by the invariance. On the other hand, and element of L2r,s (D± , G) need not be in L2r,s (D± ). We then have a natural isomorphism (directly induced by the covering map)

∼ ∞ L∞ r,s (E) = Lr,s (D− , G)

.

L2r,s (E) ∼ = L2r,s (D− , G)

(6.2.2)

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and ∗ ∼ ∞ L∞ r,s (E ) = Lr,s (D+ , G)

.

L2r,s (E ∗ ) ∼ = L2r,s (D+ , G)

(6.2.3)

Remark 6.2.6 We adhere to the convention that E is covered by D− and its double E ∗ is covered by D+ . The norms and inner products are conformally invariant, as are their straightforward generalizations to Lp spaces. These norms are found throughout complex analysis, Teichmüller theory, and number theory with various names and notation. 2 Finally, we will use the notation A∞ r (E) and Ar (E) for the subspaces of ∞ 2 holomorphic elements of Lr,0 (E) and Lr,0 (E) respectively. For example { ∞ .A2 (D+ )

}

= Q(z)dz holomorphic on D+ : sup (1 − |z| ) |Q(z)| < ∞ 2

2 2

z∈D+

is the familiar space of Nehari–bounded quadratic differentials. Also { ff A22 (D+ ) := Q(z)dz2 holomorphic on D+ :

}

.

D+

(1 − |z|2 )2 |Q(z)|2 dAz < ∞

will play an important role in the Weil–Petersson theory.

6.2.3 Deformations: Quasiconformal Maps and Quasisymmetries A quasiconformal map .f : E1 → E2 between Riemann surfaces (sometimes abbreviated as .f ∈ QC(E1 , E2 )), is an orientation-preserving homeomorphism such that the .(−1, 1)-differential .μ(f ) = ∂f/∂f satisfies ||μ(f )||∞ < 1.

.

∞ Denote the unit ball in .L∞ −1,1 (E) by .L−1,1 (E)1 . Quasiconformal maps are the generic deformations of Riemann surfaces which do not preserve the complex structures. An orientation-preserving homeomorphism h of .S1 is called a quasisymmetric mapping, if and only if there is a constant .k > 0, such that for every .α and every .β not equal to a multiple of .2π , the inequality

| | 1 || h(ei(α+β) ) − h(eiα ) || . ≤| |≤k k | h(eiα ) − h(ei(α−β) ) |

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holds. Let .QS(S1 ) denote the set of quasisymmetric maps from .S1 to .S1 . Every quasiconformal map .0 : D− → D− has a continuous extension to .S1 , and this extension is a quasisymmetry. Conversely every quasisymmetry is the boundary values of a (highly non-unique) quasiconformal map. Let .Möb(S1 ) denote the Möbius transformations which preserve the circle and its orientation. Let .E be a Riemann surface, possessing a border homeomorphic to .S1 which we denote by C. Then there is a biholomorphism .ψ : A → Ar where A is a doubly-connected region in .E bounded on one side by C and .Ar = {z : r < |z| < 1} is an annulus in the plane. Call such a biholomorphism a collar chart. By Carathéodory’s theorem .ψ extends continuously to a homeomorphism from C to one of the boundary curves of .Ar , which we assume to be .S1 . Definition 6.2.2 We say that a map .φ : C → C1 is a quasisymmetry if .ψ1 ◦φ ◦ψ −1 is a quasisymmetry of .S1 for some choice of collar chart .ψ1 and .ψ of .C1 and C respectively. The set of such maps is denoted by .QS(C, C1 ). Given a bordered Riemann surfaces .E, .E1 of type .(g, n), let .QS(∂E, ∂E1 ) denote the set of bijections .φ : ∂E → ∂E1 such that the restriction of .φ to each connected component is a quasisymmetry in the sense above. It is not hard to show that if .ψ1 ◦ φ ◦ ψ −1 is a quasisymmetry for a choice of collar charts .ψ1 and .ψ, then it is a quasisymmetry for all choices.

6.3 Three Models of Teichmüller Space In this section, we give three models of Teichmüller space: in terms of quasiconformal deformations of the surface itself; in terms of conformal deformations on the mirror image; and in terms of conformal deformations of “caps”. The first two pictures are classical, and the last picture originates in conformal field theory. The connection of this model to Teichmüller theory is due to Radnell–Schippers. Within these models, one may describe the Teichmüller space either directly on the surface itself or in terms of objects on the cover which are invariant under the Fuchsian group. We refer to these the direct picture and lifted picture respectively.

6.3.1 Definition of Teichmüller Space In the direct picture, we fix a Riemann surface .E, covered by the disk via a Fuchsian group of the second kind. Let .∂E denote the border (which is possibly empty). Definition 6.3.1 (Teichmüller Equivalence) Given two quasiconformal maps .F1 : E → E1 and .F2 : E → E2 , we say that .(E, F1 , E1 ) and .(E, F2 , E2 ) are Teichmüller equivalent if there is a biholomorphism .σ : E1 → E2 such that −1 .F 2 ◦ σ ◦ F1 is homotopic to the identity rel boundary. Here homotopy rel boundary

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means that .F2−1 ◦ σ ◦ F1 is the identity on the boundary, and this holds throughout the homotopy. The Teichmüller space is then T (E) = {(E, F1 , E1 )}/ ∼

.

where .∼ is the Teichmüller equivalence relation. Given a representative (E, F1 , E1 ), the differential .∂F1 /∂F1 is in .L∞ −1,1 (E)1 . Passing the equivalence relation over to .L∞ (E) we obtain 1 −1,1

.

T (E) ∼ = L∞ −1,1 (E)1 / ∼ .

.

In the lifted model, assume that .E = D− /G where G is a Fuchsian group of the second kind. .F1 , F2 : E → E are homotopic rel boundary if and only if, for the suitably normalized lifts .w1 , w2 to .D− , .w1 = w2 on .S1 [50]. This leads to the following equivalent definition of Teichmüller space. Theorem 6.3.2 The Teichmüller space .T (E) is in one-to-one correspondence with T (G) := {w ∈ QC(D− , D− ) : w fixes −1, i, 1, w◦g◦w −1 ∈ Möb(S1 ) ∀ g ∈ G}/ ∼

.

where the equivalence relation .∼ is defined by .w1 ∼ w2 if .w1 = w2 on .S1 . By (6.2.2) we have that T (G) ∼ = L∞ (D− , G)1 / ∼ .

.

Note that we can identify .T (D− ) with .T ({1}). There is also an important variant known as the quasi-Fuchsian model arising from the so-called Bers trick (see Sect. 6.3.2 ahead), which we will not treat here. Teichmüller space has a topology induced by the Teichmüller metric. Recall that given the points .p1 and .p2 in .T (E), the Teichmüller metric is defined by dT (E) (p1 , p2 ) = inf arctanh ||μF2 ◦F −1 ||∞ .

.

1

(6.3.1)

where the infimum is taken over all representatives .[E, Fk , Ek ] of .pk for .k = 1, 2. This can be also formulated in the lifted picture. For later use, we define here the modular group. Let .Q(E) := QC(E, E) and .Q0 (E) denote the sub-collection of .Q(E) which are homotopic to the identity rel boundary. This is a normal subgroup with respect to composition. The Teichmüller modular group is the quotient Mod(E) = Q(E)/Q0 (E)

.

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which anti-acts on the Teichmüller space via [ρ][E, F, E1 ] |→ [E, F ◦ ρ, E1 ].

.

It is of course well-known that the Riemann moduli space can be identified with T (E)/Mod(E). The following subgroups of the modular group play an important role ahead. .ModI(E) consists of those equivalence classes of .Mod(E) whose representatives are the identity on the boundary. This group is generated by two types of quasiconformal maps. A basis for the homotopy can be given by .∂1 E, . . . , ∂n E, a1 , . . . , ag , b1 , . . . , bg where .a1 , . . . , ag , b1 , . . . , bg are a basis for the homotopy of the compact surface obtained by sewing on caps to obtain a compact surface of genus g. Then .ModI(E) is generated by integer twists around these curves. The group generated by integer twists around .∂1 E, . . . , ∂n E is denoted by .DB(E). The group generated by integer twists around non-trivial simple closed interior curves .a1 , . . . , bg is denoted by .DI(E). Now by the Teichmüller equivalence relation, there is a lot of redundancy in the quasiconformal deformation. Which Beltrami differentials are “essential”? There is no global answer to this question for general surfaces, but there is a local one. Given a Riemann surface .E, define .

O−1,1 (E) = {μ ∈ D−1,1 (E) : ρE μ ∈ A∞ 2 (E)}

.

(6.3.2)

or, in the lifted picture, O−1,1 (D− , G) = {μ ∈ D−1,1 (D− ) : ρE μ ∈ A∞ 2 (D− , G)}.

.

These differentials are transverse to the Teichmüller equivalence relation at the identity. A sufficiently small ball stays within the Teichmüller space. This forms the basis for the deformation model fo the complex structure and tangent space, as we will see ahead.

6.3.2 Bers Embedding Model The underlying idea of this model comes from what is called Bers’ trick. Given a Riemann surface, rather than reflect the quasiconformal map to the double, one obtains a conformal map of the conjugate surface by extending the Beltrami differential by zero. By considering Schwarzian derivatives of these conformal maps, we obtain a model of Teichmüller space by quadratic differentials on the conjugate.

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Given any quasiconformal map .F : E → E1 , let μ=

.

∂F ∂F

be its Beltrami differential. Bers’ trick is to extend .μ by zero to the double .E d , that is, { μd (z) =

.

μz∈E 0 z ∈ E∗.

Solving the Beltrami equation on .E d , one obtains a quasiconformal map F d : E d → E1d

(6.3.3)

.

| where . F d |E ∗ is conformal. To associate a quadratic differential, represent .E as a quotient .D− /G for a ¯ be a lift of .F d . The Schwarzian derivative Fuchsian group G, and let .Fl : D+ → C ( S(Fl ) dz := 2

.

Fl''' 3 − Fl' 2

(

Fl'' Fl'

)2 ) dz2

∞ ∗ is in .A∞ 2 (D+ , G) and thus gives a well-defined element of .A2 (E ). The resulting map ∗ β : T (E) → A∞ 2 (E )

.

is called the Bers embedding. The Bers embedding depends on the base surface .E. When it is necessary to emphasize this, we will write .βE , or .βG in the lifted picture when .E ∼ = D− /G. The importance of the Bers embedding stems from the following fact. Theorem 6.3.3 .β is well-defined and injective.

6.3.3 Caps Fiber Model In this section, we consider bordered surfaces of genus g with n borders homeomorphic to .S1 ; recall that we refer to these as type .(g, n) bordered surfaces. Radnell and the first author showed that the Teichmüller space of such surfaces fibers over the Teichmüller space of compact surfaces with n punctures, and the fibers can be identified with a collection of n-tuples of conformal maps into the surface [73]. The idea is motivated by conformal field theory (see Sect. 6.10.1 ahead). This

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fibration features in one model of general Weil–Petersson Teichmüller spaces, and also relates to some of the physical motivation for the Weil–Petersson theory. The idea is as follows. Given a type .(g, n) Riemann surface .E, instead of sewing on the conjugate surface .E ∗ to obtain the double .E d , one sews a punctured disk to each boundary curve to obtain a compact surface .E P with n punctures. Instead of considering conformal maps of .E ∗ as in the Bers embedding, we consider conformal maps of the sewn disks. We will see that variations in Teichmüller space P .T (E) are spanned by variations in .T (E ) together with variations obtained by changing these conformal maps (equivalently, by changing the boundary values of the quasiconformal deformation of .E). The fibres are locally modelled by powers of the following space: Oqc = {f : D+ → C : f holomorphic and quasiconformally extendible}.

.

We then consider n-tuples of quasiconformally extendible maps into a Riemann surface Definition 6.3.4 Let .E P be a compact Riemann surface of genus g with n punctures .p1 , . . . , pn . The class .Oqc (E P ) is the set of n-tuples of maps .(f1 , . . . , fn ) where (1) .fk : D+ → E P are holomorphic and quasiconformally extendible to an open neighbourhood of the closure of .D+ for .k = 1, . . . , n; (2) .fk (0) = pk for .k = 1, . . . , n; (3) the closures of .fk (D+ ) are pairwise disjoint. To obtain a fibration of .T (E) over .T (E P ), we sew on copies of the punctured disk (“caps”). Fix a bordered surface .E of type .(g, n) and quasisymmetric parametrizations .τk : S1 → ∂k E for .k = 1, . . . , n. These can be chosen to be analytic if desired. We sew on n copies of the disk .D+ , by identifying .p ∈ S1 with P whose complex structure is the .τ (p) ∈ ∂k E. The result is a compact surface .E unique one compatible with .R and the copies of .D+ . The parametrizations .τk then extend to conformal maps .D+ → E P (the inclusion maps into the sewn surface), and we specify the punctures .pk = τk (0) for .k = 1, . . . , n. The fibre map from .T (E) to .T (E P ) is now defined as follows, assuming that ˆ : E P → E P be a quasiconformal .2g − 2 + n > 0. Given .[E, F, E1 ] ∈ T (E), let .F 1 map such that

.

∂ Fˆ (z) = ∂ Fˆ (z)

{

∂F (z)/∂F (z) z ∈ E 0 z ∈ τk (D+ )

Define C : T (E) → T (E P )

.

[E, F, E1 ] |→ [E P , Fˆ , E1P ]. Note that .C depends on .τ1 , . . . , τn . Later we will see that .C is holomorphic.

(6.3.4)

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Remark 6.3.5 One may think of the definition of .Fˆ as a natural variation on “Bers’ trick”. That is, .Fˆ plays the same role as (6.3.3). The distinction is that for .Fˆ one sews on caps, while for .F d one sews on the double. Fix any .q = [E P , F, E1P ]. Up to the action of .DB, the fibres of .C can be identified with .Oqc (E1P ). Explicitly, .f = (f1 , . . . , fn ) ∈ Oqc (E1P ), let .E1 be obtained by removing the disks .fk (D+ ). Let Ff' : E P → E1P

.

be a quasiconformal map which is homotopic to . F |E and such that .F ◦ τk = fk . (It can be shown that such a map exists [73].) Set | | Ff = Ff' | : E → E1

.

E

(6.3.5)

Then .[E, F, E1 ] is in the fibre .C −1 (q), and any two such elements are related by an element of the group .DB. We then define Fq : Oqc (E1P ) → C −1 (q)/DB

.

f |→ [E, Ff , E1 ].

(6.3.6)

Theorem 6.3.6 ([73]) Let .E be a bordered surface of type .(g, n) with .2g − 2 + n > 0, and let .C, .Fq , etc. be as above. For any point .q = [E P , F, E1P ] ∈ T (E P ), .Fq is a bijection. Remark 6.3.7 Since we assume that .n ≥ 1, the restriction that .2g − 2 + n > 0 only rules out the following two cases: .g = 0, n = 1 (the disk) and .g = 0, n = 2 (the annulus). These cases are exceptional because when disks are sewn on, the Teichmüller spaces of the resulting punctured surfaces, the once and twice punctured sphere respectively, each reduce to a point. The fiber model must therefore be adjusted slightly. The first case is just the disk, and the construction above reduces to one of the standard models of the universal Teichmüller space (cf Remark 6.3.5). Explicitly, ¯ ¯ letting .E P = C\{0}, we have .T (D− ) ∼ where G are Möbius = Oqc (C\{0})/G transformations fixing 0 (equivalently, one may apply two further normalizations ¯ to .Oqc (C\{0})). A similar fiber model occurs in the case of an annulus .A; one can identify qc ¯ .T (A)/Z with elements of .O (C\{0, ∞}) with one further normalization. This is called the Neretin–Segal semigroup, see [74].

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6.3.4 Right Translation and Change of Base Point A quasiconformal map .Fμ : E → Eμ induces a bijection between Teichmüller spaces .T (E) and .T (Eμ ). Here we follow a common notational convention which uses the Beltrami differential of F simultaneously as a label. Namely, we have RFμ−1 : T (E) → T (Eμ )

.

[E, F, E1 ] |→ [Eμ , F ◦ Fμ−1 , E1 ]. In the lifted picture, the change of base-point takes the following form. Given a quasiconformal map .wμ : D− → D− fixing .−1, i and 1, we get Rwμ−1 : T (G) → T (Gμ )

.

[w] |→ [w ◦ wμ−1 ]. Here, Gμ = {wμ ◦ g ◦ wμ−1 : g ∈ G}.

.

(6.3.7)

The Beltrami differential of .wμ ◦ g ◦ wμ−1 is zero; thus it is in .Möb(S1 ), and independent of the representative .μ. Change of base-point is a homeomorphism; indeed, when complex structures are introduced, it is a biholomorphism.

6.4 Complex Manifold and Tangent Space Structure 6.4.1 Summary of the L∞ Theory 6.4.1.1

Bers Embedding

The nature of the Bers embedding was studied for many years, and eventually understood to be a homeomorphism onto an open set. In summary, ∗ Theorem 6.4.1 The image of .β is an open set in .A∞ 2 (E ). Furthermore .β is a homeomorphism from .T (E) .(with the topology induced by the Teichmüller distance.) onto its image.

In other words, every element of the Teichmüller space can be represented by a unique quadratic differential on .E ∗ (equivalently, by a unique quadratic differential ∞ ∗ in .A∞ 2 (D+ , G)). Since the image is open and .A2 (E ) is a Banach space, this makes .T (E) a complex Banach manifold with a global coordinate.

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Let 0 : L∞ −1,1 (E)1 → T (E)

.

denote the map obtained by solving the Beltrami equation. This map depends on the base surface .E; when it is necessary to emphasize this fact we will write .0E , or .0G in the lifted picture when .E = D− /G. We will see that .0 is a submersion, that is, it has local holomorphic sections. Theorem 6.4.2 The map .β ◦ 0 is a holomorphic submersion. This is actually at the root of the deformation model.

6.4.1.2

Deformation Model

The complex structure in the deformation model is more involved, because the Beltrami differentials contain redundant information, which the Teichmüller equivalence relation removes. We also see this in the description of the tangent space in Sect. 6.4.2 ahead. At the root of the classical approach is the Ahlfors–Weill reflection. The abstract geometric definition of this reflection is quite simple: ∗ ∞ AE : A∞ 2 (E ) → L−1,1 (E)

.

1 −1 ∗ α |→ − ρE i α. 2

(6.4.1)

In the lifted picture, using the fact that the lift of i is the map .z |→ 1/¯z, this can be written explicitly as follows: ∞ AD− : A∞ 2 (D+ ) → L−1,1 (D− )

.

Q(z)dz2 |→ −

1 1 Q(1/¯z) d z¯ . 2 z¯ 4 (|z|2 − 1)2 dz

(6.4.2)

It is easily checked that .AD− preserves G-invariance. Thus we can define | AG = AD− |A

.

2 (D+ ,G)

∞ : A∞ 2 (D+ , G) → L−1,1 (D− , G)

so that .AG is the lift of .AE when .E is .D− /G. Recalling the definition of harmonic Beltrami differentials (6.3.2) we have ∗ O−1,1 (E) = A(A∞ 2 (E ))

.

O−1,1 (D− ; G) = A(A∞ 2 (D+ ; G)).

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An alternate complex structure in terms of Beltrami differentials is provided by the following. Let 0 E : L∞ −1,1 (E) → T (E)

.

denote the map obtained by solving the Beltrami equation (equivalently, quotienting by the Teichmüller equivalence relation). Denote the lifted map by .0G . The Ahlfors–Weill map is a local inverse of the Bers embedding near zero; namely, there ∗ is an open neighbourhood .UE ⊆ A∞ 2 (E ) of 0 (in fact, the ball of radius 2 centred on 0) such that .βE 0E AE = Id on .UE . Similarly, .0G AG is a local inverse of .βG on an open set .UG in .A∞ 2 (E). Together with the translation maps, we obtain a complex atlas. Theorem 6.4.3 Let .Fμ : E → Eμ . The open sets .RFμ 0Eμ AEμ (UEμ ) ⊂ T (E) and charts .βEμ RFμ−1 form a complex atlas of .T (E). In the lifted picture, the open sets .Rwμ 0G AGμ (UGμ ) and charts .βGμ Rwμ−1 form a complex atlas of .T (G). In other words, we can use the fact that .0Eμ AEμ is a biholomorphism to obtain a chart near .[E, Fμ , Eμ ]. Theorem 6.4.4 Let .E be a Riemann surface covered by the disk. (1) The two complex structures are compatible. In particular, the Bers embedding is a biholomorphism onto its image, with respect to the complex structure on .T (E) induced by the charts. (2) .β0 is a holomorphic submersion. In particular, the complex structure of ∞ (E) and that of .T (E) are compatible. .L −1,1 (3) Change of base point is a biholomorphism.

6.4.1.3

Fiber Model

We outline the basic results about the holomorphic fibration here for the case of classical .L∞ Teichmüller theory. The local model of the conformal maps is given by Oqc = {f : D+ → C : f holomorphic and quasiconformally extendible}.

.

It follows from classical function theory that χ : Oqc |→ A∞ 1 (D+ ) ⊕ C ( '' ) f ' f |→ dz, f (0) f'

.

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|| || is bounded with respect to the direct sum norm .|f ' (0)| + ||f '' /f ' dz||∞ . Furthermore, the image is open (see [72]) and thus .Oqc is a complex Banach manifold. We construct a coordinate chart in .Oqc (E P ) as follows. Let .ζ = (ζ1 , . . . , ζn ) be coordinates .ζk : Bk → C of .pk for .k = 1, . . . , n. Let .K = (K1 , . . . , Kn ) be compact sets in .Bk whose interiors are open sets containing .pk . Define Vζ,K = {f = (f1 , . . . , fn ) ∈ Oqc (E P ) : cl( fk (D+ )) ⊂ Kk , k = 1, . . . , n}. (6.4.3)

.

We then define coordinates E : Vζ,K → (Oqc )n

.

f |→ ζ ◦ f.

(6.4.4)

Using the coordinates .χ n ◦ E we obtain the following. qc P Theorem On ∞ 6.4.5 ([72]) .O (E ) is a Banach manifold locally modelled on . A1 (D+ ) ⊕ C.

The complex structure on .Oqc (E P ) is compatible with that induced by the fibration .C : T (E) → T (E P ) a complex structure compatible with that of .T (E), by the following theorem: Theorem 6.4.6 ([73]) Let .E be a bordered surface of type .(g, n) such that .2g − 2 + n > 0. (1) The map .C is holomorphic and has local holomorphic sections. In particular the fibre .C −1 (q) is a complex submanifold of .T (E) for any .q ∈ T (E P ). (2) For any point .q = [E P , F, E1P ] ∈ T (E P ), .Fq given by (6.3.6), is a biholomorphism. Using this, it is possible to obtain holomorphic coordinates on .T (E). Let .T , Vζ,K be coordinates on .Oqc (E1P ) as above, and set .Uζ,K = T (Vζ,K ). By an adaptation of Schiffer variation developed by Gardiner, there is an open set .U ⊆ Cd containing 0 where d is the dimension of .T (E P ), and a biholomorphism U → T (E P )

.

e |→ [E P , νe ◦ F, νe (E1P )] where .νe is conformal on the domain of each chart .ζi . Now, given any point .p ∈ C −1 (q), it can be written .[E, Ff , E1 ] ∈ T (E) for some .f ∈ Oqc (E P ) (since .Fq is a bijection by Theorem 6.3.6, one need only adjust by an element of .DB). One then defines G : U × Vζ,K → T (E)

.

(e, f ) |→ [E, νe ◦ Ff , νe (E1 )]. This is a local biholomorphism.

(6.4.5)

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Theorem 6.4.7 ([73]) Let .E be a bordered surface of type .(g, n) such that .2g − 2 + n > 0. .G is a biholomorphism onto its image. In particular, the collection of maps of the form .G −1 forms an atlas of a complex structure on .T (E) compatible with the standard complex structure. n By composing the second with ] .χ ◦ E, we can get charts directly to On [ component ∞ d the Banach space .C ⊕ A1 (D+ ) ⊕ C , if desired.

Remark 6.4.8 Although the notation is involved, the idea is quite simple. The Teichmüller space of .E decomposes locally into deformations of the capped surface P and the transverse deformations of the boundary values of the quasiconformal .E maps of .E. The variations of the boundary values can be modelled by conformal maps into .E P , by a modification of Bers’ trick. Remark 6.4.9 We saw in Remark 6.3.7 that the exceptional cases are .E = D− and .E = A where .A is an annulus. Tracing this through in the case of the disk just recovers the standard model in terms of the pre-Schwarzian version of the Bers embedding. Versions of Theorems 6.4.6 and 6.4.7 were given for .T (A) in [74]. Remark 6.4.10 The holomorphicity of .Fq is [73, Corollary 3.3] while the biholomorphicity of .G is [73, Theorem 4.1], with rather different notation. Remark 6.4.11 The idea for these coordinates is due to Radnell [70].

6.4.2 The Tangent Space in the Three Models Deformations Model There are many quasiconformal deformations in a given equivalence class. Thus if we are to use .L∞ −1,1 (E) in modelling the tangent space to the Teichmüller space, we must quotient out directions tangent to .L∞ −1,1 (E). These redundant directions form what is called the space of “infinitesimally trivial” differentials . Namely, define N (E) = KerD0 (βE ◦ 0E ).

.

∞ ∗ D0 (βE ◦ 0) : L∞ −1,1 → A2 (E )

.

(6.4.6)

denotes the derivative of .βE ◦ 0E at the base point .[0] = [E, Id, E]. Thus .N (E) is the set of directions tangent to the equivalence relation, by the fact that the Bers embedding is a homeomorphism onto its image. It is a fundamental result that L∞ −1,1 (E) = N (E) ⊕ O−1,1 (E).

.

(6.4.7)

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Thus if we let .[0] = [E, Id, E] denote the “origin” of Teichmüller space, the tangent space at .[0] is T[0] T (E) ∼ = O−1,1 (E).

.

The tangent space at an arbitrary point .q = [E, F, E ' ] can be obtained using right translation. Since R[F ] [E ' , Id, E ' ] = [E, F, E ' ]

.

the derivative satisfies D0 R[F ] : T[0] T (E ' ) → Tq T (E).

.

Since .O−1,1 (E ' ) models .T[0] T (E ' ) we obtain Tq T (E) ∼ = R[F ] O−1,1 (E ' ).

.

(6.4.8)

One can view the coordinates established in Sect. 6.4.1.2 as local exponentiations of this model of the tangent space. Bers Embedding Model Since the Bers embedding is a bijection from T (E) to ∞ ∗ ∗ an open subset of A∞ 2 (E ), there are no redundant directions in A2 (E ). Thus the ∞ ∗ tangent space at [0] can identified with A2 (E ), that is ∗ T[0] T (E) ∼ = A∞ 2 (E ).

.

∗ Since A∞ 2 (E ) is a linear space, we also have ∗ ∼ ∞ ∗ Tq T (E) ∼ = Tβ(q) A∞ 2 (E ) = A2 (E ).

.

Fiber Model We remark here that it is possible to get a model of the tangent space using the fibration. This will not be used elsewhere in this chapter. Let p = [E, F, E1 ] and q = C(p) = [E P , Fˆ , E1P ]. By Theorem 6.4.7, using the linearity of A∞ 1 (D+ ), .T[E,F,E1 ] T (E) ∼ = Tq T (E P ) ×

n O ( ∞ ) A1 (D+ ) ⊕ C k=1

via the derivative map of G. Since Tq T (E P ) is finite dimensional, it poses no analytic difficulties. Under G it is identified with Cd using Schiffer variation, but any model of the tangent space could be used.

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6.4.3 Weil–Petersson Teichmüller Space in a Nutshell In each of the models given in Sect. 6.3, an analytic choice leads to the complex manifold structure on Teichmüller space. The Weil–Petersson Teichmüller space is, roughly, obtained by replacing the .L∞ objects by .L2 objects. For example, in the Weil–Petersson theory the space of .L2 harmonic Beltrami differentials { } H−1,1 (E) = μ ∈ D−1,1 (E) : ρE μ ∈ A22 (E)

.

plays the role of .O−1,1 (E). Classical Beltrami differentials

∞ .L−1,1 (E)1

Deformation model coord.

.O−1,1 (E)

Bers embedding model coord.

∞ ∗ .A2 (E ) ∞ (D

Fiber model coord.

.(A1

+ ) ⊕ C)

n

Weil–Petersson ∞ .L−1,1 (E)1

∩ L2−1,1 (E)

.H−1,1 (E)

2

.A2 (E

× T (E P )

2

.(A1 (D+ )

∗)

⊕ C)n × T (E P )

The Weil–Petersson inner product of the tangent space at .[0] = [E, Id, E] is as follows. In the Bers embedding model, we will see ahead that the tangent space can be identified with .A22 (E ∗ ). In this model, the pairing is just the inner product induced by that in .A22 (E ∗ ): ff .

WP,0 := A2 (E ∗ ) = 2

E∗

−2 ρE ∗ φ1 φ2 dAhyp .

In the lifted picture we have, for .h1 (z)dz2 , h2 (z)dz2 ∈ A22 (D+ , G) / .

h1 (z)dz2 , h2 (z)dz2

\ WP,0

/ \ := h1 (z)dz2 , h2 (z)dz2

A22 (D+ ,G)

ff =

(1 − |z|2 )2 h1 (z)h2 (z) N

d z¯ ∧ dz 2i

where N is a fundamental domain of G. In the deformation model, the tangent space is modelled by harmonic Beltrami differentials. In that case, for .μ1 , μ2 ∈ H−1,1 (E), the Weil–Petersson pairing is ff .

WP,0 =

μ1 μ2 dAhyp . E

It is easily checked that up to a constant, under the Ahlfors-Weill reflection .A, this agrees with the pairing on quadratic differentials.

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Before this can be made sense of, we first need to define the Weil–Petersson Teichmüller space for some class of surfaces .E, and show that it has a Hilbert manifold complex structure. The existence and equivalence of the various complex structures on Teichmüller space was an important research question, with origins in Riemann’s work (Ji and Papadopoulos [45]). In the finite-dimensional and classical Ahlfors–Bers theory this encompasses many important results. The difficulties are no less in the Weil–Petersson theory. Remark 6.4.12 The construction of the Weil–Petersson Teichmüller space and its equivalent complex structures requires repeating all the analysis in the classical construction, but in the .L2 case. Although there are quite a few surprises, there is also a long list of “expected” theorems, some of which have technical proofs. In the initial stages, the value of the endeavour was perhaps not as obvious as it is in hindsight.

6.5 The Weil–Petersson Universal Teichmüller Space 6.5.1 History and Overview The early motivations for the Weil–Petersson Teichmüller space came from investigations of the group .Diff(S1 ) of diffeomorphisms of the circle. This group plays and important role in loop groups, representation theory, string theory, and conformal field theory [19, 20, 46, 63, 68, 100, 134]. Its coadjoint orbits have been investigated in connection with representation theory by Kirillov and Yuri’ev [47, 48]. There are two coadjoint orbits, .Diff(S1 )/Möb(S1 ) and .Diff(S1 )/S1 (where .S1 acts by the subgroup of rotations in .Möb(S1 )). Kirillov–Yuri’ev identified the symplectic forms on these orbits arising from the Kirillov–Kostant–Souriau theory [47, 48]. The tangent space at the identity of .Diff(S1 ) consists of the smooth vector fields on the circle .Vect(S1 ), generated by .Ln = −ieinθ ∂/∂θ , .n ∈ Z\{−1, 0, 1}. There is a natural complex structure J given by J Lm = −i sgn(m)Lm .

.

The unique homogeneous symplectic form on .Diff(S1 )/Möb(S1 ) is given at the identity by ω(Lm , Ln ) = a(m3 − m)δm+n,0

.

for some real constant a (see Kirillov and Yuriev [48], Kirillov [47], and Bowick and Lahiri [18]). There is thus a unique Kähler metric compatible with the complex structure J . Motivated by the work of Bowick [17] and Bowick and Rajeev [19], Nag and Verjovsky [59] observed the connection of .Diff(S1 )/Möb(S1 ) with the universal Teichmüller space. Using the deformation model of the tangent space, they showed

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that the Kähler metric agrees with the formal expression of the Weil–Petersson pairing of Beltrami differentials in .T (D− ), and that this pairing therefore converges at points in .Diff(S1 )/Möb(S1 ) ⊂ T (D− ) on directions tangent to .Diff(S1 )/Möb(S1 ). Furthermore, they showed that the inclusion of .Diff(S1 )/Möb(S1 ) in .T (D− ) is Gâteaux holomorphic. Setting geometry and algebra aside for now, the question which immediately leaps to mind is: what is the completion of .Diff(S1 )/Möb(S1 ) with respect to the Weil–Petersson pairing of Nag–Verjovsky? Since .T (D− ) is a Banach manifold, we can certainly not expect it to have a Kähler metric, and indeed the Weil–Petersson pairing does not converge in arbitrary directions tangent to .T (D− ) even at the identity. Thus one is led to find a subset or refinement of the universal Teichmüller space, such that the Weil–Petersson pairing is convergent on every tangent space. A number of researchers have developed this “Weil–Petersson universal Teichmüller space”. In this section, we describe the manifold and tangent space properties, which were developed by Cui [22] and Takhtajan and Teo [116]. The seminal paper on the Weil–Petersson Teichmüller space was that of Cui [22]. Cui defined Weil–Petersson class quasisymmetries (calling them “integrably asymptotically affine homeomorphisms”), constructed a Weil–Petersson universal Teichmüller space with a complex structure obtained from a holomorphic Bers embedding, and showed that it is the completion of .Diff(S1 )/Möb(S1 ). This completely established the Bers embedding model of the WP universal Teichmüller space. It was also shown that the WP universal Teichmüller space is complete with respect to the induced geodesic distance. Some of these results were later proven independently by Takhtajan and Teo [114], later published as the first of two chapters of [116]. They also constructed the complex structure from the deformation model, and thoroughly developed the tangent space structure. Moreover they showed that a natural infinite-dimensional analogue of the period mapping into the Siegel disk is holomorphic both in the classical and Weil–Petersson settings. This infinite-dimensional analogue was given by Segal [103], and taken up by many authors including Kirillov and Yuri’ev [48], Nag [57], Nag and Sullivan [58] and Hong and Rajeev [41]. This is described in Sect. 6.6.1 ahead. Takhtajan–Teo also established the first results on the Kähler geometry and global analysis of the WP universal Teichmüller space. This topic is deferred to Sect. 6.9.

6.5.2 Weil–Petersson Universal Teichmüller Space 6.5.2.1

Bers Embedding Model

A quasisymmetry .φ ∈ QS(S1 ) is said to be a Weil–Petersson quasisymmetry if it has a quasiconformal extension to .D− with Beltrami differential in .L2−1,1 (D− ). Denote the class of such quasisymmetries by .QSWP (S1 ). We then consider the following subset of Teichmüller space.

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Definition 6.5.1 The .WP-class universal Teichmüller space is defined to be TWP (D− ) := QSWP (S1 )/Möb(S1 ).

.

Cui obtained a characterization of Weil–Petersson quasisymmetries in terms of the Schwarzian and the pre-Schwarzian derivative. Recall that given a .φ with quasiconformal extension F , there is a corresponding map .Fˆ obtained from Bers’ trick, such that | ˆ || .f := F (6.5.1) D+

is conformal and independent of the choice of extension (up to a normalization). Define the pre-Schwarzian and Schwarzian derivatives A(f ) =

.

f '' , f'

S(f ) =

f ''' 3 − f' 2

(

f '' f'

)2 .

We have Theorem 6.5.2 ([22, 116]) The following are equivalent: (1) .φ : S1 → S1 is a .WP-class quasisymmetry; (2) f satisfies .S(f )(z) dz2 ∈ A22 (D+ ); (3) if f is normalized so that .∞ is in the interior of the complement of .f (D+ ) then 2 '' ' .(f (z)/f (z)) dz ∈ A (D+ ). 1 It was shown by Cui [22] and later by Takhtajan and Teo [116] that Theorem 6.5.3 ([22, 116]) .A22 (D+ ) ⊂ A∞ 2 (D+ ) and the inclusion is bounded. Furthermore, the following holds. Theorem 6.5.4 ([22, 116]) .β(TWP (D− )) = β(T (D− )) ∩ A22 (D+ ). In particular, .β(TWP (D− )) is open. By the fact that the image of the Bers embedding is open, .TWP (D− ) has a complex Hilbert manifold structure inherited from .A22 (D). Cui showed that the Bers embedding is holomorphic in the following sense. Modifying Cui’s notation somewhat, let .M∗ = L2−1,1 (D− ) ∩ L∞ −1,1 (D− ) with the direct sum norm ||μ||∗ = ||μ||∞ + ||μ||2 .

.

This is a Banach space. Furthermore, if M∗1 := {μ ∈ M∗ : ||μ||∞ < 1},

.

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then .M∗1 is obviously open, by the continuity of .μ |→ ||μ||∞ with respect to the direct sum norm. We then have the following theorem. Theorem 6.5.5 ([22]) The restriction of the Bers isomorphism .β ◦ 0 to .M∗1 is a holomorphic map into .A22 (D+ ). The fact that the Weil–Petersson metric on .TWP (D− ) is convergent at every point, and the induced distance is complete, was first shown by Cui. Using the complex structure obtained from .A2 (D+ ) via Theorem 6.5.4, we obtain that the tangent space at any point can be modelled by .A2 (D+ ). Given two elements .φ, ψ ∈ A2 (D+ ) ∼ = T0 A2 (D+ ), the Weil–Petersson pairing at the identity is naturally defined by the .L2 pairing in .A2 (D+ ) ff .

WP,0 =

D+

1 ψ(z)φ(z)dAz . (1 − |z|2 )2

At an arbitrary point the pairing is defined by right translation/change of base point. For .h ∈ TWP (D− ) let .Rˇ h denote the right translation induced on .β(TWP (D− )) induced by right translation .Rh on .TWP (D− ). We have Theorem 6.5.6 ([22]) .Rˇ h is a biholomorphism. Furthermore, there is a constant C (independent of h .) so that the derivative .D0 Rˇ h at the identity satisfies

.

.

1 ≤ ||D0 Rˇ h ||2 ≤ C. C

Given any pair of tangent vectors at .h ∈ TWP (D− ) represented by .ψ, φ ∈ Tβ(h) A2 (D+ ) ∼ = A2 (D+ ), we define the Weil–Petersson pairing at h by .

/ \ WP,h = (D0 Rˇ h )−1 φ, (D0 Rˇ h )−1 ψ

WP,0

.

(6.5.2)

It is immediately seen that this is a convergent Riemannian metric on every tangent space, which is smoothly varying. In other words, we have found the largest subset of the universal Teichmüller space, which is a Hilbert manifold and such that the Weil–Petersson pairing converges on each tangent space. This answers the question posed in Sect. 6.5.1. Not only are the tangent spaces complete, the manifold itself is complete with respect to the distance induced by the Weil–Petersson metric. Theorem 6.5.7 ([22]) .TWP (D− ) is complete with respect to the distance induced by the Weil–Petersson pairing. It was shown by Cui [22] that .QSWP (S1 ) is, like .QS(S1 ), a group. Takhtajan–Teo showed that it is a topological group. Theorem 6.5.8 ([116]) .QSWP (S1 )/Möb(S1 ) is a topological group. As is well-known, .QS(S1 )/Möb(S1 ) is not a topological group.

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A key aspect of the approach of Cui is that the Douady–Earle extension of an element of .QSWP (S1 ) has Beltrami differential in .L2−1,1 (D− ). The Douady– Earle extension is an explicit quasiconformal extension .E(φ) to the .D− of a quasisymmetry .φ ∈ QS(S1 ). It is conformally natural, that is, for any .M1 , M2 ∈ Möb(S1 ), E(M1 ◦ φ ◦ M2 ) = M1 ◦ E(φ) ◦ M2 .

.

Theorem 6.5.9 ([22]) If .φ ∈ QSWP (S1 ) then .E(φ) has Beltrami differential 2 .μ(E(φ)) ∈ L given .μ ∈ M∗1 let .σ (μ) ∈ M∗1 be the −1,1 (D− ). Furthermore, | Beltrami differential of .E( wμ |S1 ). Then .σ : M∗1 → M∗1 is continuous. Finally, we observe that it follows directly from Theorem 6.5.3 that Theorem 6.5.10 ([22, 116]) The inclusion from .TWP (D− ) into .T (D− ) is holomorphic. This theorem refers to the inclusion into the Banach manifold .T (D− ). As was mentioned earlier, Nag and Verjovsky [59] showed that the inclusion from 1 1 .Diff(S )/Möb(S ) into .T (D− ) is Gâteaux holomorphic. The inclusion from .TWP (D− ) into .T (D− ) endowed with Takhtajan–Teo’s Hilbert manifold structure described in the next section, is also holomorphic. This follows more or less automatically from their construction. Some of the results in this section were generalized to the .Lp case for .p ≥ 2 by Guo [39] (see also Definition 2 on page 48 of [39] for the definition of .Lp – Teichmüller space for .1 ≤ p < ∞). Tang showed that the Douady–Earle extension is in .Lp and that the Bers embedding is holomorphic with respect to the intersection norm on .Lp ∩ L∞ . For an overview of the .Lp theory as well as other refinements of the Teichmüller space see Sect. 6.7. Remark 6.5.11 In Guo [39], a paper of Cui is cited as “Teichmüller spaces and Diff(S1 )/Möb(S1 ), to appear.” This appears to refer to the paper Cui [22], and indeed the authors were once informed of this by an anonymous referee.

.

Remark 6.5.12 Although neither of [22, 39] uses the term “Weil–Petersson” in the text or title, these are as far as we know the first papers rigorously defining the Weil– Petersson Teichmüller space and developing its complex structure. At the time of the publication of [22] and [39], the journal Science in China was unfortunately still not widely available in the west.

6.5.2.2

Deformation Model

The deformation model, including the model of the tangent space by harmonic Beltrami differentials, was carried out by Takhtajan and Teo [114], later published as the first of two chapters of [116]. Their approach has a major difference: a Hilbert

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manifold structure is given to the entire Teichmüller space .T (D− ), at the cost of disconnecting the space. The connected component of .[0] ∈ T (D− ) is .TWP (D− ). Recall that in the classical case the tangent space can be modelled by the harmonic Beltrami differentials .O−1,1 (D− ), defined in terms of the Ahlfors–Weill reflection .AD− (see (6.4.2)). Takhajan and Teo define H−1,1 (D− ) = AD− A22 (D+ ),

.

(6.5.3)

which makes sense thanks to Theorem 6.5.3. As in the .L∞ case one can construct an atlas using the Ahlfors–Weill reflection. By Theorem 6.5.3 there is a sufficiently small ball .B ⊂ A22 (D+ ) centred at 0 which is contained in the ball of radius 2 in .A∞ 2 (D+ ) so that all elements of .AD− (B) are Beltrami differentials of elements of .T (D− ). For .μ ∈ L∞ −1,1 (D− ) setting .Vμ = Rwμ 0D− AD− (B), we obtain the following. Theorem 6.5.13 ([116]) Let .wμ : D− → D− be a normalized solution to the Beltrami equation. The sets .Rwμ 0D− AD− (B) and charts .βD− Rwμ−1 form an atlas for a complex Hilbert manifold structure on .T (D− ). In the topology associated to this complex structure, .T (D− ) contains infinitely many connected components. These are integral manifolds of a certain distribution, which thus coincide with the tangent spaces. These tangent spaces are precisely the directions in .T (D− ) in which the Weil–Petersson pairing converges. The connected component of the identity is .TWP (D− ). Theorem 6.5.14 ([116]) Right translation .(change of base point.) is a biholomorphism with respect to this complex structure. Remark 6.5.15 Note that this statement is stronger than the statement given in Cui’s Theorem 6.5.6 that right translation/change of base point is a biholomorphism, since it applies to all quasiconformal translations, not just Weil–Petersson class ones. If one restricts to Weil–Petersson class deformations, one obtains this part of Cui’s Theorem as a special case. On the other hand, the lower bound on the differential given in Cui’s theorem, which leads to his short proof of completeness, is not contained in Theorem 6.5.16. Recalling the decomposition of Eq. (6.4.7), we have ( ) 2 2 L∞ −1,1 (D− ) ∩ L−1,1 (D− ) = N (D− ) ∩ L−1,1 (D− ) ⊕ H−1,1 (D− ).

.

(6.5.4)

Here we have used the fact that .H−1,1 (D− ) ⊂ O−1,1 (D− ), which follows directly from Theorem 6.5.3. This shows that we can identify the tangent space at .[0] with the .L2 harmonic Beltrami differentials: T[0] TWP (D− ) ∼ = H−1,1 (D− )

.

Takhtajan–Teo showed

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Theorem 6.5.16 ([116]) For any normalized quasiconformal map .wμ : D− → D− , .D0 Rwμ is a bounded isomorphism. We then define a distribution .DT over .T (D− ) as follows: for each .[wμ ] ∈ T (D− ), we assign the Hilbert space .D0 Rwμ H−1,1 (D− ). Takhtajan–Teo’s surprising idea was to disconnect the space .A∞ 2 (D+ ) and view 2 it as a union of translates of .A2 (D+ ). That is, consider the foliation of .A∞ 2 (D+ ) ] [ 2 A∞ (D ) . φ + A + (D ) 2 (D+ ) = ∪φ∈A∞ 2 + 2

.

Each leaf is a Hilbert manifold in the obvious way, and one can view the vector space A∞ 2 (D+ ) as an uncountable collection of disconnected Hilbert manifolds (ignoring 2 the original norm). The vector spaces at .φ ∈ A∞ 2 (D+ ) tangent to .φ + A2 (D+ ) form a distribution .DA , and the leaves of the foliation are integral manifolds of this distribution. Summarizing some of their results:

.

Theorem 6.5.17 ([116]) Consider .T (D− ) and .A∞ 2 (D+ ) as Hilbert manifolds as above. Then (1) The Bers embedding .β : T (D+ ) → A22 (D− ) is a biholomorphism onto its image. (2) For .wμ ∈ T (D− ), the image of the restriction of .β ◦ Rwμ is tangent to .DA . (3) The connected component of .[0] ∈ T (D− ) is .β −1 (A22 (D+ )). In particular, we see that .TWP (D− ) is the connected component of the identity of T (D− ) with respect to Takhtajan–Teo’s complex Hilbert manifold structure. We also see that the tangent space can be modelled by

.

T[wμ ] T (D− ) ∼ = D0 Rμ H−1,1 (D− )

.

and as we saw before, by T[wμ ] T (D− ) ∼ = D0 Rˇ μ A22 (D+ ).

.

Thus we have two different expressions for the Weil–Petersson pairing; Eq. (6.5.2) and / \ −1 . WP,[w ] = (D0 Rwμ ) φ, (D0 Rwμ )−1 ψ . (6.5.5) μ WP,[0]

Tracing through the definitions shows that these are obviously equal up to a constant. This was strengthened by Matsuzaki [52], in the following way. Theorem 6.5.18 For any .[wμ ] ∈ T (D− ),

[ ] −1 2 β Rw (T (D )) = β(T (D )) ∩ β([w ]) + A (D ) . WP − − μ + 2 μ

.

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In fact, this was shown for the .Lp Teichmüller space for all .p ≥ 2. The inclusion .⊆ can be seen to follow from the arguments in Takhtajan and Teo [116] for .p = 2. Remark 6.5.19 It should be observed that the contents of Theorem 6.5.17 part (1) and Theorem 6.5.5 are quite different, even when the Bers embedding is restricted to .TWP (D− ).

6.5.3 Some Further Characterizations of Weil–Petersson Quasisymmetries In this subsection we mention some other recent characterizations of .QSWP (S1 ). Here we are brief, because there is already an excellent survey on the topic in Bishop’s paper [14]. In the same paper, Bishop gave many new and striking characterizations of the WP-class, as well as generalizations to higher dimensions. The reader is encouraged to read the original papers by Bishop [13, 14] in order to fully appreciate the scope and depth of the results that were obtained therein, and to also see the connection of those results to many branches of mathematics as well as physics. Takhtajan and Teo [116] asked a question regarding the characterization of WPclass quasisymmetries. In this connection, Shen [110] proved the following intrinsic characterization of the WP-class quasisymmetries. Theorem 6.5.20 Let f be a conformal mapping from .D+ onto a quasidisk and h = g −1 ◦ f be a corresponding quasisymmetric conformal welding on .S1 . Then the following statements are equivalent

.

(1) h is WP-class; (2) h is absolutely continuous .(with respect to the arc-length measure.) and .log h' ) 1 ( belongs to the homogeneous Sobolev space .H˙ 2 S1 . Later, based on [110], Shen in collaboration with Hu and Wu [42] showed that the smooth Hilbert manifold structure on the Weil–Petersson class Teichmüller space inherited from .H 1/2 via the pullback .f |→ log |f ' | (f is a sense-preserving homeomorphism of the unit circle .S1 ) is compatible with the standard Hilbert manifold structure of Takhtajan–Teo. These results answered the question of Takhtajan and Teo mentioned above. Recently, Shen and Liu [51] proved a characterization of the so-called p– integrable quasicircles for .p > 1. To introduce the main function spaces occurring 1 2 (S1 ) be the homogeneous Besov space of functions with in their result, let .B˙ p,p ( ||ϕ||

.

1 2 (S1 ) B˙ p,p

:=

1 4π 2

f f S1

S1

|ϕ(ζ ) − ϕ(η)|p |dζ ||dη| |ζ − η|2

)1/p < ∞.

(6.5.6)

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Let .Bp (D± ) denote the holomorphic Besov space consisting of holomorphic functions .ψ in .D± with the semi-norm (f f ||ψ||Bp (D± ) :=

.

D±

)p−2 | ' |p ( |ψ (z)| 1 − |z|2 dA(z)

)1

p

,

(6.5.7)

where .dA(z) = π1 dx dy. Given the definitions above, the result of Liu–Shen reads as follows Theorem 6.5.21 Let .p > 1 be a fixed number and let f be a conformal mapping from .D+ onto a quasidisk and( .h )= g −1 ◦ f be a corresponding quasisymmetric conformal welding for .r := f S1 . Then the following statements are equivalent (1) .r is a p-integrable quasicircle, that is, .log f ' ∈ Bp (D+ ) (2) .log g ' ∈ Bp (D− ); (3) .r is rectifiable with length l and there exists some real-valued function .b ∈ 1 2 B˙ p,p (S1 ) such that an arclength parameterization .z : S1 → r satisfies .z' (ζ ) = l ib(ζ ) ; 2π e

(4) .r is rectifiable and the unit tangent direction .τ to .r satisfies .τ (z) = eiu(z) for 1 2 (r), where the semi-norm associated with some real-valued function .u ∈ B˙ p,p 1

.

2 B˙ p,p (r) is given by

( ||f ||

.

1

2 (r) B˙ p,p

:=

1 4π 2

f f r

r

)1/p |f (ζ ) − f (η)|p |dζ ||dη| . |ζ − η|2

(6.5.8)

Note that when .p = 2, then one recovers the characterisation of WP-class 1 1 2 quasicircles for which .B˙ 2,2 = H˙ 2 (the homogeneous Sobolev space), and .B2 = D (the Dirichlet space). In [13] and [14], Bishop gave a more geometric approach to the proof of Theorem 6.5.20. He also showed the following characterization result: Theorem 6.5.22 With f and .r as in Theorem 6.5.21 one has (1) .r is a WP-class quasicircle if and only if it has finite Möbius energy, i.e., f f ( Möb(r) :=

.

r

r

1 1 − 2 |x − y| l(x, y)2

) dx dy < ∞,

(6.5.9)

where .l(x, y) = arclength distance between .x, y along curve .r. (2) .r is a WP-class quasicircle if and only if it is chord-arc and the arclength 3 parameterization is in the Sobolev space .H 2 (S1 ). Here, a rectifiable curve .r is called chord-arc if for all .x, y ∈ r one has length(γ ) = O(|x − y|) where .γ ⊂ r is the shortest sub-arc with endpoints .x, y.

.

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Remark 6.5.23 The so-called asymptotically smooth quasicircles were shown to be chord-arc in [38]. This fact also implies that the WP–class quasicircles are chordarc, since the WP–property implies asymptotic smoothness. Bishop’s result in [13], has the benefit of being a characterization result for WP–class quasicircles. Finally, we also note that the tangent space to .TWP (D− ) at the identity consists 3 precisely of the .H 2 vector fields on the unit circle with some normalization conditions (see [59, 116]).

6.6 The Period Mapping 6.6.1 Polarizations and the Siegel Disk A number of researchers have considered an embedding of the group Diff(S1 )/Möb(S1 ) in an infinite-dimensional version of the Siegel disk. Later this was extended to the Teichmüller spaces .TWP (D− ) and .T (D− ). A paper of Nag and Sullivan established the analytic point of view appropriate for Teichmüller theory. To simplify the presentation, we adopt their point of view. Along the way we will fill in some of the history of the ideas. 1/2 Let .HR (S1 ) and .H 1/2 (S1 ) denote the Sobolev .1/2 space of real- and complexvalued functions on the circle respectively. Denote spaces with constants modded 1/2 out with a dot, e.g. .H˙ R . It was independently shown by Vodopy’anov [127] and Nag and Sullivan [58] that the composition by a homeomorphism .φ

.

h |→ h ◦ φ

.

(mod constants) is a bounded operator on .H˙ 1/2 (S1 ) if and only if .φ is a quasisymmetry (Vodopy’anov formulates this result in an equivalent form on the real line). One can show that .H˙ 1/2 (S1 ) is a symplectic space with respect to the completion of the pairing ω(f, g) =

.

1 π

f S1

f · dg

(6.6.1)

for smooth functions. Thus we obtain a representation of quasisymmetries by symplectomorphisms on .H˙ 1/2 (S1 ). In summary Theorem 6.6.1 ([58, 127]) A homeomorphism .φ : S1 → S1 induces a bounded composition operator on .H˙ 1/2 (S1 ) if and only if .φ ∈ QS(S1 ). This bounded operator is a symplectomorphism. The fact that .Diff(S1 ) acts symplectically with respect to this pairing was already known. Segal [103] investigated it in conjunction with a metaplectic representation of the restricted symplectic group. The action was there restricted to smooth real-

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1/2 valued functions on the circle mod constants (the .C ∞ elements of .H˙ R ). Segal defined an infinite-dimensional analogue of the Siegel disk, which parametrizes the positive polarizations of a complexified symplectic space. Nag and Sullivan [58] showed that the action has a natural extension to .QS(S1 ), and initiated the analytic theory which is ultimately necessary to rigorously investigate the Kähler geometry of Teichmüller space, the period map, and symplectic actions. As we will see, the correct analytic setting leads to a clearer geometric and algebraic picture (in much the same way that the algebra of Fourier series in .L2 (S1 ) is clearer than the algebra of Fourier series in, say, .C ∞ (S1 )). For example, it leads to the precise connection to the moduli spaces .T (D− ) and .TWP (D− ), as we will see shortly. We now describe the infinite Siegel disk more precisely. We say that .(V C , ω, J ) is a complex symplectic Hilbert space if the following hold:

(1) .V C is the complexification of a real vector space V with complex structure; (2) J and .ω are the complex linear extensions of the complex structure and a symplectic form on V ; (3) .V C is a separable Hilbert space with respect to the pairing . := ω(v, J w); (4) .ω and J are continuous. Definition 6.6.2 Let .(V C , J, ω) be a complex symplectic Hilbert space. We say that a complex subspace W of .V C is a positive polarizing subspace if (1) W is isotropic; (2) .V C = W ⊕ W ; and (3) .ω(v, iw) is a positive-definite sesquilinear form on W . We also refer to the decomposition (2) as a polarization. It is easily verified that if .W0 and .W0 are the .−i- and i-eigenspaces of J , then W0 ⊕ W0 is a polarization, which we call the standard polarization. The set of polarizations is parametrized by the infinite Siegel disk, which we now define.

.

Definition 6.6.3 Let .(V C , J, ω) be a complex symplectic Hilbert space and let C = W ⊕ W denote the standard polarization. The infinite Siegel disk is the .V 0 0 set of bounded linear maps .Z : W0 → W0 satisfying (1) .ω(v, Zw) = ω(w, Zv) for all .v, w ∈ W0 ; (2) .I − ZZ is positive definite. The restricted Siegel disk is the set of elements of the infinite Siegel disk which in addition satisfy (3) Z is Hilbert–Schmidt. Remark 6.6.4 Condition (1) is equivalent to .Z T = Z in an orthonormal basis, or alternatively .Z ∗ = Z. Remark 6.6.5 Although we have not seen the term “restricted Siegel disk”, it is an obvious extension of terminology in use for the Shale group. In the literature the

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term “Siegel disk” sometimes agrees with the terminology here and sometimes is used for the restricted Siegel disk. It can be shown that the set of positive polarizations is in one-to-one correspondence with the infinite Siegel disk, by setting W to be the graph of Z. It is also easy to see that the group of real symplectomorphisms of V acts on the space of polarizations by complex linear extension, and thus acts on the Siegel disk. The Siegel disk was considered first by Siegel in finite dimensions [112]. Segal [103] considered the restricted Siegel disk in association with the representations of the symplectic group induced by .Diff(S1 ) as described above. The condition that Z be Hilbert–Schmidt, included by Segal, is derived from Shale [108]. Shale showed it to be necessary and sufficient that an automorphism of the CCR algebra be implementable as a unitary operator on Fock space—that is, the necessary condition to produce the projective unitary representation associated with the metaplectic group. The corresponding subgroup .Symres (V ) of the symplectic group of a vector space V is called the restricted symplectic group, and appears throughout the representation theory of infinite-dimensional groups, see e.g. Ottesen [62] or Schottenloher [100] in addition to the above references. A standard example of a polarization is given by the holomorphic and antiholomorphic decomposition of one-forms on a Riemann surface. Indeed this is the basis of the classical period map, and was a motivation for Siegel’s construction. In the literature this is usually given in terms of the equivalent Siegel upper half-plane model. Another example is provided by the positive and negative Fourier modes of functions on the circle [103], which we provide here in the setting of Nag and Sullivan. Set .V C = H˙ 1/2 (S1 ) ⊂ L2 (S1 ), where the symplectic form is the completion of (6.6.1) and the complex structure is the Hilbert transform J einθ = −i sgn(n) einθ

.

for .n ∈ Z\{0}. The standard polarization is given by the functions with only positive and only negative Fourier modes. These are boundary values of holomorphic and anti-holomorphic functions with square integrable derivatives on the disk (the ˙ + )), that is ˙ + ), .W0 = D(D Dirichlet spaces .W0 = D(D ˙ + ). ˙ + ) ⊕ D(D H˙ 1/2 (S1 ) = D(D

.

It is easily shown that .Möb(S1 ) is the precise subset of .QS(S1 ) preserving the standard polarization. Thus by Theorem 6.6.1 of Vodopy’anov, Nag–Sullivan, we obtain an embedding of the universal Teichmüller space into the infinite Siegel disk. This was considered earlier for the smooth subspace .Diff(S1 ) by many authors, including Segal [103] as mentioned above, Kirillov and Yuri’ev [48], and Nag [57].

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It was called the KYNS (Kirillov–Yuri’ev–Nag–Sullivan) embedding by Takhtajan and Teo [116].5 The finite-dimensional Siegel disk is a homogeneous complex manifold with a natural symplectically invariant Hermitian metric, which at the identity is given by ¯ 2 ) [112]. Note that by the symmetry of the Z matrices, this is the Frobenius .Tr(Z1 Z inner product for matrices. This naturally generalizes to the infinite-dimensional restricted Siegel disk, because the condition for its convergence is precisely that .Z1 and .Z2 are Hilbert–Schmidt. It’s not hard to show that diffeomorphisms induce Hilbert–Schmidt composition operators [103]. It was shown by Kirillov and Yur’iev [48], and later Nag and Sullivan [57], that the pull-back of the generalized Siegel Kähler form under the KYNS period embedding agrees with the Weil–Petersson metric for .Diff(S1 )/Möb(S1 ). The full result for .TWP (D− ) is due to Takhtajan–Teo. Now the operator Z in the KYNS embedding was shown to be the Grunsky matrix by Kirillov and Yuri’ev [48]. The Grunsky matrix takes many forms in the literature. One of these, due to Bergman and Schiffer [11], is as an integral operator on the anti-holomorphic Bergman space. Given an element of .T (D− ) ¯ be the associated normalized conformal map given by Bers let .f : D+ → C trick (6.5.1). We associate a Grunsky operator to f (and hence to an element of .T (D− )) as follows. For square integrable antiholomorphic .h(z) we define 1 π

Grf h(z) =

.

(

ff D+

1 f ' (w)f ' (z) − (f (w) − f (z))2 (w − z)2

) h(w)

d w¯ ∧ dw . 2i

(6.6.2)

Since differentiation is an isometry from the Dirichlet space mod constants to the Bergman space, one can equivalently formulate this on the homogeneous Dirichlet space. In this case we get ˙ + ) → D(D ˙ + ). Z = | |([μ]) = d −1 Grf d : D(D

.

(6.6.3)

From the integral expression (6.6.2), it is easily seen that this operator is Möbius invariant, that is .GrT ◦f = Grf for T Möbius, so that the normalization of f is irrelevant as expected. Remark 6.6.6 The Grunsky matrix can be defined for holomorphic functions f which are one-to-one in a neighbourhood of 0 [25, 66]. In [48], there is a potentially very misleading typographical error. It is stated that .I − Z f Zf > 0 is equivalent to the condition that the locally conformal map f is univalent on the disk. In fact, it is classically known that, rather, the fact that .I −ZZ ≥ 0 when applied to vectors in .Cn for any n is equivalent to univalence on .D. (In the formulation here, .I −ZZ ≥ 0 as a linear transformation on polynomials.) The condition that .I − ZZ > 0 is equivalent

5 Though Segal–Kirillov–Yuri’ev–Nag–Sullivan–Takhtajan–Teo embedding might be more appropriate.

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to univalence and quasiconformal extendibility by a result of Pommerenke and Kühnau. See for example Pommerenke [66]. The following theorem, independently obtained by Shen and Takhtajan–Teo, strongly motivates Weil–Petersson class Teichmüller theory. Theorem 6.6.7 ([109, 115, 116]) The Grunsky operator induced by .φ ∈ QS(S1 ) is Hilbert–Schmidt if and only if .φ is a Weil–Petersson class quasisymmetry. In particular, a homeomorphism induces a composition operator on .H˙ 1/2 (S1 ) which is in the restricted symplectic group if and only if it is a Weil–Petersson class quasisymmetry. Thus .TWP (D− ) is exactly the group of elements of .T (D− ) which map into the restricted Siegel disk under the period map .| |. Denote the restriction of .| | to .TWP (D− ) by .| |W P . The hierarchy of conditions is striking, in light of the geometric interpretation. .I − Z f Zf ≥ 0 implies univalence of f , but .I − Z f Zf > 0 implies in addition that f is quasiconformally extendible (and thus f is associated to an element of the universal Teichmüller space). Adding the further condition that Z be Hilbert– Schmidt then restricts to the Weil–Petersson class Teichmüller space. Remark 6.6.8 Segal–Wilson and Kirillov–Yuri’ev consider the action of .Diff(S1 ) on .L2 (S1 ), so that the extensions are in fact in the direct sum of the Hardy space and anti-holomorphic Hardy space of the disk. This is valid for smooth homeomorphisms, but quasisymmetries do not appear to be bounded on .L2 (S1 ). For quasisymmetries one has to consider the space .H˙ 1/2 (S1 ) space as Nag and Sullivan do. One may ask whether it is possible to recover the Weil–Petersson metric on finite-dimensional spaces from that on .TWP (D− ). Nag and Verjovsky [59] showed that the Teichmüller space of any Fuchsian group is transverse to the fibres. In particular, the pull-back of the Weil–Petersson metric on .TWP (D− ) to the finite-dimensional Teichmüller spaces cannot be the Weil–Petersson metric of the finite-dimensional space. However, they show how an averaging procedure due to Patterson recovers the Weil–Petersson metric on the finite-dimensional spaces from that on .Diff(S1 )/Möb(S1 ). Takhtajan–Teo use this technique to obtain Wolpert’s curvature formulas from their curvature formulas on .TWP (D− ) [116] (see Sect. 6.9 ahead).

6.6.2 Interpretation of the Period Map Among a number of interpretations, Nag and Sullivan [58] observed that this period map is analogous to the period map for compact surfaces. Historically, Riemann [86] used the period matrices and his bilinear relations to provide a necessary and sufficient condition for a set of 2g linearly independent

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(over .R) vectors in .Cg to be periods of g holomorphic differentials on a gdimensional Abelian variety. He also solved the Jacobi inversion problem for an arbitrary algebraic curve (using his Theta functions). Riemann’s work was developed further by Torelli [121], where biholomorphic equivalence of compact Riemann surfaces of genus .g ≥ 1 was connected with the so-called polarized Jacobian of the Riemann surfaces in question. The work of Torelli was developed further by Andreotti and Weil, and turned into one of the cornerstones of complex algebraic geometry. We begin with holomorphicity of the period map. Ahlfors proved that the period map of a compact surface R depends holomorphically on the surface, and that the complex structure of .T (R) is the unique one such that this holds [2]. Takhtajan and Teo [116] showed that both period maps .| | and .| |WP are holomorphic. The codomain of .| | is the Banach space .B(D(D− )) → D(D+ ) of bounded operators from .D(D− ) to .D(D+ ). We have the following. ˙ − ), D(D ˙ − )) Theorem 6.6.9 ([116]) The period embedding .| | : T (D− ) → B(D(D is injective and holomorphic. Remark 6.6.10 Their statement is in terms of .l2 . The formulation here is an easy consequence of Eq. (6.6.3), after identifying .l2 with the Bergman space. We altered the formulation for consistency with Nag and Sullivan [58] and to offer a geometric interpretation ahead. The codomain of .| |WP , the restricted Siegel disk, is a subset of the Sato–Segal– Wilson Grassmannian .S. This possesses a complex Hilbert manifold structure induced by the Hilbert–Schmidt norm. We have Theorem 6.6.11 ([116]) The KYNS period embedding | |WP : TWP (Z) → S

.

is holomorphic. The authors showed that the boundary values of an element of the homogeneous Dirichlet space of a quasidisk .K are in turn boundary values of an element of the homogeneous Dirichlet space of its complement. This “overfare” is bounded precisely for quasidisks. This allows the following result interpreting the polarizations of Nag and Sullivan. Theorem 6.6.12 ([83]) Let f be associated to an element .[μ] ∈ T (D− ) and .Z = ¯ | |([μ]). Let .K = f (D+ ) and .K ∗ = C\cl(f (D+ )). Then the graph W of Z is ∗ ˙ {h : D+ → C harmonic : h|S1 = H ◦ f for some H ∈ D(K )}.

.

It is understood in the above that H extends to .∂K ∗ = ∂K , so that this composition makes sense (there are analytic subtleties, which we suppress for the sake of brevity). In other words, the graph W of the Grunsky operator Z is the pull-

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back of the boundary values of Dirichlet-bounded holomorphic functions on .K ∗ . Thus the new polarization can be identified with the pull-back of the polarization on ∗ ∗ with an element of Teichmüller space, this is the interpretation .K . Identifying .K of the period map given in Nag and Sullivan [58] (after conjugating by d to rephrase in terms of forms). The interpretation of the polarizations as the pull-back of boundary values of holomorphic functions also arises in two-dimensional conformal field theory [43, Appendix D]. Radnell et al. discuss the relation of this result with conformal field theory in [81]; however there the theorem was stated only for Grunsky matrices associated with elements of the Weil–Petersson Teichmüller space. Later the authors discovered that, surprisingly, this extends to the entire Teichmüller space [83]. In fact it was extended there to bordered surfaces of type .(0, n).6 The theorem for qc ¯ .(0, n) is as follows in the notation used here. Let .f ∈ O (C\{p 1 , . . . , pn }) be a collection of quasiconformal maps with non-overlapping images, each taking 0 to .pn . Denote the induced Grunsky operator by n

˙ + ) → D(D ˙ + )n Zf : D(D

.

(this can be defined using the integral expression (6.6.2) for example). Setting ¯ ∪n cl fk (D+ ) E = C\ k=1

.

we have, denoting .Dn+ = D+ × · · · × D+ , that the graph of the generalized Grunsky operator is ˙ {h : Dn+ → C harmonic : h|(S1 )n = H ◦ f for some H ∈ D(E)}.

.

and Theorem 6.6.12 is a special case. In light of the above it was natural to ask whether it is possible to use the fiber/CFT model to define period matrices in such a way that the finite-dimensional period map and infinite-dimensional period map are naturally unified. The next step, extending Grunsky operators to arbitrary genus surfaces and number of boundary curves, was accomplished by Shirazi [111], and Theorem 6.6.12 was shown to hold in that setting. There, adjustments must be made for topological obstructions arising from the classical periods. The relation of the generalized period mappings with the classical period matrices for compact surfaces beyond an analogy was an open question, but recent work [98] and work in progress unifies them. Remark 6.6.13 In higher genus one needs to work with one-forms and not just functions. To this end the authors showed that the boundary values of .L2 harmonic one-forms can be identified with the Sobolev .H −1/2 space in a conformally invariant way, and developed a theory of overfare of one-forms [97, 98].

6 The

earlier result for WP-class quasicircles was given in [76], which was left unpublished.

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It is thus of interest to ask whether the extended period mappings are holomorphic maps of Teichmüller space. Radnell et al. [82] showed that Theorem 6.6.9 holds for surfaces of type .(0, n), using the atlas arising from the fiber model. (Note that there the exceptional cases .g = 0, n = 1 and .g = 0, n = 2 are taken care of as in Remark 6.3.7.) The methods in that paper should extend fairly easily to type .(g, n). A generalization of Theorem 6.6.11 to the Weil–Petersson Teichmüller space of type .(g, n) bordered surfaces (described in Sect. 6.8 ahead) is highly desirable.

