Strasbourg master class on geometry 9783037191057

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IRMA Lectures in Mathematics and Theoretical Physics 18 Edited by Christian Kassel and Vladimir G. Turaev

Institut de Recherche Mathématique Avancée CNRS et Université de Strasbourg 7 rue René-Descartes 67084 Strasbourg Cedex France

IRMA Lectures in Mathematics and Theoretical Physics Edited by Christian Kassel and Vladimir G. Turaev This series is devoted to the publication of research monographs, lecture notes, and other material arising from programs of the Institut de Recherche Mathématique Avancée (Strasbourg, France). The goal is to promote recent advances in mathematics and theoretical physics and to make them accessible to wide circles of mathematicians, physicists, and students of these disciplines. Previously published in this series: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Deformation Quantization, Gilles Halbout (Ed.) Locally Compact Quantum Groups and Groupoids, Leonid Vainerman (Ed.) From Combinatorics to Dynamical Systems, Frédéric Fauvet and Claude Mitschi (Eds.) Three courses on Partial Differential Equations, Eric Sonnendrücker (Ed.) Infinite Dimensional Groups and Manifolds, Tilman Wurzbacher (Ed.) Athanase Papadopoulos, Metric Spaces, Convexity and Nonpositive Curvature Numerical Methods for Hyperbolic and Kinetic Problems, Stéphane Cordier, Thierry Goudon, Michaël Gutnic and Eric Sonnendrücker (Eds.) AdS/CFT Correspondence: Einstein Metrics and Their Conformal Boundaries, Oliver Biquard (Ed.) Differential Equations and Quantum Groups, D. Bertrand, B. Enriquez, C. Mitschi, C. Sabbah and R. Schäfke (Eds.) Physics and Number Theory, Louise Nyssen (Ed.) Handbook of Teichmüller Theory, Volume I, Athanase Papadopoulos (Ed.) Quantum Groups, Benjamin Enriquez (Ed.) Handbook on Teichmüller Theory, Volume II, Athanase Papadopoulos (Ed.) Michel Weber, Dynamical Systems and Processes Renormalization and Galois Theory, Alain Connes, Frédéric Fauvet and Jean-Pierre Ramis (Eds.) Handbook of Pseudo-Riemannian Geometry and Supersymmetry, Vicente Cortés (Ed.) Handbook on Teichmüller Theory, Volume III, Athanase Papadopoulos (Ed.)

Volumes 1–5 are available from Walter de Gruyter (www.degruyter.de)

Strasbourg Master Class on Geometry Athanase Papadopoulos Editor with contributions by N. A’Campo and A. Papadopoulos, F. Dal’Bo, F. Herrlich, Ph. Korablev and S. Matveev, G. Link, J. Marché, C. Petronio, V. Schroeder

Editor: Athanase Papadopoulos Institut de Recherche Mathématique Avancée CNRS et Université de Strasbourg 7 Rue René Descartes 67084 Strasbourg Cedex France

2010 Mathematical Subject Classification: Primary 51-01, 51-02, 57-01, 57-02; Secondary 14H30, 14H52, 20F67, 20F69, 22E40, 22D40, 30F10, 30F20, 30F45, 30F60, 32G15, 37E30, 51F99, 51M10, 51E24, 53C21, 53C22, 53C23, 53C35, 53C70, 54C20, 57M15, 57M20, 57M27, 57M50, 57N10

ISBN 978-3-03719-105-7 Key words: Hyperbolic geometry, hyperbolic space, neutral geometry, Euclid’s axioms, hyperbolic trigonometry, spherical geometry, Khayyam–Saccheri quadrilaterals, trirectangular quadrilaterals, parallelism, horocycle, Lobachevsky parallelism function, Beltrami–Klein model, polynomial ring model, transitional geometries, coherent family, coherent model, Poincaré model, modular group, geodesic flow, horocyclic flow, Diophantine approximation, quadratic forms, Lie group, representation, symmetric space, three-manifold, triangulation, Heegaard splitting, surgery, spine, normal surface, Haken 3-manifold, algorithmic problems, gauge theory, Chern–Simons theory, geometric quantization, origami, Teichmüller space, complexity, cusped manifold, asymptotic geometry, Gromov hyperbolicity, boundary, Möbius geometry, Ptolemy space, Busemann function, quasi-metric space, quasi-geodesic The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

© 2012 European Mathematical Society Contact address: European Mathematical Society Publishing House ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the authors’ TEX files: I. Zimmermann, Freiburg Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321

Preface In 2008 and 2009, two weeks of intensive courses were organized at the University of Strasbourg under the title “Geometry Master class”. The goal was to give to fifth year students and PhD students in mathematics the opportunity to learn new topics that lead directly to the current research in geometry and topology, and also to give them the opportunity of meeting students from other universities and exchanging ideas for future involvement in research. Both goals were completely attained, and we had two memorable master classes, with more than 60 participants for each, coming from various countries. The courses were accessible to a non-specialized audience, the prerequisites being the usual mathematics that is acquired during the first five years at university and the maturity expected from a graduate student. Specifically, the program included hyperbolic geometry, three-manifold topology, representation theory of fundamental groups of surfaces and of three-manifolds, dynamics on the hyperbolic plane with applications to number theory, Riemann surfaces, Teichmüller theory, Lie groups and asymptotic geometry. Many of the students stated that the courses they heard during these two weeks were the best courses they have ever had. Some of them asked for written notes, and this was the origin of the present volume. Some courses were more specialized than others, as the reader of this book will notice, but the speakers always paid careful attention to the pedagogical needs of the participants, providing background, gentle introductory remarks and motivation. During the two intensive weeks, ten courses were given, eight of which are represented in this volume. The remaining two were given by Marc Burger (on maximal representations) and by Vladimir Fock (on higher Teichmüller spaces). The texts of these two courses are not included in this volume; they appear elsewhere in a different form. Now that these texts are written up, they can serve the whole mathematical community. The present book cannot really render the friendly and warm atmosphere of the master-classes, but it reproduces the beautiful mathematics that was taught there. I would like to take this opportunity to thank Manfred Karbe for his encouragement in publishing this book and Irene Zimmermann for her nice work on the manuscript. Strasbourg, December 2011

Athanase Papadopoulos

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Norbert A’Campo and Athanase Papadopoulos Notes on non-Euclidean geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Françoise Dal’Bo Crossroads between hyperbolic geometry and number theory . . . . . . . . . . . . . . . . . 183 Frank Herrlich Introduction to origamis in Teichmüller space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Philipp Korablev and Sergey Matveev Five lectures on 3-manifold topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Gabriele Link An introduction to globally symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Julien Marché Geometry of the representation spaces in SU.2/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Carlo Petronio Algorithmic construction and recognition of hyperbolic 3-manifolds, links, and graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 Viktor Schroeder An introduction to asymptotic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

Notes on non-Euclidean geometry Norbert A’Campo and Athanase Papadopoulos Mathematisches Institut, Universität Basel Rheinsprung 21, 4051 Basel, Switzerland email: [email protected] Institut de Recherche Mathématique Avancée CNRS et Université de Strasbourg 7 rue René Descartes, 67084 Strasbourg Cedex, France email: [email protected]

Contents 1 2

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On basic notions and axioms of the three geometries . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Hilbert’s axioms of neutral geometry . . . . . . . . . . . . . . . . 2.4 Equivalent forms of Euclid’s parallel postulate . . . . . . . . . . . 2.5 The axiom of hyperbolic geometry . . . . . . . . . . . . . . . . . 2.6 Spherical geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Euclidean trigonometric formulae obtained as limits of hyperbolic and spherical trigonometric formulae . . . . . . . . . . . . . . . . 2.8 Comments on references . . . . . . . . . . . . . . . . . . . . . . . The neutral plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Some results in neutral geometry . . . . . . . . . . . . . . . . . . 3.3 Saccheri’s Theorem and other results in neutral geometry . . . . . . 3.4 Angular deficit in neutral geometry . . . . . . . . . . . . . . . . . 3.5 Trirectangular quadrilaterals in neutral geometry . . . . . . . . . . 3.6 Khayyam–Saccheri quadrilaterals . . . . . . . . . . . . . . . . . . 3.7 Projection in neutral geometry . . . . . . . . . . . . . . . . . . . . The hyperbolic plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Some basic properties in hyperbolic geometry . . . . . . . . . . . 4.2 On quadrilaterals in hyperbolic geometry . . . . . . . . . . . . . . 4.3 Trirectangular quadrilaterals in hyperbolic geometry . . . . . . . . 4.4 Equidistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Geometric relations in quadrilaterals . . . . . . . . . . . . . . . . . Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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29 31 33 33 33 37 42 48 50 54 55 55 58 59 62 67 74