6.7 A Brief Overview of Other Refinements of Teichmüller Space Here, for the sake of completeness and comparison, we also briefly mention some other refinements of the Teichmüller space other than the Weil–Petersson. Gardiner–Sullivan’s Asymptotic Teichmüller Space The asymptotic Teichmüller space .A T(R) was introduced by Gardiner and Sullivan [34] for the upper half-plane, and by Earle et al. [26] for arbitrary hyperbolic Riemann surfaces. A univalent function f in the unit disc .D admiting a quasiconformal extension to the whole plane, with .f (∞) = ∞ is called asymptotically conformal [67] if .

| '' | | f (z) | lim (1 − |z|) || ' || = 0. f (z) |z|→1−

Now given the two quasiconformal mappings f and g on a hyperbolic Riemann surface .R, consider the following equivalence relation: .f ∼ g, if there is an asymptotically conformal mapping h of .f (R) onto .g(R) such that .g −1 ◦ h ◦ f is homotopic to the identity relative to the ideal boundary of .R. The asymptotic Teichmüller space .A T(R) is the set of equivalence classes of quasiconformal mappings of .R under this relation. Note that this definition is a variation of the definition of Teichmüller space .T (R) where the mapping h is required to be conformal. Since conformal mappings are asymptotically conformal, there is a well-defined projection .P : T (R) → AT(R). If .R is a closed Riemann surface of finite genus with a finite number of points removed, then .AT(R) consists of one point. Earle et al. [26, 27] proved that .(AT(R) has a complex manifold structure so that the quotient map .P : T (R) → AT(R) is holomorphic). For more information on this topic see Gardiner–Lakic’s monograph [33].

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Astala–Zinsmeister’s BMO Teichmüller Space This version of the Teichmüller space was introduced by Astala and Zinsmeister in [7]. To define it requires some preparation. First recall that a positive measure .λ defined in a simply connected domain .K is called a Carleson measure if { } λ(K ∩ D(z, r)) .||λ||c = sup : z ∈ ∂K , 0 < r < diameter(K ) < ∞, r (6.7.1) where .D(z, r) is the disk with center z and radius r. By .CM(K ) one denotes the set of all Carleson measures on .K . Now let .LBMO (D+ ) be the Banach space of essentially bounded measurable functions .μ on .D+ such that the measure λμ =

.

|μ|2 (z) dAz 1 − |z|2

is in .CM(D+ ). The norm on .LBMO (D+ ) is defined by 1/2

||μ||LBMO (D+ ) := ||μ||∞ + ||λμ ||c

.

|| || where .||λμ ||c is the Carleson norm of .λμ defined in (6.7.1). Set MBMO (D+ ) = {μ ∈ LBMO (D+ ) : ||μ||∞ < 1} .

.

Cui and Zinsmeister [23] extended this to surfaces associated with a Fuchsian group G as follows. Let G be a Fuchsian group, i.e. a properly discontinuous fixed point free group of Möbius transformations which keeps .D+ invariant. For such a group set { } g¯ ' ∞ .M(G) := μ ∈ L (D+ ) : ||μ||∞ < 1 and ∀g ∈ G, μ = μ ◦ g (6.7.2) . g' Now for the corresponding Riemann surface .R = D+ /G, define MBMO (G) := M(G) ∩ MBMO (D+ )

.

with the same equivalence relation as in the classical Teichmüller space. Given MBMO (G), the BMO-Teichmüller space denoted here by .BMOT(R), is defined as the quotient space associated with the equivalence relation. Note that if G is co-compact then, as for the asymptotic Teichmüller spaces, .BMOT(R) is trivial. But this is the extent of the analogy and the equivalence relation in the case of asymptotic Teichmüller space is rather different. For further developments in this context the reader is referred to [130]. .

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Guo–Tang’s .Lp -Teichmüller Space For .p ∈ [2, ∞), the .Lp -Teichmüller spaces were first introduced by Guo in [39], and developed by Tang [118] for the universal Teichmüller space, and by Yanagishita [137] for general Fuchsian groups. These are defined as follows. For a Fuchsian group G let .Lp (D+ ; G) be the space of essentially bounded measurable functions .μ ∈ M(G) on .D+ (where .M(G) is as in (6.7.2)), such that ff .

N

|μ|p (z) dAz < ∞ (1 − |z|2 )2

where N is a fundamental domain of G. Set { } Lp (D+ , G)1 = μ ∈ Lp (D+ , G) : ||μ||∞ < 1 ,

.

with the same equivalence relation as in the classical Teichmüller space. For the Riemann surface .R = D+ /G, the .Lp -Teichmüller space denoted here by .Lp T(R), is defined as the quotient space associated with the equivalence relation. In [137], Yanagishita introduced, for any .p ≥ 2, a complex structure on the G-invariant p-integrable Teichmüller space associated with an arbitrary Fuchsian group G satisfying the Lehner’s condition. For further developments in this context the reader is referred to [51, 52, 54, 55, 119, 139]. The case of .p = 2 is of course the WP-class Teichmüller space.

6.8 Weil–Petersson Teichmüller Spaces of General Surfaces 6.8.1 Overview The Weil–Petersson Teichmüller theory was extended to more general surfaces by Yanagishita and Radnell–Schippers–Staubach independently. The approaches are somewhat different, so that the results largely complement each other. In the end, the complex structures in all three models are shown to exist and be equivalent. We give a brief outline here. The precise results are given in the next section. The first step was taken by Radnell et al. [75], later published in two parts [78, 84]. There, a definition of Weil–Petersson class Teichmüller spaces of type .(g, n) was given, by restricting to equivalence classes of quasiconformal deformations whose boundary values are Weil–Petersson class quasisymmetries. The topology and a Hilbert manifold structure was given using the fiber model. Yanagishita [137] and Radnell et al. [77] (later published as [79, 80]) simultaneously and independently developed two further aspects. Yanagishita gave a complex structure on the Weil–Petersson Teichmüller space for surfaces satisfying what he calls Lehner’s condition, which puts a lower bound on the length of the simple closed geodesics. This applies to bordered surfaces of type .(g, n). His approach is

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similar to the approach of Cui [22], using the Bers embedding model of Sect. 6.4.1.1. In fact, as in Guo [39] and Tang [118] in the case of the universal Teichmüller space, the results are shown to hold for the .Lp theory for .p ≥ 2 (which in general results in a Banach manifold structure). The case .p = 2 is the WP space and one obtains a Hilbert manifold structure. On the other hand, Radnell et al. [77], later published in two parts as [79, 80], gave the Weil–Petersson Teichmüller space a complex Hilbert manifold structure based on harmonic Beltrami differentials; that is, based on the deformation model of Sect. 6.4.1.2, for type .(g, n) bordered surfaces with .2g − 2 + n > 0. Two definitions were advanced for this Weil–Petersson Teichmüller space: that the boundary values of the quasiconformal map are WP-class quasisymmetries, and that there is a representative with .L2 Beltrami differential. The latter definition agrees with Yanagishita’s definition restricted to type .(g, n) surfaces. Using sewing techniques and boundary behaviour of the hyperbolic metric, it was shown that the two definitions are equivalent. The deformation complex structure was also shown to be compatible with the complex Hilbert manifold structure obtained from the fiber model. Note that even in the classical .L∞ case, the compatibility of the complex structures derived from the fiber/CFT point of view and any of the classical structures was a fairly recent result [73]. As a consequence, for type .(g, n) bordered surfaces with .2g − 2 + n > 0, Radnell et al. [77] gave the tangent space structure and show that the Weil–Petersson pairing converges on each tangent space. Yanagishita [138] later independently gave the construction of the complex Hilbert manifold structure based on harmonic Beltrami differentials, for surfaces more generally satisfying Lehner’s condition. He then showed that this is compatible with the complex structure derived from the Bers embedding, and that the Weil– Petersson metric converges on each tangent space. He furthermore showed that the metric is Kähler, generalizing the theorem of Takhtajan–Teo from the case of the universal Teichmüller space. The results relating to Kählericity and curvature will be discussed in Sect. 6.9 ahead. In summary, for surfaces satisfying Lehner’s condition, the Bers embedding model and deformation model give equivalent complex structures. For bordered surfaces of type .(g, n) .2g − 2 + n > 0, the deformation model and fiber model give equivalent complex structures. Thus in the special case of type .(g, n) bordered surfaces with .2g − 2 + n > 0, all three complex structures are equivalent. Remark 6.8.1 One can then ask whether there is a meaningful sense in which the fiber model extends to general surfaces satisfying Lehner’s condition (or perhaps some larger subset than type .(g, n) bordered surfaces).

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6.8.2 Manifold and Tangent Space Structure 6.8.2.1

Fiber Model

The Weil–Petersson Teichmüller spaces for bordered surfaces of type .(g, n) was first defined and given a complex Hilbert manifold structure by Radnell et al. [75], later appearing in [78, 84]. This was obtained by refining the fiber model for .T (E) to Weil–Petersson class mappings. Recall that one obtains a genus g surface .E P with n punctures by sewing on caps. The idea is to restrict the construction of Sect. 6.4.1.3 to more regular Weil–Petersson class maps. The local model of the conformal maps is given by OWP = {f ∈ Oqc : ||f '' /f ' ||A2 (D+ ) < ∞}. qc

.

1

We thus define qc

χWP = χ|Oqc : OWP |→ A21 (D+ ) ⊕ C WP ( ) f |→ f '' /f ' , f ' (0) .

.

Since inclusion in .A21 (D+ ) → A∞ 1 (D+ ) is bounded by Theorem 6.5.3 of Cui and qc Takhtajan–Teo, it follows directly that the inclusion .OWP is bounded. We have qc

Theorem 6.8.2 ([84]) Let .E P be a surface of genus g with n punctures. .χWP (OWP ) is open in .A21 (D+ ) ⊕ C. qc

We say that .f ∈ OWP (E P ) if .f = (f1 , . . . , fn ) ∈ Oqc (E P ) (see Definition 6.3.4) and furthermore, there are coordinates .ζk : Bk → C of .pk for qc .k = 1, . . . , n such that .ζk ◦ fk ∈ O WP . qc We construct a coordinate chart in .OWP (E P ) by restricting the open sets (6.4.3) and charts (6.4.4) of Sect. 6.4.1.3 to the Weil–Petersson case. With notation as in that section, set qc

Vζ,K = {f = (f1 , . . . , fn ) ∈ OWP (E P ) : cl( fk (D+ )) ⊂ Kk , k = 1, . . . , n}.

.

qc

These sets form a base for a Hausdorff, secound countable topology on .OWP [75]. We then define qc

E : Vζ,K → (OWP )n

.

f |→ ζ ◦ f. qc

Theorem 6.8.3 ([84]) Let .E P be a surface of genus g with n punctures. .OWP (E P ) O is a Hilbert manifold locally modelled on . n A21 (D+ ) ⊕ C, with respect to the atlas n .χ WP ◦ E.

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We also have Theorem 6.8.4 ([84]) Let .E P be a surface of genus g with n punctures. The qc inclusion .OWP (E P ) → Oqc (E P ) is holomorphic. Recalling Definition 6.2.2 of quasisymmetries of borders, we define Weil–Petersson class quasisymmetries as follows. Definition 6.8.5 Let .E and .E1 be bordered surfaces, with components C and .C1 of the borders homeomorphic to .S1 . We say that a quasisymmetry .φ : C → C1 is Weil–Petersson class if, for any collar charts .ψ and .ψ1 of C and .C1 respectively, −1 ∈ QS .ψ1 ◦ φ ◦ ψ WP(S1 ) . If .E and .E1 are bordered surfaces of type .(g, n), we say that a map .φ ∈ QS(C, C1 ) is Weil–Petersson if its restriction to each component of .∂E is Weil– Petersson class in the above sense. Denote the set of such Weil–Petersson class maps by .QSWP (∂E, ∂E1 ). The Weil–Petersson class Teichmüller space of a type .(g, n) bordered surface was defined by Radnell–Schippers–Staubach as follows. Definition 6.8.6 Let .E be a type .(g, n) bordered surface. The Weil–Petersson class Teichmüller space of .E is TWP (E) = {[E, F, E1 ] ∈ T (E) : F |∂E ∈ QSWP (∂E, ∂E1 )} .

.

Recall the definitions of .C and .Fq given in (6.3.4), (6.3.6), and surrounding text. Let CWP = C|TWP (E) .

.

qc

For a point q let .FWP,q be the restriction of .Fq to .OWP . This is well-defined because .ModI(E), and in particular .DB(E), obviously ( qc )n preserve .TWP (E). Finally let .Uζ,K = T (Vζ,K ) be the open sets induced on . OWP ( qc ) by the open sets .Vζ,K in . OWP (E1 ) , and let GWP : U × Vζ,K → TWP (E)

.

be obtained by restricting the second component of .G to the open subset .Vζ,K of qc OWP (E1 ). .G of course depends on the choices of .ζ , and K, and the point .p ∈ C −1 (q), so one obtains a collection of such maps. Recall that the entire construction depends on a choice of quasisymmetries .τ1 , . . . , τn parametrizing the boundaries.

.

Theorem 6.8.7 ([78]) Let .E be a type .(g, n) bordered surface with .2g − 2 + n > 0, and fix an n-tuple of quasisymmetries .τ = (τ1 , . . . , τn ) where .τk ∈ QSWP (S1 , ∂k E). The collection of maps .G −1 induced by .τ as above form an atlas, so that .TWP (E) is a complex Hilbert manifold with respect to this atlas. This complex structure is independent of .τ .

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Remark 6.8.8 It was shown that the domains of maps in this atlas form the base for a Hausdorff, second countable, and separable topology. n ◦ E, the local model of .T d Remark 6.8.9 By composing with .χWP WP (E) is .C ⊕ ( 2 )n A1 (D+ ) ⊕ C .

Remark 6.8.10 With respect to this complex structure, the fibers are complex manifolds and .CWP is a holomorphic map with local holomorphic submersions. This follows more or less directly from the definitions. Remark 6.8.11 There are two exceptional cases ruled out by .2g − 2 + n > 0, because of the degeneration of .TWP (E P ) to a point, as we saw in Remark 6.3.7. The exceptional case .g = 0 and .n = 1 is just .TWP (D− ), so that the fiber model identifies qc ¯ .TWP (D− ) with suitably normalized elements of .O WP (C\{0}). Thus this case is just the pre-Schwarzian model of .TWP (D− ) already given by Cui and Takhtajan–Teo. The case .g = 0, .n = 1 can be dealt with in the same way as it was in Radnell and Schippers [74] by repeating those arguments for .TWP (A), though this has not been done. We also have Theorem 6.8.12 ([78]) Let .E be a type .(g, n) bordered surface such that .2g − 2 + n > 0, and let .F : E → E0 be a quasiconformal map such that .[E, F, E0 ] ∈ TWP (E). The change of base point map RF : TWP (E0 ) → TWP (E)

.

is a biholomorphism with respect to the complex structure obtained in Theorem 6.8.7. This of course will play a role in proving the compatibility of the different complex structures. The following theorem is also essential. Theorem 6.8.13 ([78]) Let .E be a type .(g, n) bordered surface for .2g − 2 + n > 0. The modular group .ModI preserves .TWP (E). Furthermore, it acts properly discontinuously and fixed-point-freely by biholomorphisms. In [77], later appearing in [79], Radnell et al. developed a local characterization of differentials in terms of their behaviour on collars. Let .E be a type .(g, n) Riemann surface. By definition of bordered surface, there is an atlas of charts on .E ∪ ∂E mapping into the closed disk (equivalently, the upper half plane), such that the transition maps are homeomorphisms with respect to the relative topology on the closed disk and holomorphic on the interior (see e.g. Ahlfors and Sario [6]). We call a chart an interior chart if it maps into the open disk (equivalently, its domain contains only points in .E); we call it a border chart if its domain contains a point in .∂E.

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Let .α ∈ Dl,m (E). Given a chart .ψ : U → V in this atlas, we say that .ψ ∗ α = h(z)dzl d z¯ k ∈ L2l,m (D+ ; V ) if and only if f .

V

|h(z)|2 d z¯ ∧ dz < ∞. 2 2−2l−2m 2i (1 − |z| )

In other words we demand that the pull-back under the chart be in .L2 with respect to the hyperbolic metric on .D+ . Theorem 6.8.14 ([79]) Let .E be a bordered surface of type .(g, n). The following are equivalent. (1) .α ∈ L2l,m (E); (2) .ψ ∗ α ∈ L2l,m (D+ ; V ) for all charts .ψ : U → V in the border atlas; (3) .ψ ∗ α ∈ L2l,m (D+ ; V ) for a specific collection of interior charts which cover .E and for specific collar charts .ψ1 , . . . , ψn of .∂1 E, . . . , ∂n E. Essentially, the idea is that the hyperbolic metric on a collar neighbourhood of a boundary behaves analytically the same as the hyperbolic metric on the disk near 1 .S . This is a useful tool for applying the sewing point of view. This local characterization can be used to show the following. Theorem 6.8.15 ([79]) Let .E and .E1 be bordered surfaces of type .(g, n) and let f : E → E1 be quasiconformal. Then . F |∂E ∈ QSWP (E) if and only if F is homotopic rel boundary to a quasiconformal map .F0 : E → E1 such that

.

μ(F0 ) ∈ L2−1,1 (E) ∩ L∞ −1,1 (E)1 .

.

Ahead, we will see that a result of Yanagishita shows that such a representative is obtained from the Douady–Earle extension. Remark 6.8.16 It was also shown in [77, 80] that .φk = F |∂k E ∈ QS(S1 , E) for .k = 1, . . . , n if and only if the corresponding rigging .f = (f1 , . . . , fn ), .f : D+ → qc E P obtained from the fiber map .C is in .OWP . As a consequence, Theorem 6.8.15 is a generalization of Theorem 6.5.2 of Cui and Takhtajan–Teo to type .(g, n) surfaces. In particular, Corollary 6.8.17 ([77, 79]) Let .E be a bordered Riemann surface of type .(g, n). Then TWP (E) = {[E, F, E1 ] ∈ T (E) : μ(F ) ∈ L2−1,1 (E) ∩ L∞ −1,1 (E)1 }.

.

(6.8.1)

Thus (6.8.1) gives an alternate definition of the Weil–Petersson Teichmüller space for these surfaces.

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Bers Embedding Model

Yanagishita [137] constructed a complex structure on Weil–Petersson Teichmüller space using the Bers embedding model . We outline Yanagishita’s results on the Bers embedding in the .L2 case. In fact, these results hold in .Lp for .p ≥ 1 (or in a few cases .p ≥ 2). Let G be an arbitrary Fuchsian group. Let .T p (G) denote the subset of elements p .[μ] of .T (G) with representatives .μ ∈ L −1,1 (N ) where N is a fundamental domain of G. p

Remark 6.8.18 Equivalently, .μ ∈ L−1,1 (E) on the resulting quotient surface .E = D− /G. Thus for .p = 2 this agrees with the definition given in Eq. (6.8.1) of Radnell–Schippers–Staubach, (though that definition was only stated for type .(g, n) bordered surfaces). From here on, we will restrict to .p = 2, which is the case with a complex Hilbert manifold structure and Riemannian metric. Yanagishita showed that the map σ : T 2 (G) → L∞ −1,1 (D− ; G)

.

taking .[μ] to the Douady–Earle extension of the boundary values of .wμ is in L2−1,1 (D− ; G). 2 Denote .Ael2 (D− ; G) = L∞ −1,1 (D− ) ∩ L−1,1 (D− ; G), where this is given the direct sum norm

.

|| · ||2,∞ = || · ||∞ + || · ||2

.

as in Cui (note that Cui used the notation .M1,∗ ). We also define the perturbed group Gψ for .ψ ∈ T (D− ; G) as in (6.3.7). Recall that Cui showed that the Weil–Petersson quasisymmetries form a group. This is not true for G-invariant quasisymmetries, but the following theorem generalizes the result of Cui in some sense.

.

Theorem 6.8.19 ([137]) For any Fuchsian group G, the following are equivalent. (1) .ψ ∈ T 2 (G). (2) .ψ −1 ∈ T 2 (Gψ ). 2 (3) .σ (ψ) ∈ L∞ −1,1 (D− ; G) ∩ L−1,1 (D− ; G). Written in the direct picture, setting .E and .Eψ to be .D− /G and .D− /Gψ respectively, this shows that a change of basepoint .Rψ takes .T 2 (E) to .T 2 (Eψ ). Remark 6.8.20 Note that this is a different theorem than Theorem 6.8.12 even in the type .(g, n) case, since the complex structures are not yet known to be the same. Ahead we will see that they are in fact the same.

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Yanagishita also defined the following analogue of the Teichmüller distance. For ψ1 , ψ2 ∈ T 2 (G), define

.

{|| } || || μ1 − μ2 || 2 || || .l2,∞ (G)(ψ1 , ψ2 ) = inf : μk ∈ Ael (G), [μk ] = ψk , k = 1, 2 . ||1 − μ μ || 1 2 2,∞ We also have the following theorem, partly generalizing Theorem 6.5.2 of Cui and Takhtajan–Teo. The inclusion .β(T 2 (G)) ⊆ β(T (G)) ∩ A22 (D+ ; G) is due to Yanagishita, and the reverse inclusion is due to Matsuzaki. Theorem 6.8.21 ([52, 137]) For any Fuchsian group, β(T 2 (G)) = β(T (G)) ∩ A22 (D+ ; G).

.

Remark 6.8.22 This also generalizes Theorem 6.5.2, but interestingly, it does so in a different way than observed in Remark 6.8.16. That is, Theorem 6.8.15 characterizes the .L2 elements of Teichmüller space in terms of the induced conformal maps of sewn disks, whereas the theorem above characterizes them in terms of the lifted conformal map of .D+ . In some sense, these arise from two qualitatively different applications of Bers’ trick. See Remark 6.3.5. In fact, Matsuzaki [52] showed that there is a foliation of .T (G), in the spirit of Theorems 6.5.17 and 6.5.18. Theorem 6.8.23 For any .[wμ ] ∈ T (G), [ ] −1 2 2 β Rw (T (G)) = β(T (G)) ∩ β([w ]) + A (D ; G) . μ + 2 μ

.

Again, this was shown to hold for the .Lp Teichmüller space for all .p ≥ 2. Thus one can ask about topological properties of the Bers embedding. Indeed we have the following. Theorem 6.8.24 ([137]) Let G be an arbitrary Fuchsian group. ( ) ( ) (1) .σ is continuous from . T 2 (G), l2,∞ to . Ael2 (G), || · ||2,∞ . (2) The Bers embedding .β is continuous from .(T 2 (G), l2,∞ ) to .A22 (D+ ; G). Much more can be said, if one furthermore assumes that G satisfies what Yanagishita calls “Lehner’s condition”. Definition 6.8.25 A Fuchsian group G is said to satisfy Lehner’s condition if the infimum of the lengths of all simple closed geodesics in .D− /G is finite. He then shows the following, based on results of Rajeswara, Niebur and Sheingorn, and Lehner. Theorem 6.8.26 ([137]) A Fuchsian group G satisfies Lehner’s condition if and only if the inclusion from .A22 (G) into .A∞ 2 (G) is bounded.

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Observe that a type .(g, n) bordered surface can be seen directly from the definition to satisfy Lehner’s condition. Yanagishita showed that under this condition, the Bers embedding is a homeomorphism onto its image. Theorem 6.8.27 ([137]) Let G be a Fuchsian group satisfying Lehner’s condition. Then the Bers embedding .β is a homeomorphism from .(T 2 (G), l2,∞ ) onto its image in .A22 (D+ ; G). Thus .T 2 (G) possesses a complex Hilbert manifold structure. This completes the construction of the complex structure from the Bers embedding, analogous to the classical .L∞ Banach manifold structure of Sect. 6.4.1.1. Furthermore, with respect to this complex structure, change of base point is a biholomorphism. Theorem 6.8.28 ([137]) For any Fuchsian group G and .ψ ∈ T 2 (G), the change of base point map .Rψ is a homeomorphism from .(T 2 (G), l2,∞ (G)) onto .(T 2 (Gψ ), l2,∞ (Gψ )). If G satisfies Lehner’s condition, then .Rψ is a biholomorphism with respect to the complex Hilbert manifold structure induced from .A22 (D+ ; G) by Theorem 6.8.27. On the Riemann surface, the change of base point takes .TWP (D− /G) to TWP (D− /Gψ ).

.

Remark 6.8.29 It should be noted that the statement regarding complex structure in Theorem 6.8.27 does not follow from Theorem 6.8.7 even in the case of type .(g, n) surfaces, nor does Theorem 6.8.7 follow from Theorem 6.8.27. This is because the complex structures had not yet been shown to be the same. Ultimately, it was shown that the three complex structures agree, as we will see below.

6.8.2.3

Deformation Model

The deformation model of the complex structure and tangent space was given in [77], and later published in [80], for type .(g, n) bordered surfaces such that .2g − 2 + n > 0. The theory is based on the Ahlfors–Weill reflection, as in the classical ∞ model of Sect. 6.4.1.2. .L Below, we use the notation 2 TBD(E) = L∞ −1,1 (E) ∩ L−1,1 (E).

.

Here TBD stands for “tangent Beltrami differential”, that is tangent vectors to curves 2 in both .L∞ −1,1 (E)1 and .L−1,1 (E). The following existence theorem, which was of great use in describing the tangent space, justifies considering this space. Theorem 6.8.30 ([80]) Let .E be a type .(g, n) bordered surface such that .2g − 2 + n > 0. Let .t |→ [E, Ft , Et ] be a holomorphic curve in .TWP (E) with respect to the complex structure obtained in Theorem 6.8.7, such that .[E, F0 , E0 ] = [E, Id, E].

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There is a .δ > 0 and a one-parameter family of representatives .(E, Ft Et ) for .|t| < δ such that (1) .||μ(Ft )||2 is uniformly bounded in t; (2) .t |→ μ(Ft ) is holomorphic in .L2−1,1 (E); (3) .t |→ μ(ft ) is holomorphic in .L∞ −1,1 (E). Thus one can find curves which are simultaneously holomorphic in both the .L∞ and 2 .L spaces. It was also shown that Theorem 6.8.31 ([80]) Let .E be a type .(g, n) bordered surface. We have H−1,1 (E) ⊂ O1,1 (E) and the inclusion is bounded.

.

∗ This is equivalent to the boundedness of the inclusion .A22 (E ∗ ) → A∞ 2 (E ). This also follows from Theorem 6.8.26, using the fact that type .(g, n) bordered surfaces satisfy Lehner’s condition. Alternately, one can say that these two theorems imply that type .(g, n) surfaces satisfy Lehner’s condition. As a corollary, we obtain

Corollary 6.8.32 ([80]) If .E is a type .(g, n) bordered surface, then O−1,1 (E) ∩ TBD(E) = H−1,1 (E).

.

It is natural then to define Nr (E) = N (E) ∩ L2−1,1 (E)

.

where recall that .N (E) are the infinitesimally trivial differentials, that is, those differentials which are tangent to the Teichmüller equivalence relation. Using Corollary 6.8.32, and properties of the Bergman projection on differentials, Radnell– Schippers–Staubach obtained the following characterization of .TBD(E) modulo infinitesimally trivial differentials. Theorem 6.8.33 ([80]) Let .E be a type .(g, n) bordered surface. Then we have the direct sum decomposition TBD(E) = Nr (E) ⊕ H−1,1 (E).

.

The projection from .TBD(E) onto .H−1,1 (E) is bounded. Finally, .Nr (E) is the kernel of the derivative of the Bers embedding .D0 (β ◦ 0) at .[E, Id, E]. This in turn led to a description of the tangent vectors in .TWP (E). Theorem 6.8.34 ([80]) Let v be a vector tangent to .TWP (E) at .[E, Id, E]. There is a holomorphic curve .t |→ [E, Ft , Et ] for .|t| < δ such that .[E, F0 , E0 ] = [E, Id, E] and (1) .μ(Ft ) ∈ H−1,1 (E) for .|t| < δ;

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(2) .μt is holomorphic in t with respect to .H−1,1 (E); (3) v is tangent to .[E, Ft , Et ] at .t = 0. This allows us to construct a new atlas. Recall that .0 : L∞ −1,1 (E)1 → T (E) takes a Beltrami differential to its Teichmüller space representative by solving the Beltrami equation. First, we locally invert .0. Theorem 6.8.35 ([80]) Let .E be a type .(g, n) bordered Riemann surface such that 2g − 2 + n > 0. There exists a neighbourhood U of .0 ∈ H−1,1 (E) such that . 0|U is a biholomorphism onto .TW P (E) with respect to the complex structure obtained from Theorem 6.8.7.

.

Using holomorphicity of change of base point with respect to the complex structure (Theorem 6.8.12), we have Corollary 6.8.36 ([77, 80]) Let .E be a bordered surface of type .(g, n) such that 2g − 2 + n > 0. Let .F : E → E0 be a quasiconformal map. There is an open neighbourhood B of 0 in .H−1,1 (E0 ) such that

.

w(E,f,E0 ) := RFμ 0E0

.

is a biholomorphism onto an open neighbourhood of .[E, F, E0 ]. The collection of inverses of these maps form an atlas for a complex Hilbert manifold structure on 7 .TWP (E) compatible with the complex structure obtained in Theorem 6.8.7. Remark 6.8.37 The inverses of these maps are .A−1 E0 βE0 RF −1 , so up to the obvi0

ously isometric reflection into .A22 (E0∗ ) these are the .L2 versions of the classical charts in Sect. 6.4.1.2. Note that because the Ahlfors–Weill reflection .A is obviously an isometry, we also automatically obtain that .βE0 RF −1 is an atlas of charts into 0

A22 (E0∗ ).

.

The results above show that we can identify the tangent space at an arbitrary point .[E, F, E0 ] ∈ TWP (E0 ) with T[E,F,E0 ] TWP (E) ∼ = D0 RF H−1,1 (E0 ).

.

It is an immediate consequence that the Weil–Petersson metric converges on every tangent space. In fact we can define the Weil–Petersson pairing as follows. Let .μ, ν ∈ H−1,1 (E) be elements in the tangent space of .TWP (E) at .[E, Id, E]. Then let .

7 There

[E,Id,E]

is a typographical error in the statement of this result in [80, Theorem 4.8]: .H−1,1 (E) should be replaced by .H−1,1 (E0 ) as it is here.

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denote the inner product in .H−1,1 (E). Then for any arbitrary point .[E, F, E0 ] ∈ TWP (E) and tangent vectors .v, w ∈ T[E,F,E0 ] TWP (E) we define the Weil–Petersson Riemannian metric by / .

\ D(RF )−1 v, D(RF )−1 w

[E,F,E0 ]

.

(6.8.2)

This converges at every tangent space and is immediately seen to be smooth with respect to the biholomorphicity of the change of base point map. It is the unique metric which is invariant under change of base point and agrees with the pairing in .H−1,1 (E0 ) at the identity. Later, Yanagishita extended the deformation atlas result to general surfaces .E = D− /G for Fuchsian groups G satisfying Lehner’s condition [138]. Theorem 6.8.38 ([138]) Let G be a Fuchsian group satisfying Lehner’s condition. The charts .βE0 RF −1 forms an atlas for the complex Hilbert manifold structure 0 compatible with the complex structure obtained from Bers embedding via Theorem 6.8.27. Change of base point is holomorphic. He also extended the description of the tangent space and the Weil–Petersson pairing to surfaces .E = D− /G for Fuchsian groups G satisfying Lehner’s condition. Theorem 6.8.39 ([138]) Corollary 6.8.32 holds more generally for surfaces satisfying Lehner’s condition. Theorem 6.8.40 ([138]) Theorem 6.8.33 holds more generally for surfaces satisfying Lehner’s condition. Yanagishita thus obtains the convergent Weil–Petersson pairing (6.8.2) for surfaces satisfying Lehner’s condition. The statements in Theorem 6.8.35 and in Theorem 6.8.38—that the atlas defines a complex structure—are the same in the case of type .(g, n) bordered surfaces; indeed Theorem 6.8.35 is a consequence of Theorem 6.8.38. However, the statements about the compatibility of this structure are completely different: Theorem 6.8.35 says that this atlas is compatible with the complex Hilbert manifold structure obtained from the fiber structure, whereas Theorem 6.8.38 says that this atlas is compatible with the complex Hilbert manifold structure obtained from the Bers embedding. Putting these results together, we obtain that all three agree. That is, by Theorems 6.8.7, 6.8.35, and 6.8.38 we have the following. Corollary 6.8.41 Let .E be a type .(g, n) bordered surface such that .2g − 2 + n > 0. The complex Hilbert manifold structures obtained from the fiber model, the Bers embedding, and the deformation model, are all equivalent. This was observed by the authors in our survey [96]. Remark 6.8.42 To the .L2 analogues of the two classical complex structures, we have added the new complex structure derived from the fiber/CFT point of view. In the classical .L∞ case this also required some extra work [73].

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Finally, we conclude with another interesting result of Yanagishita. In stark contrast to Cui’s completeness theorem (Theorem 6.5.7), the Weil–Petersson distance is almost never complete. Theorem 6.8.43 ([140]) Let .E be a Riemann surface satisfying Lehner’s condition. If .E is conformally equivalent to the punctured disk or the disk, then .TWP (E) is complete. Otherwise, it is not complete. Remark 6.8.44 In fact the theorem is shown to hold for .p ≥ 2; the completeness of the p-Weil–Petersson Teichmüller space of the disk for .p ≥ 2 is due to Matsuzaki [53].

6.9 Kähler Geometry and Global Analysis of the Weil–Petersson Teichmüller Space In this section we give a brief overview of some geometric results associated with the Weil–Petersson metric. The emphasis is on Kählericity, Kähler potentials, Chern classes and index theorems. Moreover, we shall also recall some of the results in the finite-dimensional case (the case of compact Riemann surfaces), both for the sake of comparison with the currently available infinite-dimensional results, and also for their intrinsic interest in the Weil–Petersson context. The presentation given here is by no means exhaustive and we confine ourselves to just a selection of results that are relevant to the WP-class Teichmüller theory.

6.9.1 Kählericity, Curvatures and Kähler Potentials In the paper [131] Weil asked whether the WP metric is Kähler and if so, what is its curvature. This problem was solved by Ahlfors [3] , who showed that for compact Riemann surfaces the Weil–Petersson metric is Kähler. More specifically he proved the following theorem: Theorem 6.9.1 Let .R be a compact genus .g > 1 Riemann surface. On the Teichmüller space .T (R) Bers’ coordinates are geodesic for the Weil–Petersson metric at the reference point. In particular this metric is Kähler. In the commentary to his collected works [5], Ahlfors mentioned that Weil also had a proof but had not published it. Later Ahlfors computed the curvatures of holomorphic sections and the Ricci curvature [4], which resulted in Theorem 6.9.2 The Ricci curvature, the holomorphic sectional curvature and the scalar curvature of the Weil–Petersson metric are negative.

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In [136], Wolpert used the classical Maaß calculus of invariant differential operators to obtain a formula for the Riemannian curvature tensor, showing that the sectional curvatures of the Weil–Petersson metric are negative. The negative curvature result was obtained independently by Royden [87, 88] and Tromba [124]. More explicitly let .R be a compact Riemann surface of genus .g > 1. Denote by .{μα ; α = 1, . . . , 3g − 3} a basis of the vector space of harmonic Beltrami differentials on .R corresponding to a set of tangent vectors .{∂/∂sα ; α = 1, . . . , 3g − 3}. Introducing the basis f gα β¯ =

.

R

(6.9.1)

μα μβ dAhyp ,

one can write the Weil–Petersson metric as ωWP = igα,β¯ dzα ∧ d z¯ β ,

.

using Einstein’s summation convention. The following result was established in [136] and [124]. Theorem 6.9.3 Let .R be a compact Riemann surface of genus .g > 1 then one has (1) The holomorphic sectional and Ricci curvatures of .ωWP on .T (R) for .g > 1 are −1 bounded from above by . 2π(g−1) . (2) The sectional curvature of .ωWP is negative. (3) The curvature tensor of the Weil–Petersson metric is equal to f Rα βγ ¯ δ¯ = − 2 f

.

−2

R

( )( ) (D − 2)−1 μα μβ μγ μδ dAz −1

R

(D − 2)

(

)

(6.9.2)

(μα μδ ) μγ μβ dAhyp

where D is the real Laplacian on .L2 -functions on .R with spectrum in .(−∞, 0]. In [101], Schumacher went beyond the negativity of the sectional curvature in Theorem 6.9.3, and proved the following result. Theorem 6.9.4 The Weil–Petersson metric on .T (R) for .g > 1 has strongly negative curvature in the sense of Siu, i.e. its Riemann tensor satisfies ( )( ) α β α β Rα βγ Aδ B γ − C δ D γ ≥ 0 ¯ δ¯ A B − C D

.

for all complex vectors .Aα , B β , C γ , D δ , and equality holds only for .Aα B β = α .C D β for all .α and .β. Recall from Sect. 6.6.1 that in finite dimensions the Siegel disk possesses a natural Kähler metric. The Kähler potential for this metric is .−Tr log (1 − ZZ).

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Recall that Segal defined an infinite Siegel disk on which this Kähler metric is defined [103]. Note that the condition that Z be Hilbert–Schmidt (part of Segal’s definition of the infinite Siegel disk) is precisely the condition that the Kähler potential converges. Kirillov and Yuriev [48] and Nag [57] showed that the pull-back of a natural Kähler metric on the infinite Siegel disk is the Weil–Petersson metric. Thus .

( ) − Tr log 1 − Z ∗ Z

(6.9.3)

is also a Kähler potential for the Weil–Petersson metric, as was observed explicitly in [41, 48]. Hong–Rajeev noted that .Diff(S1 )/Möb(S1 ) was not complete with respect to the Kähler metric, which indicated that it was not the correct analytic setting for the Weil–Petersson metric. As we saw in Secton 6.6, this analytic setting is provided by .TWP (D− ). A generalization of the results of Ahlfors, Wolpert and Tromba mentioned above in the infinite dimensional setting was Takhtajan–Teo’s Kähler–Einstein theorem for the universal Teichmüller space. To describe it consider the universal Teichmüller space .T (D− ), the universal Teichmüller curve .T (D− ), and a projection .π : T (D− ) → T (D− ). The universal Teichmüller curve .T (D− ) is a holomorphic fiber space over .T (D− ) considered by Bers. The fiber over each point .[μ] ∈ T (D− ) ¯ with the complex structure inherited from the Riemann is a quasidisk .wμ (D− ) ⊂ C sphere and .T (D− ) = {([μ], z); [μ] ∈ T (D− ), z ∈ wμ (D− )}. The Weil– Petersson universal Teichmüller space .TWP (D− ) is the connected component of the origin (i.e. the identity) of .T (D− ), and the Weil–Petersson universal Teichmüller curve .TWP (D− ) is obtained by restricting to .TWP (D− ). More explicitly, one can ( )−1 , summarize the results of Takhtajan–Teo as follows. Let .G = 12 A + 12 where .A is the Laplace–Beltrami operator of the hyperbolic metric on .D− , and let .G(z, w) denote the integral kernel of G. This kernel is explicitly given by |z−w|2 2u+1 u+1 1 .G(z, w) = 2π log u − π , where .u = u(z, w) = (1−|z|2 )(1−|w|2 ) , see e.g. [40]. Then Takhtajan and Teo [116], later published as part I of [118], proved the following ground-breaking result in Weil–Petersson Teichmüller theory of surfaces of infinite conformal type. Theorem 6.9.5 Set ff G(f )(z) =

.

D−

G(z, w)f (w) dAhyp .

(6.9.4)

then the following claims hold: (1) The Weil–Petersson metric is a Kähler metric on the Hilbert manifold .T (D− ). Moreover the Bers coordinates are geodesic coordinates at the origin of .T (D− ).

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(2) Let .μα , μβ , μγ , μδ ∈ H−1,1 (D− ) .(defined in (6.5.3) which is isomorphic to the tangent space at the origin of the universal Teichmüller space.), be orthonormal tangent vectors. Then the Riemann tensor at the origin of .T (D− ) is given by Rα βγ ¯ δ¯ = −

.

∂ 2 gα β¯ > < )> < ( = − G (μα μ¯ δ ) , μβ μ¯ γ − μα μ¯ β , G μ¯ γ μδ ¯ ∂tγ ∂ tδ

where .gα β¯ is the Weil–Petersson metric, the pairing . is the pairing in 2 .L −1,1 (D− ). (3) .TWP (D− ) and .TWP (D− ) are topological groups in the Hilbert manifold topology of .T (D− ), and the Hilbert manifold .TWP (D− ) is also a Kähler–Einstein manifold with the negative definite Ricci tensor RicWP = −

.

13 ωWP , 12π

where .ωWP is the symplectic form of the Weil–Petersson Kähler metric on T (D− ).

.

Takhajan and Teo [115], later published as Part II of [116], also introduced the so-called universal Liouville action defined by ff S1 ([μ]) =

.

D+

| ( μ )|2 |A f | dAz +

ff D−

| ( )|2 | | |A gμ | dAz − 4π log |g ' (∞)| , μ (6.9.5)

where .wμ = gμ−1 ◦ f μ is the conformal welding corresponding to .[μ] in the WPclass Teichmüller space, and .Af := f '' /f ' is the pre-Schwarzian derivative of f . They showed that this action is a Kähler potential for the WP-metric on the Weil– Petersson Teichmüller space, in fact .

i ∂∂S1 = ωWP . 2

(6.9.6)

Furthermore, if .fμ is the conformal map associated to .[μ] ∈ TWP (D− ), and Z the Grunsky map associated with f , then S2 = log det(I − ZZ)

.

(6.9.7)

is a potential for the Weil–Petersson metric. This result of Takhtajan and Teo generalizes the formula (6.9.3) for the Kähler potential obtained from pulling back that on the finite Siegel disk, and completes the picture in Nag [57], Nag and Verjovsky [59], and Kirillov and Yuriev [48]. Finally in relation to the potential above, we also mention Takhtajan–Teo’s sewing formula for the determinants in terms of the Fredholm determinant of the

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welding curve. We will say more about this in Sect. 6.10.3 (Theorem 6.10.10), where we discuss zeta-regularized determinants. Remark 6.9.6 It is interesting that the quantities .S1 [89–92] and .S2 [95] were singled out by Schiffer and Schiffer–Hawley as having special geometric significance (though with greater regularity assumed). A variational formula for .S1 was given by Schiffer and Hawley in [95], see [116, Remark 3.11]. Schiffer and Hawley also show that .S1 = S2 . Even with the stronger regularity, the argument is not entirely complete, because it is only shown to be true under a specific set of variations. The paper [95] investigates the construction of invariant domain functionals out of kernel functions and quantities arising in potential theory, using sophisticated and inspired analogies with connections and curvature in Riemannian settings. One idea of the paper is to construct canonical mappings by minimizing functionals involving the curvature of the boundary.8 Like much of the work by Schiffer and his co-authors, the paper [95] exploits ideas regarding symmetry and invariance in geometry and physics, but one gets the impression that the technology of the time was not yet adequate. Indeed Schiffer was knowledgeable in those subjects, having studied invariant theory under Schur [93], and having had a life-long interest in physics, even co-authoring a textbook on general relativity [1]. In any case, one cannot deny the persistent relevance of his techniques and ideas for complex geometry and physics. An attentive reader can even find there a Polyakov–Alvarez type formula for the Fredholm determinant of a curve. The invention of the concepts of overfare and scattering on Riemann surfaces by the authors in [98], led us to derive a formula for the Kähler potential of the WP-metric. One of the basic objects in our work is the so-called Schiffer operators whose definition we now recall. Definition 6.9.7 For a compact Riemann surface .R with Green’s function G (w, w0 ; z, q), the Schiffer kernel is defined by

.

LR (z, w) =

.

1 ∂z ∂w G (w, w0 ; z, q). πi

Let .R be a compact surface divided by a complex .r of simple closed curves into surfaces .E1 and .E2 . For .k = 1, 2, and .E1 and .E2 as above, we define the Schiffer comparison operator by Tk : A(Ek ) → A(E1 ∪ E2 ) ff α |→ LR (·, w) ∧ α(w),

.

Ek

8 This principle, with deep historical roots in complex analysis, is that there is a correspondence between extremal problems, invariant quantities, and canonical mappings.

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and the restriction operator R0k : A(E1 ∪ E2 ) → A(Ek )

.

α |→ α|Ek . Moreover, we define for .j, k ∈ {1, 2}, Tj,k = R0k Tj : A(Ej ) → A(Ek ).

(6.9.8)

.

We also have an analogue of the Cauchy operator, namely Definition 6.9.8 Let .r be a quasicircle in the Riemann sphere, whose complement ˙ has components .E1 and .E2 . Let .D(E) denote the homogeneous Dirichlet space (that is, modulo constants). Let .re approach .r from within .E1 . For fixed .q ∈ / r, the Cauchy operator is defined by .

˙ 2) ˙ 1 ) → D(E J˙ : D(E 1 lim 2π i e-0

h |→

(

f h(ζ ) re

1 1 − ζ −z ζ −q

) dζ z ∈ E2

where we discard constants. As a consequence of identities for the Schiffer operators and the properties of the Cauchy operators in [98], the authors derived the following formula for this Kähler potential. Theorem 6.9.9 ([97]) The potential for the Weil–Petersson metric on the Weil– Petersson Teichmüller space of the disk is .

( ) log det(T∗1,2 T1,2 ) = log det J˙ ∗ J˙ .

(6.9.9)

This can be compared with Theorem 6.10.10 below. Our theorem involves the complex Cauchy operator in place of the Neumann jump operator, and does not require the assumption that the curve is .C 3 . Another outcome of the scattering theory developed in [98] is the relationship between the Fredholm indices of the Schiffer comparison operators and the topological invariants of the Riemann surfaces. Theorem 6.9.10 ([98]) If .E1 , .E2 are connected, and of genus .g1 and .g2 , then index(T1,2 ) = g1 − g2 .

.

If .E2 is connected and of genus g, and .E1 consists of n disjoint simply connected regions, then index(T1,2 ) = 1 − n + g.

.

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Returning to the Kähler property of the WP-Metric, in the paper [138] Yanagishita obtained the following generalization of Takhtajan and Teo’s result. Theorem 6.9.11 ([138]) Let G be a Fuchsian group satisfying Lehner’s condition. Then the Weil–Petersson metric on .TWP (D− /G) is Kähler. Furthermore, Theorem 6.9.12 ([138]) Let G be a Fuchsian group satisfying Lehner’s condition. Then the holomorphic sectional and Ricci curvatures of the Weil–Petersson metric on .TWP (D− /G) are negative.

6.9.2 Chern Classes, Quillen Metric and Zeta Functions Next we discuss the connection of the WP–Teichmüller theory to Chern–Weil theory. In this context, Wolpert [136] computed the first Chern form .c1 (v) of the vertical tangent bundle of the universal Teichmüller curve, with the Hermitian metric induced by the hyperbolic metric on the fibers. Then since integration of powers of .c1 (v) over fibers produces characteristic classes .κn (see (6.9.13) ahead) on Teichmüller space, Wolpert showed that that .κ1 is a multiple of the Weil–Petersson Kähler form .ωWP . Another important and interesting topic is the Quillen metric on the so-called determinant line bundle and its connection with WP–Teichmüller theory. To describe this, we consider a holomorphic family .π : X → P of compact complex manifolds over a compact base P . Let also E be a vector bundle over X, with projection .π ' : E → X. Given .p ∈ P we identify the fiber .π −1 (p) with .Xp and restrict .π ' to a map ' .πp : Ep → Xp . Now there is a family of .∂-operators, denoted by .∂ p , acting on the sections of the bundle .(Ep , π ' , Xp ). This family defines an index bundle .ind ∂ over P in the sense of K-theory, where .ind ∂ = Ker ∂ − Coker ∂. If .ch(·) denotes the Chern character of a bundle and .td(Tvert X) denotes the Todd class of the vertical tangent bundle of X, then as a result of the Atiyah–Singer family index theorem [8], one has that ch(∂) = πf [ch(E) · td(Tvert X)],

.

−1

(6.9.10)

where .πf [·] : H ∗ (X) → H ∗−dim π (p) (P ) is the operation of integration along the fibers. If the bundles .Tvert X and E are also equipped with Hermitian metrics, then these bundles will carry canonical unitary connections compatible with their metrics and therefore, using these connections, .ch(E) and .td(Tvert X) will be represented by closed differential forms on X. Furthermore, if the base P is not compact and the Atiyah–Singer family index theorem (6.9.10) doesn’t apply, then if the index bundle .ind ∂ happens to be a vector bundle (this is not always true because the dimensions of the fibers .Ker ∂ p

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and .Coker ∂ p may jump), one could hope for a local version of (6.9.10), called a local index theorem. This is obtained by using a particular connection on the vector bundle .ind ∂ (if one such exists), so that (6.9.10) holds as an equality between the corresponding differential forms of its left- and right-hand sides. As it turns out, proving a local index theorem is intimately connected to the problem of expressing the Chern character and the Todd class explicitly in terms of the maps .π and .π ' defined above. But one confronts complications in producing local index theorems if .ind ∂ is not a vector bundle. So to remedy this situation, Quillen [69] suggested the study of the determinant line bundle, which in our case of study is defined by det ind ∂ = ∧max Ker ∂ ⊗ (∧max Coker ∂)−1 ,

.

and is a holomorphic line bundle over P (this of course requires the validity of certain conditions on the bundles .(X, π, P ) and .(E, π ' , X) ). If .|| · || denotes the ∗ 2 .L norm on .det ind ∂ induced by the metrics on E and .Tvert X and if .∂p ∂p is ' the associated Laplacian acting on the sections of the bundle .(Ep , π , Xp ), then ∗ .det ∂p ∂p is given by the Laplacian’s zeta function determinant regarded as a function on P , and one can define the Quillen metric as || · ||Q =

.

|| · || ∗

(det ∂ ∂)1/2

.

One of the interesting features of Quillen’s metric is that the curvature form (first Chern class) of the line bundle .(det ind ∂; || · ||Q ) on P is, up to a multiplicative constant, the canonical Kähler form on P . Now in order to obtain results that are interesting from the geometric and global analytic points of view in Teichmüller theory, the choices of P and X and a specific bundle E over X are very significant. For example in the abstract setting mentioned above let us choose .P = Tg i.e., the Teichmüller space of compact genus g Riemann −s surfaces, .X = Tg i.e. the Bers fiber space of Teichmüller curves, and .E = Tvert Tg i.e. the .−s th power of the vertical holomorphic line bundle of .Tg (the sections of this bundle are the holomorphic .(−s, 0)-differentials). Once again, there is a family of .∂ operators associated with this setting, which is denoted here by .∂ s and act on the .(s, 0)-differentials on Riemann surfaces. In this situation, Belavin and Knizhnik [9] showed that the first Chern form of the corresponding .det ind ∂ s can be expressed as f −s .c1 (det ind ∂ s ) = (ch(Tvert Tg ) · td(Tvert Tg ))2,2 , (6.9.11) π −1 (p)

where .(·)2,2 denotes the .(2, 2) component of a differential form on the Teichmüller curve, and .π −1 (p), for .p ∈ Tg , denotes the fibers of the bundle .π : Tg → Tg . Formula (6.9.11) can be regarded as a local index theorem because it establishes an equality between the .(1, 1) forms of the left and right hand sides of (6.9.11). Now if

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−s one endows .Tvert Tg with a metric in such a way that for every .p ∈ Tg each .π −1 (p) would be equipped with the Poincaré metric, then as was shown by Wolpert [136], one has the following analogue of Quillen’s result, namely

c1 (det ind ∂ s ) =

.

6s 2 − 6s + 1 ωWP , 12π 2

(6.9.12)

where .ωWP is the Weil–Petersson Kähler form on .Tg (E). If one instead takes .E = Tvert T (D− ), then the hyperbolic metric on .w μ (D− ) defines a Hermitian metric on .Tvert T (D− ). As we mentioned earlier this can be used to produce a canonical unitary connection on .Tvert T (D− ), compatible with the metric, such that .ch(Tvert T (D− )) and .td(Tvert T (D− )) will be represented by closed differential forms on .T (D− ). Thus calculating the curvature form of this connection and using integration on the fibers, one hopes to connect fiber integrals of powers of the curvature form .O (first Chern class) with some invariant quantities, such as the metric. To this end, following Wolpert [136], Takhtajan and Teo [116] considered the Miller–Morita–Mumford characteristic forms. These are .(n, n)-forms on the Hilbert manifold .T (D− ), defined by κn = (−1)n+1 πf [(c1 (Tvert T (D− )))n+1 ],

.

(6.9.13)

where .πf : H ∗ (T (D− )) → H ∗−2 (T (D− )) is the operation of integration along the fibers. Observe that (6.9.13) yields for instance that .κ1 can be calculated by fiberintegration of the square of the curvature form .O. Takhtajan and Teo showed the following. The Miller–Morita–Mumford characteristic forms .κn are right-invariant on the Hilbert manifold .T (D− ) and for .μ1 , . . . , μn , ν1 , . . . , νn ∈ H−1,1 (D− ) = T[0] T (D− ) one has that κn (μ1 , . . . , μn , ν¯ 1 , . . . , ν¯ n )

.

=

ff ) ( ) ( i n (n + 1)! E . . . G μ dAhyp , sgn(σ ) G μ ν ¯ ν ¯ 1 n σ (1) σ (n) (2π )n+1 D− σ ∈Sn

where G is given by (6.9.4). Moreover, if .T (D− ) is equipped with its Hilbert manifold structure then κ1 =

.

1 ωWP . π2

(6.9.14)

¯ In [117] Takhtajan and Zograf proved a local index theorem for a family of .∂operators on a type .(g, n) punctured surface .E, i.e., a compact surface of genus g with n punctures. This case corresponds to the general construction above with the following choices, namely take .P = Tg,n , i.e., the Teichmüller space of punctured

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Riemann surfaces of type .(g, n), .X = Tg,n , i.e., the corresponding universal family (so that the fibers of the fibration .π : Tg,n → Tg,n are Riemann surfaces of type −k th power of the vertical line bundle on .T .(g, n)), and .E = Tvert Tg,n i.e. the .k g,n . Now, in contrast with the case of compact surfaces, the Laplace operator .Ak = ∂¯k∗ ∂¯k associated with the Poincaré metric on punctured surfaces of type .(g, n) has a continuous spectrum. In connection with this fact, let us also recall the definition of Selberg’s Zeta function which for .Re s > 1 is given by the absolutely convergent product Z(s) =

.

∞ ( ) | | | | 1 − e−(s+m)|l| , {l} m=0

where .l runs over the set of all simple closed geodesics on .E with respect to the Poincaré metric, and .|l| is the length of .l. The function .Z(s) admits a meromorphic continuation to the whole complex s-plane with a simple zero at .s = 1. In the case of compact Riemann surfaces, D’Hoker and Phong [24] showed that the determinant of .Ak is defined via its Selberg zeta function .Z(s) up to a constant multiplier depending only on g and k, and is equal to .Z ' (1) for .k = 0, 1 and to .Z(k) for .k ≥ 2. The same goes also for punctured Riemann surfaces of type .(g, n). Using this, Takhtajan and Zograf calculated the first Chern form of the determinant line bundle .det ind ∂¯k on the Teichmüller space .Tg,n of type .(g, n) punctured surfaces, endowed with the Quillen metric, and showed that c1 (det ind ∂¯k ) =

.

6k 2 − 6k + 1 1 ωWP − ωcusp . 9 12π 2

(6.9.15)

Here we see that the result differs from (6.9.12) by an additional term, which turns out to be the Kähler form of a new Kähler metric on the moduli space of punctured Riemann surfaces of type .(g, n). In fact .ωcusp is the symplectic form of a Kähler metric .cusp on .Tg,n (E) which is invariant with respect to the modular group on .Tg,n (E). The metric .cusp is defined by means of the Eisenstein–Maaß series related to n punctures. In order to give a precise definition of the cusp-metric, consider a Riemann surface .E of type .(g, n) and assume that it is equipped with the Poincaré metric .Q and let .r be a torsion-free Fuchsian group such that .E = H/ r where .H is the open upper half-plane. Denote by .r1 , . . . , rn the set of non-conjugate parabolic subgroups in .r, and for every .i = 1, . . . , n fix an element .σi ∈ PSL .(2, R) such that −1 .σ i ri σi = r∞ where the group .r∞ is generated by the parabolic transformation th cusp of .z |→ z + 1. The Eisenstein–Maaß series .Ei (z, s) corresponding to the .i the group .r is defined for .Re s > 1 by the formula Ei (z, s) =

E

.

γ ∈r/ ri

( )s Im σi−1 γ z ,

i = 1, . . . , n

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It is a well-known fact that .Ei (z, s) can be meromorphically continued to the whole complex s-plane, and for .Re s = 12 the Eisenstein–Maaß series .Ei (z, s), i = 1, . . . , n, form a complete set of eigenfunctions of the continuous spectrum of the Laplace operator .A0 . Now one defines the cusp-metric as follows; first set f i =

μ, ν ∈ L2−1,1 (E),

μ¯ν Ei (·, 2)

.

E

i = 1, . . . n,

(6.9.16)

and it turns out that each scalar product .i gives rise to a Kähler metric on .Tg,n . The cusp-metric is the sum cusp =

.

n E i

(6.9.17)

i=1

which is a Kähler metric invariant under the Teichmüller modular group .Modg,n . In [85] Ratiu and Todorov constructed a determinant line bundle on + 1 1 .Diff (S )/Möb(S ), together with the corresponding Quillen norm. They further proposed that, using Quillen’s original construction of the determinant line bundle in [69], the curvature .(1, 1)-form of this Quillen metric is the Kähler form of the Weil–Petersson metric on .Diff+ (S1 )/Möb(S1 ), which would therefore be the infinite-dimensional analog of Takhtajan–Zograf’s result mentioned earlier. In connection with this circle of ideas, we would also mention the paper by Fujiki and Schumacher [32], where the authors used a significant generalization of Quillen’s work due to Bismut and Freed [15] and Bismut et al. [16] (concerning the first Chern form of a determinant bundle with Quillen metric), and a certain fiber integration formula, to construct a generalized Weil–Petersson form and to show that it is (up to a constant) the Chern form of a Hermitian line bundle on the moduli space.

6.10 Weil–Petersson Beyond Teichmüller Theory 6.10.1 Conformal Field Theory and String Theory In two-dimensional conformal field theory, Friedan and Shenker pointed out the role of moduli spaces of Riemann surfaces with punctures [30]. Vafa gave a formulation of conformal field theories using a moduli space of punctured surfaces endowed with local coordinates vanishing at the punctures [125]. Note that the “coordinates” are extra data beyond that of the conformal equivalence class of the surface. Segal sketched a definition of conformal field theory in [104] and [105]; the latter was a well-circulated preprint which was eventually published in [106]. In this definition, Riemann surfaces with boundary parametrizations (up to conformal equivalence) are the morphisms of a category whose objects are disjoint unions of circles (some labelled incoming and some labelled outgoing). Such surfaces can be

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sewn using the parametrizations. A heuristic correspondence between the moduli spaces of Vafa and Segal can be obtained if one sews on punctured disks using the boundary parametrizations, and the inclusion map becomes the inverse of the local coordinates. Remark 6.10.1 This heuristic correspondence between the two moduli spaces is one of the motivations for the fiber model of the complex structure on Teichmüller space. The idea is that the boundary values of a quasiconformal deformation .F : E → E1 of a bordered surface .E induce boundary parametrizations in the Segal model. In Vafa’s model, this data takes the form of elements of .Oqc (E1P ) for the corresponding compact surface .E1P obtained by sewing on caps. In Segal’s definition, a conformal field theory associates an operator between tensor products of Hilbert spaces associated with the incoming and outgoing boundary components. In categorical language, this is supposed to be a projective holomorphic functor. The association of the operator with the Riemann surface is required to be holomorphic, and thus one requires a complex structure on the moduli space and holomorphic sewing operation. Radnell and Schippers [71] showed that the moduli space can be identified with a quotient of Teichmüller space. To do this, the set of boundary parametrizations was enlarged to the set of quasisymmetric parametrizations; in the Vafa model, the set of coordinate data were enlarged to conformal maps of the disk with quasiconformal extensions. More precisely, and with alternate analytic choices, we make the following definitions. Definition 6.10.2 Consider pairs .(E P , f ) where .E P is a compact Riemann surface of genus g with n labelled punctures .p = (p1 , . . . , pn ) and .f = (f1 , . . . , fn ) ∈ Oqc (E P ). We say that two pairs .(E1P , f1 ) ∼ (E2P , f2 ) are equivalent if there is a biholomorphism .σ : E1P → E2P such that .σ ◦ f1 = f2 . The puncture model of the rigged moduli space is -P (g, n) = {(E P , f )}/ ∼ . M

.

The subscript P stands for “puncture” model. The rigged moduli space in the Segal border model is as follows. Definition 6.10.3 Consider pairs .(E, φ) where .E is a bordered surface of type (g, n) with ordered boundary curves .∂k E, .k = 1, . . . , n and .φ = (φ1 , . . . , φn ) is a collection of quasisymmetries .φk : S1 → ∂k E. We say that two pairs .(E1 , φ) ∼ (E2 , ψ) are equivalent if there is a biholomorphism .σ : E1 → E2 such that .ψ = σ ◦ φ. The border model of the rigged moduli space is .

-B (g, n) = {(E, φ)}/ ∼ . M

.

-B (g, n) and .M -P (g, n) [71]. There is a naturally defined bijection between .M

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This relates to the Teichmüller space as follows. Recall that .ModI is the subgroup of the modular group, whose elements fix the boundary pointwise. Theorem 6.10.4 ([71]) Let .E be a bordered surface of type .(g, n). Then there is a bijection -B (g, n)∼ -P (g, n). T (E)/ModI∼ =M =M

.

-B (g, n) and .M -P (g, n) are complex Banach manifolds, and sewing Furthermore, .M is holomorphic. Remark 6.10.5 In some citations of [71], it is stated that Radnell–Schippers extend the sewing operation to quasisymmetries, but this is a routine consequence of conformal welding. The interesting fact is rather the geometric connection between the Teichmüller space and the rigged moduli spaces. This brings insights in both directions: for example in Teichmüller theory, the CFT point of view leads directly to the fiber model of Teichmüller space. Customarily the rigging is chosen to be more regular; for example the boundary parametrizations are chosen to be analytic or smooth. The question is then what is the appropriate regularity for conformal field theory? The case for Weil–Petersson class parametrizations was advocated at length in [81]. We mention here four motivations for choosing Weil–Petersson class moduli space. The first two of these motivations are simple to state: .QSWP (S1 ) is the completion of .Diff(S1 ) (Theorem 6.5.7), and, .QSWP (S1 ) is a topological group (Theorem 6.5.8). The remaining two motivations are related to the determinant line bundle of .∂ ⊕ pr, which is part of the construction of certain CFTs satisfying the definition of Segal [43]. The definition of Segal involves the polarization of functions on the boundary .∂E induced by positive and negative Fourier modes of the pullback under the boundary parametrization. It is required that this polarization be an element of the restricted Siegel disk. Thus the boundary parametrization must be Weil–Petersson. Finally, one obtains a central extension of the group .Diff(S1 ), with specific central charge, by embedding it in the restricted general linear group. This requires that the associated composition operators be in the restricted group [43, Appendix D]. One can enlarge the group, but only if its action is still in the Shale group. We are grateful to Yi-Zhi Huang for explaining this point to us. Again, this is precisely the condition that the parametrization be a Weil–Petersson quasisymmetry (cf Theorem 6.6.7, and also [81, p 232]). One can indeed substitute Weil–Petersson class objects in the definitions of the rigged moduli spaces, and obtain the following. Theorem 6.10.6 ([80]) Let .E be a bordered surface of type .(g, n) for .2g + 2 − n > 0. Then there is a bijection ∼TWP (E)/ModI ∼ =M WPB (g, n) = M WPP (g, n).

.

Thus .M WPB (g, n) and .M WPP (g, n) are complex Hilbert manifolds.

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As we touched on in Sect. 6.6, the diffeomorphism group of the circle plays a role in representation theory. It also appears in various physical models. We indicate some literature as it connects to Weil–Petersson Teichmüller theory. In string theory, .Diff(S1 ) is the group of reparametrizations of a string, and invariance under this reparametrization can be required in different settings. The Lie algebra of .Diff(S1 ) can be identified with the smooth vector fields on the circle. The generators of this Lie algebra are .−ieinθ ∂/∂θ . The complexification is generated by elements of the form .zn−1 ∂/∂z, .n ∈ Z. The vector space of polynomial expressions in these generators, with the vector field Lie bracket, is called the Witt algebra. The Witt algebra is viewed in the physics literature as the conformal symmetry group [10, 100].9 Although the group .Diff(S1 ) has a Lie algebra .Vect(S1 ), there is no complexification. That is, there is no Lie group whose Lie algebra is the Witt algebra (or a completion) [100]. Segal [104, 105] argues that the object which is closest to filling the role of the complexification of .Diff(S1 ) is the rigged moduli space of the twicepunctured sphere with coordinates. Equivalently, it is the rigged moduli space of the annulus with parametrizations. This is a semigroup, since sewing preserves the type .(0, 2) and thus induces a multiplication, but there is no inversion. If one ¯ uses quasisymmetric parametrizations or equivalently elements of .Oqc (C\{0, ∞}) for local coordinates, one obtains from Theorem 6.10.4 that this semigroup can be identified with the Teichmüller space of an annulus .A modulo a .Z action by boundary twists, that is .

-P (0, 2) ∼ -B (0, 2) ∼ M =M = T (A)/Z

(see [74]), and sewing in this semigroup is holomorphic with respect to the complex structure obtained from .T (A). By Theorem 6.10.6 one could replace the moduli spaces above by their Weil–Petersson versions. It has been suggested that the universal Teichmüller space might play a role in a non-perturbative approach to string theory [19, 20, 41, 63]. Bowick and Rajeev [20] consider the loop space in .Rd−1,1 space-time, and .Diff(S1 )/S1 appears in association with reparametrization invariance and changes of complex structure (see Sect. 6.6.1). They state “An approach to string theory also based on complex geometry has been proposed by Friedan and Shenker [30]. It is of interest to know the relation between the ‘universal Teichmüller space’ with which they work and .Diff(S1 )/S1 .” Note that this remark does not actually refer to the universal Teichmüller space: Friedan and Shenker use the term “universal moduli space” and argue that this “cannot be the universal Teichmüller space described in the mathematics literature”—that is, .T (D− ).

9 This symmetry group is twice as large as one might expect from a differential-geometric point of view of local symmetries; local biholomorphisms fixing a point are generated by .zn ∂/∂z with .n ≥ 1.

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On the other hand, in [41], the connection of .Diff(S1 )/S1 to .T (D− ) is observed by Hong–Rajeev. A kind of sum over paths is taken over only the compact surfaces, and the fact that the universal Teichmüller space contains the moduli spaces of compact Riemann surfaces is advanced as evidence for the idea of a non-perturbative approach using .Diff(S1 )/S1 , and a possible action is advanced. The relevance to Weil–Petersson geometry is that the embedding into the infinite restricted Siegel disk of polarizations of Segal plays an explicit role. Indeed Hong–Rajeev (referring to [59]) point out that the lack of completeness of .Diff(S1 )/S1 is a technical obstacle. The results of Cui and Takhtajan–Teo (Theorem 6.5.7) have removed this obstacle. In this context, the result of Shen and Takhtajan–Teo that an element of the Teichmüller space .p ∈ T (D− ) maps into the restricted Siegel disk if and only if .p ∈ TWP (D− ) (Theorem 6.6.7), is very satisfying. Remark 6.10.7 Hong and Rajeev [41] state that .Diff(S1 )/S1 is a dense subset of the universal Teichmüller space. Setting aside different interpretations of “dense”, it is rather the quotient .Diff(S1 )/Möb(S1 ) which is a subset of the universal Teichmüller space .T (D− ) ∼ = QS(S1 )/Möb(S1 ). However, Teo [120] showed that the universal Teichmüller curve .T (D− ) (that is, the Bers fiber space over .T (D− ) mentioned earlier in Sect. 6.9.2), can be naturally identified with .QS(S1 )/S1 . Takhtajan and Teo [116] showed that .QSWP (S1 )/S1 is a Hilbert manifold and a topological group, and in fact .Diff(S1 )/S1 is a dense subset of the Weil–Petersson universal Teichmüller curve in its associated topology. This could be seen to revive the idea of Hong and Rajeev. In regards to the sum over compact surfaces, the universal hyperbolic lamination is also of interest and might provide another approach; see [21, 58, 113].