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5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Angular deficit in hyperbolic geometry . . . . . . . . . . . . 5.3 The area function . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Dissection in Euclidean geometry . . . . . . . . . . . . . . . 5.5 Dissection in non-Euclidean geometry . . . . . . . . . . . . . 6 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The function E.y/ . . . . . . . . . . . . . . . . . . . . . . . 6.3 The functional equation .x C y/ C .x  y/ D 2.x/.y/ 6.4 Pythagoras’ theorem . . . . . . . . . . . . . . . . . . . . . . 6.5 Trigonometry in an arbitrary triangle . . . . . . . . . . . . . 6.6 Geometric relations in trirectangular quadrilaterals . . . . . . 6.7 Some spherical trigonometry . . . . . . . . . . . . . . . . . . 6.8 The function E.y/ in spherical geometry . . . . . . . . . . . 7 Parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Parallelism in hyperbolic geometry . . . . . . . . . . . . . . 7.3 Parabolic motions and horocycles . . . . . . . . . . . . . . . 7.4 Horocycle contraction and applications . . . . . . . . . . . . 7.5 The functional equation f .x/f .y/ D f .x C y/ . . . . . . . 7.6 Lobachevsky’s angle of parallelism function . . . . . . . . . 8 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 A Euclidean model of the hyperbolic plane . . . . . . . . . . 8.3 A model arising from algebra . . . . . . . . . . . . . . . . . 9 Transitional geometries . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 A coherent model for the three geometries . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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74 76 78 82 87 89 89 92 95 97 102 108 110 113 115 115 116 125 131 134 135 139 139 139 150 163 163 165 173

1 Introduction These are notes on hyperbolic geometry, with many digressions on Euclidean and spherical geometry. The treatment of this subject is somehow different from the usual one because it is model-free. This is the way hyperbolic geometry was worked out by Lobachevsky, Bolyai and Gauss, the three founders of the field. The notes nevertheless contain a section on models of hyperbolic geometry (Section 8), and in the introductory part of that section, we point out several advantages of models. For instance, they provide a quick way for doing computations, by using the underlying Euclidean geometry and tools of linear algebra. They also give rise to nice pictures

Notes on non-Euclidean geometry

3

(for instance, the well-known tessellations of the Poincaré upper half-plane and disk models). Let us say a few words on why we prefer the model-free point of view approach, since this is not the usual approach. One shortcoming of models is that they all either contain some unnecessary symmetry, or on the contrary they suffer a lack of symmetry. For instance, the Poincaré disk model has a center, which is irrelevant for hyperbolic geometry. The upperhalf-plane seems to favor one direction at infinity, which is also irrelevant. More importantly, teaching hyperbolic geometry in models can give the wrong impression that the derivation of certain results which use differential calculus or integration, like the computation of length, area and volume, requires models. We mention, as examples of such results, the trigonometric formulae, the result saying that hyperbolic geometry is infinitesimally Euclidean, the study of conics in the hyperbolic plane, the structure of the boundary at infinity and so on; these and many other results are usually established in models, and about which it is important to stress that they can be completely developed in a model-free setting. Another advantage of the model-free treatment is that several methods of proof in hyperbolic geometry, when they are done in a model-free setting, can easily be transported to the realm of spherical geometry and vice-versa. One example is the proof of the Pythagorean theorem (that is, the theorem which gives the relations between the three edge lengths of a right triangle) which we give in these notes. In hyperbolic geometry, the model-free proof of the Pythagorean theorem is based on some monotonicity lemmas involving angular deficit of figures. The proof of the corresponding theorem in spherical geometry can be done along the same lines, using angular excess instead of angular deficit.1 A proof of the hyperbolic Pythagorean theorem that is established using a Euclidean model cannot be transformed into a proof of the spherical one. Finally, we believe that the model-free approach is intellectually and aesthetically more satisfying, and in any case, it is important to make it clear to students in the field that hyperbolic geometry is not only the geometry of a collection of isomorphic models, but that it can be worked out in a complete autonomous way that is independent of models. In studying hyperbolic geometry, it is natural to also talk about its two sisters, Euclidean and spherical (or elliptic).2 One reason for which Euclidean geometry is 1A parallel exposition of the trigonometric formulae in the three geometries (hyperbolic, Euclidean and spherical), based on considerations on angular deficit, is contained in the work of de Tilly [100]. 2 Spherical geometry is also called elliptic geometry (a term which, like the terms hyperbolic and parabolic, was introduced in this context by Klein, cf. his Über die sogenannte Nicht-Euklidische Geometrie [48], cf. Stillwell’s translation p. 72). Following a certain trend, we shall use the word spherical for the geometry of the sphere and the word elliptic for the geometry of the projective plane, that is, the quotient of the sphere by its canonical involution. In that context, the sphere is also sometimes called the double elliptic plane, and the projective plane the single elliptic plane. The reason for the use of the word double (respectively single) elliptic plane is probably the fact that on the sphere, two lines always intersect in two points, whereas on the projective plane, two lines always intersect in one point. Thus, compared to the sphere, the projective plane has the advantage of being a space where lines intersect in one point instead of two. It has the disadvantage of

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involved in this study is that several theorems of hyperbolic geometry are also valid in Euclidean geometry, and therefore they belong to the realm of neutral geometry, that is, the geometry in which the parallel postulate has been neutralized.3 When we work in hyperbolic (or Euclidean) geometry, it is always interesting to have a precise idea, regarding the propositions that are used, of which of them belong to the realm of neutral geometry and which do not, and sometimes this is not immediately apparent. For example, whether the three perpendicular bisectors in an arbitrary triangle meet at a common point, and other geometric statements of the same sort, belong to the realm of Euclidean geometry proper or to neutral geometry, need some thought. We also note that there are propositions (although much less numerous than those of neutral geometry) that belong to the realm of absolute geometry, that is, to the geometry that is common to the Euclidean, the hyperbolic, and the spherical worlds.4 One can also recall here the fact that there are several similarities between the formulae of elliptic and hyperbolic geometry. Such similarities were highlighted by the founders of non-Euclidean geometry, and even before them, in particular by J. H. Lambert5 and F. A. Taurinus.6 All these authors noticed that there is a passage between the being much less intuitive than the sphere since spherical geometry, besides being an axiomatic geometry (and, historically, before being an axiomatic geometry), is also the geometry of a round sphere in Euclidean space (and it is also the geometry of a round sphere in hyperbolic space). Elliptic geometry has also the advantage of including a polarity theory, which is based on a duality between points and lines. It is possible to define this duality on the sphere, but the expression of such a theory is more complicated than in the elliptic plane. For instance, on a sphere, to a line, i.e. a “great circle”, there are two associated “poles” instead of one in the projective plane. 3 In other words, neutral geometry is the geometry common to spaces of constant curvature 0 and constant curvature 1. Lobachevsky used the words imaginary geometry for neutral geometry, and pangeometry for hyperbolic geometry, see the comments on the use of these words and others in [59], p. 230–233. Neutral geometry is also sometimes called absolute geometry, but in these notes we reserve the term “absolute” to the geometry which is common to the three geometries: Euclidean, hyperbolic and spherical. 4 One example is that isosceles triangles (that is, triangles having two edges of the same length) have equal base angles. 5 Johann-Heinrich Lambert (1728–1777) was Alsatian (born in Mulhouse). He was a self-made mathematician, astronomer, physicist, poet and philosopher. We owe him the first proof of the fact that  is irrational. Lambert deduced this fact from a more general result, namely, the fact that for any rational number x, the value tan x cannot be rational; hence,  cannot be rational since tan =4 D 1. Lambert is also the author of a treatise on the theory of perspective, in which he also studied the problem of constructibility of figures using ruler alone. He made important contributions to the study of trajectories of comets, a subject on which other prominent mathematicians like Euler and Lagrange, who were Lambert’s contemporaries, also worked. Like several of the pre-non-Euclidean geometers, Lambert was convinced that Euclid’s fifth postulate follows from the others, and he tried to get a contradiction by assuming the negation of this postulate. In doing so, he ended up with interesting geometrical results in (the hypothetical) non-Euclidean geometry. In particular, Lambert’s analysis of trirectangular quadrilaterals (that is, quadrilaterals with three right angles), and his definition of area in terms of angle sum in (the hypothetical) non-Euclidean geometry can be considered as major discoveries in hyperbolic geometry. 6 Franz Adolph Taurinus (1794–1874) was a contemporary of Gauss and Lobachevsky. He used the expression “logarithmic-spherical geometry” (logarithmisch-sphärische Geometrie) for a (hypothetical) geometry in which the angle sum of any triangle is less than two right angles, that is, for the geometry that is commonly known today as hyperbolic geometry. Taurinus obtained the fundamental trigonometric formulae for this geometry by working on a sphere of “imaginary radius”. We refer the reader to the texts by Taurinus in [93] and the exposition in Bonola’s Non-Euclidean geometry [16], p. 79. Gauss corresponded with Taurinus on this subject, see Volume VIII of Gauss’s Collected Works [35]. Unfortunately, Taurinus’ life ended sadly. He was driven to despair after Gauss stopped answering his letters, and it is said that he bought all the remaining copies of a booklet he had

Notes on non-Euclidean geometry

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trigonometric formulae of spherical geometry and those of hyperbolic geometry (the latter being, in the works of Lambert and Taurinus, purely phypothetical) obtained by multiplying certain quantities by the imaginary number 1. For instance, certain trigonometric formulae for hyperbolic triangles are obtained from those of spherical triangles by replacing the trigonometric functions sin and cos of side lengths by the hyperbolic functions sinh and pcosh of these side lengths, which amounts to multiplying certain arguments by 1. Lambert pointed out this fact in his Theorie der Parallellinien (1766) [49], and Taurinus pointed it out in a work carrying the same title (1825) [93]; see also Bonola’s Non-Euclidean geometry [16], p. 79, for a general discussion. Felix Klein, in his Über die sogenannte Nicht-Euklidische Geometrie [48] made a similar observation. We also note that it is because of such a resemblance between the trigonometric formulae of spherical and hyperbolic geometry that Beltrami chose the name “pseudo-spherical geometry” for hyperbolic geometry. The ideas on hyperbolic geometry as being the geometry of a sphere of imaginary radius that were anticipated by Lambert and Taurinus can be made precise in the setting of the Minkowski model of hyperbolic space, worked out by Hermann Minkowski (1864– 1909) about 150 years after the work of Lambert. These ideas can be explained by considering p the hyperbolic plane as an “imaginary unit sphere”, that is, a sphere of radius 1, with respect to a quadratic form of signature .2; 1/ on R3 , and studying such a sphere in parallel with the usual space of spherical geometry, realised in R3 as the unit sphere with respect to the Euclidean quadratic form of signature .3; 0/. With this, the fact that the formulae of hyperbolic geometry canpbe obtained from those of spherical geometry by multiplying certain quantities by 1 can be explained in retrospect. As a final illustration of these ideas, we first recall that on a sphere of radius r, the area of a triangle with angles ˛, ˇ,  can be defined as the following multiple of angular excess: r 2 .˛ C ˇ C   /: p Taking r D 1, the above expression becomes   .˛ C ˇ C  /; which is the angular deficit (or the area) of a triangle in the hyperbolic plane. We shall dwell on these and on other similarities between the three geometries (spherical, Euclidean and hyperbolic) at several places in these notes, and in particular in Section 9, where we describe a continuous passage between them. The basic ideas that are used in this set of notes are not numerous, and it may be useful to highlight them right away. These are the following: (1) Hyperbolic geometry is the geometry that is obtained by taking the Euclidean axioms and replacing the parallel axiom by the axiom saying that for some (or, equivalently, for any) triangle, the angle sum is < . published on the theory of parallels (reproduced in [93]) and burnt them (see [110], p. 50, [87], p. 219, and [93], pp. 249–250).

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(2) In spherical geometry, the angle sum in triangles is > . (But unlike hyperbolic geometry, spherical geometry is not obtained by taking the Euclidean axioms and replacing the parallel axiom by an axiom saying that the angle sum in any triangle is > .) (3) In hyperbolic geometry (respectively, in spherical geometry), the area of a triangle is equal to its angular deficit (respectively, its angular excess), that is, the deficit to  (respectively the excess over ) of the angle sum. (4) In absolute geometry, to each straight segment of length y, there is a basic associated quantity E.y/, defined as follows.7 Construct a trirectangular quadrilateral such that y is adjacent to two edges x and e, making right angles with them, and such that the edge e is adjacent to the fourth angle of the quadrilateral, that is, the one which is not necessarily a right angle, see Figure 1. Note that it follows e y x (i)

(ii)

(iii)

Figure 1. Cases (i), (ii), (iii) represent a trirectangular quadrilateral in hyperbolic, Euclidean and spherical geometry respectively.

from Item (1) above that in the case of hyperbolic geometry, this fourth angle is acute; in the case of Euclidean geometry, it is a right angle; in the case of spherical geometry, it is obtuse. (We assume that all lengths are < =2, so that the figure, on the sphere equipped with the usual angle metric, is well defined without further explanation.) We let e E.y/ D lim : y!0 x Then, this limit exists. In hyperbolic geometry, E.y/ > 1 for all y > 0. In spherical geometry, E.y/ < 1 for all y > 0, and in Euclidean geometry the function E.y/ is constant and equal to 1. It seems that the idea of using this function occurred for the first time to de Tilly, see his work [99], [101]. (5) The formulation of the preceding item uses a trirectangular quadrilateral, and we shall study in detail such quadrilaterals in these notes. In fact, the study of quadrilaterals plays a central role in this theory, and the reader will notice that for almost all the results that are presented below, the proofs are based on monotonicity properties of edge lengths of some special quadrilaterals, namely, the following:8 7 To avoid heavy notation, we follow the classical trend of using the same letter to denote a segment and its length. 8 For the history and the names attached to these quadrilaterals, see e.g. Rosenfeld [87] and Pont [79]. We shall further comment on this in the text below.

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• Trirectangular quadrilaterals, also called Ibn al-Haytham–Lambert quadrilaterals. These are the quadrilaterals that appear in the above definition of the Function E.y/; see Figure 1. • Quadrilaterals with two opposite edges meeting a common third edge at right angles, see Figure 2. These are called Khayyam–Saccheri quadrilaterals. (6) In Item (3) above, we could have as well defined the function E using Khayyam– Saccheri quadrilaterals, instead of trirectangular quadrilaterals. e y x (i)

(ii)

(iii)

Figure 2. Cases (i), (ii), (iii) represent a Khayyam–Saccheri quadrilateral in hyperbolic, Euclidean and spherical geometry respectively.

Let us now present in more detail the content of these notes. Section 2 deals with the primary notions and axioms of the three geometries. In this section, we shall also comment at some length on the parallel axiom and on the various forms it can take. In Section 3, we discuss some notions and results in neutral geometry, namely, Saccheri’s theorem stating that in the neutral plane the angle sum of any triangle is  , angular deficit, the geometry of Khayyam–Saccheri quadrilaterals, that of trirectangular quadrilaterals, and some properties of projection. Section 4 deals essentially with hyperbolic geometry, but it also contains some remarks on analogies and differences between this geometry and the other two geometries. We review in this section the notion of equidistance and the properties of triangles, and then we discuss the geometry of quadrilaterals. We study in detail Khayyam–Saccheri and trirectangular quadrilaterals. We prove geometric monotonicity properties for edge lengths of such quadrilaterals in terms of other variables. These properties are used in Section 6 in the derivation of the hyperbolic trigonometric formulae. Section 5 concerns area. In hyperbolic and in spherical geometry, there is a notion of area which is canonical up to multiplication by a constant. We already mentioned that in hyperbolic (respectively spherical) geometry the area of a triangle is defined as its angular deficit (respectively, angular excess), that is, the deficit to (respectively the excess over)  of the angle sum. This fact was already pointed out by Lambert (for whom, as we already pointed out, hyperbolic geometry was still hypothetical) in his Theorie der Parallellinien [49]. In relation to area, we study figure dissection. We recall in this respect that Euclid did not give any definition of area, but he defined two

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figures to have the same area if they can be obtained from one another by dissection. We review dissection in Euclidean geometry and in hyperbolic geometry. In Section 6, we establish trigonometric formulae for hyperbolic geometry, in particular the so-called Pythagorean theorem that gives the relations between the three edge lengths of a right triangle. The proof of this theorem starts by performing small displacements of the triangle and using the monotonicity properties of edge lengths in Khayyam–Saccheri quadrilaterals that arise after these displacements. We then write relations between the de Tilly functions E.y/ associated to the edges of these quadrilaterals. This leads to a functional equation which, in the case of hyperbolic geometry, turns out to be the one satisfied by the hyperbolic cosine function. The same method can be used in spherical geometry, and it leads to the same functional equation, whose solution in this case is the cosine function. The method that we use for obtaining the Pythagorean theorem is model-free, and it was developed by Louis Gérard in his thesis (Paris 1892) [36], apparently based on ideas due to Battaglini. In Section 7, we study parallelism and related notions. In particular, we study horocycles and parabolic motions. Horocycles are curves that are proper to hyperbolic geometry, and parabolic motions are transformations that are also proper to hyperbolic geometry. For this reason, these notions receive special attention in these notes. We also study Lobachevsky’s angle of parallelism function. In Section 8, we discuss models of the hyperbolic plane. We start with a Euclideandisk model that will turn out to be the well known projective Beltrami–Klein model. We then study a model that arises in an unexpected manner from algebra. The points in this model are prime ideals of the ring RŒX  of polynomials with one real variable. After some work, this model will turn out to be the Poincaré upper half-plane model. This is a new illustration of the ubiquity of the Poincaré plane. We shall see that the boundary of that model also admits a description in terms of prime ideals of RŒX . The union of the hyperbolic plane with its boundary will appear as the set of all prime ideals of RŒX. In dealing with the Beltrami–Klein model, we shall describe the congruence group of hyperbolic geometry as a group of projective transformations of the disk. In dealing with the Poincaré model, we shall describe the congruence group of hyperbolic geometry as a group of inversions and as a group of linear fractional transformations. In Section 9, we study a space in which we can make a continuous transition between the three geometries: hyperbolic, Euclidean and elliptic. Besides the digressions on spherical (or elliptic) geometry, the notes contain some digressions on 3-dimensional geometry. The notes also contain several historical comments. Including historical comments in a mathematical text is useful because it is important to remember who discovered what and who proved what. It is also important to have an idea of what was known to the mathematical community at the time of a discovery. History also adds to the charm of a subject. We encourage the reader to glance through the original works, in particular Euclid in Heath’s edition [29], Bolyai [14] and Lobachevsky [56], both reprinted in Bonola [16], Gauss’s correspondence and notes on hyperbolic geometry

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that are contained in Volume VIII of his Collected works [35], and the new edition of Lobachevsky’s Pangeometry, which contains an extensive commentary [59]. An ancestor of these notes is the set of notes [1] of a course given by the first author in 1978 at Orsay, where the interest in hyperbolic geometry arose with the appearance of Thurston’s Princeton Notes [102]. The same author delivered lectures on that subject at the Strasbourg Geometry Master-class in 2008. The second author learned model-free hyperbolic geometry from the first author.