6.10.2 Fluid Mechanics A topic of interest in mathematical fluid dynamics is the study of geometric aspects of .Diff(S1 )/Möb(S1 ). Schonbek et al. [99] consider this in association with the polarizations of Segal [103] and Nag and Sullivan [58] and the embedding into the embedding into the Sato-Segal-Wilson Grassmannian [107]. A Hilbert manifold structure with tangent spaces modelled on .H 3/2 is proposed there. Motivated by Arnol’d’s approach to Euler’s equation, as well as Segal and Wilson [107], Schonbek, Todorov and Zubelli utilized the remarkable fact that the geodesics of the Teichmüller space give information about the behaviour of solutions to the periodic Korteweg–de Vries (KdV) equation, and thereby proved existence of periodic solutions to the KdV equation with initial data in the Sobolev space on the unit circle. They thereafter used infinite-dimensional Cartan–Hadamard theory, to prove the Arnol’d exponential instability of the geodesic flow. With the manifold structure of Cui and Takhtajan–Teo described in Sect. 6.5.2, 3 the tangent space at the identity is the space of functions on the circle of class .H 2 ,

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in other words, the Teichmüller space .TWP (D− ) has a (Hilbert manifold structure ) whose tangent space can be identified with the .H 3/2 S1 -vector fields. In [29] ( ) Figalli characterized the flows generated by the .H 3/2 S1 -vector fields in terms of fractional ( ) Sobolev norms. More precisely, he showed that( a flow ) generated by an 3/2 S1 -vector field belongs to the Sobolev space .W 1,p S1 for all .p ≥ 1 and .H ( ) belongs to .W 1+r,q S1 for all .0 < r < 1/2 and .1 ≤ q < 1/r. However Figalli ( ) also showed that there exists an autonomous vector field .u ∈ H 3/2 S1 such that its flow map is neither Lipschitz nor .W 1+r,1/r (S1 ) for all .r ∈ (0,(1). )In particular taking .r = 1/2 , one sees that there is a flow generated by an .H 3/2 S1 -vector field that is neither in .W 3/2,2 (S1 ) = H 3/2 (S1 ), nor in the Lipschitz space. By the aforementioned work of Takhtajan and Teo, the identity component of universal Teichmüller space takes the place of diffeomorphisms of critical Sobolev 3 class .H 2 . In [37] and [36] Gay-Balmaz, Marsden and Ratiu studied this group from the point of view of manifolds of maps by identifying it with a subgroup of the quasisymmetric homeomorphisms of the circle and proved that all elements of this 3 group are of class .H 2 −ε for all .ε > 0. Another outcome of this was that for WPclass quasisymmetries .φ on .S1 , one has that both .φ and its inverse .φ −1 are in 3 −ε 1 .H 2 (S ) for all .ε > 0. Now since the Weil–Petersson metric induces the Hilbert space topology on each tangent space, Gay-Balmaz and Ratiu were also able to show that the metric is complete, i.e., all geodesics of the Weil–Petersson metric exist for all time. They also proved that this space is Cauchy complete relative to the distance function defined by the Weil–Petersson metric, which earlier had been shown by Cui [22].10 In connection with the Sobolev regularity mentioned above, Shen [110] proved that there exists some quasisymmetric homeomorphism of the 3 Weil–Petersson class which belongs neither to the Sobolev space .H 2 nor to the 1 Lipschitz space .A . Based on this new characterization of the Weil–Petersson class, Shen introduced a metric on the Weil–Petersson Teichmüller space and used it to give a new proof of Takhtajan–Teo’s result that the Weil–Petersson Teichmüller space is a topological group. Moreover Gay-Balmaz and Ratiu obtained a new equation, the so-called Euler– 3 Weil–Petersson equation, and showed that the solutions are .C 0 in .H 2 and .C 1 1 in .H 2 . Another achievement of the Weil–Petersson techniques in the paper [35], based specifically on the long-time existence of Weil–Petersson geodesics, was the solution to a problem posed by Sharon and Mumford in [56], regarding the existence of a unique geodesic between two shapes in the plane.

10 Because

of the Hopf–Rinow theorem, given Cauchy completeness one expects geodesic completeness. However this is not necessarily true in the infinite-dimensional setting [49].

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6.10.3 Loewner Energy The Loewner energy of a Jordan curve is the Dirichlet energy of its Loewner driving term, and was introduced by Friz and Shekar [31] and Wang [128]. More precisely ¯ Loewner energy of .η, denoted let .η be a Jordan curve on the Riemann sphere .C. by .I L (η), is by definition the Dirichlet energy of this driving term. It was shown by Wang in [128] that if .η passes through .∞, then one has 1 π

I L (η) =

.

f H

|∇ log |f ' (z)||2 dAz +

1 π

f H∗

|∇ log |g ' (z)||2 dAz ,

(6.10.1)

where f and g map conformally the upper and lower half-planes .H and .H∗ onto the ¯ two components of .C\η, while fixing .∞ (i.e. .(f, g) is a welding pair). Furthermore it was shown in [128] that a Jordan curve has finite Loewner energy if and only if it is a Weil–Petersson quasicircle, i.e., its normalized welding homeomorphism belongs to the Weil–Petersson Teichmüller space. There is an interesting connection between zeta-regularized determinants and Loewner energy, which was explored by Wang in [129]. First let us briefly recall some basic facts about zeta function regularization of determinants. To this end let .Ag be the Laplace–Beltrami operator with Dirichlet boundary condition on a compact surface with conformal metric .(R, g) with smooth (say at least .C 1,1 boundary). The zeta function associated with .−Ag is defined by ζg (s) =

∞ E

.

λ−s j

j =1

1 = r(s)

f

∞

0

t s−1

∞ E

e−tλj dt

(6.10.2)

j =1

where .0 < λ1 ≤ λ2 . . . is the discrete spectrum of .−Ag . Now as a consequence of a result of Weyl [133] one has that .λj ∼ C j for a constant C, and .ζg is therefore holomorphic in .{s ∈ C; Re(s) > 1}. Thereafter .ζg could be continued meromorphically to .C (with a simple pole at .s = 1), in particular .ζg is holomorphic at .s = 0 and hence .ζg' (0) exists and is given by ζg' (0) = lim

.

s→0

ζg (s) − ζg (0) . s−0

More explicitly one has that ' .ζg (s)

( ) ∞ E − log λj = , λsj j =1

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which for .s = 0 yields that ζg' (0) = −

∞ E

.

(

)

⎛

log λj = − log ⎝

j =1

∞ | |

⎞ λj ⎠ .

j =1

Therefore we obtain '

e−ζg (0) =

∞ | |

.

) ( λj = det −Ag ,

j =1

which allows one to define the determinant of .−Ag by the formula .

) ( ' det −Ag := e−ζg (0) .

(6.10.3)

When .(S , g) is a compact surface without boundary, the Laplace–Beltrami operator Ag has a discrete spectrum and a one-dimensional kernel, and its regularized determinant denoted by .det'ζ (−Ag ) is defined in a similar way as above by considering only the non-zero eigenvalues. An interesting fact in this context is that the zeta-regularized determinant of the Laplacian depends on both the Riemannian and the conformal structures of the surface. Indeed if .g = e2λ g0 , with .λ ∈ C ∞ (R) (or in .C ∞ (S )), is a metric conformally equivalent to .g0 , then the relationship between the regularized determinants of .−Ag and .−Ag0 is given by the following Polyakov–Alvarez conformal anomaly formula [61]:

.

Theorem 6.10.8 For a compact surface .(S , g) without boundary, one has that ( ) [ f ] f ( det' −Ag ) | |2 1 1 ζ | | ( ) =− ∇g0 λ g dAg0 + log Kg0 λ dAg0 0 6π 2 S det'ζ −Ag0 S . ( A (S ) ) g + log . Ag0 (S ) (6.10.4) For a surface .(R, g) with boundary, the analogous formula is ) f f −Ag ) | |2 1 [1 | | ( ) =− ∇g0 λ g dAg0 + log Kg0 λ dAg0 0 6π 2 R detζ −Ag0 R f ] . + kg0 λ dlg0 ( det

(

ζ

∂R

−

1 4π

f ∂R

∂n λ dlg0 ,

(6.10.5)

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where .dAg0 is the area measure, .dlg0 is the boundary measure, .Kg0 is the Gauss curvature, and .kg0 the geodesic curvature on the boundary, all associated with the metric .g0 , and .∂n is the derivative along the unit outer normal. Moreover .∇g0 is the Riemannian gradient, and .Ag0 (·) is the area of the surface .(both with respect to the metric .g0 ). In [129] Wang showed that for smooth simple loops, the Loewner energy can be expressed in terms of the zeta-regularized determinants introduced above, of a ¯ with a certain Neumann jump operator. More explicitly, consider the 2-sphere .C ¯ into two Riemannian metric g, and let .γ ⊂ S 2 be a smooth Jordan curve dividing .C components .D1 and .D2 . Denote by .ADi ,g the Laplacian with Dirichlet boundary condition on .(Di , g), .i = 1, 2. Next, define the functional .H (·, g) on the space of smooth Jordan curves by ( ) ( ) ( ) ¯ − log detζ −AD ,g H (γ , g) := log det'ζ −AC,g − log volg C ¯ 1 ( ) − log detζ −AD2 ,g .

.

Given the information above, Wang [129] proved the following theorem Theorem 6.10.9 If .g = e2ϕ g0 is a metric conformally equivalent to the spherical ¯ then: metric .g0 on .C, ¯ (1) .S1 minimizes .H (·, g) among all smooth Jordan curves .γ ⊂ C. ¯ We have the identity (2) Let .γ be a smooth Jordan curve on .C. ( ) I L (γ ) = 12H (γ , g) − 12H S1 , g ( ) ( ) detζ −AD+ ,g detζ −AD− ,g ( ) ( ), = 12 log detζ −AD1 ,g detζ −AD2 ,g

.

(6.10.6)

where .D+ and .D− are the two connected components of the complement of .S1 . Moreover Wang showed that the Loewner energy equals a multiple of the universal Liouville action of Takhtajan–Teo given by (6.9.5). Prior to [129], following the work by Schiffer [89, 90], Takhtajan and Teo [116] defined the Fredholm determinant of a quasicircle. The Fredholm determinant of a curve has its origins in classical potential theory in association with Fredholm’s solution of the Dirichlet and Neumann problems via the jump formula. Schiffer [89– 92], considering .C 3 curves, showed that the classical Fredholm determinant could be expressed in terms of the Grunsky operator. Takhtajan–Teo’s work naturally extends this to quasicircles.

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Briefly, given a welding pair .(f, g) such that the corresponding operators Grf Gr∗f and .Grg Gr∗g are trace-class, the Fredholm determinant of the correspond( ) ing quasicircle .C = f S1 is defined by

.

DetF (C ) = det(I − Grf Gr∗f ) = det(I − Grg Gr∗g ).

.

Takhtajan and Teo then showed that the following sewing formula is valid for the determinants ( ) Theorem 6.10.10 Let .γ = g −1 ◦f ∈ TWP (D− ) be in .C 3 (S1 ). Then for .C = f S1 one has .

DetF (C ) =

detζ (−AK˜ ) detζ (−AK˜ ∗ ) detζ (−AD+ ) detζ (−AD− )

.

(6.10.7)

This could be interpreted as a sewing formula for the Laplace operator of a ¯ which is the metric .|dw|2 on the interior domain conformal metric on the sphere .C, |dw|2 1 ∗ .K˜ = ı (K ), with .ı(z) := z , and is the metric . |f−1 (w)|4 on the exterior domain ∗ .K˜ = ı(K ). The Fredholm determinant .DetF (C ) is the inverse of the determinant ¯ along the closed of the Neumann jump operator which corresponds to cutting of .C curve .C and considering Dirichlet boundary conditions for interior and exterior Laplace operators. Therefore putting Theorems 6.10.9 and 6.10.10 together, one also sees a relation between the Loewner energy and the Fredholm determinant of a quaiscircle. Acknowledgments We are grateful to Yi-Zhi Huang for many valuable discussions and for encouraging our work on the moduli spaces, and of course to David Radnell who has been our partner in much of this work. We are grateful to the referee for suggestions and valuable comments regarding certain important issues in the earlier draft of this chapter. Thanks also to Michael Heins for suggestions that have been incorporated in the current version of the chapter. Finally, we are indebted to Athanase Papadopoulos, for giving us the opportunity of making this contribution, and for his suggestions that have improved the overall presentation of this chapter. Eric Schippers is partially supported by the National Sciences and Engineering Research Council of Canada.

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Chapter 7

Kleinian Groups and Geometric Function Theory Hiroshige Shiga and Toshiyuki Sugawa

To the memory of Professor Yukio Kusunoki

Abstract A Kleinian group divides the Riemann sphere into two parts, the region of discontinuity and the limit set. We are interested in analytic properties of these sets from the view-point of geometric function theory. Keywords Quasiconformal mapping · Uniformly perfect set · Kleinian group · Teichmüller space 1991 Mathematics Subject Classification Primary 30C62, Secondary 30F40, 30F60, 57K32

7.1 Introduction and Preliminaries The celebrated W. Thurston [64] has made tremendous contributions to lowdimensional topology, especially to the theory of hyperbolic 3-manifolds. Moreover, broad fields in mathematics are influenced by him. Geometric function theory, which studies analytic functions and domains in the complex plane as well as Riemann surfaces from the geometric point of view, is not an exception. A hyperbolic 3-manifold may be represented as the quotient space of the hyperbolic 3-space .H3 under the action of a Kleinian group. In this article, we will survey selected topics in Kleinian groups together with the geometric function theory influenced and motivated by Thurston’s work. In particular, we are interested H. Shiga () Department of Mathematics, Kyoto Sangyo University, Kyoto, Japan e-mail: [email protected] T. Sugawa Graduate School of Information Sciences, Tohoku University, Sendai, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 K. Ohshika, A. Papadopoulos (eds.), In the Tradition of Thurston III, https://doi.org/10.1007/978-3-031-43502-7_7

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in complex analytic properties of the region of discontinuity and the limit set of a Kleinian group.

7.1.1 Hyperbolic Geometry The reader should consult [8] as a general reference for the present subsection. We first introduce two models of the hyperbolic n-space for .n ≥ 2. The upper half-space n n .H = {(x1 , . . . , xn−1 , t) ∈ R : t > 0} equipped with the Riemannian metric ds 2 =

.

2 + dt 2 dx12 + · · · + dxn−1

t2

=

|dx|2 + dt 2 t2

is called the half-space model of the hyperbolic n-space and the metric is called the hyperbolic metric. The unit ball .Bn = {x ∈ Rn : |x| < 1} equipped with the Riemannian metric ds 2 = 4

.

dx12 + · · · + dxn2 [1 − (x12 + · · · + xn2 )]2

=

22 |dx|2 (1 − |x|2 )2

is called the ball model of the hyperbolic n-space and the metric is also called the hyperbolic metric. It is well known that these two models are isometrically homeomorphic and the hyperbolic metrics induce complete metrics on .Hn and .Bn which are both called the hyperbolic distance. More explicitly,  dHn ((x, t), (y, s)) = 2 arctanh

.

|x − y|2 + (t − s)2 |x − y|2 + (t + s)2

and |x − y| dBn (x, y) = 2 arctanh  . 1 − 2 x · y + |x|2 |y|2

.

(7.1)

Here, we recall that . arctanh t = 12 log 1+t 1−t , 0 ≤ t < 1. When .n = 2, the group .PSL(2, R) acts on .H2 = H = {z ∈ C : Im z > 0} as orientationpreserving isometries. Indeed, we have the natural identifications .Aut(H) = PSL(2, R) = Isom+ (H); where .Aut(H) is the group of analytic automorphisms of .H and .Isom+ (H) is the orientation-preserving isometries of .H with respect to the hyperbolic metric. We note also that the hyperbolic metric of .B2 = D = {z ∈ C : |z| < 1} is expressed by .ds = 2|dz|/(1 − |z|2 ) and the hyperbolic distance is  z−w   = log |1 − z¯ w| + |z − w| . dD (z, w) = 2 arctanh  1 − z¯ w |1 − z¯ w| − |z − w|

.

(7.2)

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Let D be a domain in . C whose complement contains at least three points. Then, by the uniformization theorem, the universal covering space of D is conformally equivalent to the upper half-plane .H. Therefore, we can take a holomorphic universal covering projection .π of .H onto .D. The covering transformation group .𝚪 = {γ ∈ Aut(H) : π ◦ γ = π} < PSL(2, R) is a Fuchsian group, that is, a discrete subgroup of .PSL(2, R). Note that D may be expressed as the quotient space .H/ 𝚪. The hyperbolic metric .dsD = λD (w)|dw| of D is defined by λD (π(z))|π ' (z)| =

.

1 , Im z

z ∈ H.

The value of .λD (w) is independent of the choice of .z ∈ H with .π(z) = w because the hyperbolic metric .|dz|/Im z of .H is invariant under .𝚪 (< PSL(2, R)). In particular, the hyperbolic metric .dsD = λD (w)|dw| of the unit disk .D = {w ∈ C : |w| < 1} is given by .λD (w) = 2/(1 − |w|2 ) via the Cayley transform .w = f (z) = (z − i)/(z + i), mapping .H conformally onto .D. We denote by .𝓁D (γ ) the hyperbolic length of a piecewise smooth curve .γ : [0, 1] → D, that is,  𝓁D (γ ) =



1

dsD =

.

λD (γ (t))|γ ' (t)|dt.

0

γ

Using the hyperbolic metric .dsD , we may consider the hyperbolic distance dD (w1 , w2 ) between .w1 and .w2 in D by the standard manner. The injectivity radius .εD (w0 ) of D at .w0 is defined to be the maximal radius .r > 0 so that the hyperbolic disk .{w ∈ D : dD (w, w0 ) < r} is simply connected. (When D is simply connected, we define .εD (w0 ) = +∞.) It is easy to see that .εD (w0 ) is half the minimal hyperbolic length of homotopically nontrivial closed curves in D passing through the point .w0 . The hyperbolic metric is extended to any non-empty open set .Ω in . C by defining it component-wise, unless . C \ Ω contains at most two points. Likewise, the injectivity radius may be extended to .Ω. We denote them by .λΩ (w)|dw| and .εΩ (w), respectively, as before. .

7.1.2 Quasiconformal Mappings We refer to [2] and [38] for details about the material treated in this subsection. A homeomorhism f of a plane domain D onto another domain .D ' is called a quasiconformal map if f has locally square integrable partial derivatives (in the sense of distribution) and satisfies the inequality |fz¯ | ≤ k|fz |

.

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almost everywhere in .D, where k is a constant with .0 ≤ k < 1, fz = 12 (fx − ify ),

.

fz¯ = 12 (fx + ify )

and .fx , fy are partial derivatives of f with respect to .x, y, respectively, for z = x + iy. It turns out that f preserves the sets of (2-dimensional) Lebesgue measure zero and, in particular, .fz /= 0 a.e. Thus the quotient .μ = fz¯ /fz is well defined as a Borel measurable function on D and satisfies .‖μ‖∞ ≤ k < 1. This function is sometimes called the complex dilatation of f and denoted by .μf . More specifically, f is called a K-quasiconformal map, where .K = (1 + k)/(1 − k). It is known that a 1-quasiconformal map is conformal (i.e., biholomorphic) and vice versa. The composition of a .K1 -quasiconformal map and a .K2 -quasiconformal map is a .K1 K2 -quasiconformal map and the inverse map of a K-quasiconformal map is also K-quasiconformal. In particular, K-quasiconformality is preserved under composition with conformal maps. Therefore, K-quasiconformality and, hence, quasiconformality can be defined for homeomorphisms between Riemann surfaces. In particular, we can talk about quasiconformality of a homeomorphism of the Riemann sphere . C = C ∪ {∞}. More precise information about compositions of quasiconformal maps will be needed later. Let .f : D → D ' and .g : D → D '' be quasiconformal maps. Then the complex dilatation of .g ◦ f −1 is given by

.

(μg◦f −1 ◦ f )

.

μg − μf fz = . fz 1 − μf · μg

(7.3)

In particular, if .μf = μg a.e. on D then .g ◦ f −1 : D ' → D '' is conformal. The following existence and uniqueness theorem will be fundamental in this section. Theorem 7.1.1 (The Measurable Riemann Mapping Theorem) For any measurable function .μ on .C with .‖μ‖∞ < 1, there exists a unique quasiconformal map .f : C → C such that .f (0) = 0, f (1) = 1 and .fz¯ = μfz a.e. in .C. We will denote by .f μ the above normalized quasiconformal map f for a given μ.

.

A simply connected domain D in the Riemann sphere . C is called a quasidisk if it is the image of the unit disk under a quasiconformal self-map of . C. The image of the unit circle .T = ∂D under a quasiconformal map of . C is called a quasicircle. Two quasidisks .D1 and .D2 are said to be complementary if they share a quasicircle as their common boundary, that is to say, there is a quasiconformal map f of . C such that .D1 = f (H) and .D2 = f (H∗ ), where .H∗ denotes the lower half-plane.

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7.1.3 Kleinian Groups A discrete subgroup of .PSL(2, C) is called a Kleinian group. Note that .PSL(2, C)  acts conformally on the Riemann  sphere .C = C∪{∞} as the Möbius transformation az + b ab ∈ PSL(2, C). The region of discontinuity .Ω(G) .g(z) = for .g = ± cd cz + d of a Kleinian group G is defined to be the maximal open set in the Riemann sphere . C where G acts properly discontinuously, and the limit set .Λ(G) is the complement of .Ω(G). Both sets are G-invariant. It is known that .Λ(G) is the closure of the set of fixed points of all the elements in G of infinite order. A connected component of .Ω(G) is called a component of the Kleinian group G. A Kleinian group is called elementary if its limit set contains at most two points. Throughout this chapter, unless otherwise stated, we assume that a Kleinian group is non-elementary; that is, its limit set contains at least three points. A Kleinian group G is said to be of the first kind if .Ω(G) is empty. Otherwise, it is said to be of the second kind. We will be interested only in Kleinian groups of the second kind in what follows. Note that .Ω(G) carries the hyperbolic metric .λΩ(G) (w)|dw| for a non-elementary .G. By the Poincaré extension, we can extend the action of .PSL(2, C) to the upper half space .H3 as isometries with respect to the hyperbolic metric (see [8] for details). It is known that a Kleinian group G acts properly discontinuously on .H3 . So, we have a hyperbolic 3-manifold (orbifold) .N(G) = H3 /G. Since the hyperbolic metric of .H3 is invariant under the action of .PSL(2, C), it descends to .NG . It is called the hyperbolic metric of .N(G) and denoted by .dsN (G) . The induced distance is denoted by .dN (G) (p, q) for .p, q ∈ N(G). The injectivity radius is also defined similarly and denoted by .εN (G) (p). The convex hull .C(G) of G is the minimal convex set in .H3 which contains all hyperbolic geodesics connecting two points of 3 .Λ(G) in .H . Now, we define some classes of Kleinian groups. Definition 7.1.1 A Kleinian group G is called geometrically finite if the quotient of the .ε-neighborhood .Cε (G) of .C(G) by the action of G has finite volume for any .ε > 0. A geometrically finite Kleinian group G is called convex co-compact if it contains no parabolic transformations. A geometrically finite Kleinian group is called a regular b-group if it has only one simply connected invariant component. A Kleinian group G is called quasi-Fuchsian if G keeps each of two complementary quasidisks .Δ1 and .Δ2 invariant. It is known that a geometrically finite Kleinian group is finitely generated but the converse is not necessarily true. The Ahlfors Finiteness Theorem means that the quotient space .Ω(G)/G is a finite union of finitely many Riemann surfaces (orbifolds) of finite conformal type. The following result characterizes finitely generated quasi-Fuchsian groups in some sense.

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Theorem 7.1.2 (Maskit [40]) Let G be a finitely generated Kleinian group. Suppose that the region of discontinuity .Ω(G) for G consists of two simply connected invariant components .Δ1 , Δ2 . Then, G is a quasi-Fuchsian group, and both .Δ1 and .Δ2 are quasidisks. Definition 7.1.2 A Kleinian group G is said to have bounded geometry if there exists a constant .ε > 0 such that the injectivity radius satisfies the inequality .εN (G) (p) ≥ ε for any .p ∈ N(G). Finally in this subsection, we give a mapping which plays an important role in the proof of our theorems. For any point .z ∈ Ω(G) and for any .ε > 0, we define the nearest point retraction .Πε (z) ∈ ∂Cε (G). Namely, .Πε (z) is the point in .H3 where a horoball inflated at z first touches .Cε (G). From the construction, it is easily seen that Πε (g(z)) = g(Πε (z)),

.

(7.4)

for every .z ∈ Ω(G) and for every .g ∈ G. Here, we present an important theorem on the nearest point retraction. Theorem 7.1.3 (Epstein–Marden [24], Minsky [44]) For each .ε > 0, the nearest point retraction .Πε is .(cosh ε)-quasiconformal, .(4 cosh ε)-Lipschitz, and the inverse −1 : ∂C (G) → D is .(1/ sinh ε)-Lipschitz. .Πε ε

7.2 Function Theory on the Components of Kleinian Groups 7.2.1 Cannon–Thurston Maps and the Geometric Function Theory Let .ρ be a representation of a Kleinian group G to .PSL(2, C), namely, .ρ : G → PSL(2, C) is a homomorphism. We say that a continuous map .F : Λ(G) →  C is a Cannon–Thurston map for .ρ if it satisfies the relation ρ(g) ◦ F = F ◦ g

.

on Λ(G)

for any .g ∈ G. A possible construction for such a map is as follows. Suppose that a representation .ρ : G → PSL(2, C) is faithful. For an element .g ∈ G of infinite order, the image .gˆ = ρ(g) has infinite order, too. The n-th iterate .g n (z) tends to a fixed point, say .a, of g as .n → ∞ for all .z ∈  C (except for at most one point). Similarly, .gˆ n tends to a fixed point, say .a, ˆ of .g. ˆ In this way, we can define .F (a) = aˆ on the set of fixed points of all the elements in G of infinite order. Since .Λ(G) is the closure of the set, F may be extended to a mapping .Λ(G) to .Λ(ρ(G)) if F is known to be continuous. For instance, if G is a Fuchsian group of the first kind acting on the unit disk .D and if .ρ(G) is a Kleinian group of the first kind, then we may have a

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G-equivariant Peano-like map of the unit circle .T := ∂D onto the Riemann sphere  C = Λ(ρ(G)). Such a construction was first made by Cannon and Thurston [21]. The existence of Cannon–Thurston maps is a problem which attracts many authors. We consider it under a certain restriction. If G has a simply connected invariant component .Δ, then we see that .∂Δ = Λ(G). In particular, then .Λ(G) is connected. Let .ϕ : D → Δ be a Riemann map for −1 ◦ G ◦ ϕ is .Δ, where .D is the unit disk in the complex plane .C. The group .𝚪 := ϕ a Fuchsian group acting on .D and .ρϕ defined by .

ρϕ (γ ) = ϕ −1 ◦ γ ◦ ϕ

.

gives an isomorphism of .𝚪 to G. Here, we suppose that the Riemann map .ϕ can be continuously extended to the boundary .T = ∂D. Then, we see that .ϕ(T) = ∂Δ = Λ(G) and .Λ(𝚪) = T. Therefore, .ϕ|T yields a Cannon–Thurston map for .ρϕ . If the boundary .∂Δ is locally connected, then .ϕ can be continuously extended to .T (see [55, Chap. 2]). Due to the work of Mahan Mj [48], connected limit sets of finitely generated Kleinian groups are locally connected. Thus, we may take the boundary map of the Riemann map .ϕ as a Cannon–Thurston map if a finitely generated Kleinian group has a simply connected invariant component. The extendability of conformal maps and properties of the extended maps are matters in geometric function theory. In this way, geometric function theory comes into the theory of Kleinian groups. Minsky [44] shows that the limit set of a Kleinian group with bounded geometry is locally connected if it is connected. Miyachi [46] estimates the modulus of continuity of Cannon–Thurston maps for such Kleinian groups with bounded geometry. Theorem 7.2.1 (Miyachi [46]) Let G be a Kleinian group with a simply connected invariant component .Δ and .ϕ : D → Δ be a Riemann map for .Δ. Suppose that G has bounded geometry. Then, the continuous extension of .ϕ to the unit circle .T satisfies the log-Hölder condition  |ϕ(ζ1 ) − ϕ(ζ2 )| ≤ A log

.

3 |ζ1 − ζ2 |

−α ,

ζ1 , ζ2 ∈ T

for positive constants .α, A.

7.2.2 Hölder and John Domains We start with definitions of some classes of domains in .Rn . Let D be a proper subdomain of .Rn . We denote by .δD (x) the Euclidean distance from a point .x ∈ D

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to the boundary .∂D. The quasihyperbolic distance of D is defined by 

|dx| = inf γ δD (x)

kD (x1 , x2 ) = inf

.

γ

γ



1 0

|γ ' (t)|dt , δD (γ (t))

where the infimum is taken over all piecewise .C 1 curves .γ : [0, 1] → D joining .x1 and .x2 . It is known that the following inequality holds for general domains: .

log

δD (x1 ) ≤ kD (x1 , x2 ), δD (x2 )

x1 , x2 ∈ D.

n

The chordal (spherical) distance on .R = Rn ∪ {∞} is defined by 2|x − y|  σ (x, y) =  1 + |x|2 1 + |y|2

.

for .x, y ∈ Rn and it can be defined for .x = ∞ by the standard limiting procedure. The infinitesimal form is given by .dσ (x) = 2|dx|/(1+|x|2 ). The spherical analogue n of the quasihyperbolic distance is defined for a proper subdomain of .R by 

♯

kD (x1 , x2 ) = inf

.

γ

dσ (x)

γ

♯

,

δD (x)

♯

where δD (x) = inf σ (x, y), y∈∂D

♯

♯

where the infimum is taken as above. Note that .δD and .kD are comparable with .δD and .kD , respectively, for a bounded domain D in .Rn . Definition 7.2.1 (John Domains and Hölder Domains) Since plane quasiconformal mappings are locally Hölder continuous on .C, a conformal map f of the unit disk .D onto a quasidisk D is Hölder continuous up to the boundary and so is −1 : D → D. Hölder continuity of f may be guaranteed by a weaker condition. .f In this subsection, we discuss such conditions. Let D be a proper subdomain of .Rn . (i) The domain D is called a John domain if there exist .x0 ∈ D and a constant .c > 0 such that for any .x ∈ D there exists a curve .γ joining .x0 and x in D so that δD (z) ≥ c|z − x|,

.

z ∈ γ.

(7.5)

(ii) The domain D is called a Hölder domain if there exist .x0 ∈ D and constants .c1 , c2 > 0 such that kD (x0 , x) ≤ c1 log

.

δD (x0 ) + c2 , δD (x)

x ∈ D.

(7.6)

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There are many equivalent conditions for John domains (see [50, 66]). The name “Hölder domain" will be justified in Proposition 7.2.2 below. It is known that a John domain is a Hölder domain (cf. [55], p. 103), although the converse is not true in general [11]. Väisälä [66] showed that John domains in the above sense are bounded. Smith and Stegenga [59, Cor. 1] showed also that Hölder domains are bounded. Thus the notions of John domains (in the above sense) and Hölder domains are not necessarily Möbius invariant. We thus define spherical John domains and spherical Hölder domains in . C by using the spherical metric instead of the Euclidean metric in (7.5) and (7.6), respectively. Spherical John domains are introduced in [42]. By the above boundedness results, John domains are spherical John domains and Hölder domains are spherical Hölder domains. However, the converses are not true. For example, the upper half-plane .H and the slit domain .C \ [0, +∞) are spherical John domains and thus spherical Hölder domains but they are not Hölder domains because they are unbounded. Martio and Vuorinen [39] showed that the Hausdorff dimension of the boundary of a John domain in .Rn is strictly less than n. This was extended to Hölder domains by Smith and Stegenga [59, Cor. 2]. We show that this is true for spherical Hölder domains. Lemma 7.2.1 Let .D ⊂  C be a spherical Hölder domain. Then, the Hausdorff dimension of .∂D is strictly less than two. Proof We may assume that .∞ ∈ D. Then, for a small neighborhood .U ⊂ D of .∞, we see that .D ' := D \ U is a Hölder domain. Hence, the Hausdorff dimension of ' .∂D which is a subset of .∂D is strictly less than two by the Smith–Stegenga theorem [59]. ⨆ ⨅ n

The same argument works for spherical Hölder domains in .R . If a domain D is a John domain in .C, then the boundary is locally connected, and a Riemann map .ϕ : D → D can be continuously extended to .T(= ∂D). Moreover, we may control the growth of the derivative of a Riemann map for a John domain. See [55, §5.2] for details. Proposition 7.2.1 Let .ϕ : D → D be a Riemann map for a bounded John domain D in .C. Then, there exist constants .α with .0 < α ≤ 1 and .M > 0 such that 

1 − |ζ | .|ϕ (ζ )| ≤ M|ϕ (z)| 1 − |z| '

'

α−1

for any .z ∈ D and .ζ ∈ B(z) ∩ D, where .B(reit ) = {ρeiθ : r ≤ ρ ≤ 1, |θ − t| ≤ π(1 − r)}. By integration, we see that the conformal map .ϕ : D → D is Hölder continuous on the closed unit disk .D when D is a bounded John domain (see [55, Cor. 5.3]). As for Hölder domains, Becker and Pommerenke [11] show the following.

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Proposition 7.2.2 Let D be a simply connected bounded domain in .C. Then, a Riemann map .ϕ : D → D is Hölder continuous if and only if D is a Hölder domain. By a theorem of Hardy and Littlewood, an analytic function .ϕ on .D is Hölder continuous with exponent .α ∈ (0, 1]; that is, |ϕ(z1 ) − ϕ(z2 )| ≤ L|z1 − z2 |α ,

.

z 1 , z2 ∈ D

for some constant .L > 0 if and only if |ϕ ' (z)| ≤ M(1 − |z|)α−1 ,

.

z∈D

for some constant .M > 0. Let G be a finitely generated Kleinian group with an invariant component D. If D is a spherical Hölder domain, then from Lemma 7.2.1 the Hausdorff dimension of .∂D is strictly less that two. Since .∂D is nothing but the limit set of G, it follows from a theorem of Bishop and Jones [16] that G has to be geometrically finite. Taking into account this observation together with some geometric observations, we obtain the following theorem [42, 58]. Theorem 7.2.2 Let G be a finitely generated Kleinian group with an invariant component D. Then, the following conditions are equivalent. (1) D is a spherical Hölder domain. (2) D is a spherical John domain. (3) G is geometrically finite and every parabolic element stabilizes a round disk in D. Furthermore, when D is simply connected, D is a spherical Hölder domain if and only if it is a quasidisk. Hence, G is a quasi-Fuchsian group in this case.

7.2.3 Distortion Estimates of Riemann Maps In this subsection, we assume that a finitely generated Kleinian group G has an invariant simply connected component D but G is not quasi-Fuchsian. The Riemann mapping theorem guarantees the existence of a conformal mapping .ϕ from the unit disk .D onto D. We may assume that .D ϶ 0 and .ϕ(0) = 0. Then, by Theorem 7.1.2, the simply connected component D is not a quasidisk, and thus we cannot expect the Riemann mapping .ϕ to have the same growth rate of the derivative as the Riemann mappings in the previous subsection. Thus, we take a different approach. In the following, we present results from [58]. Let us describe briefly a method in the paper. We use the ball model .B3 = {x ∈ R3 : |x| < 1} as the hyperbolic 3-space so that G acts on .B3 as hyperbolic isometries. Recall that .Πε : Ω(G) → Cε (G) ⊂ B3 is the nearest point retraction for .ε > 0. We observe that .δD (z) and .1 − |Πε (z)| are

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comparable. In other words, for any .ε > 0, there exists a constant .A = Aε,G > 1 depending only on G and .ε such that A−1 δD (z) ≤ 1 − |Πε (z)| ≤ AδD (z)

.

(7.7)

for every .z ∈ D. We may assume that .0 := (0, 0, 0) ∈ ∂C(G) and .Π(0) = 0, where Π = Πε . Hence, we have

.

Π(g(0)) = g(Π(0)) = g(0)

.

(7.8)

from (7.4). For a fixed finite generating set .Σ of G, we denote by .|g| the minimal word length of .g ∈ G with respect to .Σ. We define the exponent of convergence for G by

|g|k .α(G) = sup k : sup 0 such that for any .g ∈ G

.

|g|2 ≤ A1 δD (g(0))−1 .

.

On the other hand, it is not hard to see that dD (0, g(0)) ≤ A2 |g|

.

holds for some constant .A2 > 0 independent of .g ∈ G, where .dD (·, ·) is the hyperbolic distance of D. Hence, we obtain dD (0, g(0))2 ≤ A3 δD (g(0))−1

.

(7.10)

for some constant .A3 > 0 independent of .g ∈ G. Since .D/G is a Riemann surface of finite conformal type, it follows from (7.10) that there exists a constant .A4 > 0 such that dD (0, w)2 ≤ A4 δD (w)−1

.

for any .w ∈ D. In particular, dD (0, z)2 = dD (0, ϕ(z))2 ≤ A4 δD (ϕ(z))−1

.

(7.11)

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for .z ∈ D. Here, by the Koebe distortion theorem, we have .

1 |ϕ ' (z)| 4 ≤ ≤ 2 δD (ϕ(z)) 1 − |z| 1 − |z|2

(7.12)

(see [55, p. 87]). Noting (7.2), we thus have |ϕ ' (z)| ≤

.

A , (1 − |z|)| log(1 − |z|)|2

z∈D

for a constant .A > 0 from (7.11) and (7.12). Thus, we obtain the following result on the growth of the derivative of the conformal mapping .ϕ when G is geometrically finite, namely a regular b-group (see Definition 1.1). Theorem 7.2.3 Let G be a regular b-group having a simply connected invariant component D with .∂D ⊂ C and .ϕ a conformal mapping from the unit disk .D onto D. Then there exists a constant .A > 0 depending only on .ϕ such that |ϕ ' (z)| ≤

.

A (1 − |z|)| log(1 − |z|)|2

(7.13)

holds for any z near .∂D. As was noted above, a conformal mapping .ϕ of .D onto a bounded quasidisk satisfies the much stronger inequality than (7.13) |ϕ ' (z)| ≤

.

A , (1 − |z|)κ

z ∈ D,

where .A > 0 and .0 < κ < 1 are constants. On the other hand, for a conformal map .ϕ of .D onto a bounded domain contained in the disk .|w| < M, applying the function .ϕ/M to the Schwarz–Pick Lemma yields the inequality, |ϕ ' (z)| ≤

.

M(1 − |ϕ(z)/M|2 ) M < , 1 − |z| 1 − |z|2

z ∈ D.

Thus, the estimate (7.13) in the last theorem is worse than that of a quasidisk but slightly better than the general one. Remark 7.2.1 See the next section for the definitions relevant to this remark. Gehring and Pommerenke [31] showed that if .‖Sϕ ‖ ≤ 2, then .ϕ satisfies the same inequality as that of Theorem 7.2.3. Under the assumptions of Theorem 7.2.3, the conformal mapping .ϕ : D → D represents a boundary point of the Bers embedding of the Teichmüller space of a Riemann surface of finite conformal type and therefore .‖Sϕ ‖ > 2. Hence, Theorem 7.2.3 says that our conformal mapping .ϕ still has the

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same growth of the derivative as that of Gehring–Pommerenke’s theorem while ‖Sϕ ‖ > 2.

.

Corollary 7.2.1 Let G be a regular b-group with simply connected invariant component D. Then the limit set of G is locally connected. Furthermore, .ϕ has a continuous extension to .T = ∂D, which is denoted by the same letter .ϕ, and if in addition .ΛG ⊂ C, then the inequality |ϕ(ζ1 ) − ϕ(ζ2 )| ≤

.

A log(3/|ζ1 − ζ2 |)

(7.14)

holds for any .ζ1 , ζ2 ∈ T, where .A > 0 is a constant. Remark 7.2.2 (1) Abikoff [1] shows that the limit set of G is locally connected if G is a regular bgroup. But his proof is different from ours. Also, he does not give any estimate for a Riemann mapping. (2) On the local connectivity of the limit sets, Anderson and Maskit [5] give a condition of the local connectivity of the limit sets of the structure subgroup of the Kleinian group. McMullen [43] shows that the limit set of a once punctured torus group is locally connected. The exponent 2 of .|log(1 − |z|)| in (7.13) is crucial. Actually, we may show the following. Theorem 7.2.4 Let G be a finitely generated Kleinian group having a simply connected invariant component D with .∂D ⊂ C and .ϕ be a conformal mapping from the unit disk .D onto D. Suppose that .D/G has no punctures. Then, the following conditions are equivalent: (i) There exist constants .α > 0, .A > 0 and a point .w0 ∈ D such that for any −1 (Gw ) \ ϕ −1 (∞) the following inequality holds: .z ∈ ϕ 0 |ϕ ' (z)| ≤

.

A . (1 − |z|)| log(1 − |z|)|2+α

(ii) G is a quasi-Fuchsian group. (iii) There exist constants .A > 0 and .0 < κ < 1 such that the inequality |ϕ ' (z)| ≤

.

A (1 − |z|)κ

holds for any z near .∂D. Furthermore, we have estimated the regularity of Riemann mappings of invariant components of Kleinian groups of bounded geometry as follows.

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Theorem 7.2.5 Let G be a finitely generated Kleinian group having a simply connected invariant component D with .∂D ⊂ C and .ϕ be a conformal mapping of the unit disk .D onto D. Suppose that G has bounded geometry. Then, there exist constants .A > 0 and .α > 0 such that the inequality |ϕ ' (z)| ≤

.

A (1 − |z|)| log(1 − |z|)|α

holds for any z near .∂D.

7.2.4 Uniform Perfectness of the Limit Sets Let .(X, d) be a metric space. A closed subset E of X containing at least two points is said to be uniformly perfect if there is a constant .0 < c < 1 such that {x ∈ E : cr ≤ d(x, a) ≤ r} /= ∅

.

whenever .a ∈ E and .0 < r < ( diam E)/2. If X is complete, then a uniformly perfect closed set is perfect and uncountable. A closed connected set containing at least two points (such as a continuum) is uniformly perfect. On the other hand, a totally disconnected set like a Cantor set, e. g. the middle one-third Cantor set, might be uniformly perfect. The notion of uniform perfectness was introduced essentially by Beardon and Pommerenke [10]. We refer to the survey [62] for general information about uniformly perfect sets. We give below several characterizations of uniformly perfect sets in the Riemann sphere and list some necessary notation. In what follows, for .E ⊂  C, we denote by . diam E the Euclidean diameter of E and define it to be .+∞ if .∞ ∈ E. We also denote by .D(a, r) and .D(a, r) the open disk .|z − a| < r and the closed disk .|z − a| ≤ r in .C, respectively. A domain D in . C is called a ring if it is doubly connected; that is, the complement . C \ D has exactly two components .C1 and .C2 . We say that D separates E if .D ∩ E = ∅ and if .E ∩ Cj /= ∅ for .j = 1, 2. It is well known that a ring D is conformally equivalent to either .C \ {0} or an annulus of the form .1 < |z| < r for some .r ∈ (1, +∞]. In the latter case, we define the modulus of D as . mod D = log r. The following theorem is due to Pommerenke [54]. Theorem 7.2.6 Let E be a closed set in . C containing at least three points and .Ω be its complement. Then the following conditions are mutually equivalent. (1) E is uniformly perfect with respect to the spherical metric. (2) There is a constant .0 < c < 1 such that .E ∩ (D(a, r) \ D(a, cr)) /= ∅ for every .a ∈ E \ {∞} and .0 < r < ( diam E)/2. (3) There is a constant .α > 0 such that every ring R separating E in . C satisfies . mod R ≤ α.

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(4) The hyperbolic density .λΩ (w) of .Ω is comparable with the quasihyperbolic density .1/δΩ (w). (5) There is a constant .β > 0 such that the injectivity radius satisfies the inequality .εΩ (w) ≥ β at every point w of .Ω. (6) There is a constant .0 < γ ≤ 1 such that . Cap(D(a, r) ∩ E) ≥ γ r for every .a ∈ E \ {∞} and .0 < r < ( diam E)/2. Here, . Cap stands for the logarithmic capacity. See [28] for details. It is now easy to see that an open set .Ω has uniformly perfect boundary if and only if its complement .E =  C \ Ω is uniformly perfect. Uniformly perfect sets enjoy many useful properties. In the following, Property (1) is due to Järvi–Vuorinen [33], while Property (3) is due to Fernández [25]. Theorem 7.2.7 Let E be a uniformly perfect set in . C and D be a connected C \ E. Then component of .Ω =  (1) The Hausdorff dimension of E is positive. (2) Each boundary point of D is regular in the sense of Dirichlet problem. Moreover, the boundary of D is Hölder regular. (3) The bottom of the spectrum of D is positive. (4) For a quasiconformal map f of .Ω onto another open set .Ω' , the closed set ' .E =  C \ Ω' is again uniformly perfect. Concerning the Hausdorff dimention, we have the explicit estimate .dim E ≥ log 2/ log(1 + 2/c) [56], where c is the constant appearing in Theorem 7.2.6 (2). Beardon and Pommerenke [10] showed that the limit set of a Schottky group is uniformly perfect. More generally, for a Kleinian group .G, we denote by .L(G) the infimum of the hyperbolic lengths of those loops in .Ω(G) which are not homotopically trivial. In other words, L(G) = 2

.

inf

w∈Ω(G)

εΩ(G) (w).

In view of Theorem 7.2.6(5), we conclude that .Ω(G) has uniformly perfect boundary if and only if .L(G) > 0. Let .π : Ω(G) → X(G) = Ω(G)/G be the canonical quotient map of .Ω(G) under the action of .G. Note that .X(G) has the canonical orbifold structure induced by .π and that the cone points will be regarded as ordinary points as long as we consider the topological structure of .X(G). Then the hyperbolic metric .λΩ(G) (z)|dz| projects to a metric, say, .λX(G) (w)|dw| in such a way that .π ∗ (λX(G) ) = λΩ(G) . We denote by .L∗ (X(G)) the infimum of the .λX(G) lengths of loops in .X(G) covered by hyperbolic elements in .G. The following result is found in [61, Corollary 3.3]. Theorem 7.2.8 Let G be a non-elementary Kleinian group with non-empty region of discontinuity. Then the following inequality holds: L(G) ≥ L∗ (X(G)).

.

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For instance, by the Ahlfors Finiteness Theorem, .X(G) is a finite union of disjoint Riemann surfaces of finite conformal type when G is finitely generated. In particular, .L(G) ≥ L∗ (X(G)) > 0 and thus .Ω(G) is uniformly perfect for finitely generated .G. The proof of the above theorem is simple. Indeed, let .γ be a non-trivial loop in .Ω(G). Then it is easy to see that .π∗ γ = π(γ ) is covered by a hyperbolic element in .G. Hence,   . λΩ(G) (w)|dw| = λX(G) (w)|dw| ≥ L∗ (X(G)). γ

π∗ γ

By taking the infimum over .γ , we get the inequality in the theorem.

7.3 Teichmüller Spaces and Univalent Functions The best reference for the present section is Lehto’s monograph [37]. See also [49].

7.3.1 Schwarzian Derivative The Schwarzian derivative plays an important role in the Bers embedding of Teichmüller spaces. This notion may date back to Lagrange [36]. It was efficiently used by Schwarz to describe a conformal mapping of the upper half-plane onto a Jordan domain whose boundary consists of finitely many circular arcs. Cayley named it after Schwarz. See [53] for more information. The Schwarzian derivative .Sf of a non-constant meromorphic function f on a domain D in .C is defined by  Sf =

.

f '' f'

'

−

1 2



f '' f'

2 =

f ''' 3 − f' 2



f '' f'

2 .

By its form, the Schwarzian derivative .Sf is meromorphic on .D. Note that f is locally univalent at .z0 ∈ D if and only if .Sf (z0 ) /= ∞. The following properties of the Schwarzian derivative make it very useful. (i) .Sf ≡ 0 ⇔ f is (a restriction of) a Möbius transformation. (ii) .Sf ◦g = (Sf ) ◦ g · (g ' )2 + Sg . In particular, .SA◦f = Sf and .Sf ◦A = Sf ◦ A · (A' )2 for a Möbius transformation .A(z) = (az + b)/(cz + d). A more intrinsic account on Schwarzian derivative was given by Thurston [65]. The main idea is to interpret the Schwarzian derivative as a measure of deviation of the map from Möbius transformations. Surprisingly, the Schwarzian derivative quantitively measures, as a result, the deviation of the map

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from univalent ones. The following is due to Nehari [51] (the first part was proved by Kraus [35] earlier and thus it is often called the Nehari–Kraus theorem). Theorem 7.3.1 (Nehari) Let f be a non-constant meromorphic function on the unit disk .D. If f is univalent, then .(1 − |z|2 )2 |Sf (z)| ≤ 6 for .z ∈ D. Conversely, if 2 2 .(1 − |z| ) |Sf (z)| ≤ 2 for .z ∈ D, then f is univalent. The constants 6 and 2 are sharp. Indeed, the Koebe function .K(z) = z/(1−z)2 is univalent on .D and has Schwarzian derivative .SK (z) = −6(1−z2 )−2 . The sharpness of 2 is due to Hille [32]. In a sense, the extremal univalent function is given by 1+z .L(z) = log 1−z = 2 arctanh z which maps .D univalently onto a parallel strip and satisfies .SL (z) = 2(1 − z2 )−2 . The quantity .1 − |z|2 is the reciprocal of half the density .λD (z) of the hyperbolic (or Poincaré) metric .λD (z)|dz| = 2|dz|/(1 − |z|2 ) of the unit disk .D. We define the hyperbolic sup-norm of an analytic function .ϕ on D by ‖ϕ‖D = sup 4λD (w)−2 |ϕ(w)|.

.

w∈D

We denote by .B(D) the complex Banach space consisting of analytic functions .ϕ on D with finite norm .‖ϕ‖D < +∞. Let .h : D → D be a conformal homeomorphism. Then, for a non-constant meromorphic function f on .D, −2 −2 ' 2 4λ−2 D (z)|Sf ◦h (z) − Sh (z)| = 4λD (z)|Sf (h(z))||h (z)| = 4λD (w)|Sf (w)|

.

for .w = h(z). Hence, we have .‖Sf ◦h − Sh ‖D = ‖Sf ‖D . Letting h be a Möbius transformation, we obtain the following consequence from the Nehari theorem. Corollary 7.3.1 Let D be a circular domain, that is, a Möbius image of the unit disk, and let g be a non-constant meromorphic function on .D. If g is univalent on .D, then .‖Sg ‖D ≤ 6. On the other hand, if .‖Sg ‖D ≤ 2, then g must be univalent on .D. For instance, the Cayley transform .h(z) = i(1 + z)/(1 − z) maps the unit disk .D onto the upper half-plane .H.

7.3.2 The Teichmüller Space of a Fuchsian Group We denote by .Belt(D) the open unit ball of the space .L∞ (D) for a domain .D. An element .μ of .Belt(D) is called a Beltrami coefficient on .D. Let .μ be a Beltrami coefficient on the upper half-plane .H. We extend .μ to ∗ ∈ Belt(C) by setting .μ∗ (z) = μ(¯ .μ z) on the lower half-plane .H∗ = {z ∈ ∗ C : Im z < 0}. The quasiconformal map .f μ given by the measurable Riemann mapping theorem will be denoted by .wμ . Since .wμ (z) and .wμ (¯z) both have the

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same complex dilatation .μ∗ and satisfy the same normalization condition, they must be equal, by uniqueness part of the measurable Riemann mapping theorem. In particular, .wμ (x) = wμ (x) for .x ∈ R, and consequently, .wμ maps .H onto itself. Let .𝚪 be a Fuchsian group acting on the upper half-plane .H; namely, .𝚪 is a discrete subgroup of .PSL(2, R). We denote by .Belt(H, 𝚪) the set of Beltrami coefficients .μ on .H satisfying the functional equation .γ ∗ μ := (μ ◦ γ )γ ' /γ ' = μ a.e. on .H. We now suppose that .μ ∈ Belt(H, 𝚪). Then .wμ ◦ γ and .wμ have the same complex dilatation by (7.3) and thus .γμ := wμ ◦γ ◦(wμ )−1 ∈ Aut(H) = PSL(2, R) for each .γ ∈ 𝚪. In this way, we obtain another Fuchsian group .𝚪μ = {γμ : γ ∈ 𝚪}. Let .χμ : 𝚪 → 𝚪μ denote the isomorphism defined by .χμ (γ ) = γμ = wμ ◦ γ ◦ (wμ )−1 . The group .𝚪μ (or the isomorphism .χμ ) is often called a quasiconformal deformation of .𝚪. Two elements .μ1 and .μ2 of .Belt(H, 𝚪) are called Teichmüller equivalent if .wμ1 = wμ2 on .R. This equivalence relation will be denoted by .∼ in the following. Note that .χμ1 = χμ2 if .μ1 ∼ μ2 , and indeed, this holds only if .μ1 ∼ μ2 when .𝚪 is of the first kind; namely, the limit set of .𝚪 coincides with . R := ∂H = R ∪ {∞}. The set of Teichmüller equivalence classes .[μ] of .μ ∈ Belt(H, 𝚪) is defined to be the Teichmüller space of .𝚪 (or the Riemann orbifold .H/ 𝚪) and denoted by .Teich(𝚪). When .𝚪 is the trivial group .1, we call .Teich(1) the universal Teichmüller space.

7.3.3 Bers Embedding of Teichmüller Spaces Around 1960 Ahlfors first showed that the Teichmüller space of a compact Riemann surface has a natural complex structure by using period matrices. Later, Bers invented a way to realize the Teichmüller space as a domain in a complex Banach space. This embedding is nowadays called the Bers embedding of the Teichmüller space. It is known that this works for an arbitrary Fuchsian group .𝚪. For .μ ∈ Belt(H), we define .μ˜ ∈ Belt(C) by setting .μ˜ = μ on .H and .μ˜ = 0 on .H∗ . Then we set .w μ = f μ˜ (recall Theorem 7.1.1). Note that .w μ depends on .μ holomorphically, whereas .wμ does not (see, for instance, [3] or [49]). Let .μ ∈ Belt(H, 𝚪) for a Fuchsian group .𝚪. Since .w μ ◦ γ and .w μ have the same complex dilatation on .C, by (7.3), .γ μ := wμ ◦ γ ◦ (wμ )−1 ∈ Aut( C) = PSL(2, C) for each .γ ∈ 𝚪. Hence, .χ μ (γ ) = γ μ gives a .PSL(2, C)-representation of .𝚪 and its image .𝚪 μ = χ μ (𝚪) acts discontinuously on the unbounded quasidisk .w μ (H). .𝚪 μ is a quasi-Fuchsian group and its limit set lies on the quasicircle .w μ ( R). Note that the mapping .w μ is conformal on the lower half-plane .H∗ . The construction of the Bers embedding is based on the following lemma [49]. Lemma 7.3.1 Let .μ1 , μ2 ∈ Belt(H, 𝚪). Then the following conditions are equivalent: (i) .μ1 ∼ μ2 , i.e., .wμ1 = wμ2 on .R. (ii) .w μ1 = w μ2 on .R.

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(iii) .w μ1 (H) = w μ2 (H). (iv) .Swμ1 = Swμ2 on .H∗ . Let .ϕ = Swμ for a .μ ∈ Belt(H, 𝚪). By the relation .w μ ◦ γ = γ μ ◦ w μ for ' 2 .γ ∈ 𝚪, we have the functional equation .ϕ ◦ γ · (γ ) = ϕ. This means that .ϕ is ∗ an automorphic form on .H for .𝚪 with weight .4. On the other hand, by the Nehari theorem, ‖ϕ‖H∗ = sup 4(−Im z)2 |ϕ(z)| ≤ 6.

.

z∈H∗

Hence, .ϕ ∈ B(H∗ , 𝚪), where B(H∗ , 𝚪) = {ϕ ∈ B(H∗ ) : ϕ ◦ γ · (γ ' )2 = ϕ for all γ ∈ 𝚪}.

.

Then .B(H∗ , 𝚪) is a closed subspace of .B(H∗ ) and thus a complex Banach space with the hyperbolic sup-norm .‖·‖H∗ . In this way, we can define a map .Ф : Belt(H, 𝚪) → B(H∗ , 𝚪) by assigning .Ф(μ) = Swμ . The above lemma tells us that .Ф(μ1 ) = Ф(μ2 ) if and only if .μ1 ∼ μ2 . Therefore, the image T (H∗ , 𝚪) = Ф(Belt(H, 𝚪))

.

is identified with the Teichmüller space .Teich(H, 𝚪) as a set. Bers [12] indeed showed the following result. Theorem 7.3.2 (Bers Embedding) Let .𝚪 be a Fuchsian group acting on the upper half-plane .H. Then the map .Ф : Belt(H, 𝚪) → B(H∗ , 𝚪) is a holomorphic split submersion. In particular, the image .T (H∗ , 𝚪) is a bounded domain in .B(H∗ , 𝚪). The map .Ф is called the Bers projection and the embedded space .T (H∗ , 𝚪) is often called the Bers slice (see [47]). In this way, the Teichmüller space .Teich(𝚪) is realized as a bounded domain in the complex Banach space .B(H∗ , 𝚪). The Beurling–Ahlfors extension enables us to construct a global real-analytic section of .Ф, which in turn implies that the universal Teichmüller space .Teich(1) ≡ T (H∗ , 1) is contractible. It is known, however, that there is no global holomorphic section to .Ф unless .dim Teich(𝚪) ≤ 1. However, Ahlfors and Weill [4] provided a local section of a very simple form. Theorem 7.3.3 (Ahlfors–Weill Section) Let V be the open ball .‖ϕ‖H∗ < 2 in B(H∗ , 𝚪). For .ϕ ∈ V , define .s[ϕ] ∈ Belt(H, 𝚪) by

.

s[ϕ](z) = 2(Im z)2 ϕ(¯z),

.

z ∈ H.

Then .s : V → Belt(H, 𝚪) is holomorphic and .Ф ◦ s = id on .V . In particular, we see that .T (H∗ , 𝚪) contains the ball .‖ϕ‖H∗ < 2 in .B(H∗ , 𝚪). On the other hand, the Nehari–Kraus theorem (Theorem 7.3.1) implies that .T (H∗ , 𝚪)

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is contained in the ball .‖ϕ‖H∗ < 6. Moreover, by using their extension operator, Douady and Earle [22] showed that T (H∗ , 𝚪) = T (H∗ , 1) ∩ B(H∗ , 𝚪)

.

and that .T (H∗ , 𝚪) is contractible, where .T (H∗ , 1) is the Bers embedding of the universal Teichmüller space .Teich(1).

7.3.4 Function Theory on the Unit Disk In the previous section, the Bers embedding is realized in the Banach space B(H∗ , 𝚪). Through the Möbius transformation .h(z) = −i(1 + z)/(1 − z), the unit disk .D is mapped conformally onto the lower half-plane .H∗ . Since the classical univalent function theory mainly focused on the unit disk, we can move from .H∗ to .D harmlessly. Indeed, the pull-back transformation .h∗ : ϕ I→ ϕ ◦ h · (h' )2 maps .B(H∗ , 𝚪) isometrically onto .B(D, 𝚪 ' ), where .𝚪 ' = h−1 𝚪h. Recall that the Schwarzian derivative annihilates the effect of post-composition with Möbius transformations. Thus the universal Teichmüller space is described as

.

T (1) = T (D, 1) = {Sf : f ∈ Sqc },

.

where .Sqc is the set of univalent meromorphic functions f on .D which extend to quasiconformal maps of . C. A related space is .S(1) = {Sf : f ∈ S}, where .S is the set of univalent meromorphic functions on .D. Note that by the Hurwitz theorem .S(1) is a closed set in .B(D). Therefore, .T (1) ⊂ S(1). It can be seen that a function .f ∈ S belongs to .Sqc precisely when .f (D) is a quasidisk. In particular, .fr (z) = f (rz) belongs to .Sqc for any .f ∈ S and .0 < r < 1. Bers [13] conjectured that the closure of .T (1) in .B(D) coincides with .S(1). In 1978, Gehring [30] disproved this by showing that a conformal mapping f of .D onto the domain .C\β satisfies .Sf ∈ S(1)\T (1), where .β is the double logarithmic spiral .{z = ±e(a−i)t : 0 ≤ t < +∞} for .0 < a < 1/(8π ). Let .SJ = {f ∈ S : f (D) is a Jordan domain} and set .J (1) = {Sf : f ∈ SJ }. Then .T (1) ⊂ J (1) ⊂ S(1). Gehring’s example indeed satisfies .Sf ∈ S(1) \ J (1). Flinn [26] showed that .J (1) \ T (1) /= ∅. Thurston [65] proved the more striking result that .S(1) even has an isolated point in .B(D). His brilliant idea was to construct an arc with zipper structures which are nesting finer and finer. Let .Ω be the complement of such a curve. Then we cannot “open" the zipper by a conformal mapping g of .Ω with small enough norm .‖Sg ‖Ω . More precisely, the simply connected domain .Ω is conformally rigid, that is to say, there is a constant .ε > 0 with the property that every conformal (i.e., holomorphic and univalent) mapping g of .Ω satisfying the condition .‖Sg ‖Ω < ε must be (a restriction of) a Möbius transformation. It is easy to show that .Sf is an isolated

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point of .S(1) for a conformal mapping f of .D onto a conformally rigid domain .Ω. Astala [6] made a more explicit construction of conformally rigid domains (see also [7]). Indeed, he showed that . C \ γ is conformally rigid if either .γ is a snowflake arc with .dim γ > 1 or .γ is a proper subarc of the limit set of a finitely generated quasi-Fuchsian group which is not Fuchsian.

7.3.5 The Bers Conjecture for Fuchsian Groups Though the above Bers conjecture was disproved by Gehring, it is reasonable to consider weaker versions of the conjecture. Suppose that a Fuchsian group .𝚪 is cofinite, that is, .𝚪 is a finitely generated Fuchsian group of the first kind. Then .B(𝚪) := B(D, 𝚪) is a finite dimensional complex vector space. We set .S(𝚪) = S(1) ∩ B(𝚪). It is obvious that .T (𝚪) ⊂ S(𝚪). The closure of .T (𝚪) in .B(𝚪) is compact and its boundary is called the Bers boundary of the Teichmüller space .T (𝚪). Conjecture II in Bers [13], which we call the second Bers conjecture here, reads that the Bers compactification .T (𝚪) coincides with .S(𝚪). Sullivan and Thurston extended this conjecture to the density conjecture: Every finitely generated Kleinian group is an algebraic limit of geometrically finite Kleinian groups. This conjecture was finally proved by Namazi and Souto, and by Ohshika, independently. However, the second Bers conjecture does not follow immediately from the Sullivan–Thurston density conjecture. The second Bers conjecture was proved by Minsky [45] when the Teichmüller space is one-dimensional. Bromberg [18] solved the conjecture for b-groups without parabolic elements which is an essential part for the conjecture. Indeed, the conjecture (for torsion-free Fuchsian groups at least) has been proved by recent development of the theory of Kleinian groups. Theorem 7.3.4 Let .𝚪 be a cofinite torsion-free Fuchsian group. Then .T (𝚪) = S(𝚪) in .B(𝚪). We give an outline of the proof according to the roadmap proposed by Ohshika [52] (see also §1.3 in Minsky [45]). Let .ϕ ∈ S(𝚪) and take a univalent function f on .D with .Sf = ϕ. Then the map .ρ of .𝚪 defined by .ρ(γ ) = f ◦γ ◦f −1 gives a faithful representation of .𝚪 in .PSL(2, C), whose image .G = ρ(𝚪) is a (finitely generated) Kleinian group acting on .Ω = f (D) properly discontinuously. Since .S = D/ 𝚪 ∼ = Ω/G is a Riemann surface of finite conformal type, .Ω is an invariant component of .Ω(G). Note that .N(G) = H3 /G is homeomorphic to the product space .S × R since .π1 (N (G)) ∼ =G∼ = π1 (S). 3 The nearest point retraction .Π0 : Ω(G) → C0 (G) ⊂ H induces an embedding .F : S → N (G). The embedded surface .F (S) divides .N(G) into two pieces .N+ (G) and .N− (G) with end invariants .ν+ = S and .ν− , respectively. By Marden’s tameness conjecture (see [17]), .ν− consists of geometrically tame ends. Thus we can apply Theorem 3.10 of Ohshika [52] and its proof to obtain the claim: There exists a sequence of conformal maps .fi : D → Ωi with the following properties.

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(1) The Schwarzian derivative .ϕi of .fi belongs to .T (𝚪) for each i and thus .Gi = ρi (𝚪) is a quasi-Fuchsian group, where .ρi (γ ) = fi ◦ γ ◦ fi−1 . (2) The sequence of the isomorphisms .ρi : 𝚪 → Gi converges algebraically to an isomorphism .ρ∞ : 𝚪 → G∞ of .𝚪 onto a Kleinian group .G∞ for which the end invariants of .N(G∞ ) are the same as .(ν+ , ν− ), those of .N(G). By normalizing .fi so that .fi (0) = 0, fi' (0) = 1 and .fi (D) ⊂ C, and by passing to subsequence if necessary, we may assume that the sequence .fi converges to a conformal map .f∞ locally uniformly on .D. Then .ϕi tends to .ϕ∞ = Sf∞ locally uniformly on .D as .i → ∞. Since .B(𝚪) is finite dimensional, we see that .‖ϕi − ϕ∞ ‖D → 0 as .i → ∞. Hence, .ϕ∞ ∈ T (𝚪). Since .ρi → ρ∞ algebraically, we −1 for .γ ∈ 𝚪. Thus .Ω have .ρ∞ (γ ) = f∞ ◦ γ ◦ f∞ ∞ = f∞ (D) is an invariant component of .G∞ . Now we apply the Ending Lamination Theorem for Surface Groups in Brock et al. [19] to conclude that .G∞ is Möbius conjugate to .G, which implies that .Ω∞ = M(Ω) for some .M ∈ PSL(2, C). Since .f∞ and .M ◦ f map .D onto the same domain, we obtain the relation .ϕ = ϕ∞ and thus we have shown that .ϕ ∈ T (𝚪). We should remark that Gehring [29] proved the weaker assertion than the Bers conjecture that the interior of .S(1) in .B(D) coincides with .T (1). Furthermore, Shiga [57] showed that the interior of .S(𝚪) coincides with .T (𝚪) for any cofinite Fuchsian group.

7.4 Holomorphic Motions, Quasiconformal Motions and Kleinian Groups Let E be a subset of . C containing more than three points and X be a connected C is a Hausdorff space with basepoint .x0 . We say that a map .φ : X × E →  quasiconformal motion of E over X if it satisfies the following three conditions: (1) .φ(x0 , z) = z for any .z ∈ E. (2) For any .x ∈ X, the map .φx := φ(x, ·) : E →  C is an injection. (3) Let .E4 denote the set of quadruplets of distinct points of E. Then, .{φ(·; (a, b, c, d))}(a,b,c,d)∈E4 is an equicontinuous family of mappings from X to .C \ {0, 1}, where φ(x; (a, b, c, d)) =

.

(φx (a) − φx (b))(φx (c) − φx (d)) (φx (a) − φx (c))(φx (b) − φx (d))

for .x ∈ X, .(a, b, c, d) ∈ E4 , and .C \ {0, 1} is equipped with the hyperbolic distance .dC\{0,1} . That is, for any .x ∈ X and .ε > 0, there exists a neighborhood .Ux of x such that for any .(a, b, c, d) ∈ E4 the inequality dC\{0,1} (φ(x; (a, b, c, d)), φ(x ' ; (a, b, c, d))) < ε

.

holds for any .x ' ∈ Ux .

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Let M be a connected complex manifold with basepoint .x0 . A map .φ : M ×E →  C is called a holomorphic motion of E over M if it satisfies the following three conditions: (1) .φ(x0 , z) = z for any .z ∈ E. (2) For any .x ∈ M, the map .φx := φ(x, ·) : E →  C is an injection. (3) For any .z ∈ E, .φ(·, z) : M →  C is holomorphic. It follows from the Schwarz lemma that a holomorphic motion of E over M is a quasiconformal motion. We see that both motions of a set E are extended to the motions of the closure of E. Theorem 7.4.1 Let .φ : V × E →  C be a quasiconformal motion of E over a connected Hausdorff space V with basepoint .x0 . Then, it extends to a quasiconformal motion of .E over V . If V is a complex manifold and .φ is a holomorphic motion of E over V , then it extends a holomorphic motion of .E over V . Sullivan and Thurston [63] and Bers and Royden [14] showed that for any holomorphic motion of E over the unit disk .D, there exists a neighborhood U of the basepoint of the holomorphic motion such that the restricted holomorphic motion of E over U extends to a holomorphic motion of . C over U . Sullivan and Thurston conjectured that one can take .D itself as U in the above. Finally, Slodkowski showed that any holomorphic motion of E over .D extends to a holomorphic motion of . C over .D and solved their conjecture. Moreover, Earle, Kra and Krushkal [23] showed a group equivariant version of Slodkowski’s theorem. Theorem 7.4.2 Let G be a subgroup of .PSL(2, C) and E be a closed set in . C which is invariant under the action of G. Suppose that a holomorphic motion .φ : D × E of E over .D is G-equivariant, that is, there exists a homomorphism .ρλ : G → PSL(2, C) for each .λ ∈ D such that φ(λ, g(z)) = ρλ (g)(φ(λ, z))

.

holds for any .(λ, z) ∈ D × E and .g ∈ G. Then, .φ extends to a G-equivariant holomorphic motion of . C over .D. As for quasiconformal motions, the statement of Theorem 7.4.2 does not hold even if the group G is trivial. In fact, we may construct a quasiconformal motion .φ of some closed set E over .I = [0, 1], such that it cannot be extended to a quasiconformal motion of . C over any neighborhood of 0 (cf. [34]). On the other hand, for tame quasiconformal motions which are in between quasiconformal motions and holomorphic motions (see [34] Definition 6), the statement of Theorem 7.4.2 holds for general parameter spaces. Theorem 7.4.3 ([34]) Let V be a simply connected and connected Hausdorff space with a basepoint, and .φ : V × E →  C be a G-equivariant tame quasiconformal

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motion of E over V . Then, .φ extends to a G-equivariant quasiconformal motion of  C over V .

.

Acknowledgments The authors thank Professors Katsuhiko Matsuzaki, Ken’ichi Ohshika and Matti Vuorinen for their valuable suggestions and comments.

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52. K. Ohshika, Limits of geometrically tame Kleinian groups. Invent. Math. 99(1), 185–203 (1990) 53. V. Ovsienko, S. Tabachnikov, What is . . . the Schwarzian derivative? Not. Am. Math. Soc. 56(1), 34–36 (2009) 54. C. Pommerenke, Uniformly perfect sets and the Poincaré metric. Arch. Math. 32, 192–199 (1979) 55. C. Pommerenke, Boundary Behaviour of Conformal Maps (Springer, Berlin 1992) 56. O. Rainio, T. Sugawa, M. Vuorinen, Uniformly perfect sets, Hausdorff dimension, and conformal capacity (2023). arXiv:2305.16723 57. H. Shiga, Characterization of quasi-disks and Teichmüller spaces. Tôhoku Math. J. 37, 541– 552 (1985) 58. H. Shiga, Riemann mappings of invariant components of Kleinian groups. J. Lond. Math. Soc. 90, 716–728 (2009) 59. W. Smith, D.A. Stegenga, A geometric characterization of Hölder domains. J. Lond. Math. Soc. 35, 471–480 (1987) 60. W. Smith, D.A. Stegenga, Hölder domains and Poincaré domains. Trans. Am. Math. Soc. 319, 67–100 (1990) 61. T. Sugawa, Uniform perfectness of the limit sets of Kleinian groups. Trans. Am. Math. Soc. 353, 3603–3615 (2001) 62. T. Sugawa, Uniformly perfect sets: analytic and geometric aspects. Sugaku Expo. 16, 225–242 (2003) 63. D. Sullivan, W.P. Thurston, Extending holomorphic motions. Acta Math. 157, 243–257 (1986) 64. W.P. Thurston, The Geometry and Topology of Three-Manifolds. Princeton Lecture Notes (1977/1978). http://library.msri.org/books/gt3m/ 65. W.P. Thurston, Zippers and schlicht functions, The Bieberbach Conjecture, in Proceedings of the Symposium on the Occasion of the Proof of the Bieberbach Conjecture Held at Purdue University, West Lafayette, March 11–14, 1985, ed by D. Drasin, P. Duren, A. Marden. Mathematical Surveys and Monographs, vol. 21 (American Mathematical Society, Providence, 1986), pp. 185–197 66. J. Väisälä, Uniform domains. Tôhoku Math. J. 40, 101–118 (1988) 67. T. Yamaguchi, Word length and limit sets of Kleinian groups. Kodai Math. J. 28, 439–451 (2005)

Chapter 8

Thurston’s Broken Windows Only Theorem Revisited Ken’ichi Ohshika

Abstract The “broken windows only theorem” is the main theorem of the third paper among a series of papers in which Thurston proved his uniformisation theorem for Haken manifolds. In this chapter, we show that the second statement of this theorem is not valid, giving a counter-example. We also give a weaker version of this statement with a proof. In the last section, we speculate on how this second statement was intended to be used in the proof of the bounded image theorem, which constituted a key of the uniformisation theorem. The proof of the bounded image theorem was obtained only quite recently, although a weaker version, which is sufficient for the proof of the uniformisation theorem, had already been proved. Keywords Haken manifold uniformisation · Degeneration of hyperbolic structures · Bounded image theorem 2020 Mathematics Subject Classification 57K32, 30F40

8.1 Introduction One of the most celebrated achievements of Thurston is the uniformisation of Haken manifolds, which motivated the general geometrisation theorem, proved later by Perelman. Thurston planned to publish the proof of the uniformisation theorem in a series of seven papers. Only the first one was officially published, and the second and the third papers exist as preprints, and are now included in the collected works of Thurston [37]. In the first paper, the compactness theorem for the deformation spaces of acylindrical hyperbolic 3-manifolds is proved. The third paper generalises this compactness to a general atoroidal 3-manifold with incompressible boundary as a relative convergence theorem, which is dubbed as the “broken windows only”

K. Ohshika () Gakushuin University, Mathematics Department, Tokyo, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 K. Ohshika, A, Papadopoulos (eds.), In the Tradition of Thurston III, https://doi.org/10.1007/978-3-031-43502-7_8

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theorem (Theorem 0.1 in [35]). This should have constituted the key part of the proof of what is called the bounded image theorem, one of the main ingredients of the entire proof. The theorem consists of two statements and we shall show that in fact the second statement is not valid, giving a counter-example. Although Thurston did not write a proof of the bounded image theorem, a weaker version of it, the bounded orbit theorem, which first appeared in [36], is sufficient for the proof of the uniformisation theorem as was observed by Morgan [22], and a detailed proof of this weaker version was given in Kapovich [13]. A proof of the bounded image theorem by Thurston had been unknown, except for the acylindrical case given by Kent [14] and Brock–Bromberg–Canary–Minsky [7], but quite recently a proof relying on modern techniques of Kleinian group theory (in particular the work of Brock–Bromberg–Canary–Lecuire [6] among other works) was given by Lecuire and Ohshika [16]. It is still worthwhile to consider how Thurston himself tried to prove the bounded image theorem using the tools available at that time. We shall show in the last section how this should have been done provided that the broken windows only theorem were true. We now state this broken windows only theorem and explain what is problematic. For that, we first need to explain some terms used in the theorem, whose precise definitions will be given in the following section. Let .(M, P ) be a pared manifold, that is, a compact irreducible 3-manifold M with a union P of incompressible tori and annuli on .∂M such that .∂M \ P is incompressible, and every .π1 -injective, immersed torus or properly immersed annulus in M can be properly homotoped into P . We denote by .AH (M, P ) the set of complete hyperbolic metrics on .IntM modulo isotopy such that each component of P corresponds to a parabolic end. The set .AH (M, P ) corresponds one-to-one to the set of faithful discrete representations of .π1 (M) into .PSL2 (C) sending .π1 (P0 ) to a parabolic subgroup for each component .P0 of P , modulo conjugacy. We consider the characteristic submanifold .Ф of .(M, P ) in the sense of Jaco– Shalen–Johannson, consisting of I -bundles (over incompressible surfaces which are neither discs nor annuli nor Möbius bands), solid tori intersecting .∂M along incompressible annuli on their boundaries, and thickened tori intersecting .∂M along tori and annuli. The window F of .(M, P ) is an I -bundle constructed from the characteristic submanifold as follows. All I -bundle components of .Ф are contained in F . For each solid torus or thickened torus component V of .Ф, we consider each component of .Fr V , which is an annulus. If it is not properly isotopic into the I -bundle component, we consider its thin regular neighbourhood, which is an I bundle over an annulus, and make it be included in F . Otherwise we abandon that component. If there is more than one properly homotopic frontier component, we do this procedure only for one among them. Thus obtained F , which is an I -bundles over a possibly disconnected surface, is called the window of .(M, P ). The window of .(M, P ) is denoted by .window(M, P ) and its base space by .wb(M, P ).

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Now Thurston’s “broken windows only” theorem reads as follows: Theorem 8.1 (Thurston [35]) If .𝚪 ∈ π1 (M) is any subgroup which is conjugate to the fundamental group of a component of .M ' \ window(M, P ), then the set of representations of .𝚪 in .Isom(H3 ) induced from .AH (M, P ) are bounded, up to conjugacy. Given any sequence .Ni ∈ AH (M, P ), there is a subsurface with incompressible boundary .x ⊂ wb(M, P ) and a subsequence .Ni(j ) such that the restriction of the associated sequence of representations .ρi(j ) : π1 (M) → Isom(H3 ) to a subgroup .𝚪 ⊂ π1 (M) converges if and only if .𝚪 is conjugate to the fundamental group of a component of .M \ X, where X is the total space of an interval bundle above x. Furthermore, no subsequence of .ρi(j ) converges on any larger subgroup. Thurston gave a fairly detailed proof for the first sentence. There is an alternative approach for the first part using Morgan–Shalen theory [24–26], and also a stronger version of this first part has been proved by Brock–Bromberg–Canary–Minsky [7]. We shall give a counter-example to the second sentence of Theorem 8.1. The problem lies in the fact that there may be a solid torus or a thickened tours V in the characteristic submanifold of M which intersects .∂M at more than three annuli. Such a V may bring about a 3-manifold homotopy equivalent to M which is not homeomorphic to M, by changing the order of attaching the components of .M \ V along .∂V . We call this operation shuffling along V . We shall show that even though Theorem 8.1 does not hold in the original form, a weaker version as follows can be proved making use of the Morgan–Shalen theory and the efficiency of pleated surfaces due to Thurston [34]. Theorem 8.2 Let .(M, P ) and .Ni be as given in Theorem 8.1. Then there is a pared manifold .(M ' , P ' ) obtained from .(M, P ) by shuffling around solid pairs and thickened torus pairs of the characteristic submanifold of .(M, P ) for which the following hold. Let .W ' be the characteristic submanifold of .(M ' , P ' ). Let .W be the union of its I -pairs, and .V the union of solid torus pairs and thickened torus pairs of .W ' . Let .p : W → w be the fibration as an I -bundle for .W. Then there is an incompressible subsurface x of w and a union of essential annuli (possibly with singular axes) .A in .V such that the restriction of the associated sequence of representations .ρi(j ) : π1 (M) → Isom(H3 ) to a subgroup .𝚪 ⊂ π1 (M) converges if and only if .𝚪 is conjugate to the fundamental group of a component of .M ' \ (X ∪ A) under the identification of .π1 (M) with .π1 (M ' ), where .X = p−1 (x). (When we cut M along an annulus with singular axis, we regard a regular neighbourhood of the singular axis as contained in both sides after the operation.) Furthermore, no subsequence of .ρi(j ) converges on any larger subgroup. Unfortunately, this weaker version is not sufficient for the proof of the bounded image theorem.

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8.2 Preliminaries In this section, we explain notions and terminologies which are necessary for Theorems 8.1 and 8.2 and their proofs.

8.2.1 JSJ Decompositon In this subsection, we shall review the theory of Jaco–Shalen and Johannson concerning characteristic decomposition of compact irreducible 3-manifolds by tori and annuli and homotopy equivalences between such 3-manifolds. Unless otherwise mentioned, throughout this section M will denote a compact irreducible 3-manifold with (possibly empty) incompressible boundary. We always assume 3-manifolds to be orientable. Jaco–Shalen and Johannson proved that there exists a disjoint family of embedded incompressible tori .T1 , . . . , Tp and properly embedded incompressible annuli .A1 , . . . , Aq in M with the following properties. (1) Cutting M along .(T1 ∪ · · · ∪ Tp ∪ A1 ∪ · · · ∪ Aq ), we obtain compact 3submanifolds of M each of which is one of the following. (a) A Seifert fibred manifold whose frontier in M consists of fibred tori and fibred annuli. (b) An atoroidal manifold, i.e. a manifold such that every .π1 -injective map from a torus to the manifold is homotoped into a boundary component. We call each of such manifolds a hyperbolic piece. (In particular the definition implies that each of the tori .T1 , . . . , Tp is disjoint from the interior of any hyperbolic piece.) (2) If a hyperbolic piece N contains some of the .A1 , . . . , Aq , then the 3-manifold ' .N obtained by cutting N along the annuli contained in N consists of three types of manifolds. (a) An I -bundle over a surface such that its intersection with .∂N coincides with the associated .∂I -bundle and is an incompressible subsurface of .∂N. Such a component is called an I -pair, and we denote each of them in the form of a pair .(Ф, Σ), where .Ф is an I -bundle and .Σ = Ф ∩ ∂N, or simply by .Ф if there is no fear of confusion. (b) A solid torus V whose intersection with .∂N consists of annuli which are incompressible both on .∂V and .∂N. Such a component .(V , V ∩∂N) is called a solid torus pair. (c) A thickened torus .U ∼ = S 1 × S 1 × I such that .S 1 × S 1 × {0} is contained in 1 1 .∂N and .S × S × {1} intersects .∂N along a union of incompressible annuli no two of which are homotopic in .∂N. For such a component .(U, U ∩ ∂N) is called a thickened torus pair.

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(d) An acylindrical manifold, i.e. a compact 3-manifold such that every properly embedded annulus is relatively properly homotopic into a boundary component. We note that every .π1 -injective proper map from an annulus into a hyperbolic piece is properly homotopic into either an I -pair or a solid torus pair or a thickened torus pair. We take a family of incompressible tori and annuli having the above properties which is minimal with respect to inclusion among the families with the same properties. We call a decomposition of M by such a minimal family a JSJ-decomposition. Jaco and Shalen [11] and Johannson [12] proved that a JSJdecomposition is unique up to ambient isotopy. We now assume that M is atoroidal. In this case, the union of the I -pairs, the solid torus pairs and the thickened torus pairs is called the characteristic submanifold of M. The window F of M is constructed from the characteristic submanifold as follows. We first consider the I -pairs in the characteristic submanifold, and let them be included in F . For a solid torus pair or a thickened torus pair V , we replace it with regular neighbourhoods .V1' , . . . , Vk' of the components of .Fr V , which are solid tori and can be regarded as I -bundles over annuli. If .Vj' is properly homotopic into an I -pair or another solid torus already added to F , we discard it, and otherwise we let .Vj' be included in F .

8.2.2 Homotopy Equivalences and Books of I -Bundles Each I -pair, solid torus pair and thickened torus pair in the characteristic submanifold of an atoroidal Haken manifold may give rise to a homotopy equivalence which is not realised by a homeomorphism under some conditions as will be explained below. Let .(Ф, Σ) be an I -pair in a JSJ-decomposition of an atoroidal Haken manifold M. By definition, .Ф has an I -bundle structure whose projection to its base surface we denote by .p : Ф → B. Suppose that .∂B has more than one component, and let b be a component of .∂B. Its inverse image .p−1 (b) is an annulus which is a union of fibres. We parametrise .p−1 (b) as .{(z, t) | z = eiθ , θ ∈ [0, 2π ), t ∈ [0, 1]} so that the second coordinate corresponds to fibres. We consider an involution .ι on −1 (b) taking .(z, t) to .(z, 1 − t). Now we remove N from M and paste it back .p to .M \ N by reversing the direction of each fibre in .p−1 (b) by the involution .ι just defined. Then we get another 3-manifold .M ' homotopy equivalent to M, and this operation gives a homotopy equivalence from M to .M ' . We call this operation flipping on the I -pair .(Ф, Σ), following Canary–McCullough [9], where homotopy equivalences of pared manifolds are thoroughly studied. We note that .M ' is nonorientable if the annulus .p−1 (b) is non-separating. The same kind of operation can be defined on a frontier component of a solid torus pair or a thickened torus pair in the characteristic submanifold, which we also call flipping.

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Next let .(V , C) be a solid torus pair in the JSJ-decomposition of M. We call the number of components of .V ∩ ∂M the valency of V . It is evident that the valency is also equal to the number of the components of .Fr V . Suppose that the valency of V is greater than 2. Then we can remove V from M and paste it back after changing the cyclic order of the annuli in .Fr V . In this way, we get a 3-manifold homotopy equivalent to M. We call this operation shuffling around V . We can define shuffling around a thickened torus pair .(U, U ∩ ∂M) in the same way, to be changing the order of pasting the components of .M \ U to .Fr U . Jaco and Shalen [11] and Johannson [12] proved that any homotopy equivalence between two compact irreducible 3-manifolds with (non-empty) incompressible boundaries is obtained by repeating flipping and shuffling. We now consider a special case of 3-manifolds obtained by pasting more than two product I -bundles along one solid torus. Let V be a solid torus and take a simple closed curve .α on .∂V homotopic to the p-time iteration of a core curve with .p ≥ 1. Letting n be an integer greater than 2, we consider n compact surfaces .Σ1 , . . . , Σn each of which has only one boundary component, denoted by .c1 . . . cn . We then consider product I -bundles .Σ1 × I, . . . , Σn × I , and paste them along .c1 × I, . . . , cn × I to .∂V so that all of these annuli are identified with disjoint annuli with core curves isotopic to .α which lie on .∂V in the (cyclic) order of the subscripts. In this way, we get a compact, irreducible, atoroidal 3-manifold N , and it is called a book of (product) I -bundles with n pages. Such manifolds were studied for the first time in the context of deformation spaces of Kleinian groupsby Anderson and Canary [1]. In particular when .p = 1, the book of I -bundles is said to be primitive. The characteristic submanifold of N consists of .n + 1 components, i.e. I -pairs corresponding to .Σ1 × I, . . . , Σn × I and a solid torus pair .(V , V ∩ ∂N). The window consists of n components: the characteristic submanifold with the solid torus pair .(V , V ∩ ∂N) removed. By the theory of Jaco–Shalen and Johannson mentioned above, any homotopy equivalence from N to another compact irreducible 3-manifold is obtained by shuffling along V since no flipping gives rise to a new manifold.

8.2.3 Pared Manifolds A pared manifold is a pair .(M, P ) of a compact irreducible 3-manifold M (with possibly compressible boundary) and a disjoint union of incompressible tori and annuli P lying on .∂M with the following properties. (i) Any .π1 -injective immersion from a torus into M is homotopic into P . (ii) Any .π1 -injective proper immersion of an annulus into M whose boundary is mapped into P is properly homotopic into P .

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We also assume the following: (iii) Every component of .∂M \ P is incompressible. We define the characteristic submanifold of a pared manifold .(M, P ) in the same way as before, just imposing the condition that no component of P is contained in the associated .∂I -bundle of an I -pair.

8.2.4 Deformation Spaces Let M be a compact, irreducible, atoroidal 3-manifold. We say that M admits a convex compact hyperbolic structure when there are a convex compact hyperbolic 3-manifold N and a homeomorphism from M to N . The hyperbolic manifofold N can be embedded in a complete hyperbolic manifold homeomorphic to .IntN as a convex core. Therefore a convex compact hyperbolic structure on M also induces a complete hyperbolic structure on .IntM. Each complete hyperbolic structure on .IntM corresponds to a faithful discrete representation of .π1 (M) into .PSL2 (C), which is uniquely determined up to conjugacy. The set of faithful discrete representations of .π1 (M) into .PSL2 (C) modulo conjugacy is denoted by .AH (M) and is given a topology induced from the topology of element-wise convergence of the representation space. We consider the space of convex compact hyperbolic structures on M modulo isotopy, which we denote by .QH0 (M), and regard it as a subspace of .AH (M) by using extensions of such structures to complete hyperbolic structures as mentioned above. By the theory of Ahlfors, Bers, Kra, Maskit, Marden and Sullivan, there is a covering map .q : T(∂M) → QH0 (M). (The definition of the map in this general setting is due to Bers [3], Kra [15], Maskit [18] and Marden [17]. Sullivan [31] showed that the map is surjective.) Under the present assumption that .∂M is incompressible, q is known to be a homeomorphism [3, 15, 17, 18]. In the special case when M is homeomorphic to .S × I for some closed surface S, this parametrisation is what is called the Bers ¯ → QH0 (M), simultaneous uniformisation [2], and is expressed as .qf : T(S)×T(S) ¯ denotes where .T(S) denotes the Teichmüller space of the oriented S, whereas .T(S) that of S with orientation reversed. Representations contained in .QH0 (M) in this case are called quasi-Fuchsian representations. For a pared manifold .(M, P ), we define .AH (M, P ) to be the set of faithful discrete representations of .π1 (M) into .PSL2 (C) which for each component .P0 of P send each non-trivial element of .π1 (P0 ) to a parabolic element. The set modulo isotopy of convex finite-volume complete hyperbolic structures on .(M, P ) for which P corresponds to parabolic ends forms a subspace of .AH (M, P ), which we denote by .QH0 (M, P ). As in the case of convex compact hyperbolic structures, there is a homeomorphism .q : T(∂M \ P ) → QH0 (M).

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8.2.5 Morgan–Shalen Compactification In this subsection we summarise some results contained in Morgan and Shalen [24– 26], which will be used in the proof of Theorem 8.2 in Sect. 8.4. Let M be an irreducible, atoroidal compact 3-manifold with incompressible boundary. We may also consider a pared manifold .(M, P ) as will be explained later. What interests us is the case when M has a boundary component which is not a torus. Let .{ρi } be a sequence in .AH (M). Suppose that .{ρi } does not have a convergent subsequence in .AH (M). Fix a generator system .g1 , . . . , gk of .π1 (M) such that the trace functions .trg1 , . . . , trgk generate the trace functions of elements of .π1 (M) as a ring. (See Culler and Shalen [10] for the existence of such a system.) Let .μi be .maxkj =1 {trlength(ρi (gj ))}, where .trlength denotes the translation length. By changing the ordering of the generator system and passing to a subsequence, we can assume that .μi is attained by .trlength(ρi (g1 )). Put a basepoint .xi in .H3 which is within bounded distance from the axis of .ρi (g1 ). Consider the rescaled hyperbolic space .trlength(ρi (g1 ))−1 H3 with basepoint at .xi , and take an equivariant Gromov limit. Then we get an isometric action of .π1 (M) on an .R-tree. (Morgan–Shalen [24], Paulin [29], Bestvina [4]) Here an .R-tree is a geodesic space in which for any two points, there is only one simple arc connecting them. In other words, it is a 0-hyperbolic space in sense of Gromov. An isometric action of G on an .R-tree T is said to have small edge stabilisers when for any non-trivial arc a in T , its stabiliser is a virtually abelian group. It is said to be minimal when T has no .ρG (G)-invariant proper subtree. In the case when we consider a pared manifold .(M, P ), we assume that for any component .P0 of P and any element .γ ∈ π1 (P0 ), the translation length of .ρi (γ ) is bounded as .i → ∞. (We define the translation length of any parabolic element to be 0). Then the limit action on an .R-tree has the property that for any element .γ in .π1 (P0 ) as above, the action of .γ on the tree is trivial. Theorem 8.3 (Morgan–Shalen, Paulin, Bestvina) As a rescaled Gromov limit, we obtain a minimal isometric action of .π(M) on an .R-tree T with small edge stabilisers. A codimension-1 lamination in a compact 3-manifold M is a closed subset L of M such that for any point .x ∈ M there is a coordinate neighbourhood U of M of the form .D 2 × [0, 1], where .U ∩ L = D 2 × J with a closed subset J of .[0, 1]. We can define a leaf of L in the same way as for a foliation. A codimension-1 lamination is said to be measured when there is a positive Borel measure on every arc .α transverse to the leaves which is supported on .α × L and is invariant under isotopies keeping the endpoints on the same leaves and making no extra intersections. We equip the set of codimension-1 measured laminations with a topology induced from the weak topology on the set of measures on transverse arcs. A positively weighted disjoint union of embedded surfaces is an example of a codimension-1 measured lamination. A codimension-1 measured lamination is said to be incompressible when it is a limit of positively weighted unions of properly

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embedded incompressible surfaces. An incompressible codimension-1 measured lamination is said to be annular when it is a limit of weighted properly embedded incompressible annuli. Proposition 8.1 (Morgan–Shalen [25]) Any codimension-1 annular lamination L is properly isotopic into the characteristic submanifold W of M. For each I -pair .W0 of W , the intersection .L ∩ W0 can be isotoped so that it becomes vertical with respect to the fibration. Let L be an incompressible codimension-1 measured lamination, and consider the universal cover .M˜ of M. (In our setting, .M˜ is always simply connected.) Then ˜ and the union of lifted leaves each leaf of L can be lifted to a topological plane in .M, ˜ constitute a codimension-1 measured lamination in .M. For an incompressible measured lamination L, we can construct an .R-tree dual ˜ we collapse each component to L in the following way. In the universal cover .M, ˜ ˜ of .M \ L to a point, replace isolated leaves with arcs, and put a metric coming from the lift of the transverse measure of L. Then we get an .R-tree on which .π1 (M) acts by isometries. We call this the .R-tree dual to L, and denote it by .TL . We note that we can construct an .R-tree dual to a measured geodesic lamination on a hyperbolic surface in the same way, on which its fundamental group acts by isometries. A continuous map f from an .R-tree T to another .R-tree .T ' is said to be a morphism if for each point x in T , there is an geodesic segment I in T containing x such that .f |I is an isometry. An isomorphism from T to .T ' is a surjective isometry. A morphism may fail to be an isomorphism, for f may be “folded” at some points of T . Theorem 8.4 (Morgan–Shalen [26], Morgan–Otal [23]) Let .ρ : π1 (M) → Isom(T ) be a minimal isometric action on an .R-tree T with small edge stabilisers. Then there is an annular codimension-1 lamination L in M with an .π1 (M)equivariant morphism .t : TL → T , where .TL is the .R-tree dual to L. Furthermore, let .(M, P ) be a pared manifold, and suppose that .ρ : π1 (M) → Isom(T ) as above has the property that for any component .P0 of P , its fundamental group .π1 (P0 ) lies in the kernel of .ρ. Then the annular codimension-1 lamination L above can be taken to be disjoint from P . We note that this theorem only guarantees the existence of a morphism, not an isomorphism. As we shall see in Sect. 8.3, there is an example for which we cannot have an annular codimension-1 lamination giving an isomorphism. We need to generalise slightly the notion of annular codimension-1 lamination for Theorems 8.2 and 8.6. We define an annulus with a singular axis as follows. Let V be a solid torus pair in a pared manifold .(M, P ). Then .V ∩ ∂M consists of annuli on .∂V whose core curves are homotopic in V to the p-time iteration of a core curve of V for some positive integer p. This means that there is a Seifert fibration .π : V → D for a disc D with a cone point .c of order p with respect to which each component of .V ∩ ∂M is vertical, i.e. it is expressed as .π −1 (α) for a simple arc on .∂D. We denote the union of the simple arcs corresponding to .V ∩ ∂M by A. An annulus with a singular axis is defined to be a singular annulus of the

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form .p−1 (β1 ∪ · · · ∪ βk ) for the union of simple arcs .β1 , . . . , βk with .k ≥ 2 in D connecting A to .c which intersect each other only at .c. Such an annulus has a singular axis, which is equal to .π −1 (c), around which any non-singular fibre wraps p times. We call such an annulus weighted if each of .β1 , . . . , βk , and correspondingly its preimage, has a positive weight. In Theorem 8.6, we shall allow such weighted annuli with singular axes in solid torus pairs to be included as components of an annular codimension-1 measured lamination.

8.2.6 Skora’s Theorem In the previous subsection, we talked about an action on an .R-trees dual to a codimension-1 measured lamination in a 3-manifold. In this subsection, we present a theorem saying that in the case of (possibly non-orientable) surfaces of hyperbolic type, any minimal isometric action on an .R-tree with small edge stabilisers is dual to a measured geodesic lamination on the surface. The following theorem was proved by Skora [30]. Theorem 8.5 (Skora) Let S be a closed hyperbolic surface, and suppose that ρ : π1 (S) → Isom(T ) is a minimal action with small edge stabilisers. Then there is a measured geodesic lamination .λ on S with a .π1 (S)-equivariant isomorphism between the .R-tree .Tλ dual to .λ and T .

.

We note that even in the case when S has punctures or boundaries, Theorem 8.5 holds under the assumption that the elements of .π1 (S) corresponding to loops around single punctures or boundaries act on T trivially.

8.3 Counter-Example In this section, we shall construct a counter-example to Theorem 8.1. We consider a primitive book of product I -bundles as explained in the previous section consisting of more than three pages. It is a special case of Thurston’s theorem that such a 3-manifold admits a convex compact hyperbolic structure in its interior, but even without using Thurston’s theorem, we can construct such a structure just by the Klein–Maskit combination theorem (see [19] for various versions of the combination theorem). To fix symbols, let M be a primitive book of I -bundles which is a union of product I -bundles .Σ1 × [0, 1], . . . , Σn × [0, 1] and a solid torus V , where each .Σj has only one boundary component, and .∂Σi × [0, 1] is attached to .∂V along an essential annulus .Ai whose core curve is homotopic to a core curve of V . We order the I -bundles so that the attaching annuli .A1 , . . . , An lie on .∂V in this (cyclic) order. We assume that the ordering is compatible with the parametrisation of

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[0, 1], that is, .∂Σi ×{1} and .Σi+1 ×{0} are juxtaposed. We let .Σi' be the complement of a thin annular neighbourhood of .∂Σi in .Σi . The characteristic submanifold W of M consists of a solid torus V and I -bundles .Σ1' × [0, 1], . . . , Σn' × [0, 1], and the window consists of .Σ1' × [0, 1], . . . , Σn' × [0, 1]. We call the boundary component of .Ai nearer to .Ai+1 , i.e. .∂Σi × {1} (we regard .An+1 as .A1 ) the upper boundary. For .1 ≤ i < j ≤ n, we denote by .Bi,j the closure of a component of the complement of .Ai ∪ Aj in .∂V which contains the upper boundary of .Ai . We let .Si,j be .Σi × {1} ∪ Bi,j ∪ Σj × {0} which is a closed surface embedded in M. We now define the (right) Dehn twist of M along .Aj . The right Dehn twist of M along .Aj is obtained by cutting M along .Ai into .Σj × I and the rest, and glue them back after twisting .Ai on .Σj × I by .2π in the direction of .S 1 keeping I -direction unchanged to the right when we look at it from the inside of .Σj × I . In other words, the right Dehn twist along .Aj is a product of the right Dehn twist around .Fr Σj and the identity on the fibre direction. We denote this homemorphism by .τAj : M → M. The left Dehn twist along .Aj is defined to be .τA−1 . j .

Proposition 8.2 Let M be a primitive book of product I -bundles of n pages with n ≥ 4. Fix a convex compact hyperbolic structure .m0 on M, which is also regarded as a point in .AH (M). Let .{mi ∈ AH (M)} be a sequence obtained from .m0 by pushing forward .m0 by the composition of the i-time iteration of the right Dehn twist .τA1 along .A1 and the i-time iteration of the left Dehn twist .τA−1 along .A3 , i.e. 3

.

we define .mi = (τA1 ◦ τA−1 )i (m0 ). Then the following hold: 3 ∗

(a) The sequence .{mi } does not have a convergent subsequence in .AH (M). (b) The sequence of restrictions .{mi |π1 (S1,2 )} does not have a convergent subsequence in .AH (S1,2 × I ). (c) The restrictions .mi |π1 (S1,3 ) converge up to conjugation. (d) The restrictions .mi |π1 (S2,4 ) converge up to conjugation. Proof Since (a) follows from (b), we shall first prove (b). Take a covering .M˜ 1,2 of M associated with .π1 (S1,2 ). The open 3-manifold .M˜ 1,2 has a compact core .C1,2 homeomorphic to .S1,2 × I containing homeomorphic lifts of .Σ1 × I and .Σ2 × I . Since convex compact hyperbolic structures are lifted to convex compact hyperbolic structures (see Proposition 7.1 of Morgan [22]), the sequence .{mi } is lifted to a sequence .{m ˜ i } in .QH0 (S1,2 × I ) ⊂ AH (M˜ 1,2 ). We shall show that .{m ˜ i } does not have a convergent subsequence. Recall that .mi is obtained from .m0 by the composition of the i-time iterations of the right Dehn twists along .A1 and .A3 . Putting a basepoint .x0 on .Σ1 × {0}, we see that every closed loop representing an element in .π1 (S1,2 , x0 ) can be homotoped off .A3 (fixing the basepoint). On the other hand, the annulus .A1 is lifted to an annulus .A˜ 1 properly embedded in .C1,2 , and .m ˜ i is obtained from .m ˜ 0 by the itime iteration of the right Dehn twist along .A˜ 1 . Since .m ˜ 0 is convex compact, .m ˜ 0 corresponds to a quasi-Fuchsian representation. By the Bers simultaneous uniformisation, .m ˜ 0 is expressed as .qf (g0 , h0 ), with .(g0 , h0 ) ∈ T(S1,2 ) × T(S¯1,2 ). Recall that .H3 /qf (g0 , h0 ) is homeomorphic to .S1,2 × (0, 1). The compact core .C1,2

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is homeomorphic to .S1,2 × [0, 1], and we can identify .S1,2 and .S¯1,2 with .S1,2 × {0} and .S1,2 × {1} respectively by homeomorphisms preserving markings, where for the latter the orientation of .S¯1,2 is the reverse of the natural one of .S1,2 × {1}. Let ¯1,2 isotopic to .∂ A˜ 1 ∩ S1,2 × {0} and .γ , γ¯ be simple closed curve on .S1,2 and .S ¯ .∂A1 ∩ S1,2 × {1} respectively. The simple closed curve .γ¯ corresponds to .γ on .S by the identification above. By this, .m ˜ i is expressed as .qf (τγi (g0 ), τγi (h0 )), where .τγ denotes the right Dehn twist around .γ . Then either by considering the geometric limit of the conformal structures at infinity, or just applying [27] or [6], we see that .{m ˜ i } does not have a convergent sequence in .AH (S × I ), and hence .{mi } does not have a convergent sequence in .AH (M). This completes the proof of (b), and we are also done with (a). The parts (c) and (d) can be proved by exactly the same argument. Therefore, we shall only prove (c). As in the case (a), take a covering .M˜ 1,3 associated with .π1 (S1,3 ). Let .m ˜ i be the convex compact hyperbolic structure on .IntM˜ 1,3 which is the lift of .mi . In the present case, both of the annuli .A1 and .A3 are lifted to properly embedded annuli .A˜ 1 and .A˜ 3 in .M˜ 1,3 . We can also take a compact core .C1,3 containing homeomorphic lifts of .Σ1 × [0, 1] and .Σ3 × [0, 1]. The right Dehn twist along .A1 is lifted to the one along .A˜ 1 , and the left Dehn twist along .A3 is lifted to the one along .A˜ 3 . Now the annuli .A˜ 1 and .A˜ 3 are parallel in .C1,3 , and hence the right Dehn twist along .A˜ 1 and the left Dehn twist along .A˜ 3 cancel out each other. Therefore we see that .m ˜i = m ˜ 0 for all i, which evidently implies that .{mi |π1 (S1,3 )} converges up to conjugation. ⨆ ⨅ Now we show that the sequence in Proposition 8.2 is a counter-example of the second claim of Theorem 8.1. Corollary 8.1 The sequence .{mi } in Proposition 8.2 disproves the second sentence of Theorem 8.1. Proof Recall that M is a primitive book of product I -bundles, and its window, which we denote by F , is the union of .Σ1' × I, . . . , Σn' × I . We denote the fibring of F as an I -bundle by .p : F → B. Suppose, seeking a contradiction, that the second sentence of Theorem 8.1 holds. Since .{mi } in Proposition 8.2 diverges, there must be an incompressible subsurface x of the base surface B with the property of Theorem 8.1 that the restriction of .mi to a subgroup of .π1 (M) converges if and only if .𝚪 corresponds to the fundamental group of a component of .M\X for .X = p−1 (x). Recall that for each component .Σj' × I of F , the restrictions .mi |π1 (Σj' × I ) converge. Therefore, .Σj' × I must lie outside .X = p−1 (x) for every j . This forces x to be a union of annular neighbourhoods of some of the boundaries ' ' .∂Σ (j = 1, . . . , n). If x contains an annular neighbourhood of .∂Σ , then there j 1 is no component of .M \ X corresponding the subgroup .π1 (S1,3 ), contradicting (c) of Proposition 8.2. Therefore, x cannot contain such an annulus. In the same way, by (c) of Proposition 8.2, x cannot contain an annular neighbourhood of .∂Σ2' . Now, these facts show that .S1,2 ×I is isotoped into a component of .M \X. This contradicts (b) of Proposition 8.2. Thus we are lead to a contradiction. ⨆ ⨅

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We next show that Proposition 8.2 gives an example of .R-tree action which is not dual to any codimension-1 annular measured lamination. We note that this also shows that the second sentence of Theorem 8.1 cannot be remedied just by taking into account annuli embedded in solid torus pairs, replacing the window in the statement to the entire characteristic submanifold. We recall that in the twodimensional setting, by Skora’s theorem, every isometric action of .π1 (S) on an .R-tree with small stabilisers is dual to a measured lamination. Corollary 8.2 Let .{mi } be a sequence in .AH (M) given in Proposition 8.2. Let ρ : π1 (M) → Isom(T ) be a rescaled limit of .{mi }, which is a minimal action on an .R-tree T with small edge-stabilisers as in Theorem 8.3. Then there is no codimension-1 annular measured lamination L in M with an equivariant isomorphism from the dual tree .TL to T .

.

Proof Suppose, seeking a contradiction, that there is such a codimension-1 measured lamination L. By Proposition 8.1, L can be assumed to lie in the characteristic submanifold W of M. Since the restrictions of .mi to .π1 (Σj ) converges for every .j = 1, . . . , n, we see that L must be disjoint from these pages of the book, hence must be contained in the solid torus V . By (c) of Proposition 8.2, .S1,3 × I can be homotoped so that it becomes disjoint from L. This implies that core curves of the annuli .A1 and .A3 are homotopic in .M \L. In the same way, by (d) of Proposition 8.2, core curves of .A2 and .A4 are homotopic in .M \ L. In the proof of (d), we only used the fact that .mi is defined by the i-time iterations of Dehn twists along .A1 and .A3 , and that .S2,4 can be homotoped off from them. Therefore, the same argument works also for .S2,j with .j = 5, . . . , n. This implies that .L ∩ V must be empty. Since .{mi } diverges, L itself cannot be empty. This is a contradiction. ⨆ ⨅

8.4 A Weaker Version of Thurston’s Theorem As can be seen in the proof of Corollary 8.1, the existence of a solid torus pair in M is the cause of the problem. On the other hand, such a solid torus gives rise to a homotopy equivalence by shuffling. In this section, we shall show by moving from M to a homotopy equivalent manifold, and taking into account not only the window but also solid torus pairs and thickened torus pairs of the characteristic submanifold, we can prove a weak version of Thurston’s original theorem, stated as Theorem 8.2. Before starting to prove this theorem, we prove the following theorem showing that a minimal isometric action of .π1 (M) with small edge stabilisers on an .R-tree is isomorphic to an action on an .R-tree dual to an annular measured lamination in a 3manifold homotopy equivalent to M. This should be contrasted with Corollary 8.2. Theorem 8.6 Let .(M, P ) be a pared manifold as given in Theorem 8.1. Let ρ : π1 (M) → Isom(T ) be a minimal isometric action on an .R-tree T with small edge stabilisers such that every element in .π1 (M) conjugate into the fundamental group of a component of P is mapped to the identity by .ρ. Then there are a

.

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codimension-1 annular measured lamination .L' , with weighted annuli with singular axes allowed, in a pared manifold .(M ' , P ' ) obtained from .(M, P ) by shuffling, and an equivariant isomorphism from the dual tree .TL' to T . Proof By Theorem 8.4, there is a codimension-1 annular measured lamination L in M such that there is a .π1 (M)-equivariant morphism f from the dual tree .TL to T . For any component .P0 of P , an element in .π1 (P0 ) acts trivially on T . Therefore we can choose L disjoint from P as stated in Theorem 8.4. Let .W0 be an I -pair in the characteristic submanifold W which contains a component of L. Then .W0 is an I -bundle over a surface .Σ. We denote this fibration by .p : W0 → Σ. By Proposition 8.1, .L0 := L ∩ W0 can be assumed to be vertical with respect to p. Let .W˜ 0 be a component of the preimage of .W0 in the universal cover .M˜ which is invariant under .π1 (W0 ) ⊂ π1 (M). Let .T W0 be the sub-tree of T corresponding to the rescaled limit of .W˜ 0 , and .TLW0 the subtree of .TL corresponding to .W˜ 0 . Then the restriction .f |TLW0 is a .π1 (W0 )-equivariant morphism. Since .π1 (W0 ) is isomorphic to .π1 (Σ), and an element of .π1 (Σ) corresponding to a boundary component of .Σ acts on T trivially, we can apply Theorem 8.5, and see that .T W0 ' is .π1 (W0 )-equivariantly isometric to an .R-tree .T W0 which is dual to an annular codimension-1 measured lamination .LW0 in .W0 . We replace .L ∩ W0 with .LW0 and ' hence .TLW0 with .T W0 . Then .f |TLW0 can be replaced with the equivariant isometry 'W between .T 0 and .T W0 . At the same time, every image of .TLW0 under the action of .π1 (M) and the restriction of f there are modified. Thus we get an .R-tree .T ' and a morphism .f ' : T ' → T which is an isometry on the orbit of the subtree corresponding to .W˜ 0 which is dual to .LW0 . Repeating this modification for each I -pair of W intersecting L, we get an .R-tree .Tˆ and a morphism .fˆ : Tˆ → T with an annular codimension-1 measured lamination .Lˆ to which .fˆ is dual. The morphism .fˆ is isometric outside the part corresponding to the preimage of the solid torus pairs and thickened torus pairs in W . Now let V be either a solid torus pair or a thickened torus pair in W . Since .Lˆ is annular, by isotopying it, we can assume that .V ∩ Lˆ consists of properly embedded incompressible annuli whose boundaries lie in .V ∩ (∂M \ P ), and that no two components of .V ∩ Lˆ are parallel as embeddings in .(V , V ∩ (∂M \ P )). In the case when V is a solid torus, we can also assume that it is disjoint from a core curve ˆ we can regard V as a Seifert fibred manifold with base of V . By isotoping .V ∩ L, surface D, which is a disc with singular point .c of order p when V is a solid torus and annulus when V is a thickened torus in such a way that these annuli .V ∩ Lˆ are all vertical with respect to the fibration .π : V → D. We denote the union of arcs on .∂D which is the image of .p(V ∩ (∂M \ P )) by A The subtree .TV of .Tˆ corresponding to a component of the preimage of V in .M˜ which is invariant under .π1 (V ) is a simplicial tree each of whose edges is dual to a component of .V ∩ Lˆ and has length equal to the weight given to that component. We note that the action of the group .Z corresponding to a regular fibre is trivial. In the case when V is a solid torus pair, and .p > 1, the cyclic group .Zp acts on .TV by isometries, whereas in the case when V is a thickened torus, the infinite cyclic subgroup .Z acts on .TV by isometries. Both of them are quotients of .π1 (V ) by the

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normal subgroup generated by a regular fibre. In the latter case, the stabiliser of the group .Z consists of one point since there is no leaf of .Lˆ which is a torus. We modify .TV and .fˆ|TV , by removing folds as in the proof of Skora’s theorem [30], and make .fˆ|TV an embedding to T , whose image we denote by .TV' . We can do this equivariantly with respect to the action of either .Zp or .Z. Then .TV' is a subtree of T on which either .Zp or .Z acts by isometries. We consider the quotient .T¯V of .TV' by this isometric action of .Zp or .Z, which is a finite tree. Each of the endpoints of .T¯V , i.e. each vertex of valency 1, corresponds to a component (or components) of .Fr V . Conversely, each component of .Fr V corresponds to a vertex of .T¯V , but not necessarily an endpoint. We give numbers to the components of .Fr V as .A1 , . . . , An so that the subscripts coincide with the cyclic order of these components on .∂V . Correspondingly, we number the associated vertices of .T¯V . Note that it is possible that more than one component of .∂V correspond to the same vertex, and hence the vertex has more than one number. We also note that in the case when V is a thickened torus pair, there is a vertex .vP in .T¯V corresponding to the torus component of .V ∩ ∂M which we denote by .P0 . In the case when V is a solid torus and .p > 1, there is a subtree .tc of .T¯V which is the quotient of the stabiliser of the .Zp action. We realise .T¯V in the base surface D so that the vertices numbered lie on .∂D. In the case when V is a solid torus pair with .p > 1, we collapse .tc into .c. In the case when V is a thickened torus pair, we assume that .vP lies on .P0 . Then it can be regarded as a tree dual to a partition of a disc (possibly with a cone point) or an annululs D by disjoint weighted simple arcs .α with endpoints at .A∪{c}, that is, each region of the complement corresponds to a vertex, and each arc corresponds to an edge and has weight equal to the length of the edge. If an edge has a numbered vertex, the dual arc should cut off components of A which are the images of components of .V ∩(∂M \P ) having the numbers given to the vertices. (See Fig. 8.1.) In the case when .tc contains an edge, each edge is dual to a weighted simple proper arc passing through .c, and hence corresponds to a vertical annulus containing the singular axis at the centre. Then the union of such weighted arcs is realised as a union of weighted arcs connecting .∂D to .c intersecting only at .c. We perform shuffling around V so that the order of the components of .Fr V coincides with the order of the corresponding vertices of .T¯V on D. Then we define a new annular codimension-1 lamination in V to be the union of vertical annuli (possibly with singular axes) over the weighted union of arcs .α. We note that if one vertex of .T¯V has more than one number, it does not matter how to perform shuffling among them. Repeating this operation for every torus pair and thickened torus pair ˆ we have a pared manifold .(M ' , P ' ) obtained from M by shuffling intersecting .L, without modifying the paring locus P , and an annular codimension-1 measured lamination .L' (possibly containing annuli with singular axes) in .M ' such that there is a .π1 (M ' ) = π1 (M)-equivariant isometry from the dual tree .TL' of .L' to T . ⨆ ⨅ To derive Theorem 8.2 from Theorem 8.6, we need some facts relating threedimensional hyperbolic geometry and two-dimensional one. We first prove the following rather elementary fact about degeneration of hyperbolic structures on

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A3 v1 = v3

A1

v6

A6 P0

vP

A4 A4

v4 v2

v4 A2

Fig. 8.1 The tree .TV' and its dual disc when V is a thickened torus

surfaces. This can be regarded as a two-dimensional analogue of the problematic second sentence of Theorem 8.1. Lemma 8.1 Let S be an orientable surface of hyperbolic type, and let .{mi } be a sequence in .T(S). Then, passing to a subsequence, there is a possibly disconnected (and possibly empty) subsurface .S ' of S such that .{mi |S ' } converges in .T(S ' ) and such that .S ' is maximal up to isotopy among the subsurfaces with the same property. Proof Let .αi be a sequence of shortest pants decomposition of .(S, mi ), each of which is regarded as a collection of simple closed geodesics. Passing to a subsequence, we can assume that there is a possibly empty sub-collection .βi of .αi which is constant, and no component of .αi \ βi has a constant subsequence. Since .βi is constant, we denote it by .β. We take a union of .β and all pairs of pants in .S \ αi all of whose boundary components belong to .β, and denote it by .Σ. Let .Σ0 be a component of .Σ, and denote .β∩Σ by .β0 . For a component b of .β0 , by passing to a subsequence, we can assume that either .lengthmi (b) → 0 or .lengthmi (b) converges to a positive constant. In the first case, we cut .Σ0 along b. In the second case, passing to a subsequence again, the twisting parameter (with respect to the Fenchel-Nielsen coordinates) around b can be assumed to converge or goes to .±∞. In the latter case, we cut .Σ0 along b. Repeating this for all components of .β0 , we get a subsurface .Σ1 of .Σ0 , and .mi |Σ1 converges in .T(Σ1 ). Taking the union of such ' .Σ1 for all components .Σ0 of .Σ, we obtain .S . The maximality follows easily from our construction. ⨆ ⨅ Let .(S, m) be a hyperbolic surface of finite area and M be a complete hyperbolic 3-manifold. A pleated surface .f : S → M is a continuous map such that for each point x on S, there is at least one segment s containing x in its interior such that

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f |s is totally geodesic. The set consisting of points where there is exactly one germ for such segments is called the pleating locus of f , and it constitutes a geodesic lamination on .(S, m). The following is a consequence of the “efficiency of pleated surfaces” proved in Theorems 3.3 and 3.4 in [34] combined with Lemma 8.1.

.

Proposition 8.3 Let .(M, P ) be a pared manifold, and let .ρi ∈ AH (M, P ) be a sequence. Then for every component S of .∂M \ P , there is a possibly disconnected subsurface .S ' such that .{ρi |π1 (S ' )} converges up to conjugacy, and .S ' is maximal up to isotopy among the subsurfaces of S with this property. If .S ' intersects essentially a component T of .∂M \ W for the characteristic submanifold W , then .S ' contains the entire T (up to isotopy). Proof Let .fi : (S, mi ) → H3 /ρi (π1 (M)) be a pleated surface taking each frontier of S to a cusp, which is efficient in the following sense: (*) For any simple closed curve .γ on S, the translation length of .ρi (γ ) is bounded if and only if .{lengthmi (γ )} is bounded, where .lengthmi denotes the geodesic length. The existence of such a sequence of pleated surfaces was proved in Theorem 3.3 of Thurston [34]. (We need to take a covering associated with .π1 (S) to apply this result of Thurston.) Now the first statement of our proposition follows from Lemma 8.1. Combining this with the first part of Theorem 8.1, we obtain the second statement. ⨆ ⨅ Although this surface .S ' is determined only up to isotopy, we always assume that no frontier component of .S ' is isotopic into P , and any component T of .∂M \ W intersecting .S ' is contained in .S ' . We need to refine Proposition 8.3 for twisted I -pairs in the characteristic submanifold as follows. Corollary 8.3 In the setting of Proposition 8.3, suppose that there is a twisted I -pair .(W0 , W0 ∩ ∂M) in the characteristic submanifold of .(M, P ), whose base surface we denote by .Σ0 . Let S be a component of .∂M \ P containing .W0 ∩ ∂M. Then the subsurface .S ' as obtained in Proposition 8.3 doubly covers a subsurface of .Σ0 . Proof We can consider a non-orientable efficient pleated surface .gi : Σ0 → H3 /ρi (π1 (M)). Consider a double cover .M˜ of M into which .W0 is lifted to a ˜ 0 , ni ) → product I -bundle. Then .gi is lifted to an orientable pleated surface .g˜ i : (Σ ˜ and both the length with respect to .ni and the translation length with H3 /ρi (π1 (M)), respect to .ρi are invariant under the covering involution. This implies Corollary 8.3. ⨆ ⨅ Proof of Theorem 8.2 We have only to consider the case when .{ρi } diverges in AH (M, P ). (Otherwise we can just take X to be empty.) Let W be the characteristic submanifold of .(M, P ). For every component S of .∂M \ P , there is a subsurface ' .S as was given in Proposition 8.3. Since any component of .∂M \ W that intersects .

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S ' must be entirely contained in .S ' , every frontier component of .S ' is contained in either an I -pair or a solid torus pair or a thickened torus pair. We only consider the frontier components contained in I -pairs. Let c be such a frontier component of .S ' contained in an I -pair .W0 . If .W0 is a product I -pair, then there is a simple closed curve .c' on .W0 ∩ ∂M such that .c ∪ c' bounds an essential (vertical) annulus A in .W0 . By our definition of .S ' , if .c' is contained in S, then it must also be a frontier component of .S ' . Otherwise, let .S¯ be a component of ' ¯ ' of .S¯ corresponding to .S ' in .∂M \ P containing .c . Then we have a subsurface .S ' ' ¯ Proposition 8.3, and .c is a frontier component of .S . If .W0 is a twisted I -pair, then by Corollary 8.3, either c is homotopic to .c' as above and .c ∪ c' bounds an annulus A, or c doubly covers an orientation-reversing curve in the base surface. In the latter case, c bounds an essential one-sided Möbius band, which we also denote by A. We remove from M thin regular neighbourhoods of all annuli and Möbius bands A as above for all frontier components of .S ' contained in I -pairs, and obtain a 3submanifold .M0 of M. We let the union of P and the frontier components of .M0 in M, the latter of which are all annuli, be a paring locus .P0 of .M0 . By our construction, it is easy to check that .(M0 , P0 ) is a pared manifold, and that .M \ M0 is contained in the union of I -pairs in W , and can be assumed to be vertical with respect to the fibration. We let the projection of .M \ M0 to the base surface be x in our statement, and hence X coincides with .M \ M0 . Now restricting .ρi to .(M0 , P0 ), we get a sequence .{ρi' } of .π1 (M0 ). If .{ρi } converges after passing to a subsequence, then by letting .A be empty and setting ' ' .M = M, we are done. Suppose that .{ρ } does not have a convergent subsequence. i We note that the translation length of every curve on a component of .P0 with respect to .ρi is bounded, by our definition of .S ' in Proposition 8.3. Therefore, we can apply Theorem 8.6, and there are a pared manifold .(M0' , P0 ) obtained from .(M0 , P0 ) by shuffling and a codimension-1 annular measured lamination .L0 properly embedded in .(M0' , P0 ) whose dual tree is equivariantly isometric to the Gromov limit of the action of .π1 (M0 ) on rescaled .H3 . By performing the same shuffling that gives .M0' from .M0 , we obtain a pared manifold .(Mˆ 0 , P ) homotopy equivalent of .(M, P ). By our definition of .M0 , and by Proposition 8.3, the lamination .L0 is disjoint from the I -pairs in the characteristic submanifold of .(M0' , P0 ), and hence contained in the union of solid torus pairs and the thickened torus pairs. This means that .L0 is a union of weighted incompressible annuli, possibly with singular axes. We let .A0 be the support of .L0 . Now, we cut .M0' along .A0 , and obtain a 3-manifold .M1 . In the case when .A0 contains an annulus with singular axis, we regard .M0' as containing a regular neighbourhood of the singular axis on both sides. By letting .P1 be the union of .A0 ∪ (P0 ∩ M1 ) when there are no singular axes, and when a component of .A0 has a singular axis, we let the intersection of the regular neighbourhood of the singular axis with the boundary be contained in .P1 , we get a pared manifold .(M1 , P1 ). We consider restrictions of .ρi to .π1 (M1 ), and if they converge passing to a subsequence, we let .A be .A0 , setting .(M ' , P ' ) to be .(Mˆ 0 , P ). Otherwise, we repeat the same procedure for .(M1 , P1 ) as for .(M0 , P0 ), and obtain the next pared .

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manifold .(M2 , P2 ) contained in a pared manifold .(Mˆ 1 , P ) obtained by shuffling from .(Mˆ 0 , P ). Since there are only finitely many disjoint incompressible annuli in M, this process terminates in finite steps, and there is .n0 such that .ρi |π1 (Mn0 ) 1 Aj regarded as converges. Then we set .(Mˆ n0 −1 , P ) to be .(M ' , P ' ), and .A = ∪nj =0 ' ' contained in .(M , P ). Then together with x which we defined above, also regarded as contained in .(M ' , P ' ), we are done. ⨆ ⨅

8.5 The Bounded Image Theorem To explain the significance of Theorem 8.1 in the proof of Thurston’s uniformisation theorem for Haken manifolds, we first review the overall structure of the proof. Let M be a closed atoroidal Haken manifold. In general, we should consider a pared manifold .(M, P ) and show the existence of a geometrically finite hyperbolic structure, but for simplicity, here we assume that M is closed, and hence .P = ∅. By Waldhausen’s result [38], a Haken manifold admits a “hierarchy”, that is, a finite sequence of compact 3-manifolds .M = M0 , M1 , . . . , Mn such that .Mj +1 is obtained by cutting .Mj along a non-peripheral incompressible surface, and .Mn is a disjoint union of 3-balls. The basic strategy of the proof of the uniformisation theorem is to construct a convex hyperbolic structure of .Mj from that of .Mj +1 . Topologically, .Mj is obtained from .Mj +1 by pasting an incompressible subsurface on the boundary of .Mj +1 to another incompressible subsurface disjoint from the first one. For a general step, we need to separate the boundary of .Mj +1 into two parts: the part consisting of pairs glued to each other to get .Mj , and the part which remains to be the boundary of .Mj . This makes the argument complicated. Therefore, we here consider only the last step in which each boundary component .M1 is glued to another component to get .M0 . In the logical structure of the proof given by Thurston in the form of a chart on the page 206 of [33], Theorem 8.1 is meant to be used to prove the “bounded image theorem”, which is a key step for showing that the “skinning map” defined below has a fixed point in the case when .M1 is not an I -bundle over a surface. The bounded image theorem in the original version is as below, restricted to the case which we are considering, i.e. the case when the paring locus is empty. To state the theorem, we need to define the skinning map. Let M be a 3-manifold admitting a convex compact hyperbolic metric. Let .q : T(∂M) → QH0 (M) be the parametrisation of Marden. For each element of .QH0 (M), by taking the covering associated with a component S of .∂M, we get a quasi-Fuchsian representation, ¯ and by the Bers simultaneous uniformisation, we have a point in .T(S) × T(S). ¯ and taking the product We consider the projection to the second coordinate .T(S), ¯ by considering all the components of .∂M, we have a map .σ : T(∂M) → T(∂M), where the latter denotes the Teichmüller space of .∂M with its orientation reversed. Theorem 8.7 (The Bounded Image Theorem) Let M be an atoroidal Haken manifold with incompressible boundary admitting a convex compact hyperbolic

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metric, and .τ : ∂M → ∂M be an orientation-reversing involution taking each component of .∂M to another component. Suppose that .N = M/τ is also atoroidal. ¯ Let .σ : T(∂M) → T(∂M) be the skinning map. Then there is .n0 ∈ N depending only on the topological type of M such that .(τ∗ ◦ σ )n T(∂M) is precompact for every .n ≥ n0 . There are three important books which explain the proof of the uniformisation theorem in detail: Morgan [22], Kapovich [13] and Otal [28]. In the first two of them, the bounded image theorem is mentioned, but it is noted that its proof is unknown to the authors. Instead, they gave a proof of a weaker version of the theorem, which should be called the “bounded orbit theorem”. This theorem was first stated in Thurston’s preprint [36]. To prove this, only the first part of Theorem 8.1, which is valid, and some argument involving the “covering theorem” are necessary. In the third book [28], the author invokes McMullen’s results [20, 21], which implies the existence of a fixed point for the skinning map without even using the first part of Theorem 8.1. Theorem 8.8 (The Bounded Orbit Theorem) In the setting of Theorem 8.7, let g be any point in .T(∂M). Then the sequence .{(τ ◦ σ )n∗ (g)} has a convergent subsequence in .T(∂M). Quite recently, Lecuire and the present author [16] succeeded in proving the bounded image theorem in the original general form, in particular Theorem 8.7, making use of modern techniques of Kleinian group theory, but without using the second part of Theorem 8.1. In the process of proving this, it has become possible to imagine what was Thurston’s plan to prove the bounded image theorem. As far as we understand it, his proof essentially relies on the second part of Theorem 8.1. We here try to recover the “proof” which we speculate that he should have had in mind. The “proof” is by contradiction. By the continuity of .τ∗ ◦ σ , we see that if .(τ∗ ◦ σ )n T(S) is precompact, then so is .(σ ◦ τ )m for every .m ≥ n. Therefore, what we have to exclude is the case when .(τ∗ ◦ σ )n∗ T(S) is unbounded for every n. If the image .(τ∗ ◦ σ )n T(S) is unbounded for every n, there must exist a sequence .{mi } in .T(S) such that .{(τ∗ ◦ σ )n mi }i diverges in .T(S) for every n. Then, passing to a subsequence, there are two possibilities to consider: the first is when .{(q(τ∗ ◦ σ )n (mi )}i diverges in .AH (M); the second is when .{q((τ∗ ◦ σ )n (mi )}i converges in .AH (M). For the first possibility, the second part of Theorem 8.1 is essential to reach a contradiction. We assumed that .{(q(τ∗ ◦ σ )n (mi )}i diverges in .AH (M) for every n. Then, for a fixed n, provided that the second part of Theorem 8.1 were valid, it would give us a collection .An of incompressible annuli which separates M into “converging part” and “diverging part” for every n. By an argument similar to the one employed in §16, in particular Proposition 16.5 of [13], it can be proved that this division into two parts is preserved under .τ . Therefore for each annulus in .An , its boundary is mapped by .τ to a simple closed curve which lies on the boundary of an annulus in .An+1 . We see that by continuing to connect an annulus in .An to that

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of .An+1 along a boundary starting from .n = 1, we have either an incompressible torus or an incompressible Klein bottle in N , contradicting the assumption that N is atoroidal. Now, suppose that .{q(τ∗ ◦ σ )n (mi )} converges in .AH (M) as .i → ∞ for every n. Then the argument is quite similar to that of the bounded orbit theorem explained in [13], or the proof of the bounded image theorem for the special case when M is acylindrical, explained in [14]. We can consider the limit representation n .φ∞ : π1 (M) → PSL2 (C) of .{q(σ ◦ τ ) (mi )} as .i → ∞ for each n. Then either 3 .φ∞ (π1 (M)) has a parabolic element or .H /φ∞ (π1 (M)) has a totally degenerate end, i.e., an end with a neighbourhood of a form .S × (0, ∞) for a closed surface S which is entirely contained in the convex core. In the latter case, there is a sequence of pleated surfaces .fk : S → H3 /φ∞ (π1 (M)) tending to the end as .k → ∞. These facts are proved by Bonahon [5] in a more general setting, but for the present case, there is an explanation by Thurston himself in §9 of his lecture notes [32]. In the first case when .φ∞ (π1 (M)) has a parabolic element, there must be a simple closed curve c on .∂M representing a parabolic element. (This follows from the fact that otherwise the limit .ρ∞ is a strong limit.) This will force .τ (c) to be homotopic in M to a another simple closed curve .c' on .∂M homotopically distinct from .τ (c), which implies that .τ (c) and .c' bound an incompressible annulus in M. We note here that we cannot prove this by considering the limit group .φ∞ (π1 (M)) since 3 .H /φ∞ (π1 (M)) may not be homeomorphic to the interior of M although they are homotopy equivalent. Therefore we need to work in .H3 /q(τ∗ ◦ σ )n (mi ) for very large i. The same argument can be found in §17.2.3 in [13]. Once we have such an annulus, we repeat the same argument as the one which we used for the case when n .{q(τ∗ ◦ σ ) (mi )} diverges, and reach a contradiction to the assumption that N is atoroidal. We next consider the latter case when .H3 /φ∞ (π1 (M)) has a totally degenerate end. Then we see that there must be one corresponding to a boundary component S of M, for under the assumption that there is no parabolic element in .φ∞ (π1 (M)), the convergence must be strong. By the covering theorem, which first appeared in §9 of [32] (a generalised version can be found in Canary [8]), this implies that M must have a boundary component which is homotopic to .τ (S), but is distinct from .τ (S). This is impossible since M is not a product I -bundle over a closed surface. Thus we have reached a contradiction in either case, and this would give a proof of Theorem 8.7 if the second part of Theorem 8.1 were valid. On the other hand, as we showed in Theorem 8.2, a weaker version is not sufficient for the proof of Theorem 8.7 since to construct a torus or a Klein bottle to reach a contradiction, we need annuli embedded in M, not those embedded in a homotopy equivalent manifold .M ' . This observation also highlights why a proof of the bounded image theorem given in [16] without relying on the second part of Theorem 8.1 needed much sophisticated techniques of Kleinian group theory.

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References 1. J.W. Anderson, R.D. Canary, Algebraic limits of Kleinian groups which rearrange the pages of a book. Invent. Math. 126(2), 205–214 (1996) 2. L. Bers, Simultaneous uniformization. Bullet. Am. Math. Soc. 66(2), 94–97 (1960) 3. L. Bers, Spaces of Kleinian groups, in Several Complex Variables, I (Proceedings of Conference, University of Maryland, College Park 1970) (Springer, Berlin, 1970), pp. 9–34 4. M. Bestvina, Degenerations of the hyperbolic space. Duke Math. J. 56(1), 143–161 (1988) 5. F. Bonahon, Bouts des variétés hyperboliques de dimension 3. Ann. Math. 124(1), 71–158 (1986) 6. J. Brock, K. Bromberg, R. Canary, C. Lecuire, Convergence and divergence of Kleinian surface groups. J. Topol. 8(3), 811–841 (2015) 7. J.F. Brock, K.W. Bromberg, R.D. Canary, Y.N. Minsky, Windows, cores and skinning maps. Ann. Sci. l’École Normale Supérieure 53(1), 173–216 (2020) 8. R.D. Canary, A covering theorem for hyperbolic 3-manifolds and its applications. Topology 35(3), 751–778 (1996) 9. R.D. Canary, D. McCullough, Homotopy equivalences of 3-manifolds and deformation theory of Kleinian groups. Mem. Am. Math. Soc. 172(812), xii–218 (2004) 10. M. Culler, P.B. Shalen, Varieties of group representations and splittings of 3-manifolds. Ann. Math. 117(1), 109–146 (1983) 11. W.H. Jaco, P.B. Shalen, Seifert fibered spaces in 3-manifolds. Mem. Amer. Math. Soc. 21(220), viii+192 (1979) 12. K. Johannson, Homotopy Equivalences of 3-Manifolds with Boundaries. Lecture Notes in Mathematics, vol. 761 (Springer, Berlin, 1979) 13. M. Kapovich Hyperbolic Manifolds and Discrete Groups. Modern Birkhäuser Classics (Birkhäuser, Boston, 2009). Reprint of the 2001 edition 14. R.P. Kent, Skinning maps. Duke Math. J. 151(2), 279–336 (2010) 15. I. Kra, On spaces of Kleinian groups. Comment. Math. Helv. 47, 53–69 (1972) 16. C. Lecuire, K. Ohshika, Thurston’s bounded image theorem. Geom. Top (to appear) 17. A. Marden, The geometry of finitely generated kleinian groups. Ann. Math. 99(3), 383–462 (1974) 18. B. Maskit, Self-maps on Kleinian groups. Am. J. Math. 93, 840–856 (1971) 19. B. Maskit, Kleinian Groups. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287 (Springer, Berlin, 1988) 20. C. McMullen, Amenability, Poincaré series and quasiconformal maps. Invent. Math. 97(1), 95–127 (1989) 21. C. McMullen, Iteration on Teichmüller space. Invent. Math. 99(2), 425–454 (1990) 22. J.W. Morgan, On Thurston’s uniformization theorem for three-dimensional manifolds, in The Smith Conjecture (New York, 1979). Pure and Applied Mathematics, vol. 112 (Academic Press, Orlando, 1984), pp. 37–125 23. J.W. Morgan, J.P. Otal, Relative growth rates of closed geodesics on a surface under varying hyperbolic structures. Comment. Math. Helv. 68(2), 171–208 (1993) 24. J.W. Morgan, P.B. Shalen, Valuations, trees, and degenerations of hyperbolic structures. I. Ann. Math. 120(3), 401–476 (1984) 25. J.W. Morgan, P.B. Shalen, Degenerations of hyperbolic structures. II. Measured laminations in 3-manifolds. Ann. Math. 127(2), 403–456 (1988) 26. J.W. Morgan, P.B. Shalen, Degenerations of hyperbolic structures. III. Actions of 3-manifold groups on trees and Thurston’s compactness theorem. Ann. Math. 127(3), 457–519 (1988) 27. K. Ohshika, Divergence, exotic convergence and self-bumping in quasi-Fuchsian spaces. Ann. Faculté Sci. Toulouse Math. 29(4), 805–895 (2020) 28. J.P. Otal, Thurston’s hyperbolization of Haken manifolds, in Surveys in Differential Geometry, Vol. III (Cambridge, MA, 1996) (International Press, Boston, 1998), pp. 77–194

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29. F. Paulin, Topologie de Gromov équivariante, structures hyperboliques et arbres réels. Invent. Math. 94(1), 53–80 (1988) 30. R.K. Skora, Splittings of surfaces. J. Am. Math. Soc. 9(2), 605–616 (1996) 31. D. Sullivan, Quasiconformal homeomorphisms and dynamics. II. Structural stability implies hyperbolicity for Kleinian groups. Acta Math. 155(3–4), 243–260 (1985) 32. W. Thurston, The Geometry and Topology of Three-Manifolds. Lecture Notes from Princeton University 1978-80. Mathematical Sciences Research Institute, notes Taken by Kerckhoff, S. and Floyd, W.J., (Princeton University, Princeton, 1978–1980) 33. W.P. Thurston, Hyperbolic structures on 3-manifolds. I. Deformation of acylindrical manifolds. Ann. Math. 124(2), 203–246 (1986) 34. W.P. Thurston, Hyperbolic structures on 3-manifolds II: surface groups and 3-manifolds which fiber over the circle (1998). https://arxiv.org/abs/math/9801045\/ 35. W.P. Thurston, Hyperbolic Structures on 3-manifolds, III: deformations of 3-manifolds with incompressible boundary (1998). https://arxiv.org/abs/math/9801058\/ 36. W.P. Thurston, Hyperbolic structures on 3-manifolds: overall logic, inn Collected Works of William P. Thurston with Commentary, vol. 2 (American Mathematical Society, Providence 2022) 37. W. Thurston, B. Farb, D. Gabai, S. Kerckhoff, Collected Works of William P. Thurston with Commentary, vol. 2. Collected Works Series (American Mathematical Society, Providence, 2022). 38. F. Waldhausen, On irreducible 3-manifolds which are sufficiently large. Ann. Math. 87(1), 56– 88 (1968)

Chapter 9

Geometric Structures in Topology, Geometry, Global Analysis and Dynamics Christoforos Neofytidis

Abstract Following Thurston’s geometrisation picture in dimension three, we study geometric manifolds in a more general setting in arbitrary dimensions, with respect to the following problems: (1) The existence of maps of non-zero degree (domination relation or Gromov’s order); (2) The Gromov-Thurston monotonicity problem for numerical homotopy invariants with respect to the domination relation; (3) The existence of Anosov diffeomorphisms (Anosov-Smale conjecture). Keywords Thurston geometry · Aspherical manifold · Domination · Gromov order · Monotone invariant · Simplicial volume · Kodaira dimension · Anosov diffeomorphism 2010 Mathematics Subject Classification 57M05, 57M10, 57M50, 55M25, 22E25, 20F34, 37D20, 55R10, 57R19

9.1 Introduction Thurston’s work has initiated and motivated tremendous research activity in various directions. The purpose of this survey is to present how Thurston’s geometrisation picture for 3-manifolds can be used and extended in high dimensions, including both geometric manifolds in the sense of Thurston and other non-geometric manifolds, to give a unified treatment of a diversity of problems arising in Topology, Geometry, Global Analysis and Dynamics. At the topological level, we will be dealing with an ordering of homotopy classes of manifolds of a given dimension, called Gromov order or domination relation. We shall say that a manifold M dominates N , and write .M ≥ N , if there is a map .f : M → N of non-zero degree. The domination relation has been studied

C. Neofytidis (O) Department of Mathematics, Ohio State University, Columbus, OH, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 K. Ohshika, A. Papadopoulos (eds.), In the Tradition of Thurston III, https://doi.org/10.1007/978-3-031-43502-7_9

299

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by many people and in various contexts, using a plethora of techniques and tools, such as cohomology ring structures and intersection forms in Algebraic Topology, bounded cohomology and the simplicial volume in Geometry and Global Analysis, the fundamental group in Group Theory, as well as the theory of harmonic mappings in Complex and Harmonic Analysis. In this survey, we will present an ordering in the sense of S. Wang [58] of all 3-manifolds and geometric 4-manifolds. As indicated above, the simplicial volume .|| · || is a significant tool in the study of the domination relation, and it is an example of a functorial semi-norm in homology. Namely, if .f : M → N is a map of degree d, then .||M|| ≥ |d|||N||. In dimension two, this can be restated in terms of the absolute Euler characteristic .|χ |. Pointing out again Thurston’s influence, we quote the following from Gromov’s book Metric Structures for Riemannian and Non-Remannian spaces [21, pg. 300]: “The interpretation of .|χ | as a norm originates from a work by Thurston, who used this idea to define a norm on .H2 (X3 ) using surfaces embedded into 3-manifolds”. From this more geometric and global analytic point of view, our second goal in this survey is to study the following monotonicity problem: Given a numerical invariant .ι, does .M ≥ N imply .ι(M) ≥ ι(N)? We will introduce a notion of geometric Kodaira dimension .κ g and show that .M ≥ N implies .κ g (M) ≥ κ g (N ) for all 3manifolds and geometric manifolds in dimensions four and five. We will compare our definition of .κ g with traditional notions of Kodaira dimension in Complex Geometry and establish relations to the simplicial volume. The last part of this survey has a dynamical flavor, namely the study of Anosov diffeomorphisms. A long-standing conjecture, going back to Anosov and Smale, asserts that all Anosov diffeomorphisms are conjugate to hyperbolic automorphisms of nilmanifolds [53]. Algebraic tools, such as Hirsch and Ruelle-Sullivan cohomology classes, coarse geometric methods, such as negative curvature, and many other techniques from various areas have been proven fruitful in understanding the Anosov–Smale conjecture; see for example [17, 18, 45] and their references. Here, with a more unified approach achieved via Thurston’s geometries, we will explain how to rule out Anosov diffeomorphisms from all Thurston geometric 4-manifolds that are not covered by the product of two surfaces of positive genus. Throughout this survey (and for the sake of simplicity) all manifolds are assumed to be closed, oriented and connected.