2 On basic notions and axioms of the three geometries 2.1 Introduction In this section, we give a quick glance at the axioms of the three geometries, hyperbolic, Euclidean and spherical, without any aim of being complete. The goal is to get acquainted with what is common to hyperbolic geometry and to its two sister geometries, and to stress on the differences between these three geometries. There are various versions of the list of axioms of Euclidean (and non-Euclidean) geometry. We briefly review Hilbert’s list and we make a few comments on the axioms that will be useful in the sequel. There are a number of books where these axioms are abundantly discussed; see e.g. [29], [27] and [39].

2.2 Basic notions The axioms of geometry concern a short list of primary notions (also called undefined notions, and sometimes common notions). These are points, lines, planes, alignment, betweenness, belonging, containment, congruence of segments and congruence of angles,9 and a few more notions pertaining to higher dimensions. Note that “distance” and “angle” are not primary notions. The primary notions are not defined; they are only determined by the relations they satisfy and that are given by the axioms. One sometimes makes a distinction between “undefined objects” like points, lines, etc. and “undefined relations” like alignment, congruence, etc. But this distinction is rather irrelevant for us. It is interesting to note that although the primary notions that are listed above are “undefined”, Euclid gave “definitions” of them, using words that are borrowed from intuition. For instance, in Euclid’s Elements, a point is defined as something that “has 9 Congruence is sometimes called “motion”.

Euclid, following Aristotle, avoided the use of the term “motion”, for philosophical reasons whose discussion is beyond our aim in these notes. The reader can refer to Rosenfeld’s account on this subject in [87], p. 111–112. Today, instead of congruence, we are used to talk about isometry because we think of our spaces as being metric spaces. But it is important to keep in mind that the notion of distance is not part of Euclid’s primary notions, and that geometry can be done without talking about distances. For instance, we shall see that we can do geometry in the neutral plane, and there is no natural metric on that plane.

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no part”, a line is defined as something that “has length but no width”, and so on.10 Of course, Euclid’s developments never use the fact that a point is something that has no parts or similar considerations, and, in practice, Euclid’s understanding of the primary notions is the same as ours; that is, he considers them as abstract notions. His attempt to “define” these notions should rather be considered as saying that the primary notions correspond intuitively to some familiar common-sense notions.11 One may recall in this context a remark made by David Hilbert, at a mathematical meeting in 1891, in which he expresses the axiomatic point of view on the undefined notions: “It must be possible to replace in all geometric statements the words point, line, plane by table, chair, mug.”12 We also note that although relations such as belonging and containment are primary notions, it is usual, even if one adopts an abstract (axiomatic) point of view, to imagine a line as something one-dimensional, as Euclid did, to consider that a point belongs to a line if it is an element of that line in the set-theoretic sense, and so on. But, again, this has no effect on proofs in geometry.13 Likewise, the notion of betweenness, defined on triples of distinct points that are on the same line, is an abstract notion subject to certain axioms (see Group II of Hilbert’s axioms recalled below), but it is safe to say that this notion is usually considered as a consequence of the order structure on the real numbers, a line in the plane (Euclidean or hyperbolic) being usually identified with the set of real numbers. It is however important to keep in mind that this identification is a redundancy. The construction and, a fortiori, the properties of the real numbers, are not needed in geometry. Let us also note right away that in these notes, although we adopt a classical point of view, we shall make use of the usual modern geometric terminology, e.g., of the notion of isometry, and of the language of topology. The founders of hyperbolic geometry, namely, Lobachevsky, Gauss, and Bolyai, did not have much terminology for topological notions, although they used these notions. In fact, Euclid himself implicitly used topology in the Elements, even if he did not have special words for it. For instance, he assumed that circles are continuous, that is, locally parametrized by the real numbers and not e.g. by the rationals. This is already an essential assumption in the first proposition of Book I of the Elements, which concerns the construction of equilateral triangles. This proof involves the fact that if two circles of equal radii have 10 Such a definition is due to Plato, and it concerns not only straight lines, but all lines, being understood that in the Elements, a line is either a straight line or a circle. 11 We recall for the interested reader that Aristotle gave similar definitions. For instance, in the Posterior Analytics (87a36), Aristotle wrote: “A unit is a substance without position, but a point is a substance with position: I regard the latter as containing an additional factor” ([4] Book I, p.155). In the Metaphysics (1016b25), Aristotle wrote: “That which is quantitatively and qua quantitative wholly indivisible and has no position is called a unit; and that which is wholly indivisible and has position, a point; that which is divisible in one sense a line; in two senses a plane; and that which is quantitatively divisible in all three senses, a body” ([3], Vol. I, p. 235). We can see in these texts and other similar ones an attempt to define point, line, plane and body by using the notion of dimension. 12 This remark is reported on by Hermann Weyl in [109], p. 635. 13 It is useful to remember in this respect that set theory, as a formal theory, is a modern invention (19th–20th centuries).

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their centers at a distance equal to this radius, then the two circles intersect. This uses an “axiom of continuity”, and the result would not be true for instance if the base field of the geometry were the field of rationals instead of being the field of real numbers. Another related topological property that is often used in the Elements is the fact that the complement of a line (or of a circle) in the plane has two connected components. After the primary notions, let us say a few words on the derived notions. Examples of derived notions are the notions of side of a point on a line and of side of a line in the plane, and the notions of segment, ray, triangle, angle, angle measure, area, and perpendicularity. The notion of betweenness allows one to define the notion of being on the same side of a point P on a line l. More precisely, we say that two points A and B on l are on the same side than P on l if P does not lie between A and B. One can then prove that there are two sides of a point P on a line l, and these two sides can be defined as the complementary components of the complement of P on l. Likewise, the notion of betweenness allows one to define the notion of being on the same side of a line l in the plane, by saying that two points A and B that are not on l are on the same side of l if no point on l lies between A and B. The two sides of a line can then be taken as the definition of the two connected components of the complement of that line in the plane.14 Given two points A and B in the plane, the (closed) segment AB is the union of A and B with the set of points that lie between them. Likewise, one can define the open segment associated to A and B as the set of points that lie between A and B, excluding A and B. We shall use the notation AB or ŒA; B for a closed segment, and .a; b/ for an open segment. A ray is a connected component of the complement of a point on a line, with the point added. Using topological terminology, a ray is the closure of a connected component of the point. The ray is said to start at the given point. A ray has a natural orientation, directed from the point at which it starts towards the other points on the ray. Given two distinct points A and B in the plane, we shall sometimes denote by AB the ray starting at A and containing B. We shall also use the notation r.A; B/ to denote this ray, when this is necessary to avoid confusion between the ray AB, the line AB, and the segment AB. If A and B are two distinct points in the plane, the ray r.A; B/ is also the union of the segment AB with the set of points C such that B lies between A and C . A triangle is a set of three distinct points that do not lie on the same line. The three points are called the vertices of the triangle. An edge of a triangle is a segment joining two of its vertices. A triangle is completely determined by its three vertices, but we are used to imagine a triangle as the three vertices together with the three edges that join them, and this picture is useful. For instance, it allows us to talk easily about a 14 Let us note by the way that the notion of two sides (or of two complementary components) of a polygon in the plane is more delicate to define than the notion of two sides of a line. The fact that a polygon divides a plane into two components is a theorem, Jordan’s Curve Theorem, which Camille Jordan (1838–1922) proved in 1887. Max Dehn (1878–1952) also wrote a proof of that theorem, together with a 3-dimensional generalization, in an unpublished paper that is reported on by Magnus in [65].

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line intersecting a triangle, and about other similar notions. It also allows us to make statements about the interior and the exterior of a triangle (we shall use this below). In any case, we shall usually designate a triangle by ABC , where A, B and C are the vertices. An angle, or an angular sector, is a region in the plane bounded by two rays starting at the same point, which is called the vertex of the angle. If B is the vertex of some angle and if A 6D B and C 6D B are points on the two rays defining this angle, then we denote this angle by ABC if the region between the two rays is understood. (In principle, there are two regions.) If the union of two rays starting at a given point is a straight line, then we say that the angle at the given vertex is flat, and that its measure is  (or two right angles). From this, angle measure is naturally defined using congruence and the fact that the total angle at any point is equal to 2. More precisely, one first defines angles of measure 2=n, for n D 1; 2; 3; : : : , as the congruent angles having a common vertex and whose union covers the whole plane without overlap except at the rays defining the angles. From this, one can then easily define angles of value 2m=n, where m and n are any positive integers. Finally, one can define angles of any positive measure by extending continuously the values of the form 2m=n. An interior angle (sometimes called more simply an angle) of a triangle at some vertex is the angle defined by the two rays starting at that vertex and containing the two edges that abut at that vertex, and taking the region bounded by these rays to be the component of the complement of the two rays that contains the triangle. The angle sum of a triangle is the sum of its interior angles. Angles are added and subtracted in the usual way, and there is a unique function defined on the set of angles that is continuous and additive modulo 2, invariant under congruence, and taking the value  on flat angles. All right angles are congruent. We shall see that in neutral geometry, the statement that the angle sum in an arbitrary triangle is equal to two right angles is equivalent to the parallel postulate. Note that although angles and segments are derived notions, the notions of angle congruence and segment congruence are primary notions. A circle is not a primary notion. Two points A and B are on the same circle centered at a point O if the segments AO and BO are congruent.15 Using the notion of side of a point on a line, one can define the interior of a triangle. Given a triangle ABC , a point D distinct from the three vertices is said to be in the interior of the triangle if the three following properties are satisfied:

1

15 Lobachevsky wrote in his Pangeometry (cf. [59], p. 4) that it is easier to start with the notion of circle, as a primary notion, rather than with the notion of straight line. In doing this, one defines a line in the plane as an intersection locus of a family of pairs of circles of equal radii. This is a way of defining a line as the equidistant locus from two given points without using the words “distance” or “equidistance”. The line becomes thus a derived notion, instead of being a primary notion. Avoiding putting lines at the forefront of the axiomatics is motivated by the fact that the notion of line involves in some way or another the use of infinity, whereas the definition of a circle, as an equidistance set from a point, does not. We recall that Postulate III of Euclid’s Elements ([29], p. 154) says that from an arbitrary point, and for a given arbitrary radius, one can draw a circle centered at that point and of the given radius. In the same way, in the axiomatization of 3-dimensional geometry, Lobachevsky started with the notion of sphere as an undefined notion, and he then defined a plane as the intersection locus of a family of pairs of spheres.