9.2 Domination, Monotonicity and Anosov Maps We begin our discussion with an overview of the three main topics of this survey, indicating as well their state of the art with various open questions. This section aims also to serve as an introduction for readers not familiar with these topics.

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9.2.1 The Domination Relation Definition 9.2.1 Let M, N be manifolds of dimension n. We say that M dominates N if there is a map f : M → N of non-zero degree d. We denote this by M ≥ N or by M ≥d N when we need to emphasise on the specific degree d. We also write deg(f ) for the degree. Recall that f : M → N being of degree d means that the induced map in homology Hn (f ) : Hn (M; Z) → Hn (N; Z) satisfies Hn (f )([M]) = d · [N ], where [X] denotes the fundamental (orientation) class. Domination is a transitive relation. For, if M ≥ N and N ≥ W , then M ≥ W via composition of the two dominant maps. In a lecture at CUNY Graduate Center in 1978, Gromov suggested studying the domination relation as an ordering of the homotopy classes of manifolds of the same dimension (hence the name Gromov order). In dimension two, the domination relation is indeed a total order given by the genus: Proposition 9.2.2 Let Eg , Eh be two surfaces of genus g and h respectively. Then, Eg ≥ Eh if and only if g ≥ h. Proof For the “if” part, we observe that for each g, there is a degree one map, called pinch map, given by 2 2 Eg ∼ · · #T '2 −→ T · · #T '2 ∨T 2 −→ T · · #T '2 ∨pt ∼ = Eg−1 . = 'T 2 # ·'' ' # ·'' ' # ·''

.

g

g−1

(9.1)

g−1

The “only if” part follows by the next more general lemma, since H1 (Eg ) ∼ = Z2g .

u n

Lemma 9.2.3 If M ≥ N, then bi (M) ≥ bi (N ), where bi (X) = dim Hi (X; Q) denotes the i-th Betti number of X. Proof Clearly it suffices to prove the lemma for 0 < i < n. Let f : M → N be a map of non-zero degree d and α ∈ Hi (N; Q). Consider the preimage under −1 the Poincaré duality isomorphism P DN (α) ∈ H n−i (N; Q) and then the image −1 H n−i (f )(P DN (α)) ∈ H n−i (M; Q). Let the homology class −1 −1 β := P DM H n−i (f )P DN (α) = H n−i (f )(P DN (α)) ∩ [M] ∈ Hi (M; Q).

.

Then we obtain −1 Hi (f )(β) = Hi (f )(H n−i (f )(P DN (α)) ∩ [M])

.

−1 (α) ∩ Hi (f )([M]) = P DN −1 (α) ∩ [N] = d · P DN

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C. Neofytidis −1 = d · P DN (P DN (α))

= d · α. That is, ( Hi (f )

.

) 1 · β = α, d

which means that Hi (f ) with rational coefficients is surjective and the lemma follows. u n In higher dimensions, the domination relation is not anymore a total ordering as the following example shows: Example 9.2.4 The 3-sphere S 3 is a twofold cover of the projective plane RP3 = S 3 /Z2 . For any n-manifold M, there is a degree one pinch map M ∼ = M#S n → S n (as in Proposition 9.2.2; see (9.1)). Hence, S 3 ≥2 RP3 ≥1 S 3 ,

.

(9.2)

while of course S 3 and RP3 are not homotopy equivalent. Naturally, the dominations given by (9.2) raise the following: Problem 9.2.5 Suppose that M ≥1 N ≥1 M. Are M and N homotopy equivalent? This is tightly related to the next long-standing problem of Hopf: Problem 9.2.6 ([26, Problem 5.26]) Is every self map of degree ±1 a homotopy equivalence? At the group theoretic level one has the following corresponding concept: Definition 9.2.7 A group G is called Hopfian if every surjective endomorphism of G is an isomorphism. An affirmative answer to Problem 9.2.6 holds for the class of aspherical manifolds with Hopfian fundamental groups. Recall that a manifold M is called aspherical if all its homotopy groups πk (M) vanish for k ≥ 2. Proposition 9.2.8 Let M be an aspherical manifold with Hopfian fundamental group π1 (M). Then every map f : M → M with deg(f ) = ±1 is a homotopy equivalence. Proof We begin our proof by recalling the following well-known lemma: Lemma 9.2.9 Let M, N be manifolds of the same dimension. If f : M → N is a map of non-zero degree, then [π1 (N ) : f∗ (π1 (M))] < ∞, where f∗ : π1 (M) → π1 (N ) denotes the induced homomorphism. If, moreover, deg(f ) = ±1, then [π1 (N ) : f∗ (π1 (M))] = 1.

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p

Proof Let N − → N be the covering of degree deg(p) = [π1 (N ) : f∗ (π1 (M))], which corresponds to f∗ (π1 (M)). We then lift f to f : M → N , and we have f = p ◦ f . In particular, deg(f ) = deg(p) deg(f ), which verifies both claims of the lemma. u n Since f : M → M has deg(f ) = ±1, Lemma 9.2.9 tells us that f∗ : π1 (M) → π1 (M) is surjective. By assumption π1 (M) is Hopfian, hence f∗ is an isomorphism. The proposition now follows by Whitehead’s classical theorem (see for example [22, Theorem 4.5]), since πk (M) = 0 for all k ≥ 2. u n In particular, we obtain the following partial ordering: Corollary 9.2.10 The domination relation ≥1 is a partial ordering on the class of aspherical manifolds with Hopfian fundamental groups. Note that the requirement on π1 being Hopfian might be redundant: Problem 9.2.11 Is the fundamental group of any aspherical manifold Hopfian? Problem 9.2.11 has a complete affirmative answer in dimensions ≤ 3 and in all other known cases in higher dimensions, such as for nilpotent manifolds. Finally, concerning the case of self-maps, we have the following strong version of Problem 9.2.6 for aspherical manifolds: Problem 9.2.12 ([47, Problem 1.2]) Is every self-map of an aspherical manifold either a homotopy equivalence (when the degree is ±1) or homotopic to a nontrivial covering? If moreover “homotopy equivalence” is replaced by “homotopic to a homeomorphism” (in other words, is every self-map of an aspherical manifold homotopic to a covering?), then Problem 9.2.12 becomes a strong version of the Borel conjecture, which asserts that any homotopy equivalence between closed aspherical manifolds is homotopy to a homeomorphism.

9.2.2 Monotone Invariants Definition 9.2.13 Let M be a manifold. A non-negative numerical quantity ι(M) is monotone with respect to the domination relation if M ≥ N ⇒ ι(M) ≥ ι(N).

.

Clearly such a number is a homotopy invariant. If one requires furthermore the degree of the map to be carried in the inequality, i.e., M ≥d N ⇒ ι(M) ≥ |d|ι(N),

.

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then we say that ι is functorial. Amongst the most prominent functorial homotopy invariants is the simplicial volume. Definition 9.2.14 Given a topological space X and a homology class α ∈ Hn (X; R), the Gromov norm of α is defined to be | {E } | E | .||α||1 := inf |λj | | λj σj ∈ Cn (X; R) is a singular cycle representing α . j

j

If, moreover, X is an n-manifold, then the Gromov norm or simplicial volume of X is given by ||X|| := ||[X]||1 . The functoriality of the simplicial volume follows easily by the above definition: Lemma 9.2.15 Let f : X → Y be a map between topological spaces. Then ||α||1 ≥ ||Hn (f )(α)||1 for any α ∈ Hn (X; R). In particular, if M ≥d N , then ||M|| ≥ |d|||N||. There is a tight connection between domination and monotone invariants. For if M ≥d M for d > 1, then ι(M) = 0 for all finite functorial invariants ι. Equivalently, the existence of a finite non-zero functorial invariant on M implies that M does not admit any self-maps of degree other than 0 and ±1. We record some (not necessarily mutually disjoint) examples regarding the simplicial volume: Example 9.2.16 (1) The following classes of manifolds have zero simplicial volume: (a) spheres; (b) rationally inessential manifolds, i.e., manifolds M whose classifying map cM : M → Bπ1 (M) vanishes in top degree rational homology [19]; (c) fiber bundles F → M → B, where π1 (F ) is Abelian or, more generally, amenable [19]; (d) products with at least one factor with vanishing simplicial volume [19]. (2) The following classes of manifolds have non-zero simplicial volume: (a) hyperbolic manifolds [19]; (b) irreducible, locally symmetric spaces of non-compact type [4, 32]; (c) rationally essential manifolds with hyperbolic fundamental groups [20, 37]; (d) products whose all factors have positive simplicial volume [19]. The manifolds in Example 9.2.16 (1d) (resp. (2d)) have zero (resp. non-zero) simplicial volume because of the inequalities ( ||M||||N || ≤ ||M × N || ≤

.

) m+n ||M||||N ||, n

(9.3)

where m and n denote the dimensions of M and N respectively. Finally, we remark that the study of mapping degree sets leads to knowledge about non-zero functorial invariants.

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Definition 9.2.17 Let M, N be n-manifolds. The set of degrees of maps from M to N is defined by D(M, N ) := {d ∈ Z | d = deg(f ), f : M → N }.

.

Fixing N, one can define a functorial invariant by looking at the supremum of all possible absolute degrees of maps to N (see [10]): ιN (M) := sup{|d| | d ∈ D(M, N )}.

.

(9.4)

If D(N, N ) ⊆ {−1, 0, 1}, then clearly ιN (N ) = 1. In particular, one obtains nonvanishing functorial invariants on manifolds, where classical functorial invariants are known to be zero. A prominent class given in [47] is that of not virtually (i.e., not finitely covered by) trivial S 1 -bundles over hyperbolic manifolds for which the simplicial volume indeed vanishes [19]. However, a disadvantage of (9.4) is that ιN (·) might not be finite, since the inclusion D(N, N ) ⊆ {−1, 0, 1} does not preclude the existence of a manifold M so that D(M, N ) is unbounded. It is unknown whether this is the case for the not virtually trivial S 1 -bundles over hyperbolic manifolds mentioned above, except in dimension three, where Brooks and Goldman [3] showed the existence of another -2 -manifolds, namely of the Seifert volume. non-zero functorial invariant on SL

9.2.3 Anosov Diffeomorphisms Definition 9.2.18 Suppose M is a smooth n-manifold. A diffeomorphism f : M → M is called Anosov if there exists a df -invariant splitting T M = E s ⊕ E u of the tangent bundle of M, together with constants μ ∈ (0, 1) and C > 0, such that ||df m (v)|| ≤ Cμm ||v||, if v ∈ E s , .

||df m (v)|| ≤ C −1 μ−m ||v||, if v ∈ E u ,

for all m ∈ N. The invariant distributions E s and E u are called stable and unstable distributions respectively. An Anosov diffeomorphism f is said to be of codimension k if E s or E u has dimension k ≤ [n/2], and transitive if there exists a point whose orbit is dense in M. Currently, the only known examples of Anosov diffeomorphisms are of algebraic nature, namely, Anosov automorphisms of manifolds covered by nilmanifolds; see for example [35] about the interpretation of Anosov diffeomorphisms of nilmanifolds at the level of hyperbolic automorphisms of their fundamental groups. We illustrate this with two examples, one in the Abelian case and another one for a nilpotent but not Abelian group:

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Example 9.2.19 The matrix ( A=

.

21 11

)

has no eigenvalues which are roots of unity and hence it defines a hyperbolic automorphism of Z2 . Example 9.2.20 Let G = .

.

This is a 6-dimensional torsion-free, 2-step nilpotent group. Indeed, the lower central series of G is given by c0 (G) = G, c1 (G) = [c0 (G), G] = [G, G] = , c2 (G) = [c1 (G), G] = 1.

.

In particular, the quotient of G by the isolator subgroup .

/ G c1 (G) = {x ∈ G | x k ∈ c1 (G) for some integer k ≥ 0} = c1 (G) ∼ = Z2

is isomorphic to Z4 ∼ = . = G/c1 (G) ∼ By Cassidy et al. [9] and Malfait [35], the group G admits a hyperbolic automorphism. An explicit example is given in [35, Example 3.5], namely, the automorphism φ : G → G defined by x1 |→ x12 x2−1 , x2 |→ x1−3 x22 , x3 |→ x37 x44 , x4 |→ x312 x47 , x5 |→ x52 x61 , x6 |→ x53 x62 .

.

Indeed, the restriction of φ to c1 (G) is given by ( .

) 23 , 12

and the induced automorphism φ on G/c1 (G) is given by ⎛

2 −3 ⎜ −1 2 .⎜ ⎝ 0 0 0 0

0 0 7 4

⎞ 0 0⎟ ⎟. 12 ⎠ 7

Both matrices define hyperbolic automorphisms (on Z2 and Z4 respectively), since they do not have eigenvalues which are roots of unity.

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Anosov and Smale [53] conjectured that any Anosov diffeomorphism is finitely covered by a diffeomorphism which is topologically conjugate to a hyperbolic automorphism of a nilpotent manifold. For the m-torus T m , m ≥ 2, Franks [12] proved that if f : T m → T m is an Anosov diffeomorphism, then the induced isomorphism H1 (f ) : H1 (T m ; R) −→ H1 (T m ; R)

.

has no roots of unity among its eigenvalues. Hirsch [24] extended Franks’ result to Anosov diffeomorphisms of a wider class of manifolds which includes all nilmanifolds: Theorem 9.2.21 ([24, Theorem 4]) Let M be a manifold with virtually polycyclic fundamental group, whose universal covering has finite dimensional rational homology. If M admits an Anosov diffeomorphism f : M → M, then the isomorphism H1 (f ) : H1 (M; R) −→ H1 (M; R)

.

has no roots of unity among its eigenvalues. Hirsch’s result has remarkable consequences, for instance, on polycyclic manifolds with infinite cyclic first integral cohomology group. In particular, mapping tori of Anosov diffeomorphisms do not themselves admit Anosov diffeomorphisms. Indeed, if MA = T n x A S 1 =

.

T n × [0, 1] (x, 0) ∼ (A(x), 1)

is a mapping torus, such that none of the eigenvalues of A ∈ SL(n; Z) is a root of unity, then H 1 (MA ; Z) ∼ = H1 (MA ; Z)/TorH1 (MA ; Z) ∼ = Z. Ruelle-Sullivan [50] found an interesting obstruction related to the codimension of an Anosov diffeomorphism: Theorem 9.2.22 ([50, Corollary pg. 326]) If f : M → M is a codimension k transitive Anosov diffeomorphism with orientable invariant distributions, then there is a non-trivial cohomology class α ∈ H k (M; R) and a positive real number λ /= 1 such that H k (f )(α) = λ · α. In particular, H k (M; R) /= 0.

9.3 The Gromov Order for Thurston Geometries in Dimensions ≤ 4 As explained in Example 9.2.4, the domination relation in dimensions greater than two does not define an ordering of all manifolds in the usual sense. We thus need to find an alternative natural and meaningful method to order manifolds. In dimension

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three, such a method was proposed by S. Wang [58] following Thurston’s geometrisation picture. We will first review Wang’s ordering of Thurston’s geometries, together with an extension of it to all 3-manifolds [30], and then describe an ordering of the 4-dimensional aspherical geometries. Our main reference is [43].

9.3.1 Classification of Thurston’s Geometries We begin our discussion by recalling briefly the classification of Thurston geometries in dimensions .≤ 4, together with some properties that we will need in our proofs. Suppose .Xn is a complete simply connected Riemannian manifold of dimension n. We will say that a manifold M is an .Xn manifold, or is modeled on .Xn , or carries the .Xn geometry in the sense of Thurston, if it is diffeomorphic to a quotient of .Xn by a lattice .Γ in the group of isometries of .Xn (where .Γ = π1 (M)). Two geometries n n n n .X and .Y are the same whenever there exists a diffeomorphism .ψ : X −→ Y n n n and an isomorphism .Isom(X ) −→ Isom(Y ) which sends each .g ∈ Isom(X ) to −1 ∈ Isom(Yn ). .ψ ◦ g ◦ ψ Dimension One The circle .S 1 = R/Z is the only 1-dimensional manifold and is modeled on .R. Dimension Two Surfaces .Eg , .g ≥ 0, have been already discussed in Sect. 9.2: For .g = 0 we have the 2-sphere .E0 = S 2 (modeled on .S 2 ), for .g = 1 the 2torus .E1 = T 2 = R2 /Z2 (modeled on .R2 ) and for .g ≥ 2 hyperbolic surfaces 2 2 .Eg = H /π1 (Eg ) (modeled on .H ), where π1 (Eg ) = .

.

Table 9.1 summarises the geometries in dimension two. Dimension Three Thurston proved that there exist eight homotopically unique -2 , .H2 × R, .Nil 3 , .R3 , .S 2 × R and .S 3 . In Table 9.2, we geometries: .H3 , .Sol 3 , .SL list the finite covers for manifolds in each of those geometries (see [1, 51, 55]). Dimension Four The 4-dimensional Thurston’s geometries were classified by Filipkiewicz in his thesis [11]. In Table 9.3, we list the geometries that are realised by compact manifolds, following [56, 57] and [23]. In the remainder of this paragraph we will mainly concentrate on the aspherical geometries. Table 9.1 The 2-dimensional Thurston geometries

Type Spherical Euclidean Hyperbolic

Geometry .X2 2 .S 2 .R 2 .H

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Table 9.2 Finite covers of Thurston geometric 3-manifolds Geometry .X3 3 .H 3 .Sol .SL2 3 .N il 2 .H × R 3 .R 2 .S × R 3 .S

M is finitely covered by. . . A mapping torus of a hyperbolic surface with pseudo-Anosov monodromy A mapping torus of .T 2 with hyperbolic monodromy A non-trivial .S 1 bundle over a hyperbolic surface A non-trivial .S 1 bundle over .T 2 A product of .S 1 with a hyperbolic surface The 3-torus .T 3 The product .S 2 × S 1 The 3-sphere .S 3

Table 9.3 The 4-dimensional Thurston geometries with compact representatives Type Hyperbolic Solvable

Geometry .X4 4 2 .H , .H (C) 4 4 4 4 3 3 4 .N il , .Solm/=n , .Sol0 , .Sol1 , .Sol × R, .N il × R, .R

Compact Mixed products

.S

4 , .CP2 , .S 2

.S

2

× S2 -2 × R × H2 , .S 2 × R2 , .S 3 × R, .H3 × R, .H2 × R2 , .H2 × H2 , .SL

Manifolds modeled on a geometry of type .X3 × R satisfy the following property: Theorem 9.3.1 ([23, Sections 8.5 and 9.2]) Let .X3 be a 3-dimensional geometry. A 4-manifold that carries the geometry .X3 ×R is finitely covered by a product .N ×S 1 , where N is a 3-manifold modeled on .X3 . Manifolds modeled on the geometry .H2 × H2 are either virtual products of two hyperbolic surfaces or not even (virtual) surface bundles. These two types are distinguished by the names reducible and irreducible .H2 × H2 geometry respectively; see [23, Section 9.5] for further details. A class of 4-dimensional geometries that motivates some new phenomena with respect to the domination problem, especially the property group (infinite-index) presentable by products (see Definition 9.3.3 and Proposition 9.3.4 below, as well 4 4 as Sect. 9.4.2) is that of solvable non-product geometries .Nil 4 , .Solm/ =n , .Sol0 and 4 .Sol . Let us first recall the model Lie groups of those geometries together with 1 some properties. The nilpotent Lie group .Nil 4 is the semi-direct product .R3 x R, where .R acts on 3 .R by ⎛

⎞ 1 et 0 .t → | ⎝ 0 1 et ⎠ . 0 0 1 Proposition 9.3.2 ([42, Prop. 6.10]) A .Nil 4 manifold M is finitely covered by a non-trivial .S 1 bundle over a .Nil 3 manifold and the center of .π1 (M) remains infinite cyclic in finite covers.

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Next, we give the model spaces for the three non-product solvable—but not nilpotent—geometries: Suppose m and n are positive integers and .a > b > c are reals such that .a + b + c = 0 and .ea , eb , ec are roots for the polynomial 4 3 2 .Pm,n (λ) = λ − mλ + nλ − 1. If .m /= n, the Lie group .Sol m/=n is a semi-direct product .R3 x R, where .R acts on .R3 by ⎛

⎞ eat 0 0 bt 0 ⎠ . .t |→ ⎝ 0 e 0 0 ect Note that, when .m = n, then .b = 0 and this corresponds to the product geometry Sol 3 × R. If the polynomial .Pm,n has two equal roots, then we obtain the model space for the .Sol04 geometry, which is a semi-direct product .R3 x R, where the action of .R on 3 .R is given by .

⎛

⎞ et 0 0 .t → | ⎝ 0 et 0 ⎠ . 0 0 e−2t The main result in [29] is that aspherical manifolds (more generally, rationally essential manifolds) are not dominated by direct products if their fundamental group is not presentable by products. Definition 9.3.3 A group G is called not presentable by products if for every homomorphism .ϕ : G1 × G2 −→ G with .[G : im(ϕ)] < ∞, one of the images .ϕ(Gi ) is finite. 4 4 Manifolds modeled on one of the geometries .Solm/ =n or .Sol0 fulfill the above property:

Proposition 9.3.4 ([42, Prop. 6.13]) The fundamental group of a 4-manifold which 4 4 is modeled on the geometry .Solm/ =n or the geometry .Sol0 is not presentable by products. The last solvable model space is an extension of .R by the 3-dimensional Heisenberg group ⎞ ⎛ { 1x z | } | 3 | ⎠ ⎝ .Nil = 0 1 y | x, y, z ∈ R . 001

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Namely, the Lie group .Sol14 is defined as a semi-direct product .Nil 3 x R, where .R acts on .Nil 3 by ⎞ 1 e−t x z .t |→ ⎝ 0 1 et y ⎠ . 0 0 1 ⎛

Manifolds modeled on this geometry have the following property: Proposition 9.3.5 ([42, Prop. 6.15]) A .Sol14 manifold M is finitely covered by an 1 2 3 .S bundle over a mapping torus of .T with hyperbolic monodromy (i.e., over a .Sol manifold). Every 4-manifold that carries a solvable non-product geometry is a mapping torus: Theorem 9.3.6 ([23, Sections 8.6 and 8.7]) 4 (1) A manifold modeled on the .Sol04 or the .Solm/ =n geometry is a mapping torus of 3 a self-homeomorphism of .T . (2) A manifold modeled on the .Nil 4 or the .Sol14 geometry is a mapping torus of a self-homeomorphism of a .Nil 3 -manifold.

The remaining two aspherical models are irreducible symmetric geometries, the real and the complex hyperbolic, denoted by .H4 and .H2 (C) respectively. Finally, we will need the following: Theorem 9.3.7 ([57, Theorem 10.1][28, Prop. 1]) If M and N are homotopy equivalent 4-manifolds modeled on geometries .X4 and .Y4 respectively, then .X4 and 4 .Y are the same. In particular, an aspherical geometric 4-manifold M is finitely covered by an .X4 manifold if and only if it carries the geometry .X4 .

9.3.2 Wang’s Ordering Suppose M is an aspherical 3-manifold which is not modeled on one of the six -2 , .H2 × R, .Nil 3 or .R3 . Then there is a finite aspherical geometries .H3 , .Sol 3 , .SL family of splitting tori so that M can be cut into pieces, called JSJ pieces (named after Jaco-Shalen and Johannson). M is called a non-trivial graph manifold if all the JSJ pieces are Seifert. If there is a non-Seifert JSJ piece, then this piece must be hyperbolic by Perelman’s proof of Thurston’s geometrisation conjecture. In that case, M is called a non-graph manifold.

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Fig. 9.1 Ordering 3-manifolds by maps of non-zero degree [30, 58]

Wang [58] ordered all aspherical 3-manifolds and Kotschick and I [30] extended this to include all rationally inessential 3-manifolds: Theorem 9.3.8 (Wang’s Ordering) Let the following classes of 3-manifolds: (i) aspherical and geometric, i.e., modeled on one of the six geometries .H3 , .Sol 3 , -2 , .H2 × R, .Nil 3 or .R3 ; .SL (ii) aspherical and non-geometric, i.e., .(GRAPH) non-trivial graph or .(NGRAPH) non-geometric irreducible non-graph; (iii) rationally inessential, i.e., finitely covered by .#p (S 2 × S 1 ), for some .p ≥ 0. If there exists an oriented path from a class X to another class Y in Fig. 9.1, then any manifold in Y is dominated by some manifolds in X. Otherwise, no manifold in Y can be dominated by a manifold in X. The proof of Theorem 9.3.8 for maps between (most) aspherical 3-manifolds is given in [58] and for maps from .H2 × R manifolds to manifolds modeled on -2 or .Nil 3 , or when the target manifold is finitely covered by the geometries .SL 2 1 .#p (S × S ), is given in [30]. Note also some restrictions on the diagram concerning the number of summands in .#p (S 2 × S 1 ) for domination from .Sol 3 , .Nil 3 or .R3 manifolds; see [43, pg. 4].

9.3.3 Ordering the 4-Dimensional Geometries Our goal in this section is to order in the sense of Wang all non-hyperbolic 4manifolds that carry a Thurston aspherical geometry: Theorem 9.3.9 Consider all 4-manifolds that are modeled on a non-hyperbolic aspherical geometry. If there is an oriented path from a geometry .X4 to another geometry .Y4 in Fig. 9.2, then any .Y4 -manifold is dominated by an .X4 -manifold. If there is no oriented path from .X4 to .Y4 , then no .X4 -manifold dominates a .Y4 manifold. Theorem 9.3.9 does not include the real or complex hyperbolic geometries, partially because some of the results about those geometries are well-known and because the domination relation for those geometries has been studied by other

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Fig. 9.2 Ordering Thurston geometries in dimension four

authors; see [7, 13, 29]. Similarly, the non-aspherical geometries are not included in the above theorem; those geometries are either products or their representatives are simply connected, see [39, 40] for further details. We will devote the rest of this section in sketching a proof of Theorem 9.3.9, and refer to [43] for the details.

9.3.3.1

Manifolds Covered by Products

First, we will examine 4-manifolds that are finitely covered by direct products. In other words, we will explain the right-hand side of Fig. 9.2. Non-Existence Stability Between Products Dealing with manifolds in dimension four, a natural question is whether one can extend Wang’s ordering given by Theorem 9.3.8 to 4-manifolds that are finitely covered by .N × S 1 , where N is a 3manifold as in Theorem 9.3.8. The problem is whether the non-existence results by Wang extend in dimension four, namely, whether .M > N implies .M ×S 1 > N ×S 1 . This raises the following more general stability question: Problem 9.3.10 Suppose .M > N . Does this imply .M × W > N × W for every manifold W ? This problem is of independent interest, because, for example, our current knowledge on the multiplicativity of functorial numerical invariants (such as the simplicial volume) under taking products is not enough to answer this kind of problems, even when an invariant remains non-zero under taking products; compare to (9.3).

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The next result is based on the celebrated realisation theorem of Thom [54] and gives a sufficient condition for non-domination stability for products: Theorem 9.3.11 [31, 39] Let .M, N be n-manifolds such that N is not dominated by products and W be an m-manifold. Then, .M ≥ N if and only if .M ×W ≥ N ×W . In a similar vein, we have the following: Proposition 9.3.12 ([31, 39]) Let .M, W and N be manifolds of dimensions .m, k and n respectively such that .m, k < n < m + k. If N is not dominated by products, then .M × W > N × V , for any manifold V of dimension .m + k − n. Targets that Are Virtual Products with a Circle Factor Now we apply Theorem 9.3.11 to 4-manifolds that are finitely covered by .N × S 1 , thus extending Theorem 9.3.8. In the following theorem, we shall say that a 4-manifold belongs to the class .X × R if it is finitely covered by a product .N × S 1 , where N is a 3-manifold that belongs to the class X as defined in Theorem 9.3.8. Theorem 9.3.13 Let X be one of the three classes (i)–(iii) given in Theorem 9.3.8. If there exists an oriented path from a class X to another class Y in Fig. 9.1, then any 4-manifold in .Y × R is dominated by a manifold in .X × R. Otherwise, no manifold in .Y × R can be dominated by a manifold in .X × R. Proof The existence part of Theorem 9.3.13 follows easily by the corresponding existence results for maps between 3-manifolds given in Theorem 9.3.8, hence we concentrate on the non-existence part. Note that there is no 4-manifold in the class 2 1 .(#p S × S ) × R that can dominate a manifold in the other classes, since the latter are all rationally essential. Thus, the interesting cases are when both domain and target are aspherical. We first deal with targets whose 3-manifold factor N in their finite cover .N × S 1 is not dominated by products. The proof of the following uses Proposition 9.3.12 and Theorem 9.3.11: Proposition 9.3.14 ([43, Prop. 4.4]) Suppose W and Z are 4-manifolds such that (1) W is dominated by products; (2) Z is finitely covered by .N × S 1 , where N is a 3-manifold not dominated by products. If .W ≥ Z, then there exists a 3-manifold M such that .M × S 1 ≥ W and .M ≥ N . In particular, M cannot be dominated by products. By Kotschick and Neofytidis [30, Theorem 4], only .H2 × R and .R3 manifolds are dominated by products among the aspherical 3-manifolds. Hence, Proposition 9.3.14 and the non-existence part of Theorem 9.3.8 imply the following: Corollary 9.3.15 If .Y /= H2 ×R, R3 , then the non-existence part of Theorem 9.3.13 holds true for every aspherical target in .Y × R.

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In the Thurston geometric setting, we have the following straightforward consequence of Proposition 9.3.14: Corollary 9.3.16 Let W and Z be aspherical 4-manifolds carrying product geometries .X3 × R and .Y3 × R respectively, such that .Y3 /= H2 × R, R3 . If .W ≥ Z, then every .Y3 manifold is dominated by some .X3 manifold. In order to finish the proof of Theorem 9.3.13, we need to show that manifolds -2 × R, .Sol 3 × R or .Nil 3 × R modeled on .H2 × R2 or .R4 are not dominated by .SL manifolds. For the latter two geometries, this follows by the growth of their Betti -2 ×R geometry: Note that numbers (cf. Lemma 9.2.3). We are left to deal with the .SL 4 4 each .R manifold is finitely covered by the 4-torus .T and, therefore, it is virtually dominated1 by every .H2 × R2 manifold. Thus, it suffices to show that .T 4 cannot -2 manifold. After passing to be dominated by a product .M × S 1 , where M is an .SL a finite cover, we can assume that M is a non-trivial .S 1 bundle over a hyperbolic surface .E; see Table 9.2. Suppose .f : M × S 1 −→ T 4 is a continuous map. The product .M × S 1 carries the structure of a non-trivial .S 1 bundle over .E × S 1 , by multiplying by .S 1 both the total space M and the base surface .E of the .S 1 bundle 1 .M −→ E. The .S fiber of the circle bundle S 1 −→ M × S 1 −→ E × S 1

.

has finite order in .H1 (M ×S 1 ), being also the fiber of M. Therefore, its image under 4 4 4 .H1 (f ) has finite order in .H1 (T ). Now, since .H1 (T ) is isomorphic to .π1 (T ) ∼ = 4 1 1 Z , we deduce that .π1 (f ) maps the fiber of the .S bundle .M × S −→ E × S 1 to the trivial element in .π1 (T 4 ). The latter implies that f factors through the base 1 1 1 4 .E × S , because the total space .M × S , the base .E × S and the target .T are all aspherical. This implies that the degree of f must be zero, completing the proof of Theorem 9.3.13. u n Manifolds Covered by the Product of Two Hyperbolic Surfaces We close this subsection by examining manifolds that are finitely covered by a product of two hyperbolic surfaces, i.e. reducible .H2 × H2 manifolds. Clearly, every 4-manifold modeled on .H2 × R2 or .R4 is dominated by a product of two hyperbolic surfaces. However, Proposition 9.3.12 (or Proposition 9.3.14) tells us that aspherical 4-manifolds that are finitely covered by products .N × S 1 , where N does not belong to one of the classes .H2 × R or .R3 , cannot be dominated by products of hyperbolic surfaces. Finally, we need to show that there is no manifold modeled on an aspherical geometry .X3 × R which can dominate a product of two hyperbolic surfaces. The fundamental group of a product .M × S 1 has center at least infinite cyclic (coming from the .S 1 factor), while the center of the fundamental group of a product of two hyperbolic surfaces .Eg × Eh is trivial. Therefore, every map .f : M × S 1 −→

1M

virtually dominates N if some finite cover of M dominates N .

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C. Neofytidis

Eg × Eh (which we can assume .π1 -surjective after passing to finite covers) kills the homotopy class of the .S 1 factor of .M × S 1 , and so it factors through an aspherical manifold of dimension at most three, because both .M × S 1 and .Eg × Eh are aspherical. Thus H4 (f )([M × S 1 ]) = 0 ∈ H4 (Eg × Eh ),

.

meaning that .deg(f ) = 0. Remark 9.3.17 Note that the non-domination .M × S 1 > Eg × Eh (where .g, h ≥ 2) follows also quickly by the fact that .M × S 1 has vanishing simplicial volume, whereas the simplicial volume of .Eg × Eh is positive (by the inequalities in (9.3) or more generally by Bucher [5]). However, we have chosen to give more elementary and uniform arguments for the proof of Theorem 9.3.9, revealing also the strength of algebraic considerations alone.

9.3.3.2

Finishing the Proof of Theorem 9.3.9

Thus far, we have given a proof for the right-hand side of the diagram in Fig. 9.2, i.e., concerning maps between geometric aspherical 4-manifolds that are finitely covered by products. For the remaining parts in Fig. 9.2, we need to show that each 4 4 2 2 of the geometries .Nil 4 , .Sol04 , .Solm/ =n , .Sol1 and the irreducible geometry .H × H is not comparable with any other (non-hyperbolic) geometry under the domination relation. Comparing Non-Product Solvable Geometries We begin by showing that there are no maps of non-zero degree between any two manifolds that are modeled on 4 4 4 different geometries among .Nil 4 , .Sol04 , .Solm/ =n or .Sol1 . First, we deal with .Nil 4 and .Sol1 : Proposition 9.3.18 There are no maps of non-zero degree between .Nil 4 and .Sol14 manifolds. Proof .Nil 4 manifolds and .Sol14 manifolds are finitely covered by .S 1 bundles over 3 3 .N il manifolds and .Sol manifolds respectively, and the center of their fundamental groups remains infinite cyclic in finite covers; see Propositions 9.3.2 and 9.3.5 respectively. By Theorem 9.3.8, there are no maps of non-zero degree between .Sol 3 manifolds and .N il 3 manifolds, thus the proposition follows by the next lemma. n u pi

Lemma 9.3.19 ([43, Lemma 5.1]) Let .Mi −→ Bi (.i = 1, 2) be .S 1 bundles over aspherical manifolds .Bi of the same dimension, so that the center of each .π1 (Mi ) remains infinite cyclic in finite covers. If .B1 > B2 , then .M1 > M2 . Next, we show that there are no maps of non-zero degree between .Sol04 manifolds 4 and .Solm/ =n manifolds. Recall by Theorem 9.3.6(1) that any manifold modeled on any of these geometries is a mapping torus of .T 3 , and, moreover, the eigenvalues of

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the automorphism of .Z3 induced by the monodromy of .T 3 are not roots of unity; cf. [23, pg. 164–165]. The following general result in all dimensions shows that every non-zero degree map between such mapping tori is .π1 -injective: Proposition 9.3.20 ([43, Prop. 5.3]) Suppose M and N are finitely covered by mapping tori of self-homeomorphisms of .T n so that the eigenvalues of the induced automorphisms of .Zn are not roots of unity. If .f : M −→ N is a non-zero degree map, then f is .π1 -injective. Hence, by Theorem 9.3.7, we deduce the following: 4 4 Corollary 9.3.21 Any two manifolds M and N modeled on .Solm/ =n and .Sol0 respectively are not comparable under the domination relation.

Finally, in a similar vein, using Theorem 9.3.6, as well as Propositions 9.3.2, 9.3.4 and 9.3.5, we obtain the following: Proposition 9.3.22 ([43, Prop. 5.5 and 5.6]) If M is a .Nil 4 or .Sol14 manifold and 4 4 N is a .Solm/ =n or .Sol0 manifold, then there is no map of non-zero degree between M and N. Non-Product Solvable Manifolds vs Virtual Products We now indicate why there 4 4 are no maps of non-zero degree between a .Nil 4 , .Sol04 , .Solm/ =n or .Sol1 manifold 3 2 2 and a manifold modeled on .X × R or on the reducible .H × H geometry. We need the following result, parts of which use the property group not infinite-index presentable by products, which will be defined and discussed briefly in Sect. 9.4.2: Theorem 9.3.23 ([42, Theorem F]) An aspherical geometric 4-manifold M is dominated by a non-trivial product if and only if it is finitely covered by a product. Equivalently, M is modeled on one of the product geometries .X3 ×R or the reducible 2 2 .H × H geometry. In particular, we have: Corollary 9.3.24 A 4-manifold modeled on one of the geometries .Nil 4 , .Sol04 , 4 4 .Sol m/=n or .Sol1 is not dominated by products. The proof of the converse uses again the structure theorems for the geometries 4 4 N il 4 , .Sol04 , .Solm/ =n and .Sol1 (see Sect. 9.3.1), as well as the growth of their Betti numbers (see for example [23, Sections 8.6 and 8.7] and [42, Section 6]):

.

Proposition 9.3.25 ([43, Prop. 5.8 and 5.9]) A manifold modeled on one of the 4 4 geometries .N il 4 , .Sol04 , .Solm/ =n or .Sol1 does not dominate any manifold modeled on a geometry .X3 × R or the reducible .H2 × H2 geometry. The Irreducible .H2 × H2 Geometry Finally, we deal with the irreducible .H2 × H2 geometry.

318

C. Neofytidis

Proposition 9.3.26 An irreducible .H2 × H2 manifold M is not comparable under the domination relation with any other manifold possessing a non-hyperbolic aspherical geometry. Proof Suppose first that .f : M → N is a map of non-zero degree, where N is an aspherical manifold which is not modeled on the irreducible .H2 × H2 geometry. After possibly passing to a finite cover, we may assume that f is .π1 -surjective; in particular, we have the following short exact sequence π1 (f )

1 −→ ker(π1 (f )) −→ π1 (M) −→ π1 (N ) −→ 1.

.

By Margulis [36, Theorem IX.6.14], the kernel .ker(π1 (f )) is trivial, and thus .π1 (f ) is an isomorphism. Since M and N are aspherical, we conclude that M is homotopy equivalent to N , which contradicts Theorem 9.3.7. Hence, .M > N. Conversely, we claim that M is not dominated by any non-hyperbolic geometric aspherical 4-manifold N. Since M is not dominated by products (e.g., by Theorem 9.3.23), it suffices to show that .N > M when N is modeled on one of the 4 4 geometries .Sol14 , .Nil 4 , .Solm/ =n or .Sol0 . For any of those four geometries, .π1 (N ) has a normal subgroup of infinite index, which is free Abelian of rank one (for 4 4 the geometries .Sol14 and .Nil 4 ) or three (for the geometries .Solm/ =n and .Sol0 ); see Sect. 9.3.1 and [42, Section 6]. If .f : N → M is a (.π1 -surjective) map of non-zero degree, then by Margulis [36, Theorem IX.6.14] either f factors through a lower dimensional aspherical manifold or .π1 (M) is free Abelian of finite rank. None of these cases can occur. u n We have now completed the proof of Theorem 9.3.9.

9.4 Geometric Kodaira Dimension, Monotonicity, and Simplicial Volume The Kodaira dimension is an important tool in the classification of complex manifolds, and has been generalised to various classes, such as symplectic manifolds and almost complex manifolds; we refer to [33] for a survey. Following Thurston’s geometrisation picture, we will introduce an axiomatic definition for the Kodaira dimension and show that this geometric Kodaira dimension is monotone with respect to the domination relation for manifolds of dimension .≤ 5. We will also compare the geometric Kodaira dimension with other, existing notions of Kodaira dimension, and establish a relationship to the simplicial volume.

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9.4.1 Kodaira Dimension A substantial attempt to introduce a notion of Kodaira dimension for noncomplex manifolds, in particular for odd-dimensional manifolds, was made by W. Zhang [60]. Recall that Kodaira’s original approach defines the holomorphic Kodaira dimension .κ h (M, J ) for a complex manifold .(M, J ) of real dimension 2m by ⎧ ⎨ −∞, if Pl (M, J ) = 0 for all l ≥ 1; h .κ (M, J ) = 0, if Pl (M, J ) ∈ {0, 1}, but /≡ 0 for all l ≥ 1; ⎩ k, if Pl (M, J ) ∼ cl k , c > 0.

(9.5)

where .Pl (M, J ) denotes the l-th plurigenus of the complex manifold .(M, J ) defined by .Pl (M, J ) = h0 (KJ ⊗l ), with .KJ the canonical bundle of .(M, J ). Zhang introduced the notion of geometric (or topological) Kodaira dimension for 3-manifolds and geometric 4-manifolds, following the principle suggested by (9.5) that compact geometries have the smallest value (.−∞), while hyperbolic geometries have the biggest value (half of the dimension of the manifold). Subsequently, Zhang and I [48] introduced a more unified approach which we present below. As we shall see, this unification includes as well many non-geometric situations.

9.4.1.1

Axiomatic Definition of κ g

Let .G be the smallest class of manifolds which contains all of the following: . . . . .

points; manifolds that carry a compact geometry; solvable manifolds (solvable-by-solvable); irreducible symmetric spaces of non-compact type; fibrations or manifolds that carry a fibered geometry, so that their fiber and base (geometries) belong in .G.

Definition 9.4.1 Let M be an n-manifold in .G. We define its (geometric) Kodaira dimension { } 3 n g .κ (M) ∈ −∞, 0, 1, , 2, . . . , 2 2 by the following axioms: (A0) (A1) (A2) (A3)

If M If M If M If M

is a point, then we set .κ g (M) = 0; carries a compact geometry, then .κ g (M) = −∞; is of solvable type, then .κ g (M) = 0; is irreducible symmetric of non-compact type, then .κ g (M) = n2 ;

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C. Neofytidis

(A4) If M is a fiber bundle or carries a fibered geometry .F → Xn → B, such that it does not satisfy any of (A1)–(A3), then κ g (M) = sup{κ g (F ) + κ g (B)},

.

F,B

where the supremum runs over all manifolds F and B which can occur in a fibration .F → M → B or are modeled on a geometry .F and .B respectively, and which satisfy one of (A1)-(A3). Note the following consequence of the above axioms: Lemma 9.4.2 Let .M ∈ G. If .M → M is a finite cover, then .M ∈ G and .κ g (M) = κ g (M).

9.4.1.2

Classification in Dimensions ≤ 5

We will now classify manifolds up to dimension five according to their Kodaira dimension. 0-Manifolds By (A0), the Kodaira dimension of a point is zero. 1-Manifolds The circle .S 1 = R/Z is modeled on the real line, hence .κ g (S 1 ) = 0 by (A2). 2-Manifolds Let .Eh be a surface of genus h. First, if .h = 0, then .E0 = S 2 , hence g 2 2 2 g .κ (E0 ) = 0 by (A1). Next, if .h = 1, then .E1 = T = R /Z and thus .κ (E1 ) = 0 g by (A2). Lastly, if .h ≥ 2, then .Eh is hyperbolic, and so .κ (Eh ) = 2/2 = 1 by (A3). The above are summarised in Table 9.4. 3-Manifolds The geometry .S 3 satisfies (A1), the geometries .R3 , .Nil 3 and .Sol 3 satisfy (A2), and the geometry .H3 satisfies (A3). We are left with three geometries which do not satisfy any of (A1)-(A3). For .S 2 × R, Axiom (A4) and the Kodaira dimensions for 1- and 2-manifolds imply κ g (S 2 × S 1 ) = κ g (S 2 ) + κ g (S 1 ) = −∞.

.

Table 9.4 The Kodaira dimension for surfaces

.κ

g

Geometry

.−∞

.S

0 1

.R

2 2

.H

2

9 Geometric Structures in Topology, Geometry, Global Analysis and Dynamics Table 9.5 The Kodaira dimension for geometric 3-manifolds

.κ

g

.−∞

0 1 .

Table 9.6 The Kodaira dimension for geometric 4-manifolds

.κ

g

3 2

321 Geometry 3 2 .S , .S × R 3 3 3 .R , .N il , .Sol 2 .H × R, .SL2 3 .H

Geometry

.−∞

.S

0

.R

1

× X2 , .S 3 × R 4 , .Sol 4 , .Sol 4 × R, .Solm,n 0 1 2 2 .H × R , .SL2 × R

3 2

.H

3

2

.H

4 , .H2 (C), .H2

.

4 , .CP2 , .S 2

4 , .N il 4 , .N il 3

×R × H2

-2 geometry is finitely Finally, since any 3-manifold M modeled on the .H2 × R or .SL covered by an .S 1 bundle over a hyperbolic surface .Eh , Axiom (A4), Lemma 9.4.2 and the Kodaira dimensions for .S 1 and hyperbolic surfaces imply κ g (M) = κ g (S 1 ) + κ g (Eh ) = 1.

.

Table 9.5 summarises the above values of .κ g . 4-Manifolds If a manifold M is modeled on one of the three compact geometries S 4 , .CP2 and .S 2 × S 2 , then .κ g (M) = −∞ by (A1). The geometries .R4 , .Nil 4 , 4 4 3 4 g .N il × R, .Sol , .Sol and .Solm,n satisfy (A2), hence .κ (M) = 0 for any manifold 0 1 M modeled on any of these geometries. If M is modeled on one among .H4 , .H2 (C) or the irreducible .H2 × H2 geometry, then it satisfies (A3), hence .κ g (M) = 4/2 = 2. We are left with seven geometries which fall in Axiom (A4): If M carries one among the geometries .S 2 × R2 , .S 2 × H2 or .S 3 × R, then it is finitely covered by a manifold which is a fiber bundle with fiber one of the compact manifolds .S 2 or 3 g g n .S . Thus .κ (M) = −∞, because .κ (S ) = −∞ for .n ≥ 2. If M is modeled on 2 2 -2 × R, then .κ g (M) = 1 by the corresponding one of the geometries .H × R or .SL classifications in lower dimensions. Similarly, if M is modeled on .H3 × R, then g 2 2 .κ (M) = 3/2. Finally, if M carries the reducible geometry .H × H , then it is g finitely covered by a product of two hyperbolic surfaces, hence .κ (M) = 2, since hyperbolic surfaces have Kodaira dimension one. We summarise these values in Table 9.6. .

Remark 9.4.3 The geometry .H3 ×R is one of the first examples which indicate our new approach to introduce systematically half-integer values for .κ g , distinguishing thus further the various classes of manifolds by their Kodaira dimension. In addition, this example reveals the usefulness and necessity of (A4): A 4-manifold M modeled on the geometry .H3 ×R is finitely covered by .F ×S 1 for some hyperbolic 3-manifold F . In addition, F is (up to finite covers) a mapping torus of a pseudo-Anosov diffeomorphism of a hyperbolic surface .E (see Table 9.2), hence, in particular, M

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is (covered by) a fiber bundle .E → M → T 2 . Thus, we compute by the values of g .κ in dimensions .≤ 3 that { } 3 3 g g g 1 g g 2 .κ (M) = sup{κ (F ) + κ (S ), κ (E) + κ (T )} = ,1 = . 2 2 Remark 9.4.4 The values of the geometric Kodaira dimension of 4-manifolds match with the values of the holomorphic Kodaira dimension for Kähler manifolds. However, according to (A2), the Kodaira dimension for .Sol04 and .Sol14 manifolds is zero instead of .−∞ as defined in [60], following Wall’s scheme for complex non-Kähler surfaces [57]. We could have imposed further conditions (e.g., on the virtual second Betti number) so that our Kodaira dimension for those manifolds is .−∞, however, we have chosen to keep our axiomatic approach natural with minimal assumptions. Indeed, the value .κ g = 0 here not only follows by (A2), but it is also compatible with (A4). 5-Manifolds The 5-dimensional Thurston geometries were classified by Geng [14]. According to Geng’s list, there exist fifty eight geometries, of which fifty four are realised by compact manifolds. We will only enumerate the latter geometries according to Definition 9.4.1, and refer the reader to the three papers from Geng’s thesis [14–16], as well as to the references in [14], for further details. Finally, as it is remarked in [14, Section 4], a similar classification for the Thurston geometries was partially done in dimensions six and seven. In particular, one can use Definition 9.4.1 to determine the Kodaira dimensions of those manifolds. (A1) There are three geometries of compact type, namely, the 5-sphere .S 5 , the Wu symmetric manifold .SU (3)/SO(3), and .S 2 × S 3 . If a manifold M carries any of these geometries, then .κ g (M) = −∞. (A2) There are twenty geometries of solvable type. First, there exist two nilpotent and six solvable but not nilpotent geometries of type .R4 x R, denoted by .A5,1 , A5,2 a,b,−1−a−b 1,−1,−1 −1,−1 and .A5,7 , .A1,−1−a,−1+a , A5,7 , A−1 , A−1 5,7 5,8 , A5,9 5,15 respectively. Next, 4 there are two nilpotent semi-direct products .Nil x R, denoted by .A5,5 and .A5,6 . Furthermore, there is one nilpotent and one solvable but not nilpotent geometry of type .(R × N il 3 ) x R, which are denoted by .A5,3 and .A05,20 respectively. Also,

−1,−1 . The there is a solvable but not nilpotent geometry .R3 x R2 denoted by .A5,33 last irreducible solvable-type geometry is .Nil 5 . The rest of those geometries are products of lower dimensional geometries, namely .R5 , .Nil 3 ×R2 , .Nil 4 ×R, .Sol04 × 4 × R (here .Sol 4 3 2 R, .Sol14 × R, and .Solm,n m,m × R = Sol × R ). If a manifold M is g modeled on any of the above geometries, then .κ (M) = 0 by (A2).

(A3) If a manifold M carries one of the irreducible symmetric geometries of noncompact type .H5 or .SL(3, R)/SO(3), then .κ g (M) = 52 . (A4) For the remaining geometries, we obtain a variety of values. First, suppose M is a manifold which is modeled on one among the following sixteen geometries:

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Table 9.7 The Kodaira dimension for geometric 5-manifolds .κ

g

.−∞

Geometry .SU (3)/SO(3), .S

5 , .S 2

× X3 , .S 3 × X2 , .S 4 × R, .CP2 × R,

-2 ×α S 3 , .L(a, 1) ×S 1 L(b, 1), .T 1 (H3 ) ×R S 3 , .SL 5 4 3 2 5 4 3 .R , .R x R, .R x R , .N il , .N il x R, .(R × N il ) x R, 3 .N il

0 1

4 ×R × R, .N il 3 × R2 , .Sol04 × R, .Sol14 × R, .Solm,n 3 2 3 2 2 3 2 -2 -2 , .R2 x SL -2 , .N il 3 ×R SL .H × R , .H × Nil , .H × Sol , .R × SL

3 2

.H

2

2 .H

.N il

.

.

5 2

3

4

× R2

-2 ×α SL -2 , .H2 × H2 × R, .H4 × R, .H2 (C) × R, .U (2,-2 , .SL 1)/U (2) × SL 5 3 2 .H , .SL(3, R)/SO(3), .H × H .

-2 , S 2 × H3 , S 2 × S 2 × S 2 × R, .S 2 × R3 , S 2 × Nil 3 , S 2 × Sol 3 , S 2 × H2 × R, S 2 × SL 2 3 3 2 3 2 4 3 3 3 H , S ×R , S ×H , .S ×R, CP ×R, Nil ×R S , SL2 ×α S , L(a, 1)×S 1 L(b, 1), or .T 1 (H3 ). Then M has a fiber or base which is one of the compact geometries .S 2 , 2 3 4 g .S , .S or .CP . Thus, .κ (M) = −∞ by the classification of Kodaira dimensions of manifolds of dimension .≤ 4. Next, suppose M is modeled on one of the geometries .R3 × H2 , Nil 3 × 2 -2 , or .Nil 3 ×R SL -2 × R2 , R2 x SL -2 . Then M fits into a fibration, H , Sol 3 × H2 , .SL 2 where the involved geometries are .H and some solvable-type geometry. Therefore, g .κ (M) = 1. If M carries the geometry .H3 × R2 , then .κ g (M) = 32 , where the supremum is achieved by the geometries .H3 and .R2 (compare to Remark 9.4.3). -2 , H2 × H2 × Next, suppose that M carries one of the geometries .H2 × SL -2 ×α SL -2 , .H4 × R, H2 (C) × R, or .U (2,1)/U (2). Each of these geometries R, SL fibers over one of the 4-dimensional geometries .H2 × H2 , .H2 or .H2 (C), which have Kodaira dimension two; see Table 9.6. We conclude that .κ g (M) = 2. Finally, if a manifold M carries the geometry .H2 ×H3 , then .κ g (M) = 1+ 32 = 52 . We summarise the above in Table 9.7. .

9.4.2 Monotonicity of the Kodaira Dimension One of the main motivations in [60] was to study whether the geometric Kodaira dimension is monotone with respect to the domination relation (see Definition 9.2.13, where .−∞ is allowed as well). To this end, Zhang defines the Kodaira dimension for all 3-manifolds according to Thurston’s picture: Let M be a 3-manifold. If it carries a Thurston geometry, then its Kodaira dimension is given by Table 9.5. We call each of the values .−∞, 0, 1, 32 the category to which a geometric 3-manifold belongs. If M does not carry any of the eight Thurston geometries, then consider first its Kneser-Milnor prime decomposition (which is trivial when M is prime) and then a toroidal decomposition for each prime summand of M, so that

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Table 9.8 The Kodaira dimension for 3-manifolds .κ

g

.−∞

0 1 .

3 2

For any T -decomposition of M. . . Each piece belongs to the category .−∞ There is at least one piece in the category 0, but no pieces in the category 1 or . 32 There is at least one piece in the category 1, but no pieces in the category . 32 There is at least one piece in the category . 32 (hyperbolic piece)

each piece of the toroidal decomposition carries one of the eight geometries with finite volume. We call this a T -decomposition of M. The Kodaira dimension of M is then given in Table 9.8. The main result of [60] is the following: Theorem 9.4.5 Let .M, N be 3-manifolds. If .M ≥ N , then .κ g (M) ≥ κ g (N ). Remark 9.4.6 Note that Theorem 9.4.5 is also a consequence of Theorem 9.3.8, which is sharper in the sense that it tells us the existence or not of maps between manifolds modeled on different geometries with the same Kodaira dimension, while Theorem 9.4.5 does not. Subsequently, Zhang asked whether the monotonicity result for the Kodaira dimension in Theorem 9.4.5 could be extended in higher dimensions. For geometric 4-manifolds, this is a consequence of the ordering given in Theorem 9.3.9: Theorem 9.4.7 Let .M, N be geometric 4-manifolds. If .M ≥ N , then .κ g (M) ≥ κ g (N ). In my recent paper with Zhang [48], we showed that .κ g is monotone for geometric 5-manifolds: Theorem 9.4.8 Let .M, N be geometric 5-manifolds. If .M ≥ N , then .κ g (M) ≥ κ g (N ). We will only summarise the basic steps of the proof, pointing out some techniques and phenomena, and refer to [48] for the details. Sketch of Proof We need to show that if .κ g (M) < κ g (N ), then .M > N . Hence, we need to examine the various cases according to Table 9.7. First, we observe that if .κ g (N ) = 5/2, then .||N || > 0, whereas .||M|| = 0 whenever .κ g (M) /= 5/2, and thus .M > N; these use Gromov’s results [19], the approximations given by (9.3), as well as a result by Bucher [4] for the geometry .SL(3, R)/SO(3). We are now left to examine the cases κ g (M) ∈ {−∞, 0, 1, 3/2} and κ g (N ) /=

.

5 . 2

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If .κ g (M) = −∞, then M is rationally inessential and thus it cannot dominate rationally essential manifolds. But if .κ g (N ) /= −∞, then N is aspherical, in particular rationally essential. Hence .M > N . If .κ g (M) = 0, then M is modeled on a solvable-type geometry, whereas if g .κ (N ) > 0, then .π1 (N ) is not solvable. Hence, the non-domination .M > N follows by the fact that if .ϕ : H1 −→ H2 is a group homomorphism with .H1 solvable, then the image .ϕ(H1 ) ⊆ H2 is solvable. Suppose now that .κ g (M) = 1. This is the most delicate case and requires a stepby-step examination of many geometries individually. Among the most interesting cases occur when M is a .Sol 3 × H2 manifold and N is finitely covered by a nontrivial circle bundle over a hyperbolic or an .H2 × H2 manifold, because this reveals some new group theoretic phenomena. Recall (cf. Definition 9.3.3) that a group G is called presentable by products if there exist two infinite elementwise commuting subgroups .G1 , G2 ⊆ G, so that the image of the multiplication homomorphism G1 × G2 −→ G

.

has finite index in G. If in addition both .Gi can be chosen with [G : Gi ] = ∞,

.

then G is called infinite index presentable by products or IIPP. This notion was defined in [42] and it is a sharp refinement between reducible groups (that is, groups that are up to finite-index subgroups direct products of two infinite groups), and groups presentable by products, i.e., {reducible groups} C {IIPP groups} C {groups presentable by products}.

.

The following result gives a criterion for the equivalence between IIPP and reducible for central extensions: Theorem 9.4.9 ([42, Theorem D]) Let G be a group with center .C(G) such that the quotient .G/C(G) is not presentable by products. Then, G is reducible if and only if it is IIPP. Non-elementary hyperbolic groups is a standard prominent class of groups that are not presentable by products [29]. If N is modeled on the geometry 1)/U (2), then N is finitely covered by a non-trivial .S 1 bundle over a complex .U (2, hyperbolic 4-manifold B. Since .π1 (N ) is not reducible, Theorem 9.4.9 implies that .π1 (N ) is not IIPP. Hence, any map from a manifold modeled on the geometry 3 2 .Sol × H to N has degree zero, by the next theorem:

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Theorem 9.4.10 ([42, Theorem B]) Let N be an .S 1 bundle over an aspherical manifold B, so that .π1 (N ) is not IIPP and its center remains infinite cyclic in finite covers. Then .P > N, for any non-trivial direct product P . We remark that the same argument applies when the target N is a non-trivial .S 1 bundle over a 4-manifold which is modeled on the irreducible .H2 × H2 geometry. Note, however, that Theorem 9.4.9 does not hold anymore if we remove the condition on the quotient group .G/C(G) being not presentable by products. For instance, the fundamental group of a .Nil 5 manifold N is irreducible and IIPP (see [42, Section 8]), and fits into the following central extension 1 −→ Z −→ π1 (N ) −→ Z4 −→ 1.

.

It was shown in [42, Section 8] that N does not admit maps of non-zero degree from non-trivial direct products. In a similar vein, one proves that .M > N when N is -2 ×α SL -2 , since in that case N is (finitely covered by) modeled on the geometry .SL 1 a non-trivial .S bundle over the product of two hyperbolic surfaces .Eg × Eh , its fundamental group fits into the central extension 1 −→ Z −→ π1 (N ) −→ π1 (Eg ) × π1 (Eh ) −→ 1,

.

and it is moreover irreducible and IIPP. Finally, if .κ g (M) = 32 , i.e., M is modeled on .H3 × R, then M is (finitely covered by) a product of a hyperbolic 3-manifold F and the 2-torus. Thus, we can assume that the center of .π1 (M) has rank two. On the other hand, if N has Kodaira dimension two, then it is (finitely covered by) an .S 1 bundle over a manifold which is modeled on one of the geometries .H4 , .H2 (C) or .H2 ×H2 . In particular, the center of .π1 (N ) is infinite cyclic. Then the non-domination .M > N follows by a factorization argument and the asphericity of the involved spaces (cf. Lemma 9.3.19). u n

9.4.3 Kodaira Dimension Beyond Geometries and the Simplicial Volume The notion of geometric Kodaira dimension defined here goes well beyond Thurston’s geometries. This has already been explained for 3-manifolds in the previous paragraph (Table 9.8). Below we give a complete classification for fiber bundles in dimension four:

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Theorem 9.4.11 Let M be a 4-manifold which is (finitely covered by) a fiber bundle with fiber F and base B. Then ⎧ −∞, ⎪ ⎪ ⎪ ⎪ 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1, g .κ (M) = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ 2, ⎪ ⎪ ⎪ ⎪ ⎩ 2,

if one of F, B isS 2 or finitely covered by #m S 2 × S 1 ; if F = B = T 2 , or one of F, B is a 3-manifold which is not -2 finitely covered by #m S 2 × S 1 and contains no H2 × R, SL 3 or H pieces in its torus or sphere decomposition; if one of F, B is T 2 and the other is a hyperbolic surface, -2 or one of F, B is a 3-manifold with at least one H2 × R or SL piece and no H3 pieces in its torus or sphere decomposition; if one of B, F is a 3-manifold with at least one H3 piece in its torus or sphere decomposition; if both F and B are hyperbolic surfaces.

Proof The claim follows from our axioms in Definition 9.4.1 and .κ g in lower dimensions. u n The values of the Kodaira dimension for manifolds of dimension .≤ 3 and for geometric 4-manifolds suggest that top Kodaira dimension is often equivalent to the positivity of the simplicial volume. This is again the case for the 4-dimensional fibrations of Theorem 9.4.11: Theorem 9.4.12 Let M be a 4-manifold which is (finitely covered by) a fiber bundle with fiber F and base B. Then .||M|| > 0 if and only if .κ g (M) = 2. Proof This is a consequence of Theorem 9.4.11 and [6, Corollary 1.3].

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A natural problem stemming from this study is to understand the relationship or compatibility of the geometric Kodaira dimension .κ g with existing notions of Kodaira dimension (see also Remark 9.4.4). Motivated by the above discussion, we give the following result about the holomorphic Kodaira dimension .κ h for complex 2n-manifolds, which verifies the relationship to the simplicial volume. Theorem 9.4.13 ([48, Theorem 1.5]) (1) If M is a smooth complex projective n-fold with non-vanishing simplicial volume, then .κ h (M) /= n − 1, .n − 2 or .n − 3. (2) If M is a smooth Kähler threefold with non-vanishing simplicial volume, then h .κ (M) = 3. Proof We will summarise the main steps of the proof, giving a uniform treatment for both parts of the theorem, and refer the reader to [48, Section 4] for the details. If .κ h (M) > 0, then M admits an Iitaka fibration, namely, M is birationally equivalent to a projective manifold X that admits an algebraic fiber space structure .φ : X → Y over a normal projective variety Y , such that the Kodaira dimension of a very general fiber of .φ has Kodaira dimension zero. In dimensions .≤ 3, Kollár [27] showed that the fundamental group of a smooth proper variety with vanishing

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holomorphic Kodaira dimension is virtually Abelian (and conjectured that this is true in all dimensions). Using this and Gromov’s mapping theorem [19], one can show that .||X|| = 0, whenever .dim(M) = n − 3, n − 2 or .n − 1; see [48, Theorems 4.5 and 4.6]. But the simplicial volume is a birational invariant [48, Lemma 4.1], hence we obtain that .||M|| = ||X|| = 0 if .dim(M) = n − 3, n − 2 or .n − 1. If .κ h (M) = 0 and .n ≤ 3, then .π1 (M) is virtually Abelian as mentioned above, hence M has vanishing simplicial volume. Finally, we note that .||M|| > 0 implies that M cannot be uniruled [48, Prop. 4.2]. Uniruled manifolds satisfy .κ h = −∞. In fact, Mumford conjectured that a smooth projective variety is uniruled if and only if .κ h = −∞ [2], and this is known to be true for complex projective threefolds [38]. The proof for the Kähler case follows by the fact that any compact Kähler manifold of complex dimension three is bimeromorphic to a Kähler manifold which is deformation equivalent to a projective manifold [8, 34]. u n

9.5 Anosov Diffeomorphisms The final section of this survey has its origins in Dynamics and the AnosovSmale conjecture. Our goal is to show that Anosov diffeomorphisms do not exist on geometric 4-manifolds which are not finitely covered by the product of two aspherical surfaces. The main reference for this section is [46].

9.5.1 The Main Result We will prove the following: Theorem 9.5.1 If M is a 4-manifold that carries a geometry other than .R4 , .H2 × R2 or the reducible .H2 × H2 geometry, then M does not admit transitive Anosov diffeomorphisms. The transitivity assumption in the above theorem is mild and will only be used when M is a product of the 2-sphere with an aspherical surface. Note that Theorem 9.5.1 does not exclude (transitive) Anosov diffeomorphisms on geometric 4-manifolds which are finitely covered by a product of surfaces .Eg × Eh , where .g, h ≥ 1: Problem 9.5.2 ([18, Section 7.2]) Does the product of two aspherical surfaces at least one of which is hyperbolic admit an Anosov diffeomorphism? Recently, D. Zhang [61] showed how to exclude Anosov diffeomorphisms on certain products of two hyperbolic surfaces.

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Finally, as we have done in our study thus far, we will make extended use of properties of finite covers of geometric 4-manifolds. We thus collect the following lemmas (see [18] and [17] respectively): Lemma 9.5.3 Let M be a manifold and .p : M → M be a finite covering. If f : M → M is a diffeomorphism, then there exists some integer .m ≥ 0 such that .f m lifts to a diffeomorphism .f m : M → M, that is, the following diagram commutes.

.

.

Lemma 9.5.4 If .f : M → M is a transitive Anosov diffeomorphism and .f : M → M is a lift of f for some cover .M of M, then .f is transitive.

9.5.2 Proof of Theorem 9.5.1 We will examine each of the geometries and exploit tools from Algebraic Topology, such as Hirsch and Ruelle-Sullivan classes, as well as coarse geometric properties, such as negative curvature.

9.5.2.1

Hyperbolic Geometries

We first deal with the real and complex hyperbolic geometries. The following theorem is well-known to experts, but we will present a proof for two reasons: First, the proof contains some useful facts about Anosov diffeomorphisms which will be used below as well, such as the behaviour of their Lefschetz numbers. Second, the tools used for the proof (e.g., the Gromov norm) reveal the beauty of connections between domains that initially might seem irrelevant to each other. Theorem 9.5.5 ([18, 59]) If M is a negatively curved manifold, then M does not admit Anosov diffeomorphisms. In particular, there are no Anosov diffeomorphisms on manifolds modeled on the geometry .H4 or the geometry .H2 (C). Proof The first proof due to Yano [59] shows that there are no transitive Anosov diffeomorphisms on a negatively curved manifold M. Suppose the contrary, and let .f : M → M be a transitive Anosov diffeomorphism. Since only tori admit codimension one Anosov diffeomorphisms [12, 49], we can assume that the dimension of M is at least four and the codimension k of f is at least two. By Theorem 9.2.22, there is a homology class .a ∈ Hl (M; R) such that .Hl (f )(a) = λ·a for some .λ > 1, where .l = k > 1 or .l = dim(M) − k > 1. This means

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that the Gromov norm of a is zero which is impossible because M is negatively curved [19, 25]. More generally, Gogolev-Lafont [18] showed that there are no Anosov diffeomorphisms on a negatively curved manifold M of dimension .≥ 3, using the fact that the outer automorphism group .Out(π1 (M)) is finite. By the latter property and the fact that M is aspherical (because it is negatively curved), we conclude that some power .f l of a hypothetical Anosov diffeomorphism .f : M → M induces the identity on cohomology (which already implies that f cannot be transitive by Theorem 9.2.22 or by Shiraiwa [52]). Thus, the Lefschetz numbers .A (that is, the sum of indices of the fixed points) of any power of .f l are uniformly bounded, which contradicts the growth of periodic points of .f l , by the equation |A(f m )| = |Fix(f m )| = remhtop (f ) + o(emhtop (f ) ), m ≥ 1,

.

(9.6)

where .htop (f ) denotes the topological entropy of f and r is the number of transitive basic sets with entropy equal to .htop (f ); we refer to [18, Lemma 4.1] for further details. u n

9.5.2.2

Product Geometries

We split our study into three cases: (i) Product geometries with a compact factor; (ii) Aspherical geometries .X3 × R; (iii) The irreducible geometry .H2 × H2 . Products with a Compact Factor First, we show the following: Theorem 9.5.6 There are no (transitive) Anosov diffeomorphisms on a 4-manifold that carries a geometry .S i × X, for .i = 2, 3. Proof We will examine each of the involved geometries. The Geometry .S 2 × S 2 Suppose .f : S 2 × S 2 → S 2 × S 2 is a diffeomorphism or, more generally, a map of degree .±1. The Künneth formula tells us that H 2 (S 2 × S 2 ) = (H 2 (S 2 ) ⊗ H 0 (S 2 )) ⊕ (H 0 (S 2 ) ⊗ H 2 (S 2 )).

.

Let .ωS 2 ×1 ∈ H 2 (S 2 )⊗H 0 (S 2 ) and .1×ωS 2 ∈ H 0 (S 2 )⊗H 2 (S 2 ) be cohomological fundamental (orientation) classes. We assume that .deg(f ) = 1, after possibly passing to .f 2 . Then, by f ∗ (ωS 2 × 1) = a · (ωS 2 × 1) + b · (1 × ωS 2 ), a, b ∈ Z,

.

f ∗ (1 × ωS 2 ) = c · (ωS 2 × 1) + d · (1 × ωS 2 ), c, d ∈ Z,

.

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and the naturality of the cup product, we deduce that ad + bc = 1.

.