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(1) the points D and A are on the same side of the line BC ; (2) the points D and B are on the same side of the line AC ; (3) the points D and C are on the same side of the line AB. In the spirit of Euclid’s Elements, the length of a segment is defined as its congruence class. Thus, the notion of congruence allows us to talk about two segments having the same length, without reference to a distance function. Lobachevsky, in the new geometry he invented, had the modern notion of length, and he was certainly the first mathematician to compute length, area and volume (and he made extensive such computations) in a setting which is not the Euclidean one. The congruence relation is sometimes denoted by . Thus, to denote the fact that two segments AB and CD (respectively two angles ABC and A0 B 0 C 0 ) are congruent, we write AB  CD (respectively ABC  A0 B 0 C 0 ). Using the notions of congruence and betweenness, one can define an order relation on the set of segments; that is, one can introduce the notion of a segment being shorter than another one, again without making use of a distance or of a length function. This is done as follows. If A, B and C , D are two pairs of distinct points, then we say that AB is shorter than CD, and we write AB < CD, if there exists a point E that lies between C and D satisfying AB  CE. One can also talk about sums of lengths of segments without any reference to a distance function, by defining, for four distinct points A, B, C , D the sum AB C BC to be the congruence class of a segment JL such that there exist three collinear points J , K, L with K between J and L and such that JK  AB and KL  CD. In this respect, we note the following propositions from Euclid’s Elements, which is a formulation of the triangle inequality, and that holds in neutral geometry:16 “In any triangle two sides taken together in any manner are greater than the remaining one” (Proposition 20 of Book I, Heath’s translation, [29], p. 284). When the Euclidean or hyperbolic plane is considered as a metric space, with distance function d , then betweenness is related to the distance function by the fact that if A, B, C are three distinct points, then B lies between A and C if and only if d.A; B/ C d.B; C / D d.A; C /.

1 2

1

2

2.3 Hilbert’s axioms of neutral geometry The axiomatic approach in geometry started with the Greeks, with no apparent practical need. The basic written text on the subject that reached us is Euclid’s Elements. The list of axioms is not the same in the various versions of this book that have come down to us. The axioms are sometimes referred to as Hilbert’s axioms. In Hilbert’s approach there was nothing fundamentally new compared to the one of Euclid, except that Hilbert was more systematic. His axiomatization came after a need to make Euclid’s approach more rigorous and to make the set of axioms more practical. In 16 Remember

that there is no natural metric in neutral geometry.

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particular, Hilbert also studied the question of the independence of axioms, by constructing models which satisfy certain axioms and not others. Lobachevsky, several decades before Hilbert, studied the question of the consistency of hyperbolic geometry. He raised this question at several places of his works. From the fact that he derived his hyperbolic trigonometric formulae using Euclidean geometry (namely, the geometry of the horosphere) and spherical geometry (namely, the geometry of a round sphere embedded in hyperbolic 3-space), Lobachevsky argued that if there were a contradiction in hyperbolic geometry, then there would be one in Euclidean or in spherical geometry (see e.g. [59], p. 250). But he did not have a rigorous proof of this fact. Hilbert studied the consistency question, and he eventually reduced in a rigorous way the question of the consistency of hyperbolic geometry (and of Euclidan geometry) to the question of the consistency of the real number system. Let us quote Hermann Weyl from [109]: The Greeks had conceived of geometry as a deductive science which proceeds by purely logical processes once a few axioms have been established. Both Euclid and Hilbert carry out this program. However, Euclid’s list of axioms was still far from being complete; Hilbert’s list is complete and there are no gaps in the deductions. In what follows, we give a short overview of Hilbert’s axiomatics.17 Hilbert’s list of axioms is composed of five groups, naturally ordered, and they are described in Chapter I of his Grundlagen der Geometrie [43]. These are (in Hilbert’s ordering): (1) the axioms of incidence; (2) the axioms of order; (3) the axiom of parallelism; (4) the axioms of congruence; (5) the axiom of continuity (or the Archimedean axiom). In the quick survey that we give below, we have included the parallelism axiom at the end of the list, in order to separate it from the axioms of neutral geometry which will constitute the first four groups. The first two groups of axioms have a descriptive character. They concern the relations of belonging, containment, incidence, and betweenness. These axioms are closely related to each other, and for a modern reader, they can be considered as pertaining to the realm of set theory. In any case, we adopt in these notes the usual notation of set theory, that is, we use 2 for belonging, \ for intersection, and so on. 17 Euclid made a distinction between postulates and axioms, whereas Hilbert did not. In Euclid’s sense, postulates differ from axioms in that they allow geometric constructions. Some authors considered that the difference between postulates and axioms lies in the fact that axioms are “self-evident”, whereas postulates are not. Several antiquity thinkers, including Aristotle, discussed at length the difference between an axiom and a postulate. We shall not enter into such a discussion here, and for our purposes, no distinction is made between axioms and postulates.

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Axioms 2.1. Group I: Incidence axioms. (1) Given two distinct points, there is a line containing them. (2) Given two distinct points, the line containing them is unique. (3) Every line contains at least three points. (4) There exist at least three points that are not contained in the same line. The axioms in this group are also called connection axioms, a term (in German, “Axiome der Verknüpfung”) used by Hilbert in his Grundlagen, since these axioms make connections between the various undefined notions. There are other axioms in group I, which concern planes in 3-space, and higherdimensional analogues. Since we are primarily interested here in plane geometry, we shall not worry about them. Let us make a few comments on this first group of axioms. (1) In contrast with the usual trend in mathematics where one combines properties in order to make economical statements, it is usual to state separately Axioms (1) and (2), rather than stating a combined axiom saying that given two distinct points, there is a unique line containing them. The reason is that in an approach based on axioms, it is useful to know exactly what sequence of (elementary) axioms is used in each proof. (2) Axiom (2) does not hold in spherical geometry. Thus, spherical geometry is not part of “neutral geometry”.18 (3) Regarding Axiom (3), we note that it is true that every line contains infinitely many points, but this is a theorem rather than an axiom. Remember that when we write a list of axioms, we must insure that the list is not redundant, i.e. that it contains only a minimal amount of information. There exist finite geometries that satisfy the incidence axioms, but they do not satisfy some of the other axioms below. (4) Axiom (4) implies that the geometry is not one-dimensional. Thus, the real line is not a model of neutral geometry. The axioms in the next group concern the relation of betweenness for triples of points that are on the same line. Axioms 2.2. Group II: Betweenness, or alignment axioms. (1) Given three distinct points A, B, C , if B lies between A and C , then B also lies between C and A. (2) Given a line and three distinct points on it, one of these three points lies between the other two. 18 We already noted that it is useful to consider that the space of spherical geometry is not the sphere but the projective plane, that is, the quotient of the sphere by its canonical involution. (We note that the sphere has a canonical involution and that this is a consequence of the axioms of spherical geometry, and not only of the sphere model in 3-space. We shall dwell on this fact below.) We also recall that in the context of the projective plane one talks about elliptic geometry instead of spherical geometry. Unlike the sphere, the projective plane satisfies all the axioms of Group I, but it does not satisfy some of the axioms in Group II.

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(3) Given a line and three distinct points on it, a point among the three that lies between the other two is unique with this property. (4) Given two points A and B, we can find a point C such that B lies between A and C . (5) Given a triangle  and a line l, if l intersects an edge of , then, it also intersects one and only one other edge of , unless the line l contains a vertex. We make a few remarks on the axioms of Group II. Hilbert calls the axioms in Group II the order axioms. As Hilbert puts it in his Grundlagen, these axioms were already formulated in essentially the same manner by Moritz Pasch (1843–1930). (1) Axiom (5) is usually called Pasch’s axiom. Unlike the other axioms in this group, this axiom holds in plane geometry but not in higher-dimensional geometries. (2) There are no finite geometries that satisfy the betweenness axioms. (3) We already noted that the sphere does not satisfy Group I of the axioms. The projective plane satisfies the axioms of Group I, but not those of Group II (Axiom (2) fails). We shall elaborate on this fact in Section 2.6 below. (4) The rational plane Q2 satisfies Groups I and II of the axioms. An open subset of the real plane R2 , with the geometry induced from that of the plane, also satisfies these two groups. The next group of axioms is divided into two subgroups: congruence axioms for segments and congruence axioms for angles. Axioms 2.3. Group III: Congruence axioms. Group III A: Segment congruence. (1) Given two points A, B on a line l and a point A0 on a line l 0 (l D l 0 is allowed) and given a choice of a side of A0 on l 0 , there exists a point B 0 on that side such that AB  A0 B 0 . (2) The point B 0 in the conclusion of Axiom (1) is unique. (3) Three axioms saying that AB  A0 B 0 is an equivalence relation on the set of segments. (4) (Addition axiom19 ) If A, B, C are three distinct points on the same line, with B lying between A and C , and if A0 , B 0 , C 0 are three other distinct points on a line, with B 0 lying between A0 and C 0 , then we have the implication: .AB  A0 B 0 and BC  B 0 C 0 / ) AC  A0 C 0 .

19 Note that the existence of a subtraction rule for segments, that is, a rule analogous to Axiom (4) but where C lies between A and B and C 0 lies between A0 and B 0 and with the same conclusion, is not an axiom, but a consequence of the other axioms.

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Group III B: Angle congruence.

1

(1) Given an angle ABC and given a point B 0 and a ray B 0 C 0 starting at B 0 , and given a choice of a side on the line B 0 C 0 , there exists a ray B 0 A0 starting at B 0 , with A0 being on the chosen side of B 0 C 0 , such that A0 B 0 C 0  ABC .

2 1

(2) The ray B 0 A0 in the conclusion of Axiom (1) is unique.

1 2

(3) Three axioms saying that ABC  A0 B 0 C 0 is an equivalence relation on the set of angles.

1

(4) Given two triangles ABC and A0 B 0 C 0 , if AB  A0 B 0 , CB  C 0 B 0 and ABC  A0 B 0 C 0 , then, AC  A0 C 0 .

2

A0

A

B

B0

C

C0

Figure 3

From the congruence axioms, one can prove that congruence preserves betweenness. The congruence axioms also imply that the plane is homogeneous: given any two points, there is a plane congruence taking one of them to the other one. The plane is also isotropic: given a point and two rays based at that point, there is a congruence of the plane taking one ray to the other one. When the plane is considered as a metric plane, like the Euclidean or the hyperbolic plane, then a motion can be considered as a bijection that preserves distances. Axiom 2.4. Group IV: The Archimedean axiom. Let A and C be two distinct points, let l be the line containing them, and let B be a point lying between A and C and distinct from them. Then, we can find a finite sequence of points B1 D B; B2 ; : : : ; Bn on l satisfying AB1  B1 B2  B2 B3      Bn1 Bn and such that C lies between A and Bn . Note that this is a way of saying, without using the notion of distance, that lines are infinitely extendable. Using the fact that angle congruence can be defined using triangle congruence, there is an Archimedean property (which is not an axiom) for angles that follows from the Archimedean axiom for segments. Two angular sectors .Ox; Oy/ and .O 0 x 0 ; O 0 y 0 / are congruent (or have the same measure) if there exist two triangles OAB and O 0 A0 B 0 ,

18

Norbert A’Campo and Athanase Papadopoulos x0

x A0

A O

B

O0

y

B0

y0

Figure 4

with A 2 Ox, B 2 Oy, A0 2 O 0 x 0 and B 0 2 O 0 y 0 , and a motion sending the triangle OAB to the triangle O 0 A0 B 0 (see Figure 4). The Archimedean axiom was not explicitly stated by Euclid but it was implicitly assumed. Hilbert, in his work on the axiomatization of geometry, constructed examples of non-Archimedean geometries, e.g. geometries where the field of real numbers is replaced by a non-Archimedean field. Dehn (who was a student of Hilbert) also worked on the axioms of geometry, in relation with the Archimedean property. In his paper [23], Dehn constructed a non-Archimedean geometry in which all the axioms of Euclidean geometry are valid except the parallel axiom. To the Archimedean axiom is usually added a “continuity axiom”, which we also tacitly assume. This axiom combined with the Archimedean axiom says, in modern language, that each line is identified with the real number field R. This combined axiom expresses at the same time the idea of continuity, and the fact that lines are infinitely extendable; see [44] for an account. We also note that Hilbert himself, in his Grundlagen, showed that a kind of Euclidean geometry can be developed without the axiom of continuity. The continuity issue in geometry is an old one, cf. Heath’s account [29], Vol. 1, p. 234 ff. Although the mathematical definition of continuity, as we intend it today, is a nineteenth-century invention, continuity was a major issue in Greek antiquity, and it was implicitly assumed in Euclid’s Elements. As we already recalled, Euclid’s propositions usually ask for constructions in which the points are obtained as intersections of lines and circles. In particular, there is a difference between working with the real or with the rational numbers, since in the rational numbers setting intersections of lines or of circles might not exist. Axiom 2.5. Group V: The parallel postulate. Given a line l and a point A not on l, there exists at most one line containing A and disjoint from l.

2.4 Equivalent forms of Euclid’s parallel postulate From now on, we make use of the common notions of geometry that derive from the axioms, even though we did not define them before (perpendicularity, distance and so on). We recall that in Euclidean geometry, two lines are said to be parallel if they are

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coplanar and do not intersect.20 We already recalled that this is not the usual definition of parallelism in non-Euclidean geometry. We shall dwell on this fact later on. We recall Euclid’s formulation of the “fifth postulate”. In Heath’s translation, the axiom says ([29], Vol. I, p. 202): That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” The situation is illustrated in Figure 5.

˛

l1

ˇ

l2

l3 Figure 5. Euclid’s fifth postulate: If the two l1 and l2 make with a line l3 and on the same side angles ˛ and ˇ whose sum is less than two right angles, then l1 and l2 necessarily intersect on the given side of l3 .

One can easily deduce from this postulate that given a line l and a point A not on l, there is at most one line that contains A and that is disjoint from l. This fact is a form of the parallel postulate which is probably more familiar. We state this in the theorem below, together with a list of other statements that are equivalent to the parallel axiom. We learn from the history of the announcement of “proofs” of the parallel axiom that at several instances, this axiom was implicitly or explicitly transformed into an equivalent statement whose validity (as a proposition, and not as an axiom) was taken for granted, because this new statement seemed to be common-sense, and the author did not consider it as requiring a proof. Several such statements are contained in the next theorem. Theorem 2.6. In the neutral plane, each of the following statements is equivalent to the parallel axiom. (1) For any line l and for any point A not on l, there exists a unique line l 0 containing A and disjoint from l. 20 Definition 23 of Euclid’s Elements, in Heath’s translation, says: “Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction”. (See [29], p. 190.)

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(2) There exists a line l and a point A not on l such that a line containing A and disjoint from l exists and is unique. (3) There exists a triangle whose angle sum is two right angles. (4) Any line containing a point in some angular sector whose value is strictly between 0 and  intersects at least one side of the sector. (5) The angle sum of any triangle is constant. (6) The angle sum of any triangle is equal to two right angles. (7) There exist homothetic non-congruent triangles.21 (8) There exist two distinct lines that stay at a bounded distance from each other. (In other words, given two disjoint lines l and l 0 , the length of the perpendicular from a point x on l to l 0 is bounded when x moves on l to infinity, in both directions.) (9) There exist two distinct lines that are equidistant from each other. (10) A line that intersects a second line necessarily intersects any third line which is disjoint from the second line. (11) For any K > 0 the set of points that are at distance K from a given line is a union of two lines. (12) For any K > 0, there exists a triangle whose area is greater than K. (13) There exists a quadrilateral with four right angles. (14) Given any trirectangular quadrilateral, its fourth angle is also a right angle. (15) There exist two disjoint equidistant lines. (16) Any two non-intersecting lines have a common perpendicular. (17) Any three non-collinear points lie on a circle. (18) If a triangle is inscribed in a semicircle, with one of its edges being the diameter of the semicircle, then the angle opposite to that edge is a right angle. (19) The values of the angles of any triangle are determined by the ratios of its sides. (20) Given a line l and a point P not on l, there is at most one line that is disjoint from l and that contains P . (21) Any two disjoint lines are equidistant. (22) If two distinct lines are disjoint from a third line, then the first two lines are disjoint. (23) The angle of parallelism function is constant and equal to a right angle.22 21 Note that there exist, in neutral geometry, homothetic figures, namely, homothetic circles. In neutral geometry, the existence of homothetic circles is an axiom (or it follows from the axioms, depending on what set of axioms we start with; it is Axiom 3 in Euclid’s Elements). 22 This function is defined in Section 7.6 below.