(9.7)

Moreover, since the cup product of .ωS 2 × 1 with itself vanishes, we have 0 = f ∗ ((ωS 2 × 1) ∪ (ωS 2 × 1)) = f ∗ (ωS 2 × 1) ∪ f ∗ (ωS 2 × 1) = 2ab · (ωS 2 ×S 2 ),

.

that is, ab = 0.

.

(9.8)

Similarly, since .(1 × ωS 2 )2 = 0, we have cd = 0.

.

(9.9)

If .a = 0, then by (9.7), (9.8) and (9.9) we obtain .b = c = ±1 and .d = 0. If .b = 0, then by the same equations we obtain .a = d = ±1 and .c = 0. Hence, f induces the identity in cohomology, after possibly replacing f by .f 2 . Therefore, the Lefschetz numbers of all powers of f are uniformly bounded, and so f cannot be Anosov (cf. (9.6)). The Geometry .S 2 × R2 In this case, M is finitely covered by .S 2 × T 2 [23, Theorem 10.10]. Since any map from .S 2 to .T 2 has degree zero (see Proposition 9.2.2), if f is a diffeomorphism of .S 2 × T 2 , then f ∗ (ωS 2 × 1) = a · (ωS 2 × 1) + b · (1 × ωT 2 ), a, b ∈ Z,

.

and f ∗ (1 × ωT 2 ) = d · (1 × ωT 2 ), d ∈ Z.

.

As above, we may assume that .deg(f ) = 1, and so by the naturality of the cup product we obtain ad = 1.

.

(9.10)

In particular, .a = d = ±1. Also, .b = 0 because .(ωS 2 × 1)2 = 0. Since any manifold that admits a codimension one Anosov diffeomorphism is homeomorphic to a torus, we may assume that our hypothetical Anosov f has codimension two. In that case, there is a class .α ∈ H 2 (S 2 × T 2 ; R) such that 2 .H (f )(α) = λ · α for some positive real .λ /= 1; see Theorem 9.2.22. Now α = ξ1 · (ωS 2 × 1) + ξ2 · (1 × ωT 2 ), ξ1 , ξ2 ∈ R,

.

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and by .H 2 (f )(α) = λ · α we obtain .

λξ1 = aξ1 = ±ξ1

(9.11)

λξ2 = dξ2 = ±ξ2 .

(9.12)

and .

If .ξ1 /= 0, then .λ = ±1 by (9.11), which is impossible. If .ξ1 = 0, then .ξ2 /= 0 and (9.12) yields again .λ = ±1. Hence, .S 2 × T 2 does not admit transitive Anosov diffeomorphisms. The Geometry .S 2 ×H2 If M carries the geometry .S 2 ×H2 , then it is finitely covered by an .S 2 bundle over a hyperbolic surface .E; see [23, Theorem 10.7]. The product case .S 2 × E can be treated similarly to the case of .S 2 × T 2 . Moreover, GogolevRodriguez Hertz showed that a 4n-manifold E, which is a fiber bundle .S 2n → E → B, does not admit transitive Anosov diffeomorphisms [17, Theorem 1.1] (note that this covers as well the geometry .S 2 ×R2 ). The argument in [17] uses again Lefschetz numbers (equation (9.6)) and cup products via the Gysin sequence 0 −→ H 2n (B; Z) −→ H 2n (E; Z) −→ H 0 (B; Z) −→ 0.

.

For our purposes, .2n = 2 is the only case of interest for the codimension. The Geometry .S 3 × R Finally, if a manifold carries the geometry .S 3 × R, then it is finitely covered by a product .S 3 × S 1 ; see [23, Ch. 11]. The latter does not admit Anosov diffeomorphisms because .H2 (S 3 × S 1 ) = 0 and .H1 (S 3 × S 1 ) = Z. The proof of Theorem 9.5.6 is now complete. Aspherical

Geometries .X3

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× R Next, we will prove the following:

Theorem 9.5.7 Let M be a 4-manifold that carries one of the geometries .H3 × R, 3 -2 × R or .Nil 3 × R. Then M does not admit Anosov diffeomorphisms. .Sol × R, .SL Proof As usual, we will treat the various geometries according to their algebraic properties. -2 × R Suppose M is a 4-manifold that carries The Geometries .Nil 3 × R and .SL 3 -2 × R. In this case, M is finitely covered by a one of the geometries .Nil × R or .SL 1 3 -2 manifold respectively; see product .N × S , where N is a .Nil manifold or an .SL Proposition 9.3.1. According to Table 9.2, we can moreover assume that N is a nontrivial .S 1 bundle over the 2-torus or a hyperbolic surface respectively. We conclude that the center of .π1 (N × S 1 ) has rank two. A finite power of the generator of the

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fiber of N vanishes in .H1 (N ), and thus, for any diffeomorphism .f : N × S 1 → N × S 1 , the generator of .H1 (S 1 ) maps to a power of itself (modulo torsion). Hence, we have in cohomology H 1 (f )(1 × ωS 1 ) = a · (1 × ωS 1 ), a ∈ Z.

.

Recall that N is not dominated by products by Theorem 9.3.8 (and Table 9.2). Since the degree three cohomology of .N × S 1 is H 3 (N × S 1 ) ∼ = H 3 (N ) ⊕ (H 2 (N ) ⊗ H 1 (S 1 )),

.

we obtain H 3 (f )(ωN × 1) = b · (ωN × 1), b ∈ Z;

.

see the proof of Theorem 1.4 in [41] for further details. Since .deg(f ) = ±1, we deduce that .a, b ∈ {±1}. Hence, we can assume that f ∗ (1 × ωS 1 ) = 1 × ωS 1 ,

.

after replacing f by .f 2 , if necessary. We conclude that f is not Anosov by Lemma 9.5.3 and the next theorem, which is a generalisation of Theorem 9.2.21. Theorem 9.5.8 ([24, Theorem 1]) Let .f : M → M be an Anosov diffeomorphism and a non-trivial cohomology class .u ∈ H 1 (M; Z) such that .(f ∗ )m (u) = u, for some positive integer m. Then the infinite cyclic covering of M corresponding to u has infinite dimensional rational homology. The Geometries .H3 × R and .Sol 3 × R Let M be a 4-manifold modeled on the 3 3 1 .H × R or the .Sol × R geometry. Then, M is finitely covered by .N × S , where N 3 is a hyperbolic 3-manifold or a .Sol manifold respectively; see Proposition 9.3.1. In particular, the fundamental group .π1 (N × S 1 ) has infinite cyclic center generated by the .S 1 factor, which we denote by .π1 (S 1 ) = . If .f : N × S 1 → N × S 1 is a diffeomorphism, then .f∗ () = , and therefore .H 1 (f )(ωS 1 ) = ωS 1 (replace f by .f 2 , if necessary) as above, because N does not admit maps of non-zero degree from direct products (see Theorem 9.3.8 and Table 9.2). Hence, f cannot be Anosov by Lemma 9.5.3 and Theorem 9.5.8. Remark 9.5.9 When N is hyperbolic, the main result of [18] implies also that N × S 1 does not admit Anosov diffeomorphisms, because .Out(π1 (N )) is finite and .π1 (N ) is Hopfian with trivial intersection of its maximal nilpotent subgroups. In fact, as shown in [44], the only properties needed to exclude Anosov diffeomorphisms on .N × S 1 is that .Out(π1 (N )) is finite and .π1 (N ) has trivial center. .

We have now finished the proof of Theorem 9.5.7.

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The Irreducible .H2 × H2 Geometry Finally, suppose that M carries the irreducible geometry .H2 × H2 . Then .π1 (M) has finite outer automorphism group by the strong rigidity of Mostow, Prasad and Margulis. Thus, the same argument as in the proof of Theorem 9.5.5 implies that M does not admit Anosov diffeomorphisms.

9.5.2.3

Non-product, Solvable or Compact Geometries

Finally, we prove the following: Theorem 9.5.10 There are no Anosov diffeomorphisms on a manifold carrying one 2 4 4 4 4 of the geometries .Nil 4 , .Solm/ =n , .Sol0 , .Sol1 , .S or .CP . Proof The proof will be done according to certain properties of the involved geometries. The Geometry .N il 4 Let M be a 4-manifold modeled on the geometry .Nil 4 . By Proposition 9.3.2, the fundamental group of M (after possibly replacing M by a finite cover) is given by π1 (M) = , where .k ≥ 1, .l ∈ Z, and it has infinite cyclic center .C(π1 (M)) = . Then π1 (M)/ = .

.

Let .f : M → M be a diffeomorphism. The automorphism .f∗ : π1 (M) → π1 (M) induces an automorphism on the quotient .π1 (M)/, because .f∗ () = . But .C(π1 (M)/) = , hence .f∗ (x) = zn x m , for some .n, m ∈ Z, .m /= 0. Since .[x, y] = z and by the fact that the relation .txt −1 = x is mapped to m −1 = x m , we obtain that .f (t) does not contain any powers of y. .f∗ (t)x f∗ (t) ∗ By the commutative diagram

.

where h : π1 (M) → H1 (M; Z) = π1 (M)/[π1 (M), π1 (M)]

.

denotes the Hurewicz homomorphism, we conclude that .H1 (f ) maps the homology class .t¯ ∈ H1 (M; Z)/T orH1 (M; Z) to a multiple of itself. In fact, the induced

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automorphism on .H1 (M; Z)/T orH1 (M; Z) = × ¯ = Z × Z implies that H1 (f )(t¯) = t¯, which means that f cannot be Anosov by Lemma 9.5.3 and Theorem 9.2.21 (or Theorem 9.5.8).

.

4 4 4 The Geometries .Solm/ =n , .Sol0 and .Sol1 For these geometries, the following immediate consequence of Theorem 9.2.21 will suffice:

Theorem 9.5.11 ([24, Theorem 8]) Suppose M is a manifold such that (a) .π1 (M) is virtually polycyclic; (b) the universal covering of M has finite dimensional rational homology; (c) .H 1 (M; Z) ∼ = Z. Then M does not admit Anosov diffeomorphisms. 4 Suppose M is a 4-manifold modeled on one among the geometries .Solm/ =n , 4 4 .Sol or .Sol . If M carries one of the first two geometries, then M is a mapping 0 1 torus of a hyperbolic homeomorphism of .T 3 ; see Theorem 9.3.6(1); in particular, 1 .H (M; Z) ∼ = Z. Since .π1 (M) is polycyclic and M is aspherical, Lemma 9.5.3 and Theorem 9.5.11 imply that M does not admit Anosov diffeomorphisms. If M carries the geometry .Sol14 , then, by Theorem 9.3.6(2) (see also Proposition 9.3.5), we have

π1 (M) = , where .k, l ∈ Z and the matrix ( .

ab cd

)

has no eigenvalues which are roots of unity. The Abelianization of .π1 (M) tells us that .H 1 (M; Z) ∼ = Z. Since moreover M is aspherical and .π1 (M) is polycyclic, Lemma 9.5.3 and Theorem 9.5.11 imply that M does not admit Anosov diffeomorphisms. Remark 9.5.12 Note that Theorem 9.5.11 is not applicable if M is a .Nil 4 manifold, because .H 1 (M; Z) ∼ = Z2 . The Geometries .S 4 and .CP2 The only 4-manifold modeled on .S 4 is .S 4 itself [23, Section 12.1]. Since any orientation preserving diffeomorphism .f : S 4 → S 4 induces the identity in homology, f cannot be Anosov (cf. (9.6)).

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Similarly to .S 4 , the only 4-manifold modeled on .CP2 is .CP2 itself [23, Section 12.1]. Let .f : CP2 → CP2 be a diffeomorphism. Since the cohomology groups of 2 m induces the identity on .CP are .Z in degrees 0, 2 and 4 and trivial otherwise, .f cohomology, for some .m ≥ 1. Hence, f cannot be Anosov. We have now completed the proof of Theorem 9.5.10. u n This finishes the proof of Theorem 9.5.1. Acknowledgments I would like to thank Ken’ichi Ohshika and Athanase Papadopoulos for their invitation to write this survey, as well as an anonymous referee for the careful reading and the suggestions.

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20. M. Gromov, Hyperbolic groups, in Essays in Group Theory. Mathematical Sciences Research Institute Publications, vol. 8 (Springer, New York-Berlin, 1987), pp. 75–263 21. M. Gromov, Metric Structures for Riemannian and Non-Riemannian spaces. Progress in Mathematics, vol. 152 (Birkhäuser Verlag, Basel, 1999). With appendices by M. Katz, P. Pansu and S. Semmes, translated from the French by S. M. Bates 22. A. Hatcher, Algebraic Topology (Cambridge University Press, Cambridge, 2002) 23. J.A. Hillman, Four-Manifolds, Geometries and Knots. Geometry and Topology Monographs, vol. 5 (Coventry University, Coventry, 2002) 24. M.W. Hirsch, Anosov maps, polycyclic groups, and homology. Topology 10, 177–183 (1971) 25. H. Inoue, K. Yano, The Gromov invariant of negatively curved manifolds. Topology 21(1), 83–89 (1981) 26. R. Kirby, Problems in Low-Dimensional Topology (University of California Berkeley, Berkeley, 1995) 27. J. Kollár, Shafarevich Maps and Automorphic Forms. M. B. Porter Lectures (Princeton University Press, Princeton, 1995), x+201 pp. 28. D. Kotschick, Remarks on geometric structures on compact complex surfaces. Topology 31, 317–321 (1992) 29. D. Kotschick, C. Löh, Fundamental classes not representable by products. J. Lond. Math. Soc. 79, 545–561 (2009) 30. D. Kotschick, C. Neofytidis, On three-manifolds dominated by circle bundles. Math. Z. 274, 21–32 (2013) 31. D. Kotschick, C. Löh, C. Neofytidis, On stability of non-domination under taking products. Proc. Am. Math. Soc. 144, 2705–2710 (2016) 32. J.-F. Lafont, B. Schmidt, Simplicial volume of closed locally symmetric spaces of non-compact type. Acta Math. 197, 129–143 (2006) 33. T.-J. Li, Kodaira dimension in low-dimensional topology, in Tsinghua Lectures in Mathematics. Advanced Lectures in Mathematics (ALM), vol. 45 (International Press, Somerville, 2019), pp. 265–291 34. H.-Y. Lin, Algebraic approximations of compact Kähler threefolds. arXiv:1710.01083 35. W. Malfait, Anosov diffeomorphisms on nilmanifolds of dimension at most six. Geom. Dedicata 79, 291–298 (2000) 36. G.A. Margulis, Discrete Subgroups of Semisimple Lie Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 3. Folge Bd. 17 (Springer-Verlag, Berlin-Heidelberg, 1991) 37. I. Mineyev, Bounded cohomology characterizes hyperbolic groups. Q. J. Math. 53, 5–73 (2002) 38. S. Mori, Flip theorem and the existence of minimal models for 3-folds. J. Am. Math. Soc. 1, 117–253 (1988) 39. C. Neofytidis, Non-zero degree maps between manifolds and groups presentable by products. Munich Thesis, 2014. Available online at http://edoc.ub.uni-muenchen.de/17204/ 40. C. Neofytidis, Branched coverings of simply connected manifolds. Topol. Appl. 178, 360–371 (2014) 41. C. Neofytidis, Degrees of self-maps of products. Int. Math. Res. Not. IMRN 22, 6977–6989 (2017) 42. C. Neofytidis, Fundamental groups of aspherical manifolds and maps of non-zero degree. Groups Geom. Dyn. 12, 637–677 (2018) 43. C. Neofytidis, Ordering Thurston’s geometries by maps of non-zero degree. J. Topol. Anal. 10, 853–872 (2018) 44. C. Neofytidis, Anosov diffeomorphisms of products II. Aspherical manifolds. J. Pure Appl. Algebra 224(3), 1102–1114 (2020) 45. C. Neofytidis, Anosov diffeomorphisms of products I. Negative curvature and rational homology spheres. Ergodic Theory Dynam. Syst. 41, 553–569 (2021) 46. C. Neofytidis, Anosov diffeomorphisms on Thurston geometric 4-manifolds. Geom. Dedicata 213, 325–337 (2021) 47. C. Neofytidis, On a problem of Hopf for circle bundles over aspherical manifolds with hyperbolic fundamental group. Algebr. Geom. Topol. 23, 3205–3220 (2023)

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48. C. Neofytidis, W. Zhang, Geometric structures, the Gromov order, Kodaira dimensions and simplicial volume. Pac. J. Math. 315, 209–233 (2021) 49. S.E. Newhouse, On codimension one Anosov diffeomorphisms. Am. J. Math. 92, 761–770 (1970) 50. D. Ruelle, D. Sullivan, Currents, flows and diffeomorphisms. Topology 14, 319–327 (1975) 51. P. Scott, The geometries of 3-manifolds. Bull. Lond. Math. Soc. 15, 401–487 (1983) 52. K. Shiraiwa, Manifolds which do not admit Anosov diffeomorphisms. Nagoya Math. J. 49, 111–115 (1973) 53. S. Smale, Differentiable dynamical systems. Bull. Am. Math. Soc. 73, 747–817 (1967) 54. R. Thom, Quelques propriétés globales des variétés différentiables. Comment. Math. Helv. 28, 17–86 (1954) 55. W.P. Thurston, Three-Dimensional Geometry and Topology (Princeton University Press, Princeton, 1997) 56. C.T.C. Wall, Geometries and geometric structures in real dimension 4 and complex dimension 2, in Geometry and Topology (College Park, Md., 1983/1984). Lecture Notes in Mathematics, vol. 1167 (Springer, Berlin, 1985), pp. 268–292 57. C.T.C. Wall, Geometric structures on compact complex analytic surfaces. Topology 25, 119– 153 (1986) 58. S. Wang, The existence of maps of non-zero degree between aspherical 3-manifolds. Math. Z. 208, 147–160 (1991) 59. K. Yano, There are no transitive Anosov diffeomorphisms on negatively curved manifolds. Proc. Jpn. Acad. Ser. A Math. Sci. 59, 445 (1983) 60. W. Zhang, Geometric structures, Gromov norm and Kodaira dimensions. Adv. Math. 308, 1–35 (2017) 61. D. Zhang, Anosov diffeomorphisms on a product of surfaces. Preprint. arXiv:2205.11296.

Chapter 10

Counting Problems for Invariant Point Processes Jayadev S. Athreya

Abstract Using linear algebra and the ergodic theory of .SL(2, R) actions, we survey how to solve several natural asymptotic counting problems for discrete subsets of the plane using an axiomatic perspective. Applications include counting holonomies of saddle connections, lattice points, and fine scale distribution in various contexts. This is a perspective inspired by work of Veech (Ann. Math. (2) 148(3):895–944, 1998), and developed further by, among others, Eskin– Masur (Ergodic Theory Dyn. Syst. 21(2):443–478, 2001), Athreya–Ghosh (Enseign. Math. 64(1–2), 1–21, 2018), and Marklof (Lond. Math. Soc. Newsl. 493:42–49, 2021). Keywords Point process · Saddle connection · Random lattice · Poisson process 2010 60G55, 32G15,11H60

10.1 Introduction Surface Geometry, Moduli Spaces, and Point Processes We describe how linear algebra, together with the ergodic theory of the action of linear groups, can be used to solve counting problems which arise from the geometry of surfaces. This is a circle of ideas that originates with Veech [46] in his seminal paper on Siegel Measures (which in turn was motivated by Margulis’s solution [28] of the Oppenheim conjecture). This philosophy, which associates to certain natural measures on moduli spaces point processes in Euclidean spaces, has been subsequently developed by, among others, Eskin–Masur [19], Marklof [30] and Athreya–Ghosh [7], and used extensively (see, for example [1, 2, 4]) to address problems motivated by Diophantine approximation in a wide variety of contexts.

J. S. Athreya () Department of Mathematics, University of Washington, Seattle, WA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 K. Ohshika, A. Papadopoulos (eds.), In the Tradition of Thurston III, https://doi.org/10.1007/978-3-031-43502-7_10

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We will focus on big picture ideas emerging from the comparison of these point processes to Poisson point processes. The level of detail is designed to focus on key motivating examples, and to suggest a variety of open problems and further directions (see Sect. 10.10 in particular). Organization In the rest of this chapter, we will (Sect. 10.2) define the notion of ergodic SL(2, R)-invariant point processes on the complex plane C, and give several examples (Sect. 10.3) arising from classical probability theory (Sect. 10.3.1) and from geometry (lattice points in Sect. 10.3.2 and holonomy vectors of saddle connections in Sect. 10.3.3). We will state (and give context for) our main counting results in Sect. 10.4, focusing on counting in three families of sets: circles and sectors (Sect. 10.4.1); regions bounded by hyperbolas (Sect. 10.4.2); and triangles (Sect. 10.4.3). In Sect. 10.5, we will recall how to get counting results using classical probability theory in the setting of planar Poisson processes, and provide context and motivation for our results in the geometric context. In Sect. 10.6, we give the necessary background on the ergodic theory of SL(2, R)-actions. In Sect. 10.7, we will recall arguments of Veech [46] and Eskin–Masur [19] to address counting problems in circles and sectors, and state an ergodic theorem (Theorem 10.6.2) of Nevo’s [35] which provides the key dynamical input. We also describe recent developments due to Athreya–Fairchild–Masur [12] and Bonnafoux [15] motivated by understanding correlations. In Sect. 10.8, we prove our results on counting in regions bounded by hyperbolas, using the classical Birkhoff ergodic theorem for the action of the diagonal subgroup of SL(2, R), and in Sect. 10.9, we show how to use equidistribution of unipotent orbits to understand counting in triangles. Finally, in Sect. 10.10 we discuss how these approaches can be generalized to higher dimensions (Sect. 10.10.2), pose some problems on the construction of invariant point processes in more general contexts (Sect. 10.10.3), and state an intriguing question connecting geometry, probability, and representation theory (Sect. 10.10.5).

10.2 SL(2, R)-Invariant Point Processes on C Linear Actions We start by defining the notion of an .SL(2, R)-invariant point process on .C. Our starting point is the .R-linear action of .SL(2, R) on .C, via  .

 ab · (x + iy) = (ax + by) + i(cx + dy). cd

This action induces an action on .P(C), the set of locally finite Radon Borel measures on .C. This action preserves the subset .D(C) of locally finite atomic Radon Borel measures on .C, which we can view as the collection of (weighted) discrete

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subsets of .C. Note that if .Λ ⊂ C is discrete, and .w : Λ → R+ is a weight function, we have the associated atomic Borel measure  .ρΛ,w = w(λ)δλ . λ∈Λ

Vice-versa, associated with each atomic measure .ρ ∈ D(C), we have its discrete support .Λρ and the weight function .w = wρ : Λ → R+ defined by w(λ) = ρ(λ), λ ∈ Λ.

.

Definition 10.2.1 An .SL(2, R)-invariant point process on .C is an .SL(2, R)invariant measure .P on the set .D = D(C) of locally finite atomic Radon Borel measures on .C, that is, a probability measure on the set of (weighted) discrete subsets of .C. We say .P is ergodic if the action of .SL(2, R) on .(D, P) is ergodic, that is, if all .SL(2, R)-invariant subsets .A of .D have .P(A) = 0 or .P(A) = 1. General Invariant Point Processes We will discuss the more general notion of invariant and ergodic point process for actions of a group G on a space X in Sect. 10.10, but essentially, the definition is the same as above with G in place of SL(2, R) and X in place of C (this requires certain assumptions on X and the action of G). Until we discuss these further in Sect. 10.10, we will use the shorthands invariant point process for SL(2, R)-invariant point process on C, and ergodic point process for ergodic SL(2, R)-invariant point processes on C. As in Definition 10.2.1, we will also write D for D(C). Siegel–Veech Transforms Following Veech [46], given a bounded compactly supported complex-valued function f (we denote the collection of such functions Bc (C)), we define the Siegel–Veech transform f : P(C) → C by the integral of f with respect to the measure, that is f(ρ) =



.

C

f dρ.

When ρ ∈ D(C), this is a weighted sum, that is, f(ρ) =



.

C

f dρ =



wρ (λ)f (λ).

λ∈Λρ

Temperedness Given an invariant point process P, and k > 0, we say that it is ktempered if for f ∈ Bc (C), f ∈ Lk (D, P). We will be most interested in, for ϵ > 0, k = 1 + ϵ (which we will simply call tempered) or k = 2 + ϵ (which we will call 2+-tempered). Measure-Valued Processes Many of the results we state will hold for more general measure-valued processes, that is, SL(2, R)-invariant measures μ on P(C) (under

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appropriate integrability conditions), but for our applications, point processes seem to be the natural setting. For this reason, and the sake of exposition, we will work with point processes, and, unless otherwise specified, those with trivial weight function w = 1. The Siegel–Veech Formula Before turning to our examples of invariant point processes, we record an axiomatic result (an observation due to Veech [46]): suppose P is a tempered invariant point process. Then there are constants aP and cP such that  .

D

fdP = cP f (0) + aP

 C

f dm,

(10.2.1)

where m denotes Lebesgue measure on C. To see why this is true, note that by invariance and temperedness,  TP (f ) =

.

D

fdμ

is a well-defined, SL(2, R)-invariant linear functional on Bc (C), and so there must (by the Riesz representation theorem) be a SL(2, R)-invariant measure ηP on C such that  .TP (f ) = f dηP . C

But the only SL(2, R)-invariant measures on C are linear combinations of δ measure at 0 and Lebesgue measure, so we must have ηP = cP δ0 + aP m.

.

For the sake of exposition, we will consider only point processes P where cP = 0. Of course, the real interest in this result comes from checking the temperedness condition, and trying to identify the constants a and c in a variety of examples. We now turn to giving some concrete examples of tempered invariant point processes.

10.3 Examples Probability and Geometry We start with an important motivating example from classical probability theory before turning to examples from geometry.

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10.3.1 Poisson Point Processes A Poisson point process .Pτ on .C is a measure on .D characterized by a positive real parameter .τ > 0 (the intensity) and the following property: for any (finite) collection of disjoint Borel measurable sets .A1 , . . . , An ⊂ C with .0 ≤ m(Ai ) = ai < ∞, and non-negative integers .ki , Pτ (ρ ∈ D : ρ(A) = # (Λ ∩ Ai ) = ki , i = 1, . . . , n) =

n 

.

e−τ ai

i=1

(τ ai )ki . ki ! (10.3.1)

That is, the random variables .ρ(Ai ) = # (Λ ∩ Ai ) are independent Poisson random variables with mean .τ ai . SL(2, R)-Invariance Since the distribution of the number of points

.

N(Λ, A) = ρ(A) = #Λ ∩ A

.

in a (finite area, Borel) set A only depends on the Lebesgue measure .m(A), and Lebesgue measure is .SL(2, R)-invariant, we have that .Pτ is .SL(2, R)-invariant. It is a exercise to show that .aPτ = τ , that is,  .

D

fdPτ = τ

 C

f dm.

Ergodicity It is a nice exercise in geometric probability to show that this is an ergodic point process (in fact, this is a ergodic invariant measure for the affine action on .D(C) induced by the affine action of .SL(2, R) ⋉ C on .C). Exponential Temperedness It is also a nice exercise to show that .Pτ is .∞ tempered, that is, .f ∈ Lp (D, Pτ ) for any .p > 0, and in fact, we have .et f ∈ L1 (D, Pτ ) for any .t ∈ R. Motivations Many of the questions about the examples we consider below are motivated by comparison to Poisson point processes, which are, in some sense, the most truly random possible point processes, since there are no correlations between their behavior in disjoint regions.

10.3.2 Lattice Points We introduce the space of unimodular lattices in .C, which we denote by .X2 . A unimodular lattice in .C is a discrete subgroup .Λ ⊂ C of covolume 1. It is an exercise to see that any unimodular lattice .Λ can be written as .Λ = gZ[i] for .g ∈ SL(2, R),

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and so we identify the space of unimodular lattices with the homogenous space SL(2, R)/SL(2, Z) by

.

gSL(2, Z) I−→ gZ[i].

.

The Haar measure .μ2 on .SL(2, R) can be normalized so that the induced measure μ2 (X2 ) = 1, and we have, associated with this space and measure, two interesting ergodic point processes coming from maps from .X2 to .D, and the pushforward of .μ2 . .

All Lattice Points We start with the most natural map, which is to associate to gSL(2, Z) the full lattice .gZ[i] ⊂ C. This is a discrete subset of .C, and we use .PL to denote the pushforward of the measure .μ2 to .D. We note that .PL is 1-tempered, and .2 − ϵ tempered, but not 2-tempered (see, for example, [22, 38]). This process is ergodic since the action of .SL(2, R) on .X2 is transitive, and therefore ergodic. The Siegel–Veech formula in this setting has .aPL = 1, cPL = 1: intuitively, the expected number of (random) non-zero unimodular lattice points in a set is the Lebesgue measure of the set, and since we have not excluded 0, we get the value of the function at 0.

.

Primitive Lattice Points A natural subset of a lattice is the set of primitive or visible points. Define Zprim [i] = {m + ni : m, n ∈ Z, gcd(m, n) = 1},

.

and for .Λ = gZ[i] Λprim [i] = gZprim [i].

.

We do not consider 0 to be a primitive lattice point. We denote the pushforward of μ2 under the map

.

gSL(2, Z) I−→ gZprim [i]

.

by .PpL . Classical results from the geometry of numbers [38, 40] show that .PpL is 1 = π62 . For details on how to compute higher in fact .∞-tempered, and .aPpL = ζ (2) moments, see, for example, the recent work of Fairchild [22]. This process is also ergodic by the transitivity of .SL(2, R) on .X2 . Affine Lattice Points A natural, and closely related example is the space of affine unimodular lattices, translates of unimodular lattices. An affine unimodular lattice .Λ is given by Λ = gZ[i] + w,

.

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where .g ∈ SL(2, R), and w is defined up to translation by .gZ[i], that is, it can be thought of as valued in .C/gZ[i]. The space of affine unimodular lattices .AX2 is then identified with the homogeneous space ASL(2, R)/ASL(2, Z) = (SL(2, R) ⋉ C) / (SL(2, Z) ⋉ Z[i]) ,

.

via [g, w] I→ gZ[i] + w.

.

(10.3.2)

The space .AX2 is a fiber bundle over .X2 , with fiber over .gZ[i] being the torus C/gZ[i]. There is a natural Haar probability measure on .AX2 , which we denote by .α2 (whose push-forward to .X2 is .μ2 ). We write .PαL for the pushforward of .α2 to .D under the map (10.3.2). This process is ergodic, since the action of .SL(2, R) on .AX2 is ergodic, and 2-tempered (see, for example [3]), and we have .aPαL = 1. .

10.3.3 Saddle Connection Holonomies Associated with any unit covolume lattice .gZ[i], .g ∈ SL(2, R), we have a flat torus .C/gZ[i], together with a holomorphic one-form, dz, since dz is preserved by translations. This one-form induces a flat (Euclidean) metric on the torus. The set of primitive points .gZprim [i] has a natural geometric interpretation, namely, they are in one-to-one correspondence with simple closed geodesics in this flat metric passing through 0. Translation Surfaces We can generalize the picture of discrete sets in the plane associated with families of curves (or arcs) on surfaces by considering saddle connections on higher-genus translation surfaces. For a more detailed background on translation surfaces, see, for example [8]. We recall that a (compact) genus g translation surface is a pair .(X, ω), where X is a compact genus g Riemann surface and .ω a holomorphic 1-form. We will usually refer to .ω for the pair, with X being implicit. Polygons Concretely, we can present .ω as a (collection) of polygons .P = (P1 , . . . , Pn ) in .C with parallel sides identified by translations (see Fig. 10.1 for an example of a genus 2 translation surface), where the one-form is the pull-back of dz. The one-form .ω will vanish at a finite set of points .Σ, and the order of the zeros of .ω will sum to .2g − 2, by the Riemann-Roch theorem. Away from the zeros, .ω determines a flat metric on the surface X. Saddle Connections A saddle connection .γ is a geodesic in this flat metric which connects points (connecting a point to itself is allowed) in .Σ without passing through any such points in its interior. To each saddle connection .γ we associate

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Fig. 10.1 A genus 2 translation surface with a double zero at the point .•, obtained via side identifications. The dashed lines are a selection of saddle connections

●

●

●

●

the holonomy vector .zγ = 𝓁γ eiθγ ∈ C which records the length .𝓁γ = |zγ | and direction .θγ of .γ by  zγ =

ω.

.

γ

Discreteness We denote the set of saddle connection holonomies on .ω by .Λω ⊂ C, that is, Λω = {zγ : γ is a saddle connection on ω}.

.

This set is a discrete subset of .C. By considering .SL(2, R)-invariant probability measures on a moduli space .H of unit-area translation surfaces .ω, we will build invariant point processes with this assignment. Cylinders We will also consider the set .C(ω) of core curves of embedded cylinders cyl in our surface, and the associated set .Λω of holonomy vectors. This will be a setting where weights become interesting, in particular, the weight function which associates to each cylinder .σ ∈ C the quantity .area(σ ), where the area form on i .(X, ω) is given by . ω ∧ ω. 2 Strata  If we fix the genus g and the orders of the zeros .α = (α1 , . . . , αk ) ∈ Nk , where . αi = 2g − 2, we can consider the stratum of translation surfaces .Ω(α), where we say that two translation surfaces are equivalent if there is a biholomorphism between the underlying Riemann surfaces which maps the oneforms to each other. Kontsevich–Zorich [26] showed that these strata contain at most 3 connected components. We will work with a fixed connected component .Ω. Period Coordinates On each stratum of translation surfaces, we have local period coordinates, charts to .C2g+k−1 . In a neighborhood of a fixed .ω0 ∈ Ω, we can consider the homology .V = H1 (X, Σ, C) of the surface relative to the zeros .Σ. Fixing an integral basis for V (which is .2g + k − 1 dimensional), we have local

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coordinates by integrating a nearby .ω against this basis, that is, viewing .ω as an element of relative cohomology. Pulling back Lebesgue measure from .C2g+k−1 gives a notion of Lebesgue measure .ν on .Ω (which is well-defined since changes of cooridinates are by integral invertible matrices). The .GL+ (2, R)-Action The group .GL+ (2, R) of orientation-preserving linear maps acts on .Ω, via its .R-linear action on .C. This is easiest to see in the polygon picture, since linear maps take parallel line segments of the same length to parallel segments of the same length, so if .ω is defined by a collection of polygons .P = (P1 , . . . , Pn ) with side identifications, .g · ω is the translation surface defined by .gP = (gP1 , . . . , gPn ). Area 1 Surfaces The area of the surface .

area(ω) =

i 2

 ω∧ω X

is a quadratic form in period coordinates. We consider the subset H = area−1 (1) ⊂ Ω

.

of area 1 surfaces, which we will also refer to as a stratum. This is preserved by the action of .SL(2, R). We define the Masur–Smillie–Veech masure .μ on .H by a cone construction, similar to how Haar measure on .SL(n, R)-can be defined. For .A ⊂ H, we define .μ(A) by taking the Lebesgue meausre .ν of the cone C(A) = {tω : ω ∈ A, 0 ≤ t ≤ 1}

.

over A in .Ω, that is, μ(A) = ν({tω : ω ∈ A, 0 ≤ t ≤ 1}).

.

This measure is, by construction, .SL(2, R)-invariant, and by results of Masur [31], Veech [45], and Masur–Smillie [32], is finite, and ergodic, and thus can be normalized to be an ergodic invariant probability measure. Temperedness Results Given .ω ∈ H, we define the measure .ρω on .C by ρω =



.

δz ,

z∈Λω

and we consider the measure .PH as the pushforward of the probability .μH , that is, we consider the random subset .Λω , where .ω is selected at random from .μH . Ergodicity of the action on .H implies the ergodicity of this process, and we have a series of temperedness results, starting with Veech [46], who showed that .PH is tempered. Athreya–Cheung–Masur [11] showed that .PH is .2+-tempered.

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Areas and Weights We also have point processes associated with the set .Λω of holonomy vectors of core curves of embedded cylinders in our surface, where as we have mentioned, weights become interesting. Given a function .H : (0, 1] → R+ , we can define the measure  cyl .ρ H (area(σ ))δzσ . H,ω = σ ∈C(ω)

We denote by .PH,H the pushforward of .μ on .H under this assignment. The temperedness properties of .PH,H of course depend on H . Of particular interest is the weight .H (x) = x, for which .PH,H is tempered, and the constant .aPH,H , which is known as the area Siegel–Veech constant [21].

10.4 Counting Asymptotics We now turn to our main results, which give asymptotic counting results for invariant point processes in varying families of growing regions, under natural temperedness assumptions. The common thread in the proofs is the use of the ergodic theory of .SL(2, R) and its subgroups.

10.4.1 Circles and Sectors We start by considering perhaps the most natural geometric counting problem, of counting in the ball around the origin .B(0, R) ⊂ C. We define the counting function + → R+ by .N : D × R N(ρ, R) = ρ(B(0, R)).

.

Given an invariant point process .P, we are interested in the almost-sure asymptotic (as .R → ∞) behavior of the function .N(·, R). Building on work of Veech [46], Eskin–Masur [19] proved the following result (we will explain below how to translate their language into ours): Theorem 10.4.1 ([19, Theorem 2.1]) Suppose .P is a tempered ergodic invariant point process, and for every .ρ ∈ supp(P), there is an .a = aρ , such that N(ρ, R) ≤ aR 2 ,

.

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and that .aρ can be chosen uniformly as .ρ varies in a compact set (with respect to the compact-open topology on .D(C)). Then for .P-almost every .ρ ∈ D(C), .

N(ρ, R) = aP . R→∞ π R 2 lim

Sectors Given a fixed arc .A ⊂ S 1 , we can also consider the angular counting function NA (ρ, R) = ρ(z ∈ C : arg(z) ∈ A, |z| ≤ R).

.

Theorem 10.4.1 can be modified to show for .P-almost every .ρ ∈ D(C), .

lim

R→∞

NA (ρ, R) |A| = aP . 2π π R2

Further results on shrinking sectors, where the width of the arc A is allowed to shrink with R, can be found in Athreya–Chaika [5], and Chaika–Fairchild [16].

10.4.2 Hyperbolas Our next result is motivated by classical Diophantine approximation. For .T > 0, let H (T ) := {z ∈ C : 1 < Im(z) < eT /2 , Im(z)| Re(z)| < 1},

.

(10.4.1)

be the region bounded by the hyperbolas .Im(z) Re(z) = 1 and .Im(z) Re(z) = −1, with .Im(z) ∈ [1, eT /2 ]. We have the following result (see, for example, Athreya– Parrish–Tseng [10] for this in the lattice and translation surface settings): Theorem 10.4.2 Suppose .P is a tempered ergodic invariant point process. Then for P-almost every .ρ,

.

.

ρ(H (T )) = aP . T →∞ T lim

Quantitative Diophantine Approximation This problem is motivated by a quantitative version of Dirichlet’s theorem, which states that for any irrational .α ∈ R, there are infinitely many .p/q ∈ Q (with .gcd(p, q) = 1) such that p . α − < 1/q 2 . q

(10.4.2)

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If we denote by .N(α, T ) the number of solutions .p/q to (10.4.2) with .q ≤ T , this is the same as counting the number

 # Λα,prim ∩ H (T )

.

of primitive lattice points in the intersection between the primitive points of the lattice .Λα = u−α Z[i] with the set .H (T ) where  u−α =

.

 1 −α , 0 1

(10.4.3)

since p . α − < 1/q 2 , q ≤ T ⇐⇒ q |p − qα| ≤ 1, q ≤ T q ⇐⇒ u−α (p + qi) ∈ H (T ). We note that Theorem 10.4.2 does not directly imply any counting results for N (α, T ), but using extra structural information about the ergodic theory on the space of lattices, Athreya–Parrish–Tseng [10] obtain this result (which recovers earlier work of W. Schmidt [39]), and several generalizations.

.

10.4.3 Triangles Our next series of results is on the intersection of ergodic point processes with triangles. For .s > 0, let Ts = {w ∈ C : 0 < Im(w) < s Re(w) < s}.

.

(10.4.4)

We have: Theorem 10.4.3 Suppose .P is an ergodic point process. Then .

lim

s→∞

1 1 ρ (Ts ) = aP . s 2

(10.4.5)

Counting Farey Fractions This result is informed by a perspective outlined in Athreya-Cheung [6] on how to count Farey fractions using the dynamics of the horocycle flow.

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10.5 Counting for Poisson Processes Independence While our results work for planar Poisson point processes, there are easier proofs using classical probability theory (laws of large numbers, central limit theorems, large deviations results) by partitioning our sets into disjoint sets, so expressing the corresponding quantities as sums of independent random variables. For example, in the setting of Poisson processes, all of Theorem 10.4.1, Theorem 10.4.2 and Theorem 10.4.3 follow from the following general result: Theorem 10.5.1 Let A1 , A2 , . . . An , . . . be a sequence of Borel measurable sets with disjoint interiors A◦i , whose boundaries ∂Ai have Lebesgue measure m(∂Ai ) = 0, and whose Lebesgue measures ai = m(Ai ) < ∞. If Pτ is a Poisson point process with intensity τ > 0, and Xi = Pτ (Ai ), we have, with probability 1, n i=1 Xi . lim n = τ. n→∞ i=1 ai

(10.5.1)

Proof Since the random variables Xi are almost surely equal to Yi = ρ(A◦i ), we can replace Xi with Yi . The Yi ’s are independent random variables with mean ai < ∞, so (10.5.1) follows from the strong law of large numbers. Counting in Regions To derive Theorem 10.4.1, Theorem 10.4.2 and Theorem 10.4.3 from Theorem 10.5.1 for Poisson processes, we leave as an (easy) exercise how to partition the respective regions into sets with disjoint interiors.

10.5.1 Further Limit Theorems In fact, the independence of the intersection of Poisson process with disjoint sets allows one to apply more classical results from probability theory to obtain, for example, central limit theorems. Central Limit Theorem Suppose for the sake of simplicity, that in the setup of Theorem 10.5.1, all the sets have the same measure, say .ai = 1 > 0, and .τ = 1. Then we have the central limit theorem: for all .T ∈ R,   n  ∞ 1 2 i=1 Xi − n >T = √ . lim P1 e−x /2 dx. (10.5.2) √ n→∞ n 2π T We will discuss what is known about central limit theorems in the other settings we consider in Sects. 10.7.5 and 10.8.4.

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10.6 Ergodic Theory of SL(2, R) Subgroups We record some important classical results about the ergodic theory of SL(2, R)-actions, and that of several key subgroups, including the positive diagonal subgroup

.

 t/2 

e 0 .A = gt = :t ∈R , 0 e−t/2 +



(10.6.1)

the upper triangular unipotent subgroup 

1s :s∈R ; .U = us = 01



(10.6.2)

and the lower triangular unipotent subgroup 

 1 0 :s∈R . N = hs = −s 1

.

(10.6.3)

10.6.1 Moore Ergodicity Recall that a measure-preserving action of a group G on a probability measure space (X, μ) is ergodic if any G-invariant set has measure 0 or measure 1. From this definition, it is clear that if H is a subgroup of G, and the H -action is ergodic, then the G-action is also ergodic, since any G-invariant set is certainly H -invariant. An interesting phenomenon, known as Moore ergodicity [34] says that for .G = SL(2, R) (and more generally, any connected semisiple Lie group), the ergodicity of a G-action implies the ergodicity of any of its non-compact subgroups.

.

10.6.2 SL(2, R)-Ergodic Theorems We will extensively use ergodic theorems for the action of subgroups of .SL(2, R). These describe the convergence of averaging over (pieces) of the orbit of an action to the average over the whole space X. Diagonal and Unipotent Flows The first, which we will use for counting in hyperbolas and triangles, is a direct implication of Moore ergodicity and the Birkhoff Ergodic Theorem (see, for example, Walters [47]), and applies to the action of the non-compact one-parameter subgroups .A+ , U , and N , stating that averages long pieces of orbits converge to the average over X.

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Theorem 10.6.1 For .r ∈ R, let .φr = hr , .ur , or .gr . Suppose .SL(2, R) acts ergodically on a probability measure space .(X, μ), and .f ∈ L1 (X, μ). Then for .μ-almost every .x ∈ X, .

1 R→∞ R



lim

R

 f (φr x)dr =

0

f dμ.

(10.6.4)

X

Circle Averages A more subtle ergodic theorem, due to Nevo [35] holds for averaging over circles. Let .K ⊂ SL(2, R) be the maximal compact subgroup given by 

 cos θ − sin θ : 0 ≤ θ < 2π . K = rθ = sin θ cos θ

.

(10.6.5)

Note that .rθ z = eiθ z for .z ∈ C. For an .SL(2, R) action on a space X, and .t ∈ R, we define the averaging operator .At on functions on X to be the average over the .gt translates of K, that is, (At )(f )(x) =

.

1 2π



2π

f (gt rθ x)dθ. 0

K-Finite Functions A function f defined on X is K-finite if the span of the rotated functions .{f ◦ rθ : θ ∈ [0, 2π )} is finite-dimensional. Theorem 10.6.2 ([35, Theorem 1.1]) Let .SL(2, R) act ergodically on a probability measure space .(X, μ). Let .f ∈ L1+κ (H, μ) for some .κ > 0, and assume f is K-finite. For any continuous non-negative bump function .η ∈ Cc (R) with compact support and of unit integral, and for .μ-almost every .x ∈ X, the .η-normalized averages of the averaging operators .At converge to the average of f with respect to .μ, that is,  .

lim

∞

t→±∞ −∞

 η(t − s)(As f )(x)ds =

f dμ. X

Sectors Morally, we should think of this statement as saying that for almost all x ∈ X, and sufficiently “nice” test functions f , the averages of f along circles in the .SL(2, R) orbit converge to the average of f , that is, as .t → ±∞, .(At f )(x) → X f dμ. We remark that the integral can be taken to be over any arc .(a, b) ⊂ [0, 2π ) and the conclusion of the theorem is not affected.

.

10.7 Circles and Sectors Linear Algebra We now turn to showing how linear algebra and the ergodic theory discussed above in Sect. 10.6 yield our main counting results. We start with the most natural counting problem (circles, Theorem 10.4.1) which is perhaps the most subtle

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reduction to ergodic theory of the three main results. These ideas were developed by Veech [46] and Eskin–Masur [19], building on ideas of Eskin–Margulis–Mozes [20] in their work on quantitative versions of Margulis’ [28] resolution of the Oppenheim conjecture on quadratic forms. We follow the exposition of Eskin–Masur [19], see also the survey article of Eskin [18] and the book [8, §6].

10.7.1 Circle Averages and Counting The key idea in the translation from counting in circles to averaging over group orbits is the following calculus/linear algebra observation: recall from (10.4.5) that we have the triangle T1 = {z ∈ C : 0 < Im(z) < Re(z) < 1},

.

which has vertices at the origin, 1, and .1 + i. The image of .T1 is a long, thin triangle with vertices at .0, et/2 , et/2 + e−t/2 i, namely, gt T1 = {z ∈ C : 0 < Im(z) < e−t Re(z) < e−t/2 }.

.

For t large, this long, thin, triangle is very close to a small piece of a circular arc, where the circle has radius .et/2 , and the width of the angular sector is .e−t . For .w ∈ C, let Θt (w) = {θ ∈ [0, 2π ) : rθ w ∈ gt T1 } = {θ : [0, 2π ) : g−t rθ w ∈ T1 }.

.

Observe that |w| ≥



.

et + e−t = |gt (1 + i)| =⇒ Θt (w) = ∅

(10.7.1)

and |w| ≤ et/2 =⇒ Θt (w) = (0, arctan(e−t ))

.

(10.7.2)

so for .t >> 0, since .arctan(e−t ) ∼ = e−t , |Θt (w)| ∼ = e−t χB(0,et/2 ) (w).

.

(10.7.3)

Letting .f = χT1 , we can rewrite this as 1 . 2π



2π 0

1 −t f (g−t rθ w)dθ ∼ e χB(0,et/2 ) (w). = 2π

(10.7.4)

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Putting .R = et/2 , and recalling that for .ρ ∈ D(C), f(ρ) =

 f dρ = ρ(T1 ),

.

C

we have, by integrating (10.7.4) with respect to .ρ, (A−t f)(ρ) =

.

=

 2π

1 2π

1 2π

∼ =

0

 2π 0

f(g−t rθ ρ)dθ

(10.7.5)

(g−t rθ ρ)(T1 )dθ

1 ρ(B(0, R)) 2π R 2

=

1 N(ρ, R). 2π R 2

10.7.2 Applying Nevo’s Ergodic Theorem If Nevo’s ergodic theorem could be directly applied to conclude that for an SL(2, R)-invariant ergodic measure .P on .D(C), for .P-almost every .ρ, as .R → ∞,

.

.

 1 ∼  N(ρ, R) (A f )(ρ) −→ fdP, = −t 2π R 2 D (C)

we would then have, via the Siegel–Veech formula (10.2.1), .

1 N(ρ, R) = aP 2π R 2

 C

f dm =

1 aP , 2

yielding the conclusion (after multiplying by 2) of Theorem 10.4.1.

10.7.3 Technical Details Of course, we cannot directly apply Nevo’s theorem in this fashion, as we need to account for the normalizing bump function .η, and the lack of K-finiteness of the function . h. The temperedness and boundedness assumptions, together with careful approximation arguments, allow one to conclude Theorem 10.4.1. We refer the reader to the original paper of Eskin–Masur [19] for these technical details, and also urge the reader to consult the work of Nevo–Ruhr-Weiss [36] for how to make this result quantitative under certain spectral gap assumptions for the .SL(2, R)-action. Sectors The counting result for sectors follows from Nevo’s ergodic theorem in sectors, together with the technical arguments mentioned above. As remarked above,

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it is of interest to understand the growth in sectors of shrinking angular width, and this seems, in general, to be a subtle problem. See 10.10.1 for precise formulations of questions in this direction.

10.7.4 Counting Pairs There are also recent results on counting pairs of saddle connections with conditions on the angles between them (in terms of their length), most easily formulated in terms of the virtual area of the pair, that is, the area of the parallelogram they span. See Athreya–Fairchild–Masur [12] and Bonnafoux [15] for these results in the setting of translation surfaces, which use crucially the 2-temperedness established by Athreya–Cheung–Masur [11].

10.7.5 Central Limit Theorems It would be of great interest to understand the precise (appropriately normalized) distribution of the error terms .E(ρ, R) = (N (ρ, R) − aP π R 2 ). For Poisson processes, the appropriate normalizing function would be of the order R, coming from computing the variance of .N(ρ, R), and the limiting distribution is normal (Sect. 10.5.1). In general both the computation of the variance and the existence and form of the limit distribution seem to be interesting and non-trivial problems. More generally, one can ask about the distribution of the averages .(At f ) as .t → ∞ for ergodic .SL(2, R)-actions, perhaps with spectral gap conditions.

10.8 Hyperbolas and Geodesic Flow 10.8.1 Proving Theorem 10.4.2 We now turn to the ideas behind the proof of Theorem 10.4.2. Recall that H (T ) := {z ∈ C : 1 < Im(z) < eT /2 , Im(z)| Re(z)| < 1}.

.

We write H = H (1) = {z ∈ C : 1 < Im(z) < e1/2 , Im(z)| Re(z)| < 1}.

.

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Then note that gt z ∈ H (2) ⇐⇒

.

  z ∈ H (T + 1) \H (T ) = z ∈ C : eT /2 < Im(z) < e(T +1)/2 , Im(z)| Re(z)| < 1 .

In particular, we have {t ∈ R : gt z ∈ H } = [2 log Im(z) − 1, 2 log Im(z)],

.

and putting .f = χH , .fT = χH (T ) , we have, for .w ∈ C, 

T

.

0

f (gt w) ∼ = fT (w).

(10.8.1)

Given .ρ ∈ D(C), we integrate (10.8.1) with respect to .ρ, and normalize by T , to obtain .

1 T

 0

T

1 1 ρ(H (T )). f(gt ρ)dt ∼ = f T (ρ) = T T

(10.8.2)

Now, if .P is a tempered ergodic point process, we can apply Theorem 10.6.1 (that is, the Birkhoff ergodic theorem) and the Siegel–Veech formula (10.2.1) to obtain, as desired, that for .P-almost every .ρ ∈ D(C), .

1 1 ρ(H (T )) = lim T →∞ T T →∞ T



T

lim

0

f(gt ρ)dt =

 D (C)

fdP = aP m(H ) = aP . (10.8.3)

10.8.2 Counting in Hyperbolas and Diophantine Approximation This is a simplified, continuous version of an argument from Athreya–Parrish– Tseng [10], who studied this problem for lattices and translation surface holonomies, motivated by counting solutions to Dirichlet’s theorem.

10.8.3 Probabilistic Diophantine Approximation We mention that there is a significant connection between probabilistic Diophantine approximation and this kind of ergodic theory. We mention, as an example, the resolutions and generalizations of problems in probabilistic Diophantine approximation

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posed by Erdos–Szusz–Turan [17] and Kesten [25] by Marklof [29] and Athreya– Ghosh [7] using equidistribution on appropriate moduli spaces.

10.8.4 Central Limit Theorems The distribution of the error terms EH (ρ, T ) = ρ(H (T )) − aP T

.

is once again a very interesting question. In the setting of lattices, BjorklundGorodnik [14] have Central Limit Theorems, using the (exponential) mixing of the diagonal flow .gt on the space of lattices .Xd . It would be extremely interesting to study this question for the space of translation surfaces.

10.9 Triangles and Horocycles 10.9.1 Slopes and Horocycles We now turn to the ideas behind the proof of Theorem 10.4.3. Recall from (10.4.4) that we define .

Ts = {w ∈ C : 0 < Im(w) < s, Re(w) < s}

to be the triangle with vertices at .0, 1, and .1 + is, consisting of complex numbers with real part at most 1 and slope at most s. Put .T = T1 . Note that for .w = x + iy ∈ C, .u ∈ R, hu w = x + (y − ux)i,

.

so we have, {u ∈ R : hu w ∈ T } =

.

y x

− 1,

Putting .f = χT and .fu = χTu , we have, for .w ∈ C,  s .fs (w) ∼ f (hu w)du = 0

y . x

(10.9.1)

(10.9.2)

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and integrating this with respect to .ρ ∈ D(C), and dividing by s we have .

1 1 1 ρ(Ts ) = fs (ρ) ∼ = s s s



s

f(hu ρ)du.

(10.9.3)

0

We are now in a position to apply Theorem 10.6.1, to conclude that, if .P is a tempered ergodic point process, we have that for .P-almost every .ρ, 1 1 . lim ρ(Ts ) = lim s→∞ s s→∞ s



s

f(hu ρ)du =

0

 D (C)

fdP

(10.9.4)

Finally, applying the Siegel–Veech formula 10.2.1, we have  .

1 fdP = aP m(T ) = aP , 2 D (C)

as desired.

10.9.2 BCZ-Type Maps We mention that similar ideas to this counting argument have been used to construct cross-sections to the horocycle flow in a variety of settings, starting with the work of Athreya–Cheung [6]. We mention, among others, work of Athreya–Chaika– Lelievre [9], Uyanik–Work [44], Taha [41], Heersink [24], Kumanduri–Sanchez– Wang [27]. In all of these settings there are corresponding counting problems to which Theorem 10.4.3 applies.

10.10 Further Questions/Directions We outline a series of further questions (in addition to those raised above) that this axiomatic approach suggests.

10.10.1 Shrinking Sectors As discussed in Sect. 10.7.3 above, when counting in circles, we can consider the question of counting in shrinking angular sectors. A general question is as follows:

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Question 1 Given a tempered invariant point process .P, and a function .σ : R+ → R+ such that .σ (R) → 0 as .R → ∞ , understand the almost sure behavior of ρ({z = reiθ ∈ C : r ≤ R, θ ∈ (−s(R), s(R)}).

.

Under what conditions on .P and .σ is this asymptotic to .2π R 2 aP s(R)?

10.10.2 Siegel Measures on Rn As mentioned in the introduction, we can study, more generally, as suggested by Veech [46], .SL(n, R)-invariant point processes on .Rn . Natural examples include Poisson processes on .Rn , and measures pushed forward from the natural Haar probability measure on the space of unimdular lattices .Xn = SL(n, R)/SL(n, Z). Many of the results we describe in this survey have analogs coming from dynamics on homogeneous spaces, motivated by Diophantine approximation.

10.10.3 General G-Invariant Point Processes More generally, we can consider, for a group G acting on a space X, the idea of invariant point processes on X. For concrete examples, it can be very interesting to ask whether such point processes can be built. A question motivated by surface geometry is as follows: let .Σg be a compact genus g surface, and .Mod(g) its mapping class group, and .MF the space of measured foliations. The concept of measured folations was introduced by Thurston [43], see also Fathi-LaudenbachPoenaru [42] and its excellent English translation [23] for more detailed exposition. The mapping class group .Mod(g) acts on .MF, leading to the natural question: Question 2 Can we define a natural .Mod(g)-invariant point process on .MF? Comparison to Simple Closed Curves Since the set of simple closed curves can be thought of as a kind of set of lattice points in .MF (see, for example, the work of Mirazakhani [33] on counting curves for this perspective), it would be interesting, if the answer to the question is yes, to compare the statistics of simple closed curves with the statistics of such a point process.

10.10.4 Classification Questions We have seen a collection of invariant point processes drawn from probability, the geometry of numbers, and the geometry of surfaces. A kind of meta-question is, then: Question 3 Can one, in any reasonable sense, classify (tempered) invariant point processes?

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10.10.5 Representation Theory Finally, we turn to some extremely speculative questions where the subject of invariant point processes connects to representation theory. For concreteness, we stick to the case of .SL(2, R) acting on .D(C), though the question we pose would make sense for general G-invariant point processes on a space X. A probability measure-preserving action of .SL(2, R) on a measure space .(X, μ) yields a representation of .SL(2, R) on .L20 (X, μ) (square integrable functions orthogonal to constants), and understanding the decomposition of this representation into irreducible representations can yield crucial dynamical insights (see, for example, Ratner [37] for general results, or Avila–Gouezel [13] for insights in the translation surface setting). This motivates the following: Question 4 If .P is an invariant point process, when can we obtain meaningful information about the decomposition of the representation of .SL(2, R) on .L20 (D(C), P) into irreducible representations? Acknowledgments The author was partially supported by National Science Foundation Division of Mathematical Sciences Grant 2003528; the Pacific Institute for the Mathematical Sciences; the Royalty Research Fund and the Victor Klee fund at the University of Washington. This work was concluded during his term from July–December 2023 as the Chaire Jean Morlet at the Centre International de Recontres Mathématiques (CIRM) Luminy. The author thanks Anish Ghosh, Jens Marklof, and Mahan Mj for useful discussions and the Tata Institute of Fundamental Research (TIFR) for its hospitality in January 2023 when this work started to emerge.

References 1. A. Artiles, The Minimal Denominator Function and Geometric Generalizations (2023). Available at 2308.08076 2. J. S. Athreya, Cusp excursions on parameter spaces. J. Lond. Math. Soc. (2) 87(3), 741–765 (2013). https://doi.org/10.1112/jlms/jds074. MR3073674 3. J.S. Athreya, Random affine lattices. Geometry, Groups and Dynamics. Contemporary Mathematics, vol. 639 (American Mathematical Society, Providence, 2015), pp. 169–174. https:// doi.org/10.1090/conm/639/12793. MR3379825 4. J. S. Athreya, Gap Distributions and Homogeneous Dynamics, Geometry, Topology, and Dynamics in Negative Curvature. London Mathematical Society Lecture Note Series, vol. 425 (Cambridge University Press, Cambridge, 2016), pp. 1–31. MR3497256 5. J.S. Athreya, J. Chaika, The distribution of gaps for saddle connection direc- tions. Geom. Funct. Anal. 22(6), 1491–1516 (2012). https://doi.org/10.1007/s00039-012-0164-9. MR3000496 6. J.S. Athreya, Y. Cheung, A Poincaré section for the horocycle flow on the space of lattices. Int. Math. Res. Not. IMRN 10, 2643–2690 (2014). https://doi.org/10.1093/imrn/rnt003. MR3214280 7. J.S. Athreya, A. Ghosh, The Erd˝os–Szüsz–Turán distribution for equivariant processes. Enseign. Math. 64(1–2), 1–21 (2018). https://doi.org/10.4171/lem/64-1/2-1. MR3995711 8. J.S. Athreya, H. Masur, Translation Surfaces (American Mathematical Society, Providence, 2023)

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Chapter 11

Orbifolds and the Modular Curve Juan Martín Pérez and Florent Schaffhauser

Abstract We provide an account of the construction of the moduli stack of elliptic curves as an analytic orbifold. While intimately linked to Thurston’s point of view on the subject (discrete groups acting properly and effectively on differentiable manifolds), the construction of the modular orbi-curve and its universal family of elliptic curves ends up requiring a bit more technology, in order to allow for noneffective actions. Keywords Stacks and moduli problems · Complex-analytic moduli problems 2020 Mathematics Subject Classification Primary 14D23, Secondary 32G13

11.1 Introduction Orbifolds were first introduced by Satake, under the name V -manifolds [11], as a way to handle the ramification phenomena that occur in the general theory of group actions on manifolds. And it is generally accepted in the community that Satake chose that name in reference to the German word Verzweigung, which means ramification.1 Now, ramification is a concept that is also present in the analytic or algebraic contexts, in particular to talk about morphisms that are not étale. And it is in fact remarkable that some of the concepts introduced by Satake and Thurston

1 The second-named author believes he heard this explanation of the term in a talk by Professor Kaoru Ono, who asked Satake himself directly about it.

J. M. Pérez Freie Universität Berlin, Berlin, Germany e-mail: [email protected] F. Schaffhauser () Mathematisches Institut, Universität Heidelberg, Heidelberg, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 K. Ohshika, A. Papadopoulos (eds.), In the Tradition of Thurston III, https://doi.org/10.1007/978-3-031-43502-7_11

365

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for differential-geometric purposes, namely orbifolds and their fundamental groups [12, 16], have natural algebro-geometric counterparts that can in turn be used to enrich our perspective and the scope of our methods in differential and analytic geometry. One of the goals of the present chapter is to illustrate this point of view through the example of the moduli space of elliptic curves. It is well-known that elliptic curves can be classified by their j -invariant and this is the basis for the construction of the so-called modular curve, whose points parameterise isomorphism classes of elliptic curves [9]. However, there are various reasons why we cannot quite content ourselves with the classical construction of this space. The main one, perhaps, being that the existence of a universal family of elliptic curves can only be obtained in the context of stacks, not manifolds. In this chapter, we will present a construction of the moduli stack of elliptic curves and show that it is in fact an analytic orbifold. This illustrates the need to go beyond the original definition of an orbifold because, in Thurston’s definition of the quotient orbifold .[X/G], the group G is supposed to act effectively on the manifold X, but to prove that the moduli stack of elliptic curves is analytic, one ends up describing it as the quotient stack .[h/SL(2; Z)], where .h is the upper half-plane in .C and .SL(2; Z) acts on it by homographic transformations. This action is not effective, and in fact its kernel is precisely .{±I2 }, so .PSL(2; Z) acts effectively on .h. While .h/SL(2; Z) and .h/PSL(2; Z) are homeomorphic topological spaces, the quotient stacks .[h/SL(2; Z)] and .[h/PSL(2; Z)] are not isomorphic, and only the first one is isomorphic to the moduli stack of elliptic curves, so the fact that .h/PSL(2; Z) is an orbifold in the sense of Thurston is not entirely satisfying here, even though it does get us the correct coarse moduli space. Resolving this issue by showing that .h/SL(2; Z) carries a canonical structure of non-effective orbifold is not quite good enough either, because of the difficulties that arise with the universal family. So the only way out of this seems to be considering the quotient stack .[h/SL(2; Z)] and showing that it is an orbifold in the stacky sense. Of course, there are deeper implications to this discussion, for instance the fact that the fundamental group of the moduli space of elliptic curves should be .SL(2; Z), not .PSL(2; Z), because elliptic curves always have an Abelian group structure, so in particular an inversion map .x I−→ −x, which is the map coming from the element .−I2 ∈ SL(2; Z) acting trivially on .h. The example of the moduli space of elliptic curves illustrates how thinking about non-effective orbifolds as a particular class of stacks may be useful, even for classical problems. This suggests the need to go beyond Satake and Thurston’s original definitions, using the input from algebraic geometry to enrich the differentialgeometric perspective on the notion of orbifold. Note that this is not the only occurrence of such a phenomenon. While no evidence suggests that the theory of the étale fundamental group of schemes may have been of inspiration to Thurston (except perhaps his construction of the universal cover as an inverse limit in [16, Proposition 13.2.4, pp. 305–307]), the analogy is clear, and in fact illustrates a common trend between the two fields, which is to include more maps in the definition of a covering space in order to have a richer notion of fundamental group. Indeed, consider for instance the orthogonal projection from the unit sphere in .R3

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to the closed unit disk in .R2 × {0} ⊂ R3 . In the usual topology of .R3 , this map is not a covering map (in fact, it is not even locally injective at points of the unit circle). But in algebraic geometry, it is the map induced on the set of closed points by the canonical projection .P1C → P1R = P1C /Gal(C/R), which is an étale morphism of schemes (in particular, the étale fundamental group of .P1R is not trivial, and since 1 is simply connected, it is in fact exactly .Gal(C/R)). If we view this example .P C through the orbifold prism, the orbifold fundamental group, in the sense of Thurston, of the quotient of .S 2 ⊂ R3 by the group generated by the reflection through the equator is indeed .Z/2Z, just like in algebraic geometry. More generally, this leads to a richer notion of fundamental groups for all Klein surfaces (= quotients of Riemann surfaces by anti-holomorphic involutions), for which we refer for instance to [13]. In analytic and algebraic geometry too, it is useful to allow orbifolds: the fundamental group of the quotient of .C by the finite group .μn (C) ≃ Z/nZ acting by multiplication by a primitive n-th root of unity should be .μn (C), not the trivial group. As a matter of fact, the canonical projection .C −→ C/μn (C) is the typical case of a map which is ramified when seen as a morphism between smooth algebraic or analytic varieties (in which case it can be identified with the map .z I−→ zn ), but which can be viewed as a covering map in the orbifold sense (see Example 11.4.9). Recall that a complex-analytic elliptic curve is a pair .(C , e) where .C is a compact Riemann surface of genus 1 and e is a point in .C . A morphism between two elliptic curves .(C1 , e1 ) and .(C2 , e2 ) is a holomorphic map .f : C1 −→ C2 such that .f (e1 ) = e2 . We shall denote by .f : (C1 , e1 ) −→ (C2 , e2 ) such a morphism. The main result that we wish to discuss in this chapter is stated as follows. Theorem (Theorem 11.6.23) The moduli stack of complex elliptic curves is a complex analytic stack, isomorphic to the orbifold .[h/SL(2, Z)], where .h := {z ∈ C | Im (z) > 0} is the hyperbolic plane and  SL(2, Z) :=

.

a c bd



 : a, b, c, d ∈ Z, ad − bc = 1

is the so-called modular group, acting to the right on .h by homographic transformations   az + b a c = .z · . bd cz + d This moduli stack admits the (non-compact) Riemann surface .h/SL(2, Z) ≃ C as a coarse moduli space. In Sect. 11.2, we recall basic results on elliptic curves. Sections 11.3 and 11.4 contain the formal definition of a stack and an explanation of what it means for a stack defined on the category of complex analytic manifolds to be analytic, and to be an orbifold. In Sects. 11.5 and 11.6, we discuss the notion of moduli spaces for a

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complex analytic stack and we prove that the stack of elliptic curves is an analytic orbifold. This chapter is entirely expository and makes no claims to originality: its main goal is to be self-contained enough in order to be useful to young researchers who are entering the field and are interested in the interactions between differential and algebraic geometry, all the while celebrating the vitality and scope of Thurston’s geometric vision.

11.2 Complex Elliptic Curves 11.2.1 Complex Tori Let us start by recalling the definition of a complex elliptic curve (in the analytic setting). We recall that the (geometric) genus of a compact connected Riemann surface .C is the dimension of the complex vector space .H 0 (C ; Ω1C ) which, as .C is 1-dimensional and compact, is the space of holomorphic 1-forms on .C . Definition 11.2.1 An analytic complex elliptic curve is a pair .(C , e) where .C is a compact Riemann surface of genus 1 and e is a point in .C . A morphism of elliptic curves between .(C , e) and .(C ' , e' ) is a holomorphic map .f : C −→ C ' such that ' .f (e) = e . We now give the two sources of examples of complex elliptic curves that will be of interest to us: complex tori (Example 11.2.2) and smooth plane cubics (Example 11.2.3). Example 11.2.2 Let us see .C ≃ R2 as a two-dimensional real vector space (equipped with its usual topology and additive group structure) and let .Λ be a discrete subgroup of .C such that the quotient group .C/Λ is compact (such a subgroup .Λ ⊂ C will be called a (uniform) lattice). In particular, the group .Λ is a free .Z-module of rank 2. Note that there is a canonical marked point in .C/Λ in this case, namely .0 mod Λ, the neutral element for the group law. Moreover, there is a (unique) Riemann surface structure on .C/Λ that makes the canonical projection .p : C −→ C/Λ holomorphic. A holomorphic 1-form on .C/Λ is necessarily closed, so it pulls back to an exact global 1-form .f (z) dz on the simply connected Riemann surface .C, with .f (z) holomorphic and .Λ-periodic (in the sense that, if .z ∈ C and .λ ∈ Λ, then .f (z + λ) = f (z); this holds because dz is .Λ-invariant for the action of .Λ on .C by translation). Since .C/Λ is compact, the holomorphic function f has to be bounded, therefore constant by Liouville’s theorem, and this proves that the space of holomorphic .1−forms on .C/Λ has complex dimension 1, hence that we have constructed a complex elliptic curve .(C/Λ, 0 mod Λ), which we will refer to as a (1-dimensional) complex torus.

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The complex tori of Example 11.2.2 can be embedded onto non-singular complex projective sets using the Weierstrass .℘-function of the lattice .Λ, namely ℘Λ (w) =

.

   1 1 1 + − w2 (w − λ)2 λ2 λ∈Λ\{0}

which, if we pick a basis .(λ, μ) of .Λ as a .Z-module, satisfies the equation   ' (℘Λ (w))2 = f(λ,μ) ℘Λ (w) ,

(11.1)

.

where f(λ,μ) (x) = 4(x − v1 )(x − v2 )(x − v3 )

.

with v1 = ℘Λ

.