Notes on non-Euclidean geometry

21

There are other statements that are equivalent to the parallel axiom. Note that most of the above statements do not use the words “disjoint” or “parallel”. Some of the above statements seem self-evident because the space that surrounds us seems to be Euclidean, and it is not surprising that many geometers, over the centuries, took for granted one of these statements, and, using it, they thought they had a proof of Euclid’s parallel postulate. Likewise, several of the equivalent forms of the axiom of non-Euclidean geometry that replaces Euclid’s parallel axiom (see Theorem 2.7 below) may seem unnatural, and at first sight these statements contradict our intuition of the space that surrounds us. The attempts for proving the parallel postulate that were made by various mathematicians usually consisted in negating this postulate and searching for a contradiction. Falling on a statement such as the one stating that there do not exist equidistant straight lines was then considered an absurdity. Let us make a few more historical comments. Most of the equivalences listed above were known long before non-Euclidean geometry was discovered. For instance, Proclus,23 in his Commentary on the first book of Euclid’s Elements [80], mentioned that Posidonius 24 defined parallel lines as lines that are coplanar and that “come neither near nor apart”; that is, these lines are at a constant distance from each other. The medieval mathematician Ibn al-Haytham25 gave a definition of parallelism which is equivalent to the one of Posidonius. He called a parallel to a given line the locus of points that are the endpoints of a moving segment that stays perpendicular to that line, and he “showed” that this locus is a line. Later on, Omar Khayyam expressed his disagreement with Ibn al-Haytham’s definition of parallelism in his Commentaries on the Difficulties Encountered in Certain Postulates in the Book of Euclid; cf. the French translation with commentaries in [81], p. 310. Ibn al-Haytham gave a “proof” of the parallel postulate that is based on the existence of rectangles, that is, quadrilaterals having four right angles; see Youschkevitch [111], p. 117. This proof is also reproduced in Rosenfeld [87], p. 59 and in Pont [79], p. 169. Nasir al-Din al-Tusi (1201–1274), another medieval mathematician, and later on, Renaissance mathematicians including Christopher Clavius (1538–1612), Giovanni Alfonso Borelli (1608–1679) and Giordano Vitale da Bitonto (1633–1711) also wrote on the relation of Euclid’s postulate to the notion of equidistance. They tried to prove that Euclid’s axioms, without the parallel postulate, imply that the equidistant set to a 23 Proclus (412–485) was a neo-Platonic philosopher who became the head of Plato’s Academy. The work of Proclus is gigantic, and several of his writings survive, including an extensive commentary on Euclid’s Book I of the Elements and commentaries on various dialogues of Plato, highlighting the mathematical reasoning they contain. (Remember that Plato was a mathematician in the first place.) Proclus, referring to a work by Geminus of Rhodes (1st century b.c.) conceived the possibility that two straight lines in the plane might be asymptotic, see the account in Bonola [16], p. 3. (Geminus was an astronomer and mathematician. An astronomy book he wrote, the Introduction to the Phenomena, survives.) Because of the fact that he conceived lines that could be asymptotic, Proclus has been considered by some as a precursor of hyperbolic geometry; see Mansion [66]. Proclus, nevertheless, thought that the parallel axiom should be a theorem. 24 Posidonius (135–51 b.c.) was a Greek mathematician native of Apamea in Syria. 25Abu Ali Ibn al-Haytham (965–1039 ca.), also known as al-Hazen, was an Egyptian mathematician, physicist and astronomer. Two of his works that contain commentaries on Euclid’s Elements reached us, namely, the Commentary on the premises to Euclid’s book of the Elements and the Book on the resolution of doubts in Euclid’s book of Elements. The impressive work of Ibn al-Haytham was edited by R. Rashed, see [81] and [82].

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line is a line, cf. [16], p. 13, and [79], p. 200 & 368, by making an assumption that was equivalent to that postulate. For instance, Clavius gave in 1574 a “proof” of the parallel postulate under the assumption that a curve coplanar to a straight line and equidistant from it is a straight line. Halsted, in the introduction to his translation of Saccheri’s Euclides ab omni naevo vindicatus [89] reported that Borelli in 1658 proposed the following definition: “Parallels are coplanar straights with a common perpendicular.” Halsted also mentioned a “proof” of the parallel postulate given 1756 by Robert Simson, based on the assumption that “a straight line cannot approach toward, and then recede from, a straight line without cutting it.” He also reported that Geminus (100 b.c. ca.) defined parallel lines as equidistant straight lines, and that Giordano Vitale in 1680 saw that this presupposed the assumption of Clavius that a line coplanar with a straight and everywhere equidistant from it is itself straight. We also mention that Joseph Fourier (1768–1830) made several attempts to prove Euclid’s parallel postulate using the fact that an equidistance set to a line is the union of two lines (cf. [79], p. 554). There are several other such examples. Saccheri26 showed that the parallel postulate may be replaced by a postulate saying that there exist two non-homothetic congruent triangles. John Wallis (1616–1703) showed that the Euclidean postulate is equivalent to the existence of a triangle having homothetic triangles of an arbitrary size, and he proposed that in the teaching of geometry, the parallel postulate be replaced by such a similarity postulate. Lambert, whom we already mentioned and whom we shall mention at several other occasions below, studied trirectangular quadrilaterals, and he proved that the parallel postulate is equivalent to the fact that the fourth angle of such a quadrilateral is also a right angle. Johann Friedrich Lorenz (1738–1807) obtained the equivalence of the parallel axiom with Item (4) in Theorem 2.6. The result saying that Euclid’s parallel postulate can be replaced by a postulate saying that any three non-collinear points lie on a circle (Statement (17) in Theorem 2.6) is usually attributed to Farkas Bolyai (1775–1856) (the father of János Bolyai, the co-discoverer of hyperbolic geometry). The equivalence between (13) and the parallel postulate is linked to the work of Alexis Claude Clairaut (1713–1765). Axiom (1) is usually called Playfair’s axiom, in reference to the mathematician John Playfair (1748–1819). This axiom was already stated in practically the same terms by Ibn al-Haytham, in his Book on the resolution of doubts in Euclid’s book of Elements, see Pont [79], p. 171. To give a concrete example of how the equivalent forms in Theorem 2.6 were used, let us review the way Wallis used the axiom of existence of homothetic triangles to deduce the parallel axiom.27 The proof is as follows. Consider in the neutral plane two lines l1 and l2 meeting a third line l3 at points A and B respectively, with angles ˛ and ˇ satisfying ˛ C ˇ < . (See Figure 6.) If we move continuously the point B on the line l3 towards the point A, carrying during that motion the line l2 in such a way that the angle it makes with the line l3 is constant, then, 26 Giovanni Girolamo Saccheri (1667–1733) is considered, together with Lambert, whom we already mentioned, as a precursor of hyperbolic geometry who was very close to the discovery of that field. We shall mention Saccheri several times in these notes. 27 The proof by Wallis is contained in [106], and reproduced in [93], pp. 21–30, and it is also reported on in [79] and in [16].

23

Notes on non-Euclidean geometry l1

l4

l2

C0 ˛ A

ˇ B0

B

l3

Figure 6. The figure illustrating proof by Wallis of the parallel postulate

when the point B reaches a position B 0 which is close enough to A, the new position of l2 (this is the line denoted by l4 in Figure 6) necessarily cuts l3 . (This is a result of neutral geometry, which Wallis proved beforehand.) Let C 0 be the intersection point of l4 with l3 . We consider the triangle AB 0 C 0 , and we now follow the construction backwards; that is, we move continuously the point B 0 towards B, carrying during this motion the line l4 in such a way that the angle it makes with the line l3 remains constant. By the similarity axiom, there is a triangle ABC homothetic to AB 0 C 0 , with l3 carrying the edge AB 0 and we can assume that the triangle ABC is situated on the same side of the line l3 than A0 B 0 C 0 . This shows that l1 and l2 intersect (at the point C ).

2.5 The axiom of hyperbolic geometry Keeping all the axioms of Euclidean geometry except the parallel postulate, and replacing it by a statement saying that this postulate fails (for instance, a statement negating any one of the equivalent statements given in Theorem 2.6), we obtain a set of axioms for non-Euclidean geometry. In the next theorem we display some of the equivalent statements that characterize non-Euclidean geometry: Theorem 2.7. Keeping all the axioms of planar Euclidean geometry except the parallel axiom, the following additional statements are all equivalent. (All these statements hold in the plane.) (1) For some line l and for some point A, there exist at least two lines containing A and disjoint from l. (2) For any line l and for any point A not on l, there exist at least two lines containing A and disjoint from l. (3) For some line l and for some point A, there exist infinitely many lines containing A and disjoint from l. (4) For any line l and for any point A not on l, there exist infinitely many lines containing A and disjoint from l.

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(5) Two lines that have a common perpendicular diverge on either side of this perpendicular. (6) There exists a triangle whose angle sum is < . (7) There exists a triangle whose angle sum is 6D . (8) The area of triangles is uniformly bounded. (9) The angle sum of triangles is not constant. (10) If a triangle is inscribed in a semicircle with one of its edges being the diameter, then the angle opposite to that edge is acute. (11) Any two triangles with pairwise congruent angles are congruent. (12) There are no distinct lines that are at a bounded distance from each other. (13) The set of points that are at a given distance from a given line and situated on the same side is not a line. (14) There exists an angular sector whose value is strictly between 0 and  and which contains a line. (15) Any angular sector contains a line. (16) The angle of parallelism function is not constant. (17) The angle of parallelism function takes all values in 0; =2Œ.