λ

∈ C, v2 = ℘Λ

2

μ 2

∈ C and v3 = ℘Λ



λ+μ 2



∈ C,

thus inducing a holomorphic map .

C/Λ −→ CP2  ' (w) : 1 w mod Λ I−→ ℘Λ (w) : ℘Λ

(11.2)

whose image, because of Eq. (11.1) and up to normalisation of the basis .(λ, μ), is a smooth projective curve of the type described in Eq. (11.3). Example 11.2.3 Let .u ∈ C \ {0, 1} and consider the smooth projective curve

 Cu := [x : y : z] ∈ CP2 | y 2 z = x(x − z)(x − uz) .

.

(11.3)

By the genus-degree formula, the smooth cubic .Cu has genus .g = (3−1)(3−2) = 1. 2 And since the point .e := [0 : 0 : 1] belongs to .Cu for all .u ∈ C \ {0, 1}, we have defined a family of elliptic curves .(Cu , e)u∈C\{0,1} , depending on the parameter u. This family is called the Legendre family (see also Example 11.6.2). Building on Example 11.2.3, let us set .fu (x) = x(x − 1)(x − u). The classical theory of elliptic curves says that .Cu is a compactification of the plane cubic of equation .y 2 = fu (x). Moreover, the map .(x, y) I−→ x induces a ramified double covering .p : Cu −→ CP1 , with four branch points, so an application of the Riemann-Hurwitz formula shows that .Cu is homeomorphic to .S 1 × S 1 . If we pick a square root of .fu , the differential form . √fdt(t) on .CP1 pulls back to a holomorphic u

1-form .ωu := p ∗ √fdt(t) that can be extended holomorphically to .Cu . Then we can u integrate this 1-form along loops based at .e ∈ Cu and, since .ωu is closed, obtain a

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group morphism Фωu :

.

π1 (Cu , e) −→ C [γ ] I−→ γ ωu

(11.4)

whose image .Λωu := Im (Фωu ) is in fact a lattice in .C, called the period lattice. This induces a holomorphic map .

Cu −→ C/Λωu x x I−→ e ωu mod Λωu

(11.5)

which is an isomorphism of elliptic curves .(Cu , e) ≃ (C/Λωu , 0 mod Λωu ). Note that if we choose a different non-zero holomorphic 1-form .ω on .Cu , it will satisfy ∗ .ω = αωu for some .α ∈ C , because the space of such forms is 1-dimensional. Consequently, the lattices .Λωu and .Λω satisfy .Λω = αΛωu , so multiplication by .α in .C induces an isomorphism of elliptic curves .(C/Λωu , 0 mod Λωu ) ≃ (C/Λω , 0 mod Λω ). So, from the existence of the projective embedding (11.2) and the isomorphism (11.5), we see that there is a bijection   1-dimensional complex tori isomorphism   ≃ smooth plane cubics isomorphism .

.

And as a matter of fact, every elliptic curve .(C , e) in the sense of Definition 11.2.1 is of this form, meaning that it is isomorphic to a one-dimensional complex torus. One way to show this is to generalise the construction of the period subgroup (11.4) to the abstract setting: choose a generator .ω of the 1-dimensional complex vector space .H 0 (C ; Ω1C ) and show that the image of the group morphism .Фω is a lattice .Λω ⊂ C and that there is an isomorphism of elliptic curves .(C , e) ≃ (C/Λω , 0 mod Λω ). The issue here is that, with our definition of the genus, we do not yet know that .C is homeomorphic to .S 1 × S 1 , which makes it harder to prove that the period subgroup is a lattice in .C. The way to solve this difficulty is to use Abel’s theorem, which gives a necessary and sufficient condition for a Weil divisor D on a compact Riemann surface .C to be a principal divisor, i.e. for D to be the divisor of zeros and poles of a meromorphic function on .C . More precisely, one can follow the approach in [4, pp. 163–173] to show the following: (1) A Weil divisor .D ∈ Div(C ) is the divisor of zeros and poles of a meromorphic function f on .C if and only if D is of degree 0 and there exists a chain .c ∈ C1 (C ; Z) such that .∂c = D and . c ω = 0 (Abel’s theorem in the .g = 1 case, [4, Theorem 20.7, p. 163]). (2) The period subgroup .Λω := Im Фω ⊂ C is a lattice [4, Theorem 21.4, p. 168]. This uses the previously stated version of Abel’s theorem, to prove that .Λω is a discrete subgroup of .C.

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One gets in this way a 1-dimensional complex torus .Jac(C ) := C/Λω , namely the Jacobian variety of the elliptic curve .(C , e). It depends on .ω, but if we choose a different basis .αω of the complex vector space .H 0 (C ; Ω1C ), we obtain an isomorphic complex torus, as we have already seen. Moreover, the Riemann-Roch theorem has the following consequence. Lemma 11.2.4 Let .(C , e) be an elliptic curve and let .Pic0 (C ) := Div0 (C )/ DivP (C ) be the group of degree 0 divisors modulo principal divisors. Then the map Ψ:

.

C −→ Pic0 (C ) x I−→ [x] − [e]

is bijective. Proof The map .Ψ is injective because, if .Ψ(x) = Ψ(x ' ) with .x /= x ' , then the Weil divisor .[x] − [x ' ] would be principal, so there would exist a meromorphic function f on .C with exactly one pole, of order 1, so the corresponding holomorphic map .f : C −→ CP1 would be an isomorphism, contradicting the fact that 1 0 .dimC H (C ; Ω ) = 1. And the map .Ψ is surjective because every divisor D of C degree 0 on .C is equivalent, modulo a principal divisor, to a divisor of the form ' ' .[x] − [e]. Indeed, if we set .D = D + [e], then .deg(D ) = deg(D) + 1 = 1 so, 0 ' by the Riemann-Roch theorem, .dim H (C ; OC (D )) ⩾ deg(D ' ) = 1, meaning that there exists a non-zero meromorphic function f on .C whose poles are bounded by ' ' ' .D , i.e. in terms of divisors, .(f ) + D ⩾ 0. Since .(f ) + D is an effective Weil divisor of degree 1, there exists a point x in .C such that .(f ) + D ' = [x], hence ' .D = D − [e] ∼(f ) [x] − [e]. ⨆ ⨅ Remark 11.2.5 Lemma 11.2.4 shows in particular that an elliptic curve .(C , e) has ≃ a natural structure of Abelian group, induced by the isomorphism .Ψ : (C , e) −→ (Pic0 (C ), 0) and the group structure on .Pic0 (C ). In particular, an elliptic curve always has non-trivial automorphisms (namely the inversion map .D I−→ −D, seen as a morphism of marked Riemann surfaces). Finally, one can deduce from all the above that a complex elliptic curve .(C , e) is isomorphic to a complex torus, namely the Jacobian variety .Jac(C ) := C/Λω , defined by the period lattice. This proves in particular that .π1 (C , e) ≃ Λω ≃ Z2 . So we see from the classification of compact connected orientable surfaces that .C is homeomorphic to .S 1 × S 1 . Theorem 11.2.6 Let .(C , e) be a complex elliptic curve and let .ω be a non-zero holomorphic 1-form on .C . Let .Λω ⊂ C be the period lattice and let .Jac(C ) :=

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C/Λω be the associated complex torus, i.e. the Jacobian variety of .(C , e). Then the map F :

.

C −→ Jac(C ) = C/Λω x x I−→ e ω mod Λω

is an isomorphism of elliptic curves. We refer to [4, Theorems 21.7 p. 171 and 21.10 p. 172] for a complete proof of Theorem 11.2.6. Note that the map .F : C −→ Jac(C ) is the composition of the map .Ψ : C −→ Pic0 (C ) of Lemma 11.2.4 with the map J :

.

Pic0 (C ) −→ Jac(C )  D = ∂c I−→ c ω mod Λω

which is well-defined by definition of the period lattice (i.e. independent of the choice of the chain .c ∈ C1 (C ; Z) such that .∂c = D), and injective in view of Abel’s theorem. The surjectivity of J is proven in [4, Theorem 21.7 p. 171] and for the fact that the map .F : C −→ Jac(C ) is holomorphic, we refer to [4, Theorem 21.4 p. 168]. As an application, Theorem 11.2.6 gives us the structure of the automorphism group of a complex elliptic curve and of a compact Riemann surface of genus 1 without marked point. To obtain it, one just needs to observe that two complex tori .C/Λ1 and .C/Λ2 are isomorphic as complex elliptic curves if and only if there exists ∗ .α ∈ C such that .αΛ1 = Λ2 , which can be deduced by lifting such an isomorphism to the universal cover .C of these tori and using the classification of automorphisms of .C. In particular, the automorphism group of the elliptic curve .(C/Λ, 0) can be identified with the group of all .α ∈ C∗ such that .αΛ = Λ. As .Λ is a discrete subgroup of .C, the set of such .α is in fact a discrete subgroup of the circle group 1 .S , hence a finite rotation group, generated by a root of unity. If we do not ask to preserve the base point, then translations by an element .[z] ∈ C/Λ also induce automorphisms of .C/Λ. Corollary 11.2.7 Let .(C , e) be a complex elliptic curve. Then there exists a natural number .n ⩾ 1 such that the automorphism group of the marked Riemann surface .(C , e) is .

Aut(C , e) ≃ Z/nZ.

(11.6)

Moreover, the automorphism group of the Riemann surface .C is the semi-direct product .

Aut(C ) ≃ C ⋊ Aut(C , e)

where the group .C ≃ Pic0 (C ) acts on itself by translations and the structure of semi-direct product is given by the action of .Aut(C , e) on .C . In particular, the

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group .Aut(C ) is a 1-dimensional compact analytic Lie group over .C, whose identity component is isomorphic to .C . Remark 11.2.8 We will see in Corollary 11.2.18 that, in fact, the natural number n in Equation (11.6) can only be 2, 4 or 6 (never 1!). In particular, we will give an alternate, more detailed proof of Corollary 11.2.7. So the conclusion we can draw from the present paragraph is two-folded: firstly, the classification of complex elliptic curves up to isomorphism can be reduced to the classification of complex tori, and secondly, such objects have non-trivial automorphisms as marked Riemann surfaces. As a consequence, even though we can always find, given a complex elliptic curve .(C , e), an isomorphism of elliptic curves with a complex torus .C/Λ, this isomorphism will never be unique, because we can compose it with an automorphism of .C/Λ. This is one of the reasons why constructing a moduli space of complex elliptic curves requires additional notions, namely framings, in order to get rid of non-trivial automorphisms.

11.2.2 Framed Elliptic Curves When faced with a moduli problem for objects that have non-trivial automorphisms, an idea that has proven useful is to add an extra structure to these objects. This is because the corresponding notion of isomorphism will then be more restrictive (since the extra structure needs to be preserved, too), so in particular there will be fewer automorphisms. In the best case scenario, there is a group action on these objects with extra structure, and two such objects with extra structure lie in the same orbit if and only if they are isomorphic as objects without extra structure, so the moduli space of objects without extra structure can be constructed as a quotient of the moduli space of objects with extra structure (provided the latter moduli space actually exists, of course). We will now illustrate this more concretely in the case of elliptic curves. We begin with the definition of an extra structure. Definition 11.2.9 Let .(C , e) be a complex elliptic curve. A framing of .(C , e) is a basis .([a], [b]) of the free .Z-module .H1 (C , Z) ≃ π1 (C , e) ≃ Z2 , with the additional property that    ω for all non-zero ω ∈ H 0 C ; Ω1C , Im a ω > 0 .

.

(11.7)

b

Since .H 0 (C ; Ω1C ) is one-dimensional by assumption, it suffices to test Condition (11.7) on a single non-zero holomorphic 1-form .ω. A morphism of framed elliptic curves     f : C , e, ([a], [b]) −→ C ' , e' , ([a ' ], [b' ])

.

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is a morphism of elliptic curves .f : (C , e) −→ (C ' , e' ) such that .f∗ ([a]) = [a ' ] and .f∗ [b] = [b' ], where .f∗ : H1 (C ; Z) −→ H1 (C ' ; Z) is the group morphism induced by f in homology. Example 11.2.10 If .(C , e) = (C/Λ, 0) for some lattice .Λ ⊂ C and .(λ, μ) is a basis of the .Z-module .Λ that satisfies the additional positivity condition .Im μλ > 0, then .(λ, μ) defines a framing of .(C/Λ, 0) by letting .λˆ and .μˆ be the 1-cycles in .C/Λ corresponding to the paths .t I−→ tλ and .t I−→ tμ in .C. A lattice .Λ ⊂ C equipped with such a direct basis .(λ, μ) is also called a framed lattice, and we shall sometimes identify a framing of .Λ in this sense with the induced framing of .C/Λ. So, when we write .(C/Λ, 0, (λ, μ)), we mean the framed elliptic curve ˆ μ)). .(C/Λ, 0, (λ, ˆ Note that if a non-zero holomorphic  1-form .ω on.C is fixed, then we can define .(λ, μ) from .([a], [b]) by setting .λ := a ω and .μ := b ω, but without ˆ μ). ˆ Regardless, if .Λ' = αΛ for some such an .ω, we cannot recover .(λ, μ) from .(λ, ∗ .α ∈ C , then the map .z I−→ αz induces an isomorphism of framed elliptic curves  .

   C/Λ, 0, (λ, μ) ≃ C/(αΛ), 0, (αλ, αμ) .

In particular, if we set .τ := μλ , then .τ is a complex number with positive imaginary part, i.e. .Im (τ ) > 0, and there is an associated lattice .Λ(τ ) := Z1 ⊕ Zτ ⊂ C with direct basis .(1, τ ) such that the framed elliptic curve .(C/Λ, 0, (λ, μ)) is isomorphic to the framed elliptic curve .(C/Λ(τ ), 0, (1, τ )). In view of Theorem 11.2.6 and Example 11.2.10, we see that a framed complex elliptic curve .(C , e, ([a], [b])) is isomorphic to a framed complex torus of the form .(C/Λ(τ ), 0, (1, τ )) where τ ∈ h := {z ∈ C | Im (z) > 0}.

.

Indeed, it suffices to show that a complex elliptic curve admits a framing, which comes from the simple observation that if a basis .([a], [b]) for .H1 (C ; Z) does not satisfy the positivity condition (11.7), then the basis .([b], [a]) does. Note that, a priori, given a framed elliptic curve .(C , e, ([a], [b])), there could be more than one .τ ∈ h such that .(C , e, ([a], [b])) ≃ (C/Λ(τ ), 0, (1, τ )). But Lemma 11.2.11 shows that this is in fact not the case. In particular, a framed elliptic curve of the form .(C/Λ(τ ), 0, (1, τ )) has no non-trivial automorphisms. Lemma 11.2.11 The framed complex tori .(C/Λ(τ ), 0, (1, τ )) and .(C/Λ(τ ' ), 0, (1, τ ' )) are isomorphic if and only if .τ = τ ' . Proof An isomorphism between such complex tori is induced by .z I−→ αz for some .α ∈ C∗ , which respects the framing if and only if .α = 1 and .τ ' = τ . ⨅ ⨆ Remark 11.2.12 It is not entirely apparent from the proof, but the positivity condition .Im (τ ) > 0 is necessary if we want a uniqueness result such as the one in Lemma 11.2.11 to hold. Indeed, if we had defined a framing of .Λ simply as a basis of the .Z-module .Λ, then, even if we normalise the lattice so that the first vector in

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the basis be equal to 1, two normalised framings of the complex torus  .(C/Λ(τ ), 0) do not have to be equal: of the two bases .(1, τ ) and .(τ, 1) ∼ 1, τ1 of the lattice .Λ(τ ) = Z1 ⊕ Zτ = Zτ ⊕ Z1, only the first one satisfies the required positivity condition. We then have the following classification result for framed elliptic curves, whose proof follows directly from the discussion above and Lemma 11.2.11. Theorem 11.2.13 Let .(C , e, ([a], [b])) be a framed elliptic curve. Then there exists a unique   τ ∈ h = z ∈ C | Im (τ ) > 0

.

and a unique isomorphism of framed elliptic curves     C , e, ([a], [b]) ≃ C/Λ(τ ), 0, (1, τ ) .

.

As a consequence, framed elliptic curves have no non-trivial automorphisms: .

Aut(C , e, ([a], [b])) = {idC }.

In particular, the points of the Riemann surface h := {z ∈ C | Im (z) > 0}

.

represent isomorphism classes of framed elliptic curves. This parameter space of framed elliptic curves is called the Teichmüller space of elliptic curves, for reasons that will be explained next. Note that Theorem 11.2.13 constitutes an improvement on Theorem 11.2.6 since we can now compare two given elliptic curves in a more formal sense because (at least when they are framed) these elliptic curves are represented by points of a same space, namely the upper half-plane .h. The next step is to ask ourselves: when are two framed elliptic curves isomorphic as plain elliptic curves? This corresponds to forgetting the framing, so answering this question should also give us a sense of what the possible framings are on a given elliptic curve. More precisely, we will construct the set of isomorphism classes of elliptic curves as a quotient of the Teichmüller space .h, so it will come equipped with a map from .h to it, and the fibres of that map will be in bijection with the possible framings (the map in question is the one that forgets the frame on a given framed elliptic curve). Theorem 11.2.14 Let .(C , e, ([a], [b])) and .(C ' , e' , ([a ' ], [b' ])) be framed elliptic curves and let .τ, τ ' ∈ h be the corresponding points in the Teichmüller space of elliptic curves. Then the unframed elliptic curves .(C , e) and .(C ' , e' ) are isomorphic

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as elliptic curves if and only if there exists a matrix A=

.

  aτ + b a c ∈ SL(2; Z) such that τ ' = , bd cτ + d

meaning that .τ and .τ ' are related by a direct Möbius transformation with integral coefficients. Proof In view of Theorem 11.2.13, the framed elliptic curves .(C , e, ([a], [b])) and (C ' , e' , ([a ' ], [b' ])) can be replaced by the framed elliptic curves .(C/Λ(τ ), 0, (1, τ )) and .(C/Λ(τ ' ), 0, (1, τ ' )) and the latter are isomorphic as unframed elliptic curves if and only if there exists .α ∈ C∗ such that .αΛ(τ ) = Λ(τ ' ) = Z1 ⊕ Zτ ' . So there exist .a, b, c, d ∈ Z such that .τ ' = α(aτ + b) and .1 = α(cτ + d). This implies that

.

α=

.

1 cτ + d

τ' =

and

Applying the same argument to .Λ(τ ) =

aτ + b . cτ + d

1 ' α Λ(τ ), 

we see that the (Möbius)  a c is invertible, therefore transformation .z I−→ az+b is bijective, so the matrix . cz+d bd has determinant .±1 ∈ Z. And since it sends .τ such that .Im (τ ) > 0 to .τ ' such that ' .Im (τ ) > 0, it must have determinant .+1. ⨆ ⨅ So the framed elliptic curves .(C , e, ([a], [b])) and .(C ' , e' , ([a ' ], [b' ])) become isomorphic when forgetting the frame if and only if the points .τ, τ ' ∈ h lie in a same orbit of the .SL(2; Z)-action on .h defined by  z·

.

a c bd

 =

az + b . cz + d

(11.8)

Note that this is a right action. Thanks to this, we can construct a set whose points are in bijection with isomorphic classes of elliptic curves. Corollary 11.2.15 The quotient set .h/SL(2; Z) is the set of isomorphism classes of elliptic curves and we have a forgetful map F : h −→ h/SL(2; Z)

.

sending the isomorphism class of the framed elliptic curve .(C/Λ(τ ), 0, (1, τ )) to the isomorphism class of the elliptic curve .(C/Λ(τ ), 0). In particular, the orbit of a point .τ ∈ h under the .SL(2; Z)-action (11.8) is in bijection with the set of all framings on the elliptic curve .(C/Λ(τ ), 0), and the stabiliser group of .τ in .SL(2; Z) is isomorphic to the automorphism group of the elliptic curve .(C/Λ(τ ), 0).

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11.2.3 The Modular Curve The modular curve is the quotient .h/SL(2; Z), of the Teichmüller space .h by the action of the modular group .SL(2; Z). Note that Corollary 11.2.15 tells us that this quotient set should be the underlying set of the moduli space of elliptic curves. In this section, we recall what the canonical structure of Riemann surface on .h/SL(2; Z) is. But first, the mere fact that it is a quotient set already gives us insight into the classification of complex elliptic curves: by studying the action of the modular group on the Teichmüller space (stabilisers and fundamental domain), we can determine what the possible automorphism groups of complex elliptic curves are, and what the quotient may be as a Riemann surface, which turns out to be intimately related to the structure of the modular group itself. Note for instance that the centre of the modular group is the subgroup .{±I2 } ≃ Z/2Z and that this subgroup acts trivially on the Teichmüller space. So every elliptic curve admits an order two automorphism. This is already apparent in Theorem 11.2.6, which endows every Riemann surface of genus 1 with a marked point with a (noncanonical) structure of Abelian group: the associated order-two automorphism is just the inversion map .D I−→ (−D) in this Abelian group. And if the elliptic curve is given by a smooth cubic equation .y 2 z = f (x, z) in the projective plane, then the involution is induced by the involution .[x : y : z] I−→ [x : (−y) : z] of .CP2 . This also shows that the action of .SL(2; Z) on .h is not faithful, as the centre of the modular group acts trivially on .h. Hence an induced action of .PSL(2; Z) := SL(2; Z)/{±I2 }, which turns out to be faithful. The important observation here is that the orbits of the .PSL(2; Z)-action are the same as those of the .SL(2; Z)action but the stabilisers are not the same. In particular, the .PSL(2; Z)-stabiliser of a point .τ ∈ h is not isomorphic to the automorphism group of the elliptic curve .(C/Λ(τ ), 0). The following result is classical (see for instance [14, Chapter VII.1 pp. 77–79] or [3] for an exposition). Theorem 11.2.16 The modular group .SL(2; Z) is generated by the matrices  S=

.

0 −1 1 0



 and

T =

11 01

 ,

 0 −1 hence also by the matrices S and .R := ST = . This induces the following 1 1 presentations of .SL(2; Z) and .PSL(2; Z) by generators and relations: 

   SL(2; Z) ≃ S, R  S 2 = R 3 , S 4 = R 6 = 1 ≃ Z/4Z ∗Z/2Z Z/6Z

.

and    PSL(2; Z) ≃ [S], [R]  [S]2 = [R]3 = 1 ≃ Z/2Z ∗ Z/3Z.

.

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Theorem 11.2.16 is about the algebraic structure of .SL(2; Z) and .PSL(2; Z). The matrices S and R show that this group is generated by an element of order four and an element of order six, which themselves satisfy the relation .S 2 = R 3 (equal to .−I2 ). So .SL(2; Z) is the amalgamated coproduct of the two Abelian groups .Z/4Z and .Z/6Z over .Z/2Z, with the non-trivial element of .Z/2Z being sent to .S 2 in 3 .〈S〉 ≃ Z/4Z and to .R in .〈R〉 ≃ Z/6Z. As a consequence, the group .PSL(2; Z) = SL(2; Z)/{±I2 } is generated by the classes .[S] and .[R], which are respectively of order 2 and 3 with no other relations between them, so .PSL(2; Z) is the coproduct of ∗ .Z/2Z and .Z/3Z. This implies for instance that a character .χ : SL(2; Z) −→ C has image contained in the subgroup generated by a fourth root of unity and a sixth root of unity, so .Im χ ⊂ μ12 (C). Geometrically, though, it is the action of the generators S and T that is the most convenient to visualise. Indeed, S acts on .h via the Möbius transformation .[S] : z I−→ − 1z , which is a direct rotation of angle . π2 about i, and T acts via .[T ] : z I−→ z + 1, which is a translation. As a consequence, .R = ST 1 acts via .[R] : z I−→ 1−z , which is a rotation of angle . π3 about .eiπ/3 . This leads to the usual construction of a fundamental domain for the action of .SL(2; Z) (or equivalently .PSL(2; Z), since the orbits are the same) on .h. Theorem 11.2.17 Let  F := τ ∈ h | |τ | ⩾ 1 and

.

1 2

⩽ Re (τ ) ⩽

1 2



.

Then the following properties hold: (1) Every .SL(2; Z)-orbit in .h intersects .F in at least one point and at most two points. (2) If an orbit intersects .F in two points, then these two points lie on the topological boundary .∂F of .F. (3) The points of .F with non-trivial stabiliser in .SL(2; Z) are the points .i = eiπ/2 , whose stabiliser is the cyclic group of order 4 generated by S, and the points i2π/3 and .−j 2 = − 1 = eiπ/3 = j · T , whose stabilisers are cyclic groups .j = e j of order 6, generated respectively by .T RT −1 = T S and R. Note that the point at infinity of .CP1 = C ∪ {∞} may be added to .h ⊂ C and that the action of .SL(2; Z) on .h by biholomorphisms extends to a conformal action on .hˆ := h ∪ {∞}. The stabiliser of the point at infinity is then the infinite cyclic group generated by T . Algebraically, the point at infinity no longer corresponds to an elliptic curve (smooth plane cubic with a marked point) but to a nodal plane cubic with a marked point (given in an affine chart by an equation of the form 2 2 .y = x (x − 1)). Corollary 11.2.18 Let .τ ∈ F and let .Λ(τ ) := Z1 ⊕ Zτ be the associated lattice in .C. Let .G := Aut(C/Λ(τ ), 0) be the automorphism group of the elliptic curve .(C/Λ(τ ), 0). Then the following properties hold: / i, j, −j 2 , then .G ≃ Z/2Z. (1) If .τ = (2) If .τ = i, then .G ≃ Z/4Z.

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(3) If .τ = j or .−j 2 , then .G ≃ Z/6Z. The shape of the fundamental domain .F = gives us an indication of what the quotient space .h/PSL(2; Z) might look like: a topological space homeomorphic to .C, and this identification can be used to put a Riemann surface structure on .h/PSL(2; Z). Thinking about .h/PSL(2; Z) in terms of .F also suggests that there are two special points, namely the orbits of i and j , which are the points with nontrivial stabiliser in .PSL(2; Z), and this makes .h/PSL(2; Z) an orbifold in the sense of Thurston. We will now show that .h/PSL(2; Z) is also the quotient of a Riemann surface by the action of a finite group (Theorem 11.2.20). Remark 11.2.19 We will recall in Corollary 11.4.15 why .SL(2; Z) acts properly on .h. By Theorem 11.2.16, the group .SL(2; Z) is a finitely generated subgroup of .GL(2; C). So, by Selberg’s Lemma, it contains a torsion-free normal subgroup of finite index, say .π ⊲ SL(2; Z). The discrete group .π , being a subgroup of .SL(2; Z), acts properly on .h. Since .π is torsion-free, it must therefore act freely on .h (see Theorems 11.4.11 and 11.4.12 for further details on that). So the canonical projection .h −→ h/π =: M˜ is an analytic covering map between Riemann ˜ 𝚪, surfaces, and the topological space .M := h/SL(2; Z) is homeomorphic to .M/ where .𝚪 := SL(2; Z)/π is a finite group. A priori, there could exist several pairs ˜ 𝚪) consisting of a Riemann surface .M˜ and a finite group .𝚪 acting on M by .(M, ˜ 𝚪 ≃ M. An explicit such presentation is given in automorphisms, such that .M/ Theorem 11.2.20. Theorem 11.2.20 The set .M := h/SL(2; Z) of isomorphism classes of elliptic ˜ 𝚪 where .M˜ = C \ {0; 1} and .𝚪 is the curves is in bijection with the set .M/ permutation group .S3 = 〈σ, τ 〉, generated by the permutations .σ = (1 2) and .τ = (1 3) and acting to the right on .u ∈ C \ {0; 1} via .fσ (u) := (1 − u) and 1 .fτ (u) := u. Proof As seen after Example 11.2.2, for all .τ ∈ h, the framed elliptic curve (C/Λ(τ ), 0, (1, τ )) can be embedded onto the smooth plane cubic of equation

.

y 2 = 4(x − v1 z)(x − v2 z)(x − v3 z)

.

in .CP2 , with pairwise distinct coefficients v1 = ℘Λ(τ )

.



1 2

, v2 = ℘Λ(τ )

τ  2

and v3 = ℘Λ(τ )



1+τ 2



∈ C.

2 Let us set .u := vv31 −v −v2 ∈ C \ {0; 1}. This corresponds to sending .(v1 , v2 , v3 ) to z−v2 1 .(1, 0, u) via the homographic transformation .z I−→ v1 −v2 , preserving .∞ ∈ CP . 1 So we may also view u as a point of .CP \ {0; 1; ∞}, with limit cases .u = 0, 1, ∞ corresponding respectively to .v3 = v2 , .v3 = v1 and .v1 = v2 . The point is that, if .v1 and .v2 are tranposed by .σ = (1 2) ∈ S3 , then u is changed to .fσ (u) = 1 − u, and if 1 .v1 and .v3 are transposed by .τ = (1 3) ∈ S3 , then u is changed to .fτ (u) = u . In fact,

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the group .〈fσ , fτ 〉 is the group of automorphisms of .CP1 which preserve the subset 1 .{0; 1; ∞}. Note that this defines a right action of .S3 on .CP \ {0; 1; ∞} because ' ' 1 .fσ τ = fτ ◦ fσ . Moreover, if .τ ∈ h, then the corresponding .u ∈ CP \ {0; 1; ∞} ' lies in the same .S3 -orbit as u if .C/Λ(τ ) ≃ C/Λ(τ ) as ellipticcurves.  In view of a c +b Theorem 11.2.14, this occurs if and only if .τ ' = aτ cτ +d for some . b d ∈ SL(2; Z). Thus, we have indeed proved that   h/SL(2; Z) ≃ C \ {0; 1} /S3 .

.

(11.9) ⨆ ⨅

Remark 11.2.21 The parameter .u ∈ C \ {0; 1} that appears in the proof of Theorem 11.2.20 coincides with the parameter u in the Legendre family of Example 11.2.3. Based on the bijection (11.9), we can identify this set with the Riemann surface C in two equivalent ways. Either by defining an .SL(2; Z)-invariant surjection .jˆ : h −→ C whose fibres are precisely the .SL(2; Z)-orbits in .h, or by defining a .S3 -invariant surjection .λˆ : C \ {0; 1} −→ C whose fibres are precisely the .S3 orbits in .C \ {0; 1}. The first method induces a bijection .j : h/SL(2; Z) −→ C sending the .SL(2; Z)-orbit of .τ ∈ h to the j -invariant of the framed elliptic curve .(C/Λ(τ ), 0, (1, τ )). The latter is constructed via the theory of modular forms and defined by converting Eq. (11.1) into the equation .

℘ '2 (z) = 4℘ 3 (z) − g2 (τ )℘ (z) − g3 (τ )

.

where .g2 (τ ) = 60G4 (τ ), .g3 (τ ) = 140G6 (τ ) and .Gk (τ ) is the Eisenstein series  1728 g2 (τ )3 1 . (m,n)∈Z2 ,(m,n)/=(0,0) (mτ +n)k , then setting .j (τ ) = g2 (τ )3 −27g3 (τ )2 . For the former approach and the definition of a bijection .λ : C \ {0; 1}/S3 −→ C, recall that we want a surjective map .λˆ : C \ {0; 1} −→ C with generic fibres having six elements, and that is invariant under the transformations .fσ : u I−→ (1 − u) and (1+u(u−1))3 1 ˆ .fτ : u I−→ u . This leads to .λ(u) := (u(u−1))2 , whose numerator is a polynomial of degree 6 which is non-zero when .u = 0 or .u = 1. It can be checked that, indeed, ˆ Finally, note that if we reduce the cubic equation ˆ ◦ fσ = λˆ and .λˆ ◦ fτ = λ. .λ 2 2 3 .y = x(x − 1)(x − u) to an equation of the form .y = 4x − g2 x − g3 , we find that √ 3

1 g2 = 34 (1 + u(u − 1)) and .g3 = 27 (u + 1)(2u2 − 5u + 2), so the j -invariant of the elliptic curve defined by the Legendre equation .y 2 = x(x − 1)(x − u) is equal ˆ to .256 × λ(u).

.

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11.3 Stacks 11.3.1 Prestacks Prestacks formalise the notion of families of geometric objects by building upon the intuition that a family is a special kind of map, one that you should be able to pull back. For instance, if a group G acts on a space X, there should be: (1) For all space S, a notion of family of G-orbits in X, parameterised by S, and a notion of isomorphism of such families. (2) For all morphism .f : T −→ S and all family P parameterised by S, a family ∗ ∗ .f P parameterised by T , such that there are natural isomorphisms .id P ≃ P S ∗ ∗ ∗ and .(f ◦ g) P ≃ g (f P ) for all morphism .g : U −→ T . Example 11.3.1 If G is a Lie group acting by automorphisms on a manifold X, a family parameterised by S of G-orbits in X is defined to be a pair .(P , u) where P is a principal G-bundle on S, and .u : P −→ X is a G-equivariant map. In particular, the image of u is a union of G-orbits in X, and the intersection of the fibre of u above a point .x ∈ X with a G-orbit in P , if non-empty, is in bijection with the stabiliser of x in G. Moreover, if .f : T −→ S is a morphism, then .f ∗ P is indeed a principal G-bundle on T , equipped with a G-equivariant map to X:

.

An isomorphism between two families .(P , u) and .(Q, v) parameterised by the ≃ same space S, is an isomorphism .λ : P −→ Q of principal G-bundles over S, such that .v ◦ λ = u. Note that the definitions above make sense when .X = • (a single point), with G acting trivially on it. We are now ready to give the formal definition of a prestack. Since we are mainly interested in the moduli stack of elliptic curves, our spaces S will be complex analytic manifolds. We shall denote by .An the category of such spaces. Definition 11.3.2 A prestack on .An is a 2-functor .M : Anop −→ Groupoids, meaning an assignment, • for all complex analytic manifold .S ∈ An, of a groupoid .M(S); ∗ : M(S) −→ • for all morphism .f : T −→ S in .An, of a pullback functor .fM ∗ M(T ), often denoted simply by .f ;

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• for all pair of composable morphisms .g : U −→ T and .f : T −→ S in .An, of a commutative diagram

(11.10)

.

such that: (1) .id∗S = idM(S) , and (2) for all morphism .h : V −→ U in .An, the following diagram commutes:

(11.11)

.

In other words, the dashed arrow .(f ◦ g ◦ h)∗ : M(S) −→ M(V ) is welldefined. In practice, the identification .(f ◦ g)∗ ≃Ф M g ∗ ◦ f ∗ in Diagram (11.10) is often f,g

canonical: this was for instance the case in Example 11.3.1 where, for all principal bundle .P −→ S, it follows from the construction of the pullback bundle that there is indeed a canonical isomorphism .(f ◦ g)∗ P = g ∗ (f ∗ P ). Likewise .h∗ ((f ◦ g)∗ P ) = (g ◦ h)∗ (f ∗ P ) , which means that Condition (11.11) is automatically satisfied in this case. So Example 11.3.1 indeed defines a prestack, usually denoted by .[X/G] and called the quotient stack of X by G. If .M is a prestack and S is a manifold, the groupoid .M(S) is called the groupoid of S-points of .M. For instance, the groupoid of S-points of .[X/G] is

.

When .X = • is a point, the prestack .[•/G] is usually denoted by BG, and called the classifying stack of G. The case when .G = {1} is also interesting, for it enables us to see manifolds as prestacks (Example 11.3.3). Finally, as we shall see in Sect. 11.6.1, families of elliptic curves parameterised by a complex analytic manifold .S ∈ An indeed define a prestack.

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Example 11.3.3 A manifold X defines a prestack, denoted by .X, by setting X(S) := Mor(S, X) = {u : S −→ X}, with morphisms between .u : S −→ X and .u' : S −→ X given by

.

MorX(S) (u, u' ) :=

.



{idS } if u = u' , ∅ if u /= u' .

Note that this is the standard way in which groupoids generalise sets: the groupoid associated to a set is the category whose objects are the elements of the set and the only morphisms are the identity morphisms. The pullback functor corresponding to .f : T −→ S is given by .f ∗ u = u ◦ f , which is indeed a functor given the way morphisms are defined in .X(S) and .X(T ). As we shall see in Lemma 11.3.6, this defines a fully faithful functor .X I−→ X from manifolds to stacks. A prestack .X for which there exists a manifold X such that .X ≃ X is called a representable stack. By Lemma 11.3.6, a manifold X satisfying .X ≃ X is defined up to canonical isomorphism. To conclude this subsection, we give the definition of a morphism of prestacks. Definition 11.3.4 Let .M, N : Anop −→ Groupoids be prestacks on .An. A morphism of prestacks .F : M −→ N is a natural transformation between the 2functors .M and .N, meaning an assignment for all space .S ∈ An, of a functor .FS : M(S) −→ N(S) such that, for all morphism .f : T −→ S in .An, the following diagram is 2-commutative (i.e. commutative up to a natural transformation in the category .N(T ))

.

(11.12)

M ) = ФFNU (f ),FU (g) as and moreover, for all .g : U −→ T in .An, we have .FU (Фf,g functors from .N(U ) to .N(U ). We shall denote by .PSt the 2-category of prestacks.

Example 11.3.5 If X is a manifold, morphisms of prestacks .F ∈ MorPSt (X, M) correspond to objects .PX ∈ M(X). Indeed, the morphism F is entirely determined by the object .PX := FX (idX ) ∈ M(X) and the fact that, as .idX ◦ u = u, the morphism F must satisfy, for all .S ∈ An, FS :

.

X(S) = MorAn (S, X) −→ M(S) . u I−→ u∗M PX

This sets up an equivalence of categories .MorPSt (X, M) ≃ M(X), obtained by sending F to .PX := FX (idX ). For instance, if .M = BG := [•/G], the groupoid .BG(S) is the category of principal G-bundles on S and, given a principal bundle

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P −→ X, there is a natural functor

.

FS :

.

MorAn (S, X) −→ BG(S) u I−→ u∗ P

and the 2-commutativity of Diagram (11.12) holds because of the canonical identification .(u ◦ f )∗ P ≃ f ∗ (u∗ P ) for all .f : T −→ S. Unwrapping the details of Example 11.3.5, we get the following stacky version of the Yoneda lemma. Lemma 11.3.6 Let .X, Y be objects in .An and let .X, Y be the corresponding prestacks. Then MorPSt (X, Y ) ≃ MorAn (X, Y ).

.

In particular, .X ≃ Y in .PSt if and only if .X ≃ Y in .An.

11.3.2 Stacks There is an obvious analogy between Definition 11.3.2 and that of a presheaf on An. Based on that analogy, we can define stacks as prestacks that satisfy a descent condition with respect to open coverings of objects .S ∈ An. The only subtlety is that we have to glue not only morphisms but also objects. In what follows, given a prestack .M on .An, a morphism .f : T −→ S in .An, and a morphism .ϕ : A −→ B in .M(S), we denote by .A|T and .B|T the objects .f ∗ A and .f ∗ B of .M(T ), and by ∗ .ϕ|T : A|T −→ B|T the morphism .f ϕ in .M(T ). M .

Definition 11.3.7 A stack on .An is a prestack .M : Anop −→ Groupoids such that, for all .S ∈ An and all open covering .(Si )i∈I of S, the following two conditions are satisfied: (1) (Gluing of morphisms). For all pair of objects .A, B ∈ M(S) and all family of morphisms ϕi : A|Si −→ B|Si in M(Si )

.

such that, for all .i, j in I , .ϕ|Si ∩Sj = ϕj |Si ∩Sj , there exists a unique morphism ϕ : A −→ B in .M(S) such that, for all .i ∈ I , .ϕ|Si = ϕi . (2) (Gluing of objects). For all family of objects .(Ai )i∈I ∈ i∈I M(Si ) equipped with isomorphisms .

≃

ϕij : Aj |Si ∩Sj −→ Ai |Si ∩Sj in M(Si ∩ Sj )

.

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satisfying the cocycle conditions .ϕii = idAi and .ϕij ◦ ϕj k = ϕik in .M(Si ∩ Sj ∩ Sk ) for all triple .(i, j, k), there exists an object .A ∈ M(S) together with isomorphisms .ϕi : A|Si −→ Ai in .M(Si ), such that, for all .(i, j ), one has −1 .ϕi ◦ ϕ j = ϕij in .M(Si ∩ Sj ). A morphism of stacks .F : M −→ N is a morphism of prestacks from .M to .N. We shall denote by .St the 2-category of stacks. Remark 11.3.8 It is perhaps not easy to note at first, but an important feature of the definition of stacks is that objects are required to glue but without any requirement of uniqueness: in Point (2) of Definition 11.3.7, the object .A constructed from the .Ai is “unique up to isomorphism", by a direct application of Condition (1), but the isomorphisms .ϕi : A|Si −→ Ai are not unique, which is the familiar condition from the theory of fibre bundles and is in fact what allows objects with non-trivial automorphisms to define stacks. We will for instance study the stack of elliptic curves from that point of view in Sect. 11.6. Proposition 11.3.9 The prestacks .[X/G], BG and .X introduced in Examples 11.3.1 and 11.3.3 are stacks on .An. Proof The prestacks BG and .X are special cases of the prestack .[X/G]. We leave the proof that the prestack .[X/G] is a stack as an exercise (it follows from the fact that principal G-bundles on .S ∈ An, and morphisms between them, can be constructed by gluing). ⨆ ⨅ In line with the intuition that stacks are designed to formalise the idea of families as mathematical objects that can be pulled back (prestack) and glued (stacks), the 2-category of stacks admits fibre products. Definition 11.3.10 Let .F : M −→ X and .G : N −→ X be morphisms of stacks on .An. Let us denote by .M ×X N the 2-functor sending a space .S ∈ An to the groupoid whose objects are

.

(11.13) and whose morphisms between the objects .(α, β, ϕ) and .(α ' , β ' , ϕ ' ) are given by ≃ triples .(λ, μ, ψ) of morphisms .λ : M −→ M, .μ : N −→ N and .ψ : X −→ X such that .F ◦ (λ ◦ α) = ψ ◦ (F ◦ α) and .G ◦ (μ ◦ β) = ψ ◦ (G ◦ β) as morphisms from .S to .X, and .ϕ ' ◦ ψ = ψ ◦ ϕ as morphisms from .X to .X (i.e. .(λ, μ, ψ) induces an isomorphism between diagrams of the form appearing in (11.13)).

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Definition 11.3.10 is not easy to digest, because it involves a lot of hidden compatibility conditions. It is instructive to compare it to Thurston’s definition of the fibre product of two orbifolds, which appears in the course of the proof of Proposition 13.2.4 in [16, pp. 305–307]. If we think of the groupoid .MorPSt (S, M ×X N) as families parameterised by S, we get the following more concrete description: (M ×X N)(S)   ≃ := (A, B, ϕS ) | A ∈ M(S), g ∈ N(S) and ϕS : FS (A) −→ GS (B) in X(S) (11.14)

.

with morphisms .(A, B, ϕS ) −→ (A' , B ' , ϕS' ) being pairs .(uS , vS ) of morphisms ' ' .uS : A −→ A in .M(S) and .vS : B −→ B in .N(S) making the following diagram commute in .X(S):

.

Let us show how we can use (11.14) as a definition of the fibre product .M ×χ N. We simply observe that, because of the fact that, for all morphism .f : T −→ S in .An, we have .FT (A|T ) = FS (A)|T and .GT (BT ) = GS (B)|T in .X(T ), we must add the compatibility condition that .ϕT = ϕS |T as functors from .X(T ) to .X(T ), to ensure that we have a well-defined pullback morphism ∗ fM× : XN

.

(M ×X N)(S) −→ (M ×X N)(T ) . (A, B, ϕS ) I−→ (A|T , B|T , ϕT )

This compatibility condition, which needs to be added here, is encapsulated in Definition 11.3.10 as the fact that the arrow .ϕ : X −→ X appearing in Diagram (11.13) is a morphism of prestacks. Example 11.3.11 Let us compute the fibre product of a morphism .M −→ BG with the morphism .• −→ BG, where .M ∈ An and .• is a single point. By Example 11.3.5, the first morphism corresponds to a principal G-bundle .P −→ M, while the second morphism corresponds to the G-bundle .G −→ •. Then, by definition of the stacks .M, .• and .M ×BG •, and using the fact that a principal bundle is trivial if and only if it admits a global section, we have:   (M ×BG •)(S) ≃ (u, ϕ) | u : S −→ M and ϕ is an isomorphism u∗ P −→ S × G

.

≃ {(u, σ ) | u : S −→ M and σ is a section of u∗ P }.

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But, by the universal property of .u∗ P , giving a section .σ : S −→ u∗ P is equivalent to giving a morphism .f : S −→ P .

.

So .(M ×BG •)(S) ≃ Mor(S, P ), which proves that the fibre product .M ×BG • is isomorphic to the stack .P , where .P −→ M is the principal G-bundle defining the given morphism .M −→ BG. In particular, the stack .M ×BG • is representable by the manifold P . Proposition 11.3.12 The 2-functor .M ×X N defined in (11.13) is a prestack and it comes equipped with two morphisms .pM : M×X N −→ M and .pN : M×X N −→ N making the following diagram a 2-cartesian square:

.

Proof We deduce from the discussion following Definition 11.3.10 that .M ×χ N is a prestack. By (11.13), the prestack .M ×χ N satisfies the universal property of a 2-fibre product for all prestacks of the form .S where .S ∈ An. Therefore, it satisfies it for all prestacks on .An. ⨆ ⨅ We omit the proof that the prestack .M ×X N is a stack. Note that if .M, N , and .X are representable stacks, the stack .M ×X N is not representable in general, because .M −→ X and .N −→ X do not admit a fibre product in .An in general.

11.3.3 Analytic Stacks An analytic stack is a stack .X on .An that admits a presentation (or atlas) .p : X −→ X, where .X is the stack associated to a space .X ∈ An (henceforth, we shall often write simply X for the stack .X associated to the space .X ∈ An). Before we can define more precisely what a presentation is, we need to introduce the following notion. Definition 11.3.13 A morphism of stacks .F : M −→ X is called representable if, for all morphism .S −→ M with .S ∈ An, there exists a complex analytic manifold .MS ∈ An such that .S ×X M ≃ MS .

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In other words, for all .S −→ X, the morphism of stacks .F : M −→ X basechanges to a morphism of complex analytic manifolds .fS : MS −→ S, as per the following 2-Cartesian square:

.

(11.15)

Example 11.3.14 The computation of .M ×BG • in Example 11.3.11 shows that the canonical morphism .• −→ BG is representable. Since the pullback of the trivial bundle .(X × G) −→ X to an arbitrary manifold S mapping to X is trivial, the same proof works for the canonical morphism .X −→ [X/G], corresponding to the trivial bundle .X × G (with G-equivariant map from .X × G to X given by the action map .(X × G) −→ X, where G acts on X to the right). Note that the forgetful morphism .BG −→ • (sending a principal G-bundle .P −→ S to the constant map .S −→ •) is not a representable morphism as soon as .• ו BG ≃ BG is not a representable stack. Working with representable morphisms enables one to carry over certain local properties of morphisms from .An to .St. Most prominently for us, the following one. Definition 11.3.15 Let .M and .X be stacks over .An. A morphism of stacks .F : M −→ X is called a submersion if it is representable and, for all morphism of stacks .S −→ X with .S ∈ An, the induced morphism .fS : MS −→ S appearing in Diagram (11.15) is a submersion in .An. Intuitively, this is a meaningful definition because, for manifolds, a morphism f : Y −→ X is a submersion in .An if and only if for all open coverings .X = ∪i∈I Xi and .Y = ∪j ∈J Yj , the induced morphisms .fij : Yj ∩ f −1 (Xi ) −→ Xi are submersions. Next, we define surjective morphisms of stacks (which bears an analogy to the notion of surjective morphism of sheaves).

.

Definition 11.3.16 Let .M and .X be stacks over .An. A morphism .F : M −→ X is said to be surjective if, for all .S ∈ An and all .P ∈ X(S), there is an open covering .(Si )i∈I of S such that .P |Si is in the essential image of the functor .FSi : M(Si ) −→ X(Si ), i.e. there exists a family .(Pi )i∈I of objects .Pi ∈ M(Si ) and, for all .i ∈ I , an isomorphism .FSi (Pi ) ≃ P |Si in .X(Si ). We can now define a presentation (or atlas of an analytic stack) of a stack .X on An as a surjective representable morphism .p : X −→ X which is a submersion and whose domain is a manifold .X ∈ An. This leads to the following definition of an analytic stack.

.

Definition 11.3.17 An analytic stack is a stack .X on .An equipped with an analytic manifold .X ∈ An and a morphism .p : X −→ X such that: (1) p is representable (Definition 11.3.13).

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(2) p is surjective (Definition 11.3.16). (3) p is a submersion (Definition 11.3.15). A morphism between analytic stacks .M −→ X is a morphism of stacks from .M to X.

.

The same definition works for a differentiable stack .X; it is a stack on the category of differentiable manifolds which is equipped with a representable surjective morphism .p : X −→ X such that X is a differentiable manifold and p is a submersion. Example 11.3.18 It is straightforward to check that, if .X ∈ An, the identity morphism .idX gives a presentation .X −→ X. So .X is an analytic stack in a canonical way. In fact, we could also take, as a presentation for .X, the analytic manifold .⨆i∈I Ui , where .(Ui )i∈I is an analytic atlas of X in the traditional sense. Let us now prove that if G is a complex Lie group acting on an analytic manifold X, then the quotient stack .[X/G] admits a canonical structure of analytic stack, given by the morphism .X −→ [X/G] defined by the trivial principal G-bundle on X. We saw in Example 11.3.14 that this morphism is representable. It is surjective because principal G-bundles on a manifold S are locally trivial. And it is a submersion because the projection map of a principal bundle .P −→ S is a submersion. In particular, the classifying stack BG of a complex Lie group G admits a canonical structure of analytic stack. Note that a fibre product of analytic stacks is a stack over .An but not necessarily an analytic stack.

11.4 Orbifolds We will eventually define an analytic orbifold as an analytic stack admitting an open covering by quotient substacks of the form .[U/ 𝚪], where U is an analytic manifold and .𝚪 is a finite group acting on U by automorphisms. Before we can do that, we need to define open substacks. Definition 11.4.1 Let .X be a stack on .An. A substack of .X is a pair .(Y, F ) where Y is a stack on .An and .F : Y −→ X is a morphism of stacks. If .X is an analytic stack and the substack .Y is also analytic, we will call .Y an analytic substack of .X.

.

Example 11.4.2 Let G be a complex Lie group acting to the right on an analytic manifold X and let .Y ⊂ X be an open submanifold of X that is invariant under the action of a subgroup .H ⊂ G. The H -equivariant map .Y ͨ→ X induces a canonical morphism of stacks .F : [Y /H ] −→ [X/G]. Explicitly, the morphism F is defined,

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for all .S ∈ An, by the functor

.

(11.16)

sending a principal H -bundle P on S equipped with an H -equivariant map to Y to the extension of structure group .P ×H G, equipped with the induced G-equivariant map to .Y · G ⊂ X . This morphism F makes .[Y /H ] a substack of .[X/G]. Note that .[X/G](S) is in general strictly larger than .[Y /H ](S), since it is possible that not all principal G-bundles on S admit a reduction of structure group on H . Also, the functor .FS is in general neither full nor faithful: given two principal G-bundles on S admitting a reduction of structure group to H , a morphism between them does not necessarily come from a morphism between the two reductions and, even if it does, the morphism it comes from is not necessarily unique (for instance, a given principal G-bundle may admit non-isomorphic reductions of structure group to H ). Note that if we study the analogous situation with Y closed in X, we need to be careful with the manifold structure on .Y · G in order to get a morphism .Y · G −→ X in .An (which may or may not be a closed immersion). It is easy to make sense of the notion of open substack for representable morphisms (Definition 11.4.3). And for analytic stacks, it suffices to check the condition with respect to an atlas (Proposition 11.4.6). Definition 11.4.3 Let .X be a stack on .An. A substack .F : Y −→ X is called an open substack of .X if: (1) F is representable (Definition 11.3.13). (2) For all morphism of stacks .M −→ X with .M ∈ An, the morphism .fM : Y ×X M −→ M is an open morphism in .An. Note that Condition 2, which says that .fM : Y ×X M −→ M is open in .An, makes sense because the stack .Y ×X M is representable by a manifold (in this case because F is assumed to be a representable morphism; see Remark 11.4.7 for further comments). Example 11.4.4 Let us retake Example 11.4.2. We will show that, if .Y ⊂ X is open and .G/H is a discrete topological space, then .[Y /H ] is an open substack of .[X/G]. This will apply in particular when G is a discrete group (see also Example 11.4.9). Firstly, we have to show that the morphism (11.16) is representable and secondly, that it is open (this is where we shall use that .G/H is discrete). For the first part, we compute the fibre product .[Y /H ] ×[X/G] M for all morphism of stacks .M −→ [X/G] with .M ∈ An. Such a morphism is given by a principal G-bundle P over M, equipped with a G-equivariant morphism .P −→ X in .An. Likewise, an arbitrary morphism .S −→ [Y /H ] with .S ∈ An is given by a principal H -bundle Q over S,

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equipped with an H -equivariant morphism .Q −→ Y in .An. So, given a morphism u : S −→ M in .An, a morphism of stacks .S −→ [Y /H ] ×[X/G] M is given by an isomorphism of principal G-bundles between .u∗ P and .Q ×H G over S, i.e. a reduction of structure group from G to H for the principal G-bundle .u∗ P −→ S. It is well-known that reductions of structure group are in bijection with sections of the associated fibre bundle .u∗ P ×G (G/H ) ≃ u∗ (P ×G (G/H )) over S. Then, as in Example 11.3.11, such sections are in bijection with morphisms of analytic manifolds from S to .P ×G (G/H ), which proves that the canonical morphism .F : [Y /H ] −→ [X/G], defined in (11.16) by extension of structure group and inclusion of .Y · G in X, is representable (in particular, when .M = X and .P = X × G, then .P ×G (G/H ) = X × (G/H )). Since we now have a 2-Cartesian diagram .

.

we see that the morphism of stacks .[Y /H ] −→ [X/G] pulls back to the fibre bundle P ×G (G/H ) −→ M in .An, which is indeed an open map when the fibre, which is .G/H , is discrete. .

Remark 11.4.5 Note that, if we take .H = G in Example 11.4.4, then the canonical morphism .[Y /G] −→ [X/G] is an open embedding in the sense that, for all .M ∈ An, the morphism .[Y /G]M −→ M is an open embedding in .An (this is obvious because .[Y /G]M ≃ M when .H = G). Proposition 11.4.6 Let .X be a stack on .An and assume that .X is an analytic stack with presentation .p : X −→ X. A substack .F : Y −→ X is an open substack of .X if: (1) F is representable. (2) The morphism .fX : Y ×X X −→ X is an open morphism in .An. Proof If .(Y, F ) is an open substack of the analytic stack .X, Conditions (1) and (2) of Proposition 11.4.6 are indeed met. To prove the converse, it suffices to show that, if .fX : YX −→ X is open, then .fM : YM −→ M is open for all .M ∈ An, where, for all .S ∈ An, we use the notation .YS ∈ An for the representable stack .Y ×X S. To prove what we need to, note that, given a morphism of stacks .M −→ X, we have a commutative diagram

.

where the first vertical arrow is open because .fX : YX −→ X is open by the assumption made on .F : Y −→ X and since the property of being open is invariant

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by base change. The commutativity of the left square then implies that the second vertical arrow is open. Since .F : Y −→ X is a representable morphism of stacks, the stack .Y ×X M is representable by a manifold, and since the middle square is a pullback diagram, the morphism .Y ×X XM −→ Y ×X M is also a surjective submersion in .An. In particular, it is open and surjective. So the fact that .Y ×X XM −→ XM is open implies that .f : Y ×X M −→ M is indeed open. ⨆ ⨅ We leave it as an exercise to show that, if .F : Y −→ X is an open substack of an analytic stack .X, then .Y is an analytic stack. Remark 11.4.7 Some comments are in order. For the sake of simplicity, we have omitted certain aspects of the definition of an analytic stack or of an open substack. For instance, in Definition 11.3.17, it is often required that, for a stack .X over .An to be an analytic stack, every morphism .M −→ X with M in .An should be representable. So Condition (1) of Definition 11.3.17 becomes unnecessary. But of course it then becomes more difficult to show that a given stack over .An is an analytic stack. In practice, one uses the fact that the following two conditions are equivalent (we refer to [10, Remark 8.1.6 p. 170] for a proof of this): (1) Every morphism .M −→ X with M in .An is representable (meaning that, for all .N −→ X with .N ∈ An, the stack .M ×X N is representable by a manifold). (2) The diagonal morphism .Δ : X −→ X × X is representable. So the more official version of Definition 11.3.17 is that a stack .X on .An is called an analytic stack if the diagonal morphism .Δ : X −→ X × X is representable and there exists a manifold .X ∈ An and a (necessarily representable) surjective submersion .p : X −→ X. The stack .X is called separated if the representable morphism .Δ is closed. In the same way that we defined what it means to be an open substack of a stack X, we can define what it means to be a cover of such a stack. Comments similar to Proposition 11.4.6 and Remark 11.4.7 apply when .X is an analytic stack, but we omit them here and only give the elementary definition.

.

Definition 11.4.8 Let .X be a stack on .An. A morphism of stacks .F : Y −→ X is called a cover or covering stack of .X if: (1) F is representable. (2) For all morphism of stacks .M −→ X with .M ∈ An, the morphism .fM : Y ×X M −→ M is a cover in .An. Such a cover is called a finite cover if, for all M, the covering map .fM is a finite cover in .An. Example 11.4.9 If G is a discrete group acting by automorphisms on the space X ∈ An, then the canonical morphism .X −→ [X/G] (defined by the trivial cover .(X × G) −→ X) is a covering stack of .[X/G]. Indeed, if .M ∈ An and .P −→ M is the principal G-bundle associated to a morphism .M −→ [X/G], then .X×[X/G] M is isomorphic to P (see Examples 11.3.11 and 11.3.14), and this is indeed a cover of M .

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because G is discrete. A subgroup .H < G then defines an intermediate cover .X −→ [X/H ] −→ [X/G]. And if Y is a G-equivariant cover of X, then .[Y /G] −→ [X/G] is a cover in the stacky sense. Mixing these two examples together, we get the following commutative diagrams, all of whose arrows are covers (the first vertical arrow being a G-equivariant cover in .An).

.

The proof of these claims follows from the same techniques as the ones used in Example 11.4.4. And for a concrete example, the group .μn (C) of n-th roots of unity 2π acts (non-freely) on .C by multiplication by .ei n , and we have a covering stack (or stacky cover) .C −→ [C/μn (C)]. The same construction works if we replace .C by an open disk .D(0, ε) ⊂ C. Note that a covering stack is in particular an open morphism of stacks because the covering maps .fM : Y ×X M −→ M are all open morphisms in .An. We are now ready to give the definition of an orbifold. The intuition if that an analytic stack is an orbifold if it admits an open covering of the form .[U/ 𝚪], where .U ∈ An and .𝚪 is a finite group acting on U by automorphisms. Definition 11.4.10 An analytic orbifold is an analytic  stack .X (Definition 11.3.17) equipped with a family of substacks of the form . Fi : [Ui / 𝚪i ] −→ X i∈I such that: (1) For all .i ∈ I , the pair .(Ui , 𝚪i ) consists of a manifold .Ui ∈ An and a finite group .𝚪i acting on .Ui by automorphisms. (2) For all .i ∈ I , the substack .Fi : [Ui / 𝚪i ] −→ X is an open substack (Definition 11.4.3). (3) The morphism of stacks .F : ⨆i∈I [Ui / 𝚪i ] −→ X is surjective (Definition 11.3.16). The in Definition 11.4.10 is defined, for all .S ∈ An,  stack .⨆i∈I [U   i / 𝚪i ] appearing  by . ⨆i∈I [Ui / 𝚪i ] (S) = ⨆i∈I [Ui / 𝚪i ](S) . An ultimately equivalent but slightly more abstract definition of an analytic orbifold would be that of an analytic stack equipped with a (surjective) finite cover .F : M −→ X with .M ∈ An. For instance, all manifolds have canonical orbifold structures. Either way, the main source of examples is obtained as an application of the following result, which is due to Thurston in the case when the group G acts effectively on the manifold X [16, Proposition 13.2.1, p. 302]. Theorem 11.4.11 Let X be an analytic manifold and let G be a discrete group acting to the right on X by automorphisms. For all .x ∈ X, we denote by Gx := {g ∈ G | x · g = x}

.

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the stabiliser of x in G. Assume that the following properties are satisfied: (1) For all .x ∈ X, there exists an open neighbourhood .Ux of x such that g ∈ Gx ⇒ (Ux · g) ⊂ Ux

.

and

g∈ / Gx ⇒ (Ux · g) ∩ Ux = ∅.

(2) For all .x ∈ X, the group .Gx is finite. Then the quotient stack .[X/G] is an orbifold. Proof Since, for all .x ∈ X, the subset .Ux ⊂ X is .Gx -invariant and open in X, we know from Example 11.4.4 that the quotient stack .[Ux /Gx ] defines an open substack of .[X/G]. And since the .(Ux )x∈X form an open covering of X, the canonical morphism of stacks .F : ⨆x∈X [Ux /Gx ] −→ [X/G] is surjective. Indeed, if .S ∈ An and .(P , u) is a pair consisting of a principal G-bundle P on S and a G-equivariant map .u : P −→ X, we can take an open covering .(Si )i∈I of S such that, for all i, the principal G-bundle .Pi := P |Si is trivial. Let us fix a trivialisation .Pi ≃ (Ui × G) in which .u|Pi : Pi −→ X is the action map, and choose .x ∈ X such that .x ∈ u(Pi ). Set then .Qi,x := (Si × Gx ). By Condition (1), an element .ξ ∈ u−1 (Ux ) ∩ Pi satisfies .g · ξ ∈ u−1 (Ux ) ∩ Pi if and only if .g ∈ Gx , so .u−1 (Ux ) ∩ Pi ≃ Qi,x . So .Pi lies in the essential image of the canonical functor Fx,Si : [Ux /Gx ](Si ) −→ [X/G](Si ),

.

hence also in the image of .FS : ⨆x∈X [Ux /Gx ](Si ) −→ [X/G](Si ), which proves that the morphism .F : ⨆x∈X [Ux /Gx ] −→ [X/G] is surjective in the sense of Definition 11.3.16. Finally, since each .Gx is finite by Condition (2), we have indeed shown that .[X/G] is an orbifold. ⨆ ⨅ Example 11.4.12 If G is finite to begin with, we can simply use the finite cover X −→ [X/G] to prove that .[X/G] has a canonical orbifold structure. For instance, the quotient stack .[C/μn (C)] of Example 11.4.9 is canonically an orbifold. We will see in Corollary 11.4.15 that the action of the discrete group .SL(2; Z) on the upper half-plane .h := {z ∈ C | Im z > 0} satisfies the conditions of Theorem 11.4.11, so the quotient stack .[h/SL(2; Z)] is also an orbifold, in a canonical way.

.

Remark 11.4.13 Note that we are not assuming that the action of G on X is effective, only that it has finite stabilisers. In particular, if G is a finite group, then the classifying stack .BG = [•/G] is an orbifold. When the action of the group G on the space X is effective, the group G can be identified with a subgroup of .Aut(X) and the quotient stack .[X/G] is called an effective orbifold. To construct orbifolds, it suffices to find examples of actions of discrete groups on manifolds for which Conditions (1) and (2) of Theorem 11.4.11 are satisfied. An important tool to do that is the following result. We state it for a real Lie group H , but we are mostly interested here in the case where H is a Hermitian group, meaning that the symmetric space .X := K\H has a canonical structure of complex

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analytic (in fact, Kähler) manifold, in which case the quotient stack .[X/G] is indeed an analytic orbifold. Theorem 11.4.14 ([15, Corollary 3.5.11]) Let H be a real Lie group whose Lie algebra admits an H -invariant positive definite scalar product. Fix a maximal compact subgroup .K ⊂ H and let us endow the homogeneous space .X := K\H with its canonical manifold structure. Let .G ⊂ H be a discrete subgroup of H . Then the canonical right action of G on X satisfies Conditions (1) and (2) of Theorem 11.4.11. In particular, the quotient stack .[X/G] has a canonical orbifold structure. Corollary 11.4.15 Since the upper half-plane .h = {z ∈ C | Im z > 0} can be described as the homogeneous space .SO(2)\SL(2; R), the quotient stack .[h/SL(2; Z)] has a canonical orbifold structure. Let us make a few remarks on the proof of Theorem 11.4.14. Remarks 11.4.16 and 11.4.17 show that proper, isometric actions of discrete subgroups of Lie groups on Riemannian manifolds give rise to orbifolds. Together with Remark 11.4.18, they can be formalised into a proof of Theorem 11.4.14 and Corollary 11.4.15. Remark 11.4.16 What is actually proven in [15, Corollary 3.5.11] is that, if .G ⊂ H is a discrete subgroup of the Lie group H , the action of G on .K\H (or, as a matter of fact, on all manifold X on which H acts transitively with compact stabilisers .Hx ) satisfies the following condition: (P) For all compact subset .C ⊂ X, the set .{g ∈ G | C · g ∩ C /= ∅} is finite. An (effective) action of a group G by homeomorphisms on a locally compact space X is called properly discontinuous if it satisfies Property (P); see [15, Definition 3.5.1.(v), p. 153]. If G is endowed with the discrete topology, this is equivalent to asking that the continuous map .X × G −→ X × X given by .(x, g) I−→ (x, x · g) be proper. So giving a properly discontinuous action on a locally compact space is equivalent to giving a proper, continuous action of a discrete group on that space [15, Exercise 3.5.2, p. 154]. Note that, for such an action, the stabilisers .Gx are finite, so Condition (2) of Theorem 11.4.11 is indeed satisfied. Remark 11.4.17 To prove Theorem 11.4.14, we still need to show that if a discrete group G acts continuously and properly on a manifold X, then, as stated in the proof of [16, Proposition 13.2.1, p. 302], for all .x ∈ X, there exists an open neighbourhood .Ux of x in X which is .Gx -invariant and disjoint from its translates by elements of G that are not in .Gx (i.e. Condition (1) of Theorem 11.4.11 is also satisfied). In fact, to prove Theorem 11.4.14, it suffices to do it in the case when there is a G-invariant distance d on X (i.e. the group G acts by isometries on X). And in that case we can take .Ux to be a Dirichlet domain Ux = DG (x) := {y ∈ X | ∀ g ∈ / Gx , d(y, x) < d(y, x · g)}.

.

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Such a set is .Gx -invariant because, if .y ∈ DG (x) and .h ∈ Gx , the G-invariance of d gives, for all .g ∈ / Gx , d(y · h, x · g) = d(y, x · gh−1 ) > d(y, x) = d(y, x · h−1 ) = d(y · h, x).

.

The set .DG (x) is also disjoint from its translates by elements of G that are not in Gx because, if .g ∈ / Gx , then we cannot have both .y ∈ DG (x) and .(y · g) ∈ DG (x), as this would imply that

.

d(y, x) < d(y, x · g −1 ) = d(y · g, x) < d(y · g, x · g) = d(y, x).

.

So there only remains to prove that .DG (x) is open. For the sake of completeness, we sketch here the arguments to be found for instance in [1, Theorem 9.4.2, pp. 227–228]. Recall that x is fixed throughout. For all .g ∈ / Gx , set Hg (x) := {y ∈ X | d(y, x) < d(y, x · g)}

.

and

Lg (x) := {y ∈ X | d(y, x) = d(y, x · g)}.

In particular, .Hg (x) is open, .Lg (x) is closed and .DG (x) = ∩g ∈G / x Hg (x). Since G is a discrete subgroup of a Lie group, it is countable. And since G acts properly on X, a compact set .C ⊂ X can only intersect a finite number of .Lg (x). Indeed, this is a direct consequence of the fact that, if .G = {g0 , g1 , . . .}, then     ∀ n | gn ∈ / Gx , d x, Lgn (x) = 12 d x, x · gn −→ +∞.

.

n→+∞

Now let us fix .y ∈ DG (x) = ∩g ∈G / x Hg (x) and consider the compact disk .D[y; ε] centred at y. As we have just seen, the set {g ∈ / Gx | D[y; ε] ∩ Lg (x) /= ∅} =: {g1 , . . . , gr }

.

is finite. Since .y ∈ Hg (x), we have that, for all .i ∈ {1, . . . , r}, there exists an integer .ni ⩾ 1 such that the open disk .D(y, 2εni ) has an empty intersection with .Lgi (x), and is therefore contained in .Hgi (x). So, we can find an open neighbourhood r D(y , ε ) such that, for all .g ∈ .Uy := ∩ / Gx , .Uy ⊂ Hg (x), i.e. .Uy ⊂ DG (x), i 2ni i=1 proving that .DG (x) is indeed open. Remark 11.4.18 Note that, in the situation of Theorem 11.4.14, a G-invariant distance exists on .X = K\H because H possesses a left-invariant Riemannian metric. Indeed we have assumed that the Lie algebra of H admits an H -invariant positive definite scalar product (this holds for instance if H is a real form of a reductive complex Lie group, such as .H = SL(2; R) in Corollary 11.4.15). Note that the complex analytic structure of .h indeed comes from its description as the homogeneous space .SO(2)\SL(2; R) because of the exceptional isomorphism between .SL(2; R) and the Hermitian real form .SU(1, 1) of .SL(2; C).

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Remark 11.4.19 As explained after the proof of Corollary 3.5.11 in [15, p. 157], if X is a space of the type considered in Theorem 11.4.14, Condition (P) in Remark 11.4.16 admits equivalent characterisations, all of them given in [15, Definition 3.5.1.(ii)–(v), pp. 153–154]. For instance the fact that G is a discrete subgroup of H , or the fact that the action is wandering, in the following sense: (W) For all .x ∈ X, there is an open neighbourhood .Ux of x such that the set .{g ∈ G | Ux · g ∩ Ux /= ∅} is finite. If X is locally compact, then (P) .⇒ (W). And if X is a metric space such that G acts on X by isometries, then (W) .⇒ (P). We shall not need this notion in what follows.

11.5 Moduli Spaces 11.5.1 Coarse Moduli Spaces and Fine Moduli Spaces It is sometimes desirable to approximate an analytic stack by a space, in an appropriately optimal sense that we now define. Definition 11.5.1 Let .X be an analytic stack. A coarse moduli space for .X is a pair (M, π ) where .M ∈ An and .π : X −→ M is a morphism of stacks satisfying the following universal property:

.

For all morphism of stacks ϕ : X −→ N with N ∈ An, ∃! ϕ : M −→ N | ϕ◦π = ϕ.

.

Note that the morphism .ϕ : M −→ N is a morphism of stacks between two manifolds, so by the Yoneda lemma 11.3.6 it is induced by a uniquely defined morphism of analytic manifolds. The condition that .(M, π ) be a coarse moduli space for .X is usually summed up by the commutativity of the following diagram.

.

Sometimes, an extra condition is added, namely the fact that the functor .π• : X(•) −→ M(•) ≃ M be surjective, or in other words, that there be a surjective map from the set .X(•)/ ∼ of isomorphism classes of objects parameterised by the stack .X to the underlying set of the manifold M. Note however that asking for this map to be bijective (as opposed to just surjective) might be too restrictive (insofar as, in some examples, the universal property and the surjectivity condition are satisfied, but not the bijectivity condition). Typically, this happens when .X = [X/G] and the action of G on X has non-closed orbits (see Example 11.5.6). To understand this,

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we first need to study the existence of coarse moduli spaces for quotient stacks. The key observation is the following result. Proposition 11.5.2 Let G be a complex Lie group acting to the right on an analytic manifold X. Then, the canonical morphism .p : X −→ [X/G] defines a bijection .ϕ I−→ ϕ ◦ p between morphisms of stacks .ϕ from .[X/G] to a manifold .N ∈ An and G-invariant morphisms .ψ : X −→ N in .An: .

{morphisms of stacks ϕ : [X/G] −→ N} ≃ {G-invariant morphisms ψ : X −→ N in An}

More precisely, given a G-invariant morphism .ψ : X −→ N in .An, there is a unique morphism of stacks .ϕ : [X/G] −→ N such that .ϕ ◦ p = ψ. Proposition 11.5.2 says that the quotient stack .[X/G] satisfies the following universal property with respect to G-invariant morphisms .ψ : X −→ N .

.

Proof of Proposition 11.5.2 First note that, as in Example 11.4.4, we have a 2Cartesian diagram

(11.17)

.

whose commutativity can be taken as a definition that the morphism of stacks .p : X −→ [X/G] is G-invariant. The commutativity property also implies that, if .ϕ : [X/G] −→ N is a morphism of stacks whose target stack is a manifold .N ∈ An, then the morphism .ψ := ϕ ◦ p : X −→ N is G-invariant as a morphism in .An. Now assume that a G-invariant morphism .ψ : X −→ N has been given in .An. We will show that, conversely, there is a unique morphism of stacks .ϕ : [X/G] −→ N such that .ϕ ◦ p = ψ, .i.e. that the universal property depicted in Diagram (11.18) holds for the quotient stack .[X/G].

.

(11.18)

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Recall that to define .ϕ is to define .ϕS : [X/G](S) −→ N (S) for all .S ∈ An. But an object in .[X/G](S) is a pair .(P , u) consisting of a principal G-bundle P on S and a G-equivariant map .u : P −→ X. So we have a G-invariant morphism .ψ ◦ u : P −→ N in .An. Since P is a principal G-bundle on S, the morphism .ψ ◦ u factors into a unique morphism .f : S ≃ P /G −→ N (by the universal property of the quotient .P /G).

.

The morphism .ϕS : [X/G](S) −→ N(S) is then defined by .ϕS (P , u) := f .

⨆ ⨅

As an immediate consequence of Proposition 11.5.2, we get the following special case of Definition 11.5.1: for quotient stacks, the notion of coarse moduli space coincides with the classical notion of categorical quotient. In particular, depending on the action, they may or may not exist. Corollary 11.5.3 Let G be a complex Lie group acting to the right on a complex analytic manifold X. The quotient stack .[X/G] admits a coarse moduli space if and only if the action of G on X admits a categorical quotient, i.e. if there exists a pair .(M, ρ) where .M ∈ An and .ρ : X −→ M is a morphism of analytic manifolds such that: (1) The map .ρ is G-invariant: for all .g ∈ G and all .x ∈ X, .ρ(x · g) = π(x). (2) For all G-invariant morphism of analytic manifolds .ψ : X −→ N , there exists a unique morphism of analytic manifolds .ψ : M −→ N such that the following diagram is commutative:

.

(11.19)

A particularly good case is when .M = X/G, i.e. when the orbit space .X/G is a categorical quotient. This happens when the orbit space satisfies the conditions of the following definition. Definition 11.5.4 Let G be a complex Lie group acting to the right on an analytic manifold X and denote by .ρ : X −→ X/G the canonical projection. If the orbit space .X/G, endowed with the quotient topology, admits a structure of complex analytic manifold with respect to which the holomorphic functions on an open subset .U ⊂ X/G correspond, via pullback by .ρ, to .G-invariant holomorphic functions on the open subset .ρ −1 (U ) ⊂ X, then .X/G is called a geometric quotient for the .G-action on .X.

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It can be checked that a geometric quotient is a categorical quotient: it necessarily satisfies the universal property (11.19). An example of a situation in which a geometric quotient exists is the case when G acts freely and properly on .X (see Example 11.5.5). Note that, since the points of .X/G are closed and the canonical projection .ρ : X −→ X/G is continuous, a geometric quotient can only exist if the G-orbits in X are closed, and if there are non-closed orbits, then the points of a categorical quotient (if it exists) cannot be in bijection with the orbits of the action (see Example 11.5.6). There are cases in which the orbits are closed and the action is not free but a geometric quotient exists: this is in fact what happens for the action of the modular group .SL(2; Z) on the upper half-plane .h (see also Example 11.5.13). Example 11.5.5 If a complex Lie group .G acts freely and properly on a complex analytic manifold .X, then the orbit space .X/G admits a structure of complex analytic manifold that turns the canonical projection .ρ : X −→ X/G into a holomorphic principal .G-bundle. In particular, the map .ρ has local sections and holomorphic functions on .X/G do indeed correspond to .G-invariant holomorphic functions on .X, so .X/G is indeed a geometric quotient when .G acts freely and properly on .X. As a matter of fact, we will see that more is true in that case: when the action is proper and free, the orbit space .X/G is a fine moduli space for the quotient stack .[X/G] (see Definition 11.5.7 and Proposition 11.5.8). Example 11.5.6 Let .G = C∗ act on .X = C2 via .t · (z1 , z2 ) := (t · z1 , t −1 · z2 ). The closed orbits of this action are the origin .{(0, 0)}, which is a fixed point of the action, and the hyperbolas of equation .z1 z2 = λ, for all .λ ∈ C∗ . The origin is contained in the closure of the only two non-closed orbits, which are the real and imaginary axes minus the origin. In particular, a G-invariant morphism .ψ : X −→ N will take the same value on these three orbits, which will therefore be identified in the categorical quotient. This gives us the intuition that the categorical quotient in this example should be the set of closed orbits of the G-action, which is in bijection with .C via the .G-invariant function .ρ : C2 −→ C sending .(z1 , z2 ) to .z1 z2 . By construction, this set has strictly less elements than the whole set of orbits of the .G-action. Sometimes, it can even have drastically less elements: if instead of the above action, we let .G = C∗ act on .X = C2 via .t · (z1 , z2 ) = (t · z1 , t · z2 ), then the only closed orbit is .{(0, 0)}, which gives us the intuition that the categorical quotient will be a point in this case. In Sect. 11.5.2, we will give sufficient conditions for the existence of categorical and geometric quotients for actions of reductive complex Lie groups. As we shall see, in certain cases, a geometric quotient .X/G can exist even if the action of .G on .X is not free: a simple example of this is given in Example 11.5.13. We end this section with the notion of fine moduli space, which is the case when the stack .χ is not only approximated by a manifold M, but actually represented by it. Definition 11.5.7 Let .X be an analytic stack. A fine moduli space for .X is a complex analytic manifold M such that .X ≃ M.

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It is readily checked that a fine moduli space is also a coarse moduli space for .X. As an example, let us study the existence of fine moduli spaces for a quotient stack .X = [X/G]. Proposition 11.5.8 Let G be a complex analytic group acting to the right on a complex analytic manifold X. The quotient stack .[X/G] admits a fine moduli space if and only if G acts freely and properly on .G. In that case, the canonical projection .p : X −→ X/G defines a holomorphic principal G-bundle. Proof Let us first assume that .[X/G] admits a fine moduli space M. Then the canonical morphism .X −→ [X/G] ≃ M is a principal G-bundle, so G acts freely and properly on X and .M ≃ X/G. Conversely, if G acts freely and properly on X, let us consider the principal G-bundle .X −→ X/G. For all .S ∈ An, an object in .[X/G](S) is a principal G-bundle .P −→ S equipped with a G-equivariant map .u : P −→ X. Since .X/G is a manifold, the G-equivariant morphism u induces a morphism of complex analytic manifolds .uˆ : P /G ≃ S −→ X/G ≃ M. Conversely, such a map .f : S −→ M induces, by pullback of the principal G-bundle .X −→ X/G ≃ M, a principal G-bundle .f ∗ X −→ S, equipped by construction with a G-equivariant map .P −→ X. These two maps set up natural isomorphisms .[X/G](S) ≃ M(S), hereby proving that .[X/G] is representable by M. ⨆ ⨅ Example 11.5.9 The action of .C∗ on .C2 \{(0, 0)} given by .t ·(z1 , z2 ) := (t ·z1 , t ·z2 ) is proper and free. So the quotient stack .[ (C2 \ {(0, 0)}) / C∗ ] admits .CP1 as its fine moduli space. We can reformulate Proposition 11.5.8 by saying that, for quotient stacks, a fine moduli space exists if and only if the action is free and admits a geometric quotient.

11.5.2 GIT Quotients To make sense out of Example 11.5.6 from a general point of view, we can use the Kempf-Ness formulation of Geometric Invariant Theory [7, 8]. The work of Kempf and Ness applies to analytic actions of reductive complex Lie groups .G (i.e. groups that are isomorphic to the complexification of their maximal compact subgroup) and, at least when .X can be embedded equivariantly and linearly onto a closed submanifold of some .CN (meaning G-equivariantly with respect to a .G-action on N that is induced by a linear representation .G −→ GL(N ; C)), it enables us to .C say us two things: (1) If .O ⊂ X is a .G-orbit, then its closure .O contains a unique closed orbit. As a consequence, the relation .O1 ∼ O2 if .O1 ∩ O2 /= ∅ is an equivalence relation on the set of .G-orbits (the previous result on the closure of an orbit being used to prove that this relation is transitive).

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(2) The set of closed .G-orbits in .X, denoted by .X//G, which by definition is the quotient of .X/G by the previous equivalence relation, is a categorical quotient for the .G-action on .X. In particular, the holomorphic functions on .X//G are the G-invariant holomorphic functions on .X. This quotient is called the GIT quotient for the action of .G on .X. There is in fact a more general situation, in which .X can be embedded .Gequivariantly onto a locally closed submanifold of some .CPN −1 with respect to a linearisable action of G, meaning that .G is required to act on .CPN−1 via a linear representation .G −→ GL(N; C) and the canonical action of .GL(N; C) on .CPN−1 . In that case, one has to restrict the action to the so-called semistable locus .Xss ⊂ X, which is a G-invariant open subset of .X that can be defined as follows: Xss = {x ∈ X ⊂ CPN −1 | 0 /∈ G · x, ˜ where x˜ ∈ CN is a lift of x}.

.

If .Xss /= ∅, then one can look at .G-orbits that are closed in .Xss and proceed as before to form the categorical quotient .Xss //G. The result, however, can depend strongly on the linearisation (see Example 11.5.10). In practice, it suffices to test semistability with respect to all 1-parameter subgroups .λ : C∗ −→ G, which is done via the Hilbert-Mumford criterion:  Xss = x ∈ X ⊂ CPN−1 | ∀ λ : C∗ −→ G, ∃ μ ⩾ 0,    lim t μ λ(t) · x˜ exists in CN and is /= 0 .

.

t→0

Example 11.5.10 Let us equip .C2 with the .C∗ -action given by .t · (z1 , z2 ) = (tz1 , tz2 ), and identify it with an open subset .X ⊂ CP2 via .(z1 , z2 ) I−→ [1 : z1 , z2 ], then: (1) With respect to the linearisation .t · [z0 : z1 : z2 ] := [tz0 : tz1 : tz2 ], it can be checked (via the Hilbert-Mumford criterion) that .Xss = ∅, so there is no categorical quotient in this case. (2) With respect to the so-called trivial linearisation .t · [z0 : z1 : z2 ] := [z0 : tz1 : tz2 ], the semistable locus is   Xss = [z0 : z1 : z2 ] ∈ CP2 | z0 /= 0 = X ≃ C2 .

.

The only closed orbit is .{(0, 0)} so the categorical quotient is a point in this case. (3) With respect to the linearisation .t · [z0 : z1 : z2 ] := [t −1 z0 : tz1 : tz2 ], the semistable locus is     Xss = [z0 : z1 : z2 ] ∈ CP2 | z0 /= 0 and (z1 , z2 ) /= (0, 0) ≃ C2 \ (0, 0) .

.

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All .C∗ -orbits are then closed in .Xss , so in this case the categorical quotient is   Xss //C∗ ≃ C2 \ {(0, 0)} / C∗ ≃ CP1 .

.

For a presentation of geometric invariant theory in a complex analytic setting (using the results of Kempf and Ness), we refer for instance to [18], the point being that, via that theory, linearisable actions of reductive complex Lie groups on quasiprojective analytic manifolds give rise to special categorical quotients called GIT quotients. More precisely, we have the following result. Proposition 11.5.11 Let .G be a reductive complex Lie group acting linearly on a quasi-projective analytic manifold .X ⊂ CPN −1 . If .Xss /= ∅, then the GIT quotient Xss //G := {closed G-orbits in Xss }

.

is a coarse moduli space for the analytic stack .[Xss /G]. Proof Geometric invariant theory tells us that, as .G is reductive, the space of closed semistable orbits .Xss //G carries a structure of complex analytic manifold that makes it a categorical quotient for the action of .G on .Xss . By Corollary 11.5.3, the GIT quotient .Xss //G is therefore a coarse moduli space for the analytic stack .[Xss /G]. ⨆ ⨅ Remark 11.5.12 When a reductive group .G acts on a locally closed analytic submanifold .X ⊂ CN , we can embed .X equivariantly in .CPN−1 with respect to the trivial linearization .g · [z0 : w] = [z0 : g · w], where .w ∈ CN . Then .Xss = X, so in that case the quotient stack .[X/G] admits a coarse moduli space. Example 11.5.13 Let us retake the example of the orbifold .[C/μn (C)], where the group .μn (C) ≃ Z/nZ of n-th roots of unity acts on .C via .ζ · z = ζ z (Examples 11.4.9 and 11.4.12). This action is proper but not free. Since .μn (C) is a finite group, it is reductive. So we get a GIT quotient whose orbits are all closed, and we note that such a GIT quotient is necessarily a geometric quotient in the sense of Definition 11.5.4, because by construction the holomorphic functions on a GIT quotient .Xss //G correspond to .G-invariant functions on .Xss . In the present example, we can identify this geometric quotient with .C as a complex analytic manifold, via the holomorphic map .j : C/μn (C) −→ C induced by the .μn (C)-invariant map n .z I−→ z (see Diagram 11.20). The same occurs if we replace .C by an open disk centred at the origin.

.

(11.20)

To sum up, when .G is a reductive complex analytic Lie group acting linearly on a quasi-projective complex analytic manifold .X with non-empty semistable locus

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Xss , GIT quotients .Xss //G exist in .An and satisfy the following properties:

.

GIT quotient

.

 ⇒ coarse moduli space/categorical quotient for the quotient stack Xss /G

and  GIT quotient + closed orbits ⇒ geometric quotient for the quotient stack Xss /G .

.

By definition, the points of the GIT quotient .Xss //G are the closed .G-orbits in .Xss , so saying that all orbits are closed is exactly saying that .Xss //G = Xss /G (i.e. that the GIT quotient is an orbit space). Finally: GIT quotient + closed orbits + free action

.

 ⇒ fine moduli space for the quotient stack Xss /G .

11.6 The Moduli Stack of Elliptic Curves 11.6.1 Families of Elliptic Curves Given a complex analytic manifold .S ∈ An, we define .M1,1 (S) to be the category of analytic families of elliptic curves parameterised by S, in the following sense. Definition 11.6.1 A family of elliptic curves parameterised by a complex analytic manifold .S ∈ An is a triple .(E, π, e) where: (1) (2) (3) (4)

E is a complex analytic manifold. π : E −→ S is a submersion in .An. .e : S −→ E is a section of .π. For all .s ∈ S, the pair .(Es := π −1 ({s}), e(s)) is a complex elliptic curve, i.e. a compact connected Riemann surface .Es of genus 1, equipped with a marked point .e(s) ∈ Es . In particular, the map .π is surjective. .

We define a morphism of families from .(E1 , π1 , e1 ) to .(E2 , π2 , e2 ) to be an isomorphism .u : E1 −→ E2 in .An such that .π2 ◦ u = π1 and .u ◦ e1 = e2 .

.

Example 11.6.2 The Legendre family .E = ⨆u∈C\{0,1} Cu defined in Example 11.2.3 is a family of elliptic curves in the sense of Definition 11.6.1, parame-

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terised by .S := C \ {0, 1}. Indeed,

 E := (u, [x : y : z]) ∈ S × CP2 | y 2 z = x(x − z)(x − uz)

.

is a complex analytic submanifold of .S × CP2 , mapping to .S = C \ {0, 1} via the projection .π to the first factor, which is a submersion. The fibre of .π : E −→ S above .u ∈ S is the genus 1 Riemann surface .Cu constructed in Example 11.2.3, and the points .[0 : 0 : 1] ∈ Cu define a global section .e : s I−→ (s, [0 : 0 : 1]) of .π. We shall often drop .π from the notation and simply write .(E, e) for a family of elliptic curves parameterised by S. Since we have restricted ourselves to isomorphisms between analytic families, the category .M1,1 (S) is a groupoid, for all .S ∈ An. To check that this defines a prestack M1,1 : Anop −→ Groupoids

.

in the sense of Definition 11.3.2, we must check that, for all morphism .f : T −→ S in .An, we can define a natural pullback functor .f ∗ : M1,1 (S) −→ M1,1 (T ). To do this, we set, for all .(E, π, e) ∈ M1,1 (S): f ∗ E := {(x, t) ∈ E × T | π(x) = f (t)}

.

(11.21)

Since .π : E −→ S is a holomorphic submersion, the topological space .f ∗ E admits a structure of complex analytic manifold, turning the canonical projection ∗ ∗ .f E −→ T into a submersion. Moreover, for all .t ∈ T , the fibre .(f E)t ≃ Ef (t) has a canonical structure of complex elliptic curve, with marked point .e(f (t)) = (f ∗ e)(t). This proves that .(f ∗ E, f ∗ e) is an analytic family of elliptic curves parametrised by T and that .f ∗ E if the fibre product of E and T over S in .An. So .E I−→ f ∗ E is a functor with respect to E and, if .g : U −→ T is a morphism in .An, we can identify canonically .g ∗ (f ∗ E) and .(f ◦ g)∗ E in .M1,1 (U ), which is what we need for Definition 11.3.2. Remark 11.6.3 In the general definition of a family of complex analytic manifolds, the condition that the holomorphic submersion .π : E −→ S also be proper is often incorporated in the definition. In particular, with that definition, the fibres of .π are automatically compact. Definition 11.6.4 The prestack .M1,1 defined, for all complex analytic manifold S ∈ An, by

.

  M1,1 (S) = analytic families of elliptic curves parameterised by S

.

is called the moduli stack of elliptic curves. The term stack used in Definition 11.6.4 is justified by the following result.

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Proposition 11.6.5 The prestack .M1,1 is a stack. Proof We have to check that the conditions of Definition 11.3.7 are satisfied by the prestack .M1,1 . This means showing that analytic families of elliptic curves parameterised by .S ∈ An, and morphisms between them, can be constructed by gluing. Explicitly, we want to show that, given an analytic manifold .S and an open covering .(Si )i∈I of .S, the following two properties hold: (1) If .E, E ' −→ S are families of elliptic curves parameterised by .S and .ϕi : E|Si −→ E ' |Si are morphisms between the pullbacks of these two families to .Si , satisfying the compatibility conditions .ϕi |Si ∩Sj = ϕj |Si ∩Sj over .Si ∩ Sj , then there exists a unique morphism .ϕ : E −→ E ' in .M1,1 (S) such that, for all .i, .ϕ|Si = ϕi . (2) If, for all .i in .I , we are given a family of elliptic curves .πi : Ei −→ Si , as well as isomorphisms ≃

ϕij : Ej |Si ∩Sj −→ Ei |Si ∩Sj

.

satisfying the cocycle conditions .ϕii = idEi and .ϕij ◦ ϕj k = ϕik over .Si ∩ Sj ∩ Sk , then there exists a family of elliptic curves .E −→ S together with isomorphisms .ϕi : E|Si −→ Ei over .Si , such that, for all .i, j in .I , one has −1 .ϕi ◦ ϕ j = ϕij over .Si ∩ Sj . The first property holds because we can simply define .ϕ(x) := ϕi (x) if .x ∈ E|Si and this gives a well-defined map that is a morphism of families of elliptic curves from ' .E to .E , with the property that .ϕ|Si = ϕi . And the second property holds because the complex analytic manifold E := ⨆i∈I Ei

.



∼

where .(i, x) ∼ (j, y) if .x = ϕij (y) satisfies the required properties by construction: it is equipped with a holomorphic submersion    π : E −→ ⨆i∈I Si ∼ = S

.

(11.22)

whose fibres are elliptic curves because the gluing procedure guarantees that .E comes equipped with isomorphisms between the open subsets .E|Si ⊂ E and the family .Ei . ⨆ ⨅ Remark 11.6.6 Note that if we had asked, in Definition 11.6.5, that the holomorphic submersion .π : E −→ S also be proper, Property (2) in the proof of Proposition 11.6.5 would be more difficult to show: we would also need to prove that the map (11.22), constructed by gluing, is proper, which is in fact the case when the morphisms .πi : Ei −→ Si , from which .π : E −→ S is constructed, are also assumed to be proper.

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We ultimately want to argue that the reason why the stack .M1,1 is called a moduli stack is because of the existence of a universal family on it. This will require an explanation, but the idea is that there exists, because of the geometric content behind the definition of .M1,1 , a family of elliptic curves .U1,1 −→ M1,1 with the following property: For all S ∈ An and all E ∈ M1,1 (S), there exists a unique morphism f : S −→ M1,1 such that f ∗ U1,1 ≃ E.

.

(11.23)

Note that, for Property (11.23) to make sense, we must provide a definition of a family of elliptic curves parameterised by a stack instead of a manifold. Once this is done, we can get an intuition of how we can define a universal family. Definition 11.6.7 Let .N be a stack over .An. A family of elliptic curves over .N is a morphism of stacks .χ : E −→ N such that: (1) The morphism .χ : E −→ N is representable. (2) For all .S ∈ An and all morphism .f : S −→ N, the pullback f ∗ E ≃ S ×N E ∈ An

.

is a family of elliptic curves over .S, in the sense of Definition 11.6.1. In Definition 11.6.7, the role of Condition (1) is to ensure that Condition (2) makes sense. Indeed, Condition (1) guarantees that .f ∗ E is a manifold for all morphism .f : S −→ N with .S ∈ An. In view of Definition 11.6.7, we can extend .M1,1 to the category of stacks by defining groupoids  M 1,1 (N) := {E −→ N : family of elliptic curves on N} .

.

(11.24)

Note that a family of elliptic curves .E −→ N induces a morphism of stacks .F : N −→ M1,1 , defined, for all .S ∈ An, by taking .(u : S −→ N) to .u∗ E ∈ M1,1 (S). For this to be compatible with the Yoneda embedding .M1,1 I−→ MorSt (−, M1,1 ), it is necessary and sufficient to assume that the the identity morphism .idM1,1 comes, in the same fashion, from a family .U1,1 −→ M1,1 . If this is the case, then by the same argument as in Example 11.3.5, the family .U1,1 must indeed satisfy the universal property (11.23). So in order to define a universal family of elliptic curves, it suffices to be able to interpret .idM1,1 as a family of elliptic curves over the stack .M1,1 . Remark 11.6.8 The preceding discussion suggests that all stacks can be abstractly thought of as moduli stacks: if .M is a stack on .An, we can consider the 2-functor .MorSt (−, M) : St/An −→ Groupoids and define an extension of .M : An −→  : St/An −→ Groupoids by setting .M(N)  Groupoids to .M := MorSt (−, M) for  all .N ∈ St/An. The role of the universal family .N ∈ M(M) is then played by the

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morphism .idM . But ultimately we want to have a more concrete (and geometrically interpretable) description of universal families, which will depend on the precise stack .M being studied. When .M = M1,1 , we will describe the universal family of elliptic curves in Corollary 11.6.24. For the moment, we content ourselves with two elementary but fundamental examples of universal families. Example 11.6.9 Let G be a complex analytic group and let .BG := [•/G] be the classifying stack of principal G-bundles. By Example 11.3.5, for all manifold .S ∈ An, the groupoid .BG(S) is the category of principal G-bundles on X, and by Example 11.3.18, the stack BG is an analytic stack. The universal family is then the morphism .EG := • −→ BG = [•/G]. Indeed, it is a representable morphism by Example 11.3.14 and, by Example 11.3.11, it satisfies the universal property (11.23). This means that, for all .S ∈ An and all principal G-bundle P on S, there exists a unique morphism of stacks .f : S −→ BG such that .P ≃ f ∗ EG. Because of this universal property, the principal G-bundle .EG −→ BG is called the universal bundle over the classifying stack BG. Note that EG is indeed a principal G-bundle over the stack BG, where this notion is to be taken in a sense similar to the covering stacks of Definition 11.4.8: a representable morphism .EG −→ BG whose pullback to a manifold .S ∈ An along a morphism of stacks .S −→ BG is a principal G-bundle in .An. Also, comments similar to Proposition 11.4.6 and Remark 11.4.7 apply: since BG is an analytic stack, to check that the representable morphism .EG −→ BG is a principal G-bundle, it suffices to check it with respect to the atlas .EG −→ BG, over which the pullback EG is just .EG × G, the trivial principal G-bundle over .EG = •, as seen by taking .X = • in Diagram (11.17). Remark 11.6.10 Note that BG is not a representable stack as soon a G is nontrivial, but that the universal family .EG = • is (representable by) a manifold. Example 11.6.11 As a generalisation of Example 11.6.9, consider the principal Gbundle .X −→ [X/G], where G is a complex analytic group acting on a complex analytic manifold X. To say that this is the universal family over the stack .[X/G] defined in Example 11.3.1 is to say that, for all .S ∈ An and all object .(P , u) ∈ [X/G](S), there exists a unique morphism of stacks .f : S −→ [X/G] such that ∗ ∗ .P ≃ f X and that the G-equivariant morphism .f X −→ X coincides with .u : P −→ X. In other words, it means finding an .f : S −→ [X/G] such that the diagram

.

is a pullback diagram. This is possible because the principal G-bundle P is locally trivial over S and we have already noted that, if .P = X × G and .u : X × G −→ X is the action map, then Diagram (11.17) is a pullback diagram.

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11.6.2 Families of Framed Elliptic Curves We now wish to study the representability of the moduli stack .M1,1 . As we shall see, this stack can be seen not to be representable for rather general and abstract reasons, which have to do with the fact that the objects that it parameterises, namely elliptic curves, have non-trivial automorphisms (Corollary 11.2.18) and this prevents the moduli stack of elliptic curves from being representable (Proposition 11.6.14). Since one can get rid of automorphisms by adding a framing to elliptic curves (Theorem 11.2.13), a natural route to study .M1,1 is to introduce a moduli stack fr of framed elliptic curves .M1,1 and relate it to .M1,1 . We will ultimately show fr that .M1,1 , in contrast to .M1,1 , is representable (by the Teichmüller space .h) and this will imply that the moduli stack .M1,1 is isomorphic to the quotient stack .[h/SL(2; Z)]. This last step is what shows that .M1,1 is an orbifold in the sense of Definition 11.4.10 (in particular, an analytic stack with atlas .h) and that it admits the modular curve .h/SL(2; Z) as a coarse moduli space. But before we can get to all this, let us make a couple of remarks. Remark 11.6.12 Let us assume that there exists a family of elliptic curves U1,1 −→ M1,1 (in the sense of Definition 11.6.7) such that the associated morphism .M1,1 −→ M1,1 (defined, for all .S ∈ An, by sending a morphism of stacks .f : S −→ M1,1 to the family of elliptic curves .f ∗ U1,1 ∈ M1,1 (S)) is the identity morphism. Then, as we have noted, the stack .U1,1 is a universal family in the sense that it satisfies the universal property (11.23). As suggested by Example 11.6.10, the representability of the universal family need not imply the representability of the moduli stack. In the reverse direction, however, the representability of .M1,1 (by a manifold .M1,1 , say) would imply that the universal family .U1,1 is representable (by the fibre product .M1,1 ×M1,1 U1,1 , which is indeed a manifold because the morphism of stacks .U1,1 −→ M1,1 is by assumption representable).

.

Remark 11.6.13 More generally, if we can prove that .M1,1 is an analytic stack, then the universal family .U1,1 (which we have not yet constructed), is automatically an analytic stack. Indeed, since .U1,1 −→ M1,1 is representable, the fibre product of .U1,1 with an atlas .X −→ M gives a manifold .X ×M1,1 U1,1 , which is an atlas for .U1,1 because it is straightforward to check that all morphisms in the pullback diagram

.

are representable surjective submersions. Independently of these remarks, we can prove the following result.

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Proposition 11.6.14 The moduli stack of elliptic curves .M1,1 is not representable by a manifold. Proof Let us assume that the stack .M1,1 is representable by a manifold .M1,1 ∈ An, i.e. that .M1,1 ≃ MorAn (−, M1,1 ). The contradiction will come from a special property of representable functors such as .MorAn (−, M1,1 ), namely that if .f : X −→ Y is a covering map in .An, then the induced map f ∗ : MorAn (Y, M1,1 ) −→ MorAn (X, M1,1 )

.

(11.25)

is injective. In fact, it suffices to know that this property holds for Galois covers Y = X/G where G is a finite group acting by automorphisms on Y . In this case, the image of the pullback map (11.25) is the set .FixG MorAn (X, M1,1 ) of G-invariant complex analytic maps from X to .M1,1 and since such a map induces a well-defined complex analytic map from Y to .M1,1 , the pullback map .f ∗ is injective. Thus, to prove that .M1,1 is not representable, it suffices to find a Galois covering map .π : X −→ X/G and two non-isomorphic families of elliptic curves .E1 , E2 on .X/G such that .π ∗ E1 ≃ π ∗ E2 as families of elliptic curves over X. Indeed, this will contradict the injectivity of .π ∗ : M1,1 (Y ) −→ M1,1 (X) which has to hold if we assume that .M1,1 ≃ MorAn (−, M1,1 ), i.e. that .M1,1 is representable. And this is when the fact that there are elliptic curves with non-trivial automorphisms comes in handy (to construct the families .E1 , E2 ). Let .C be the elliptic curve .C := C/(Z1 ⊕ Zi) and set .X := C . We consider the action of the group .G := Z/2Z induced on .E := X × C by the involution

.

    [x], [z] I−→ [x + 12 ], [−z]

.

(11.26)

and the quotient .E1 := E/G. Since G acts freely on X, the quotient .Y := X/G is a Riemann surface and the canonical projection .π : X −→ Y is a two-to-one cover. Note that Y has genus 1 but does not carry a preferred marked point because G does not act on X by automorphisms of elliptic curves. Moreover, there is an induced complex analytic map .E1 −→ Y , which is locally trivial with fibres isomorphic to .C . Since the action of G on the fibres of E is an action by automorphisms of elliptic curves, the map .E1 −→ Y is in fact a family of elliptic curves. The point is that .E1 is not trivial but pulls back to the trivial family .E = X × C . Since the product family like .E2 := Y × C also pulls back to E, we see that the map ∗ .π : M1,1 (Y ) −→ M1,1 (X) is not injective. Note that, to see that .E1 is not trivial, it suffices to show that it does not carry a non-vanishing holomorphic volume form (which a product of two Riemann surfaces of genus 1 would), because otherwise the pullback of such a volume form to .E = X × C would be G-invariant and equal to .cdx ∧ dz for some constant .c /= 0, contradicting the fact that .dx ∧ dz is G-anti-invariant with respect to the involution (11.26). ⨆ ⨅ Remark 11.6.15 The technique used in the proof of Proposition 11.6.14 can be adapted to show the non-representability of various moduli stacks. It rests on the fact

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that a representable stack .M ≃ MorAn (−, M) is a sheaf in various Grothendieck topologies on .An, in particular the étale topology. So, to prove that a stack .M is not representable, it suffices to find an étale covering .π : Y −→ X such that .π ∗ : M(X) −→ M(Y ) is not injective. Indeed, this injectivity is part of the conditions of being a sheaf: the restriction map (or pullback) to an open covering (or more generally an étale cover) is an injective map (if two sections of a sheaf coincide on every open of the cover, they are equal). Then, in order to show a lack of injectivity, the usual technique (as in the proof of Proposition 11.6.14) consists in finding a nontrivial family whose pullback to a finite cover is trivial. The construction of such a family (called an isotrivial family) is then typically obtained by taking the quotient of a trivial family by a finite group acting by automorphisms on the fibre and freely on the base. We now want to introduce families of framed elliptic curves. Recall that, in Definition 11.2.9, we defined a framed elliptic curve to be a triple .(C , e, (α, β)) consisting of an elliptic curve .(C , e) equipped with a direct basis .(α, β) of .H1 (C ; Z), i.e. a basis satisfying the positivity condition (11.7), and this was convenient in order to classify framed elliptic curves and prove that they have no non-trivial automorphisms (Theorem 11.2.13). While we can generalise this point of view to families of elliptic curves, it is not well-adapted to the more general moduli theory of higher genus curves (see also Remark 11.6.21). So, before we proceed, we give an equivalent of a framed elliptic curve, based on the following result. Lemma 11.6.16 Let .(C0 , e0 , (α0 , β0 )) and .(C1 , e1 , (α1 , β1 )) be framed elliptic curves. Then there exists an orientation-preserving homeomorphism .f : (C0 , e0 ) −→ (C1 , e1 ) such that .f∗ (α0 ) = α1 and .f∗ (β0 ) = β1 . Note that the condition of being an orientation-preserving homeomorphism from C0 to .C1 makes sense because complex analytic elliptic curves, like all Riemann surfaces, are canonically oriented topological surfaces. In the compact connected case, this means that a group isomorphism .H2 (Ci ; Z) ≃ Z has been chosen, and for a homeomorphism .f : C0 −→ C1 to be orientation-preserving means that the induced group isomorphism .f∗ : H2 (C0 ; Z) −→ H2 (C1 ; Z) commutes to these identifications. Equivalently, the orientation of .Ci gives a notion of direct basis on .H1 (Ci ; Z), and the induced group isomorphism .f∗ : H1 (C0 ; Z) −→ H1 (C1 ; Z) sends a direct basis to a direct basis. .

Proof of Lemma 11.6.16 By Theorem 11.2.13, we can assume that .(C0 , e0 , (α0 , β0 )) and .(C1 , e1 , (α1 , β1 )) are framed elliptic curves of the form .(C/Λ(τi ), 0, (1, τ )) for some .τ0 , τ1 ∈ C such that .Im(τi ) > 0. Then let us consider the map .f˜ : C −→ C defined by z = x + iy I−→

.

(τ1 − τ0 )z − (τ1 − τ0 )z Re(τ1 − τ0 ) Im(τ1 ) =x+ y+i y τ0 − τ0 Im(τ0 ) Im(τ0 )

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which is the orientation-preserving invertible .R-linear map from .C ≃ R2 to itself defined by the matrix  1 Re(τ1 − τ0 )/Im(τ0 ) 0 Im(τ1 )/Im(τ0 )

 .

(11.27)

and is the unique .R-linear map from .R2 to itself sending the basis .(1, τ0 ) to the basis .(1, τ1 ). By construction, we have .f˜(m + nτ0 ) = m + nτ1 , so .f˜ induces an orientation-preserving homeomorphism .f : C/Λ(τ0 ) −→ C/Λ(τ1 ), sending 0 to 0 while also taking the framing .(1, τ0 ) to the framing .(1, τ1 ). ⨆ ⨅ Note that, by the Cauchy-Riemann equations, the map .f˜ : C −→ C is holomorphic if and only if the matrix (11.27) is a similitude matrix, which happens if and only if .τ0 = τ1 . This is consistent with the fact that framed elliptic curves have no non-trivial holomorphic automorphisms. More importantly for us, Lemma 11.6.16 implies that there is a bijection between framings of an elliptic curve .(C0 , e0 ) and homotopy classes of orientation-preserving self-homeomorphisms .f ∈ Homeo+ (C0 , e0 ), where the latter notation means homeomorphisms .f : C0 −→ C0 that preserve the canonical orientation of .C0 and that satisfy .f (e0 ) = e0 . Indeed, the action of the modular group Mod(C0 , e0 ) := Homeo+ (C0 , e0 ) / homotopy

.

(11.28)

on the set of direct bases of .H1 (C ; Z) ≃ Z2 is transitive by Lemma 11.6.16 and free because, as .C has genus 1, its universal cover is .C so an orientation-preserving self-homeomorphism of .(C , e) lifts to a .π1 (C , e)-equivariant orientation-preserving self-homeomorphism of .(C, 0) and, as .C is contractible, two such transformations are .π1 (C , e)-equivariantly homotopic precisely when they induce the same group automorphism of .H1 (C ; Z), which can be seen as a lattice in .C. Thus, in the situation of Lemma 11.6.16, two orientation-preserving homeomorphisms .f, g : (C0 , e0 ) −→ (C1 , e1 ) that both send .(α0 , β0 ) to .(α1 , β1 ) will satisfy .g −1 ◦f ∼ idC0 , so g will indeed be homotopic to f in .Homeo+ (C0 , e0 ). Note that the fact that the modular group of a complex analytic elliptic curve .(C0 , e0 ) acts freely and transitively on the set of direct bases of .H1 (C0 ; Z) ≃ Z2 also proves that Mod(C0 , e0 ) ≃ SL(2; Z)

.

(11.29)

so Definition 11.28 is consistent with the terminology introduced in Sect. 11.2.3. We can now redefine framed elliptic curves as follows (compare Definition 11.2.9). Definition 11.6.17 Let us fix a complex analytic elliptic curve .(C0 , e0 ). Then, given a complex elliptic curve .(C , e), we define a framing .[f ] of .(C , e) to be a homotopy class of orientation-preserving homeomorphism, i.e.   [f ] ∈ Homeo+ (C0 , e0 ), (C , e) / homotopy.

.

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The set of all framings of .(C , e) will be denoted by .Fr(C0 ,e0 ) (C , e). It is acted upon freely and transitively by the modular group .Mod(C0 , e0 ), via the right action defined, for all .[g] ∈ Mod(C0 , e0 ) by [f ] · [g] := [f ◦ g].

.

(11.30)

Thanks to Definition 11.6.17, we will define a framing of a family of elliptic curves .(E, π : E −→ S, e) in the sense of Definition 11.6.1 essentially as a framing of the elliptic curve .(Es , e(s)) for all .s ∈ S, i.e. as the data, for all .s ∈ S, of a framing .σ (s) ∈ Fr(C ,e ) (Es , e(s)), where .(C0 , e0 ) is a fixed elliptic curve. In order for .σ (s) 0 0 to depend continuously on s, it suffices to construct a fibre bundle .Fr (E, e) −→ S whose fibre over .s ∈ S is .Fr(C0 ,e0 ) (Es , e(s)) and to define .σ as a continuous section of that fibre bundle. We will follow Grothendieck’s construction in [5]. So let us start with a family of elliptic curves .(E, π : E −→ S, e). By Definition 11.6.1, this means in particular that .π : E −→ S is a surjective holomorphic submersion with compact connected fibres over the manifold S. By the so-called monotone-light factorisation (see for instance [17, p. 102]), such a map is necessarily proper. But then, the fact that .π is a submersion implies that, for all .z ∈ E, there exists an open neighbourhood U of .x := π(z) in S, an open neighbourhood .V ⊂ π −1 (U ) of z in E, an open subset .Ω ⊂ π −1 ({x}) and a biholomorphic map .V ≃ U × Ω over U [6, Proposition 6.2.3 p. 240]. Such a map is called simple in [5], where the base of a family .E −→ S is a general complex analytic space, not necessarily a manifold. Note that, unless .V = π −1 (U ) and −1 ({x}), this is strictly weaker than .π being a locally trivial holomorphic .Ω = π vector bundle. However, a simple holomorphic map .π : E −→ S is a locally trivial topological fibre bundle. When S is a manifold, this is Ehresmann’s theorem, since a simple map is submersive, and for the case when S is an analytic space, which we will not require, we refer to [5, Lemma 2.1, p. 3]. The point here is that a family of elliptic curves is in particular a locally trivial continuous map .π : E −→ S, whose fibres are oriented compact connected topological surfaces of genus 1 equipped with a marked point, meaning that for all .s ∈ S, the set .Homeo+ ((C0 , e0 ), (Es , e(s))) is non-empty. In particular, we may view .π : E −→ S as a topological fibre bundle equipped with a continuous section .e : S −→ E, with typical fibre .(C0 , e0 ) and structure group .Homeo+ (C0 , e0 ). Then, by taking homotopy classes of the transition functions of E, we can associate to it the locally trivial topological fibre bundle with discrete fibres .p : Fr (E, e) −→ S defined by Fr (E, e) :=



.

  Fr(C0 ,e0 ) Es , e(s)

s∈S

where   Fr(C0 ,e0 ) (Es , e(s)) = Homeo+ (C0 , e0 ), (Es , e(s)) / homotopy

.

(11.31)

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so .Fr (E, e) −→ S is actually a principal covering space of S, with structure group the discrete group .Mod(C0 , e0 ) introduced in (11.28). We will call .Fr (E, e) the frame bundle of the family .(E, π, e). Its construction only depends on the homeomorphism class of the fixed complex elliptic curve .(C0 , e0 ), which is that of a torus .S 1 × S 1 , equipped with a marked point. This construction is functorial, in the sense that a morphism of families of elliptic curves .u : (E1 , e1 ) −→ (E2 , e2 ) (Definition 11.6.1) gives rise to a morphism of principal bundles Fr (u) : Fr (E1 , e1 ) −→ Fr (E2 , e2 )

.

induced fibrewise (for all .s ∈ S) by the map

.

      



Homeo+ C0 , e0 , E1 |{s} , e1 (s) −→ Homeo+ C0 , e0 , E2 |{s} , e2 (s) f (s) − I → us ◦ f (s)

where .us : E1 |{s} −→ E2 |{s} is the isomorphism of elliptic curves induced by the isomorphism .u : (E1 , e1 ) −→ (E2 , e2 ) (recall that, by Definition 11.6.1, all morphisms of families are isomorphisms). Definition 11.6.18 Let us fix a complex analytic elliptic curve .(C0 , e0 ). A family of framed elliptic curves is a quadruple .(E, π, e, σ ) where .(E, π : E −→ S, σ ) is a family of elliptic curves parameterised by .S ∈ An in the sense of Definition 11.6.1 and .σ : S −→ Fr (E, e) is a continuous section of the frame bundle .Fr (E, e) of the family .(E, π, e), meaning that, for all .s ∈ S, we can think of .σ (s) as (the homotopy class of) a homeomorphism .(C0 , e0 ) −→ (Es , e(s)). A morphism between families of framed elliptic curves (E1 , π1 , e1 , σ1 ) −→ (E2 , π2 , e2 , σ2 )

.

is a morphism of families of elliptic curves .u : (E1 , e1 ) −→ (E2 , e2 ) such that the induced morphism of principal covering spaces Fr (u) : Fr (E1 , e1 ) −→ Fr (E2 , e2 )

.

satisfies .Fr (u) ◦ σ1 = σ2 .

.

In particular, all morphisms of framed elliptic curves are isomorphisms, by definition. We shall often drop .π from the notation and simply write .(E, e, σ ) for a family of framed elliptic curves parameterised by S.

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Remark 11.6.19 Note that passing to homotopy classes in the definition of .Fr (E, e) gives us a locally trivial fibration, with the topological group + .Homeo (C0 , e0 ) replaced by the discrete group .Mod(C0 , e0 ) as a fibre. In particular, we can endow .Fr (E, e) with a canonical structure of complex analytic manifold: the one with respect to which the local homeomorphism .p : Fr (E, e) −→ S is holomorphic. It then becomes a local biholomorphism (.= an étale analytic map) and the continuous sections of .Fr (E, e) −→ S are exactly the holomorphic sections with respect to that holomorphic structure. If .f : T −→ S is a morphism in .An and .(E, e, σ ) is a family of framed elliptic curves on S, then the pullback family f ∗ E := {(x, t) ∈ E × T | π(x) = f (t)}

.

(11.32)

defined in (11.21) is canonically framed by the section .f ∗ σ of the frame bundle ∗ ∗ ∗ .f Fr (E, e) = Fr (f E, f e). The next result is then proved similarly to Proposition 11.6.5. Proposition 11.6.20 The functor fr M1,1 :

.

Anop −→ Groupoids   S I−→ analytic families of framed elliptic curves parameterised by S

is a stack, called the moduli stack of framed elliptic curves. It is also clear that there is a morphism of stacks from the moduli stack of framed elliptic curves to the moduli stack of elliptic curves, defined, for families parameterised by .S ∈ An, by forgetting the frame, F :

.

fr M1,1 (S) −→ M1,1 (S) (E, e, σ ) I−→ (E, e)

since this is compatible with taking pullbacks. fr Remark 11.6.21 We could also have defined the objects in .M1,1 (S) as triples ' .(E, e, σ ) where .(E, e) is a family of elliptic curves parameterised by .S ∈ An and .σ ' is a section of the bundle .Fr (E, e) ×Mod(C0 ,e0 ) H 1 (C0 , e0 ), associated to the principal .Mod(C0 , e0 )-bundle .Fr (E, e) via the action of the modular group on .H 1 (C0 , e0 ), such that, for all .s ∈ S, the element .σ ' (s) = (α(s), β(s)) is a direct basis of .H 1 (C0 , e0 ). The upshot is that this is the direct generalisation of Definition 11.2.9 to families of elliptic curves. The fact that one can use one bundle or the other is because, in genus 1, the action of .Mod(C0 , e0 ) on .H 1 (C0 , e0 ) is faithful, which is no longer the case in higher genus. Definition 11.6.18, in contrast, generalises immediately to higher genus (where it is no longer necessary to provide the curves with a base point).

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11.6.3 Orbifold Structure and Coarse Moduli Space Contrary to the moduli stack .M1,1 , which is not representable by Proposifr tion 11.6.14, the moduli stack .M1,1 is representable. To understand why, recall that, for all .S ∈ An,   fr M1,1 (S) = (E, e, σ ) : families of framed elliptic curves on S

.

so, for all .s ∈ S, the framed elliptic curves .(Es , e(s), σ (s)) should be isomorphic, by Theorem 11.2.13, to the framed elliptic curve .(C/Λ(τs ), 0, (1, τs )), for a unique .τs ∈ h. This defines a map fσ :

.

S −→ h s I−→ τs

(11.33)

corresponding to the family .(E, e, σ ), which is in fact a morphism of complex analytic manifolds. By naturality with respect to S, the correspondence .(E, e, σ ) I−→ fσ in (11.33) induces a morphism of stacks fr M1,1 −→ MorAn (−, h)

.

(11.34)

which will be seen to be an isomorphism if we can define a universal family of framed elliptic curves fr fr U1,1 −→ M1,1 .

.

fr Indeed, if for all .τ ∈ h, the framed elliptic curve .(U1,1 )|{s} is isomorphic to .(C/Λ(τs ), 0, (1, τs )), then by construction of the map .fσ introduced in (11.33), we fr have .fσ∗ U1,1 ≃ (E, e, σ ) as families of framed elliptic curves, and .fσ is the unique fr map with that property. So .U1,1 is indeed a universal family of framed elliptic curves and the morphism of stacks defined, for all .S ∈ An, by

.

fr MorAn (S, h) −→ M1,1 (S) fr ∗ (f : S −→ h) I−→ f U1,1

is an inverse to the morphism of stacks .(E, e, σ ) I−→ fσ introduced in (11.34). fr To check that the previous considerations are correct and that .M1,1 is indeed representable by .h, we still need to: (1) Prove that the map .fσ : S −→ h constructed in (11.33) is indeed holomorphic. (2) Construct a universal family of framed elliptic curves over .h, meaning a family fr fr .U 1,1 −→ h such that, for all .τ ∈ h, the fibre of .U1,1 at .τ is isomorphic to the framed elliptic curve .(C/Λ(τ ), 0, (1, τ )).

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For the first part, recall that, because of Definition 11.6.18, the definition of the moduli stack of framed elliptic curves requires fixing an arbitrary complex elliptic curve .(C0 , e0 ) and that it is in fact only the homeomorphism type of that complex elliptic curve that matters. But the latter is uniquely determined: it is the homeomorphism type of the topological torus .S 1 × S 1 , with an arbitrary base point, for instance .(1, 1). The point is that this topological torus comes equipped with a canonical direct homology basis .(α0 , β0 ) for .H1 (S 1 ×S 1 ; Z), and that all other bases can be brought onto that one by an orientation-preserving self-homeomorphism of .S 1 × S 1 . Alternately, we can also choose .(C0 , e0 ) to be the complex torus corresponding to the lattice .Λ(i) := Z1 ⊕ Zi ⊂ C, which comes equipped with the direct basis .(1, i). The point is that we can fix not only .(C0 , e0 ) but also a direct homology basis .(α0 , β0 ) for .H1 (C0 ; Z) and, given a family of framed elliptic curves .(E, π : E −→ S, e, σ ), use that direct basis .(α0 , β0 ) to define the map .ϕσ explicitly as follows: ϕσ :

.

S −→ h



s I−→ τs :=

σ (s)∗ β0 σ (s)∗ α0

ωs ωs



=

 β0 α0

σ (s)∗ ωs

(11.35)

σ (s)∗ ωs

where .σ (s) : C/Λ(i) −→ E(s) is the orientation-preserving homeomorphism determined by .σ , and .ωs is a continuous family of holomorphic 1-form on the elliptic curve .E(s). This last part means that .ω is a section of the sheaf .π∗ Ω1E/S , the push-forward of the sheaf of relative holomorphic differential forms of degree 1 of E over S. It makes sense to see these relative differentials as families of differential forms along the fibres because, as .π : E −→ S is a submersion by assumption, the canonical sequence of .OE -modules 0 −→ π ∗ Ω1S −→ Ω1E −→ Ω1E/S −→ 0

.

is exact and locally split (see for instance [2, Proposition 5, p. 37]). It then follows that the map .ϕσ defined in (11.35) is indeed holomorphic. fr It therefore only remains to construct a universal family .π : U1,1 −→ h. Again, we do this explicitly. First we consider the (free and proper) action of the discrete Abelian group .Z2 on the product .h × C defined as follows and which we write on the right. For all .(m, n) ∈ Z2 and all .(τ, z) ∈ h × C, set (τ, z) · (m, n) := (τ, z + mτ + n).

.

(11.36)

Next we take the quotient manifold fr U1,1 := (h × C)/Z2

.

(11.37)

fr and consider the map .π : U1,1 −→ h induced by the first projection .(τ, z) I−→ τ , which is a well-defined holomorphic submersion. The point is that, for all .τ ∈ h,

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the identity map of .C induces an isomorphism of complex analytic manifolds   π −1 {τ } ≃ C/Λ(τ )

.

fr −→ h has a where .Λ(τ ) = Z1 ⊕ Zτ . Moreover, the family of complex tori .U1,1 fr canonical section .e : h −→ U1,1 , induced by the section .τ I−→ (τ, 0) of the first projection .h × C −→ h. Finally, for all .τ ∈ h, we can let

σ (τ ) : C/Λ(i) −→ C/Λ(τ )

.

be the orientation-preserving homeomorphism defined as in the proof of Lemma 11.6.16, by setting .τ0 = i and .τ1 = τ in (11.27). We have therefore defined fr a family of framed elliptic curves .(U1,1 , e, σ ) in the sense of Definition 11.6.18. And, by construction, we have, for all .τ ∈ h an isomorphism of framed elliptic curves

   fr . U1,1  , e(τ ), σ (τ ) ≃ C/Λ(τ ), 0, (1, τ ) {τ } fr is indeed a universal family. We summarise the previous so the family .U1,1 considerations as follows. fr Theorem 11.6.22 The moduli stack of framed elliptic curves .M1,1 is representable by the Teichmüller space .h. fr is compatible with It remains to show that this representability theorem for .M1,1 the action on .h of the modular group .Mod(C0 , e0 ) which, as in (11.29), we identify with .SL(2; Z). First note that, for all .S ∈ An, all family of elliptic curves .(E, e) over S and all .s ∈ S, the modular group .Mod(C0 , e0 ) acts on the set of framings of the elliptic curve .(Es , e(s)). Indeed, according to Definition 11.6.17, a framing of .(Es , e(s)) is a homotopy class of orientation-preserving homeomorphism .f : C0 −→ Es such that .f (e0 ) = e(s), and we have seen in (11.30) that there is a right action of .Mod(C0 , e0 ) on the set of such framings, defined by .[f ] · [g] = [f ◦ g]. This induces an action of .Mod(C0 , e0 ) on the set .MorAn (S, h), under which the morphism .ϕσ : S −→ h corresponding to the family .(E, e, σ ) is sent to the morphism .ϕσ ·g corresponding to the family .(E, e, σ · [g]), where, for all + .g ∈ Homeo (C0 , e0 ), we denote by .σ · [g] the following section of the frame bundle .Fr(E, e) introduced in (11.31):

(σ · [g])(s) = σ (s) · [g] .

.

(11.38)

There only remains to compute .ϕE·[g] more explicitly. Note that .σ · [g] is a framing of the same family of elliptic curves .(E, e). So, in view of Definition 11.35 and the

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fact that .(f ◦ g)∗ = f∗ ◦ g∗ in homology, we can use Theorem 11.2.14 to conclude that ϕσ ·[g] (s) = ϕσ (s) · [g],

.

(11.39)

where .σ · [g] is defined as in (11.38) and the action on the right-hand side is the SL(2; Z)-action on .h defined in (11.8). We are now in a position to prove that there is an isomorphism of stacks .M1,1 ≃ [h/SL(2; Z)], which by Corollary 11.4.15 will show that the moduli stack of elliptic curves is an orbifold, and to identify the universal family .U1,1 as an orbifold itself. In order to define a morphism of stacks .M1,1 −→ [h/SL(2; Z)], note that, if .S ∈ An and .(E, π : E −→ S, e) ∈ M1,1 (S) is a family of elliptic curves parameterised by S, then there is a morphism of analytic varieties .u : Fr(E, e) −→ h from the frame bundle of .(E, π : E −→ S, e) to the Teichmüller space .h, which is defined as follows. Recall that an element .[f ] ∈ Fr(E, e) is, by Definition 11.6.17 and Eq. (11.31), the homotopy class of an orientation-preserving homeomorphism .f : (C0 , e0 ) −→ (Eπ([f ]) , e(π([f ]))), where .(C0 , e0 ) is our fixed elliptic curve .(C/Λ(i), 0) and .p : Fr(E, e) −→ S is the canonical projection of the frame bundle. So we can define an analytic map .u : Fr(E, e) −→ h by sending .[f ] ∈ Fr(E, e) to the complex number .τp([f ]) defined in (11.35). For the same reason as in Eq. (11.39), the map .u : Fr(E, e) −→ h thus constructed is .SL(2; Z)-equivariant with respect to the .SL(2; Z)-action on .Fr(E, e) defined in (11.30) and the usual .SL(2; Z)-action on .h. And since .Fr(E, e) is a principal .SL(2; Z)-bundle on S and the construction we have just given is compatible with pullbacks with respect to a morphism .T −→ S, we have indeed a morphism of stacks .M1,1 −→ [h/SL(2; Z)]. .

Theorem 11.6.23 Let .S ∈ An. The natural correspondence

.

≃

induces an isomorphism of stacks .Ф : M1,1 −→ [h/SL(2; Z)] over .An. In particular, the moduli stack of complex elliptic curves is a complex analytic orbifold in the sense of Definition 11.4.10, admitting the geometric quotient .h/SL(2; Z) ≃ C as a coarse moduli space in the sense of Definition 11.5.1. Proof We will define an inverse to .Ф : M1,1 −→ [h/SL(2; Z)] in the following way. Given a pair .(P , u), consisting of a principal .SL(2; Z)-bundle .P −→ S and an .SL(2; Z)-equivariant map .u : P −→ h, we consider the pullback family .Eˆ := fr fr over P . We claim that .SL(2; Z) acts on the universal family .U1,1 and that u∗ U1,1 ˆ this induces a free and proper action on the family .E, so we can consider the quotient

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ˆ family .E := E/SL(2; Z) over .S = P /SL(2; Z). The correspondence .(P , u) I−→ E then defines a natural transformation .ΨS : [h/SL(2; Z)](S) −→ M1,1 (S), which provides an inverse to .Ф. In order to check some of the details, we spell out the fr action of .SL(2; Z) on the universal family .U1,1 . fr 2 Recall from (11.37) that .U1,1 = (h × C)/Z , where .Z2 acts on .h via (11.36). In fact, this .Z2 -action extends to an action of the semi-direct product .Z2 ⋊ SL(2; Z), where .SL(2; Z) acts to the left on .Z2 , via the standard representation. Explicitly, if we consider the semi-direct product .Z2 ⋊ SL(2; Z) with group law defined as follows  ' '       ' '   a c a c m ,n , . m, n , bd b' d '    ' '      ' a c ac ' ' ' = m, n + am + cn , bm + dn ) , , bd b' d ' then the formula          aτ + b z + mτ + n a c := m, n , . τ, z · , bd cτ + d cτ + d

(11.40)

ˆ := Z2 ⋊ SL(2; Z) on .h × C. There is therefore an defines a right action of .G ˆ 2 on the universal family induced action of the quotient group .SL(2; Z) = G/Z fr 2 .U 1,1 = (h × C)/Z . Going back to the proof of Theorem 11.6.23, we see that the action (11.40) fr induces an action of .SL(2; Z) on .u∗ U1,1 which, by construction, makes the proper, fr ∗ surjective map .u U1,1 −→ P equivariant. Since the .SL(2; Z)-action on P is free fr and proper, it is also free and proper on .u∗ U1,1 , as proved for instance in [15, Proposition 3.5.8, p. 156]. ⨆ ⨅ As a corollary, we obtain a description of the universal family of (unframed) elliptic curves .U1,1 −→ M1,1 . Contrary to the universal family of framed elliptic fr curves .U1,1 , it is no longer (representable by) a manifold. We can nonetheless identify it explicitly as an analytic orbifold. Corollary 11.6.24 Let .U1,1 −→ M1,1 be the universal family of elliptic curves over the moduli stack .M1,1 . Then there is an isomorphism of stacks   2  U1,1 ≃ (h × C) Z ⋊ SL(2; Z)

.

where the semi-direct product .Z2 ⋊ SL(2; Z) acts on .(h × C) as in (11.40). In particular, the universal family of elliptic curves is an analytic orbifold. The isomorphism .M1,1 ≃ [h/SL(2; Z)] appears as Theorem 4.1 in [5, p. 12], and the fact that .h/SL(2; Z) is a coarse moduli space for .M1,1 is discussed in [5,

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Theorem 5.2, p. 15]. The link with V -manifolds in the sense of Satake is mentioned on p. 16 of [5].

References 1. A.F. Beardon, The Geometry of Discrete Groups, vol. 91. Graduate Texts in Mathematics (Springer, New York, 1995). Corrected reprint of the 1983 original 2. S. Bosch, W. Lütkebohmert, M. Raynaud, Néron Models, vol. 21. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] (Springer, Berlin, 1990) 3. K. Conrad, SL2 (Z). https://kconrad.math.uconn.edu/blurbs/grouptheory/SL(2,Z).pdf 4. O. Forster, Lectures on Riemann Surfaces, vol. 81. Graduate Texts in Mathematics (Springer, New York, 1991). Translated from the 1977 German original by Bruce Gilligan, Reprint of the 1981 English translation 5. A. Grothendieck, Techniques de construction en géométrie analytique. I: Description axiomatique de l’espace de Teichmüller et de ses variantes. Sém. H. Cartan 13 (1960/61), Fasc. 1, No. 7–8, 33 pp. (1962) 6. J.H. Hubbard, Teichmüller Theory and Applications to Geometry, Topology, and Dynamics. Vol. 1 (Matrix Editions, Ithaca, 2006). With forewords by William Thurston and Clifford Earle 7. G.R. Kempf, Instability in invariant theory. Ann. Math. (2) 108(2), 299–316 (1978) 8. G.R. Kempf, L. Ness, The length of vectors in representation spaces, in Algebraic Geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), vol. 732. Lecture Notes in Math. (Springer, Berlin, 1979), pp. 233–243 9. H. McKean, V. Moll, Elliptic Curves (Cambridge University Press, Cambridge, 1997). Function theory, geometry, arithmetic 10. M. Olsson, Algebraic Spaces and Stacks, vol. 62. American Mathematical Society Colloquium Publications (American Mathematical Society, Providence, 2016) 11. I. Satake, On a generalization of the notion of manifold. Proc. Natl. Acad. Sci. USA 42, 359–363 (1956) 12. I. Satake, The Gauss-Bonnet theorem for V -manifolds. J. Math. Soc. Jpn. 9, 464–492 (1957) 13. F. Schaffhauser, Lectures on Klein surfaces and their fundamental group, in Geometry and Quantization of Moduli Spaces. Adv. Courses Math. CRM Barcelona (Birkhäuser/Springer, Cham, 2016), pp. 67–108 14. J.-P. Serre, A Course in Arithmetic, vol. 7. Graduate Texts in Mathematics (Springer, New York-Heidelberg, 1973). Translated from the French 15. W.P. Thurston, Three-Dimensional Geometry and Topology. Vol. 1, vol. 35. Princeton Mathematical Series (Princeton University Press, Princeton, 1997). Edited by Silvio Levy 16. W.P. Thurston, Geometry and topology of three-manifolds (2002). http://library.msri.org/ books/gt3m/. Electronic edition of the 1980 lecture notes (Princeton University) 17. G. Whyburn, E. Duda, Dynamic Topology. Undergraduate Texts in Mathematics (Springer, New York-Heidelberg, 1979). With a foreword by John L. Kelley 18. C.T. Woodward, Moment maps and geometric invariant theory. Les cours du CIRM 1(1), 55–98 (2010)

Chapter 12

Some Footnotes on Thurston’s Notes The Geometry and Topology of 3-Manifolds Athanase Papadopoulos

J’aimais cette résistance coriace dont je ne venais jamais à bout; mystifié, fourbu, je goûtais l’ambiguë volupté de comprendre sans comprendre : c’était l’épaisseur du monde J.-P. Sartre, Les mots

Abstract These are a few historical remarks, addenda and references with comments on some topics discussed by Thurston in his notes The geometry and topology of three-manifolds. The topics are mainly hyperbolic geometry, geometric structures, volumes of hyperbolic polyhedra and the so-called Koebe–Andreev– Thurston theorem. I discuss in particular some works of Lobachevsky, Andreev and Milnor, with an excursus in Dante’s cosmology, based on the insight of Pavel Florensky. Keywords Geometric structure · Hyperbolic structure · .G-structure · (G, X)-structure · Hyperbolic volume · Volume of a hyperbolic tetrahedron · Volume of an ideal tetrahedron · Geometric convergence · Convex co-compact group · Koebe–Andreev–Thurston theorem · Dante · P. A. Florensky · N. I. Lobachevsky

.

AMS Codes 57-03, 57-06, 57K32, 57K35, 53C15, 01-160, 01A75, 01A20

A. Papadopoulos () Institut de Recherche Mathématique Avancée and Centre de Recherche et d’Expérimentation sur l’Acte Artistique (ITI CREAA), Université de Strasbourg and CNRS, Strasbourg, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 K. Ohshika, A. Papadopoulos (eds.), In the Tradition of Thurston III, https://doi.org/10.1007/978-3-031-43502-7_12

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12.1 Introduction William Thurston’s notes The geometry and topology of three-manifolds recently appeared in print, as Volume IV of his Collected Works. The notes are associated with the courses he gave at Princeton University during the academic year 1977– 1978 and the fall semester of 1978–1979. Steve Kerckhoff writes in the Preface of this volume: “Thurston started the first year by writing up notes himself, but after a while he enlisted Bill Floyd and me to help. [. . . ] What was written was a reasonable facsimile of the way Thurston was presenting his amazing mathematics to us, in real time.” The plan of the notes consists of 13 chapters, but Chaps. 10 and 12 are missing, and Chap. 11 contains only a short section on extensions of vector fields. The lectures corresponding to Chap. 7 were given by John Milnor. The overall subject is geometric structures and the topology of three-manifolds. Thurston’s notes require almost no prerequisites from the reader, they use only elementary mathematics, but they are difficult to read: one needs time, visualisation and imagination. On the other hand, reading every section and trying to understand the underlying ideas is always rewarding. It is not an exaggeration to say that hundreds of young mathematicians wrote their PhD thesis by trying to understand some section of these notes. In this chapter, I shall make a few comments which I consider as footnotes to these notes, adding references here and there, sometimes mentioning some related work that preceded the notes, and, more rarely, some developments and examples over the subsequent decades, needless to say, without proofs. My footnotes are divided into 6 sections, numbered 2 to 7. Section 12.2, which follows this introduction, consists of notes on the situation of non-Euclidean geometry at the time where Thurston’s notes were written, and how these notes resurrected the subject. Section 12.3 starts with a sentence of Thurston: “It would nonetheless be distressing to live in elliptic space, since you would always be confronted with an image of yourself, turned inside out, upside down and filling out the entire background of your field of view.” This sentence reminds me of a passage in Dante’s Divine Comedy. Section 12.3 is then an excursion in the world of Dante, who imagined the universe as non-Euclidean, non-orientable and projective. Section 12.4 is a note on the origin of Thurston’s notion of geometric structure, highlighting the work of Ch. Ehresmann. Section 12.5 starts again with a sentence of Thurston on horospheres, and leads to an exposition of spheres and horospheres in hyperbolic space, in the work of N. I. Lobachevsky. Section 12.6 consists of a few remarks on the work of E. M. Andreev to whom Thurston refers in his notes, regarding the classification of discrete isometry groups of hyperbolic space generated by reflections along hyperplanes with compact fundamental domain. This is the basis of what became later the Koebe–Andreev–Thurston theorem. Section 12.7 is concerned with the part of Thurston’s notes that is due to Milnor, on the computation of volume of hyperbolic polyhedra.

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12.2 Non-Euclidean Geometry Chapter 2 of Thurston’s Notes is an introduction to the two non-Euclidean geometries of constant curvature, namely, spherical (or elliptic) and hyperbolic. Let me start by a few words on spherical geometry. Spherical geometry is a classical subject, extensively studied in Greek Antiquity, in particular by Autolycus of Pitane (c. 360–c. 290 BC), Theodosius of Tripoli (c. 160–90 BC) and Menelaus of Alexandria (c. 70–140 AD). In these works, the sphere is considered as space having the same status as Euclidean 2-space, with lines, angles, etc. On the sphere, the lines are the great circles (that is, the intersection of the sphere with planes passing through the origin) and the angle between two lines is the dihedral angle between the planes that define them. The length of a segment joining two points is defined as the angle between the two Euclidean lines that join these points to the origin. With this, one defines triangles on the sphere and studies their properties, in particular trigonometry. After Greek Antiquity, spherical geometry was extensively studied by the medieval Arabs, who developed a complete trigonometric system and who discovered in particular duality and the notion of polar triangle. The modern period of spherical geometry starts with Euler and his collaborators and followers (Johann Lexell, Theodor von Schubert, Nikolaus Fuss, etc.), who worked systematically on analogues in spherical geometry of theorems in Euclidean geometry due to Pappus and others. The references are [4, 54, 80, 84]. Thurston writes in Chap. 2 of his notes: “In the sphere, an object moving away from you appears smaller and smaller, until it reaches a distance of .π/2. Then, it starts looking larger and larger and optically, it is in focus behind you. Finally, when it reaches a distance of .π , it appears so large that it would seem to surround you entirely.” Rather than the sphere, Thurston considers the projective space, the quotient of the sphere by its canonical involution. Unlike the sphere, the projective space is a uniquely geodesic metric space: two points are joined by a single geodesic. The properties to which Thurston refers need a mental effort. After a glimpse into the mystery of elliptic geometry, on which I will comment in the next section, Thurston presents several models of hyperbolic geometry and he explains how to work with them, in particular the sphere of imaginary radius, i.e., the hyperboloid model. The importance of right angled hexagons is highlighted, and Thurston derives the basic trigonometric formulae for triangles, using this model. One might remember that at the time Thurston wrote his notes, non-Euclidean geometry was a dormant subject; there were almost no textbooks on hyperbolic or spherical geometry, except for old books mostly dating back to the nineteenth or the first years of the twentieth century. Let me mention here the interesting PhD thesis, defended in 1892, by L. Gérard, a student of Paul Appell. The president of the jury of this thesis was Henri Poincaré. The dissertation is titled Sur la géométrie non euclidienne. It was published in a book form [30]. In this book, the author derives the non-Euclidean trigonometric formulae by a model-free method, using essentially the fact that the angle sum of any triangle in the hyperbolic plane is .