2.6 Spherical geometry The sphere, equipped with what is called today its “canonical” metric, enters into the story, since it can also be defined axiomatically. The set of axioms that gives rise to the sphere gives a geometry that is at the same level as Euclidean or hyperbolic geometry. The parallel postulate does not hold in spherical geometry. There are other Euclidean axioms that do not hold in spherical geometry, for instance the Archimedean axiom which expresses the fact that lines are unbounded. In spherical geometry the angle sum of a triangle is > . The introduction of the sphere as a new geometry which is at the same level than the other two, is usually attributed to Riemann, with a reference to his Über die Hypothesen, die der Geometrie zugrunde liegen (On the foundations of the hypotheses that are at the foundation of geometry) [86]. In some works of the late nineteenth and beginning of the twentieth century, the term “Riemann’s geometry” refers to the geometry of the sphere. But spherical geometry is a very old subject. We mention in this respect the works of Eudoxus of Cnidus (410–355 b.c. ca.) and Menelaus of Alexandria (70–140 a.d. ca.), who both wrote treatises entitled the Sphaerics. The treatise by Menelaus survived in Arabic translations. In this work, the author studied spherical triangles whose edges are segments of great circles, in the way Euclid did it for Euclidean triangles. Euclid’s Elements were systematically transposed by Menelaus to the case of the geometry of the sphere. For instance, we can find in

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Menelaus’ Sphaerics the propositions saying that the base angles of an equilateral triangle are equal, that the sum of the lengths of two edges in a triangle is greater than the length of the third edge, that if two triangles have congruent edges then they have congruent angles, and so on. “Menelaus’ Theorem”, as we know it today, is proved for the Euclidean and the spherical case in Menelaus’ treatise. Ptolemys’ Almagest also contains a proof of the spherical case of Menelaus’ Theorem. Menelaus’ treatise also contains the bases of spherical trigonometry. During the Greek period, spherical geometry was linked to astronomy.28 After the Greeks, this subject was developed by the medieval Arabic and Arabicspeaking mathematicians. According toYouschkevitch, the first treatment of spherical trigonometry as a research field in itself, that is, without being subject to astronomy, was done by Nasir al-Din al-Tusi ([111], p. 142). Euler wrote several memoirs on spherical geometry, see e.g. [30] and [32]. According to Rosenfeld [87], the modern form of spherical geometry, as well as of all trigonometry, is due to Euler; see [87], p. 31 ff. for a short account of Euler’s work on spherical geometry and trigonometry. In the nineteenth century, Trigonometric formulae for spherical geometry were worked out by Taurinus (see his works in [93]). According to Bonola [16], p. 82, the recognition of spherical geometry as the geometric system in which the angle sum of triangles is greater than two right angles is an eighteenth-century discovery, mainly due to Lambert, who, as we already mentioned, is also one of the most important precursors of hyperbolic geometry. Before Lambert, spherical geometry was considered as a chapter in Euclidean geometry, namely, the study of the geometry of a sphere in a Euclidean space. We shall dwell on this fact in Section 6.7 below. For an exposition of the axioms of spherical geometry, we refer the reader to [39] and to [19] and to the review we make below. A sphere embedded in Euclidean space (or in hyperbolic space) is a model of spherical geometry, where the lines are the great circles, that is, the intersections of the planes passing through the center with the sphere, and where the angles between two lines are the dihedral angles made by the planes that define these lines. Some geometric theorems belong to the realm of “absolute geometry”, that is, the geometry that is common to the Euclidean, spherical and hyperbolic worlds. But there are also severe differences between spherical geometry on the one hand and Euclidean and hyperbolic geometry on the other hand. For example, the space of spherical geometry is compact, and the lines in that space are homeomorphic to circles. In spherical geometry, the geodesic lines (great circles) are also circles in a geometric sense; that is, they are equidistant loci from a point, the center of these lines, which is also the center of the great circle. On the sphere, every great circle, seen as a geometric circle, has two centers, which are also its poles.

28 It is significant in this respect that the term sphaerica, in the Pythagorean literature and in the Greek literature that followed it, was used for astronomy. Euclid’s Elements do not contain anything on the sphere, except in Book XII (Propositions 16–18) where it is proved that the ratio of volumes of two spheres is equal to the cube of the ratios of their diameters, and in the use of the circumscribed spheres in the study of regular convex polyhedra.

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We already mentioned elliptic geometry, represented by the projective plane. This is the space whose elements are pairs of antipodal points on the sphere. It is also the space whose points are the set of lines in R3 that pass through the origin, and whose lines are the planes in R3 that pass through the origin. In elliptic geometry (whose model is the projective plane), every great circle has a unique center, which is also called the “pole” of the great circle. This remark is at the basis of a duality theory in elliptic geometry, also called polarity theory, in which to each line (which is the quotient of a great circle on the sphere by the canonical involution of the sphere) corresponds to a “pole”. Thus, the pole of a line in elliptic space can be defined in several equivalent ways: (1) The pole of a line is the point that is farthest away from this line. (2) The pole is the equivalence class of the two centers of this line, when this line is considered as the quotient of a great circle on the sphere, and where the term “center” refers to this great circles being considered as a geometric circle on the sphere. (3) Expressed in the usual coordinate description x 2 C y 2 C z 2 D 1 of the unit sphere, the pole of a great circle obtained as the intersection of the plane of equation ax C by C cz D 0 with the sphere is the (equivalence class of the points) ˙.a; b; c/. (4) Again, in the usual coordinates, the pole of a great circle is the equivalence class of the two points of intersection with the sphere of the Euclidean perpendicular to its plane passing through the center of R3 . Polarity theory in the elliptic plane is powerful because it gives new statements by interchanging in any statement lines (great circles) with their poles. An example can be seen on the following two trigonometric formulae, for a spherical triangle whose angles are denoted by A, B, C , with opposite edge lengths a, b, c respectively: cos a D cos b cos c C sin b sin c cos A and cos A D  cos B cos C C sin B sin C cos a: The formal analogy between the two formulae is striking, and the second one is indeed obtained from the first one by exchanging the roles played by the edges and the angles. We shall elaborate more on duality in spherical (or elliptic) geometry later on in these notes. Another difference between spherical (or elliptic) geometry and neutral geometry is that the notion of betweenness, which expresses the fact that for three aligned distinct point, one of these points lies between the other two, does not hold in spherical geometry. This notion is replaced by a notion called “separation” that we discuss below. Lobachevsky noted at several occasions that spherical geometry, which was considered as the geometry of a sphere embedded in the 3-dimensional Euclidean space,

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is also the geometry of a sphere embedded in the 3-dimensional hyperbolic space (see e.g. [59], p. 22), and he described this fact as “truly remarkable”. Lobachevsky drew this conclusion from the fact that one can define lines and angles on a sphere in hyperbolic space in the same way as in Euclidean space using intersections of planes passing trough the origin and so on, and he noticed that the trigonometric formulae for such spheres in hyperbolic geometry coincide with those of “ordinary” spherical geometry, that is, of the geometry of a sphere in Euclidean space. He concluded that in this sense, spherical geometry is independent of Euclid’s parallel axiom. A similar fact was highlighted by Beltrami, several years after Lobachevsky, in his paper Teoria fondamentale degli spazii di curvatura costante [9].29 We already recalled that hyperbolic geometry has the same set of undefined notions than Euclidean geometry (points, lines, planes, containment, betweenness, congruence, etc.) and that the axioms of hyperbolic geometry are obtained from those of Euclidean geometry by removing a single axiom, the parallel axiom, and replacing it by another one saying that given a line and a point that is not on that line, there exist more than one parallel to the given line containing the given point. To summarize, unlike the case of hyperbolic geometry, the set of axioms of spherical geometry is not obtained from the set of axioms of Euclidean geometry by simply replacing one axiom by another one on the sphere or on the elliptic plane. Let us reformulate some of the features of spherical (or elliptic) geometry. • We cannot draw any parallel line to any given line, since there are no nonintersecting great circles. Thus, the parallel axiom does not hold in spherical geometry. • In spherical geometry, lines cannot be identified with the real line, since they are homeomorphic to circles. • There is a canonical notion of length for lines, since any two lines are congruent, and therefore they have the same length, which is finite. • The relation of betweenness is not useful on the geometry of the sphere. Given three points on the same line on the sphere (or on the projective plane), there is no consistent way of saying which point lies between the other two. The notion of betweenness is replaced by a notion of separation involving quadruples instead of triples of points. The relation is represented in Figure 7, in which we say that A and C separate B and D. To see how this relation is used, we note six separation axioms that are used for the geometry of the sphere, following the exposition and the notation in Greenberg [39], where .A; C jB; D/ means that the pair .A; C / separates the pair .B; D/. (1) If .A; BjC; D/ then the four points A; B; C; B are collinear and distinct. 29 The

conclusion of Beltrami’s paper is as follows: “The geodesic spheres of radius , in an n-dimensional  1 2 . space of constant negative curvature  R12 , are the .n1/-dimensional spaces of constant curvature R sinh  R

Therefore spherical geometry can be regarded as part of pseudo-spherical geometry.” (“Pseudo-spherical geometry”, in the sense of Beltrami, is hyperbolic geometry).

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Norbert A’Campo and Athanase Papadopoulos C

D

B

A Figure 7. In spherical (or elliptic) geometry, the notion of betweenness for triples of points is replaced by the notion of separation of quadruples. In this figure, the points A and C separate the points B and D.

(2) If .A; BjC; D/ then .C; DjA; B/ and .B; AjC; D/. (3) If .A; BjC; D/, then .A; C jB; D/ does not hold. (4) If the points A; B; C; D are collinear and distinct, then either .A; BjC; D/ or .A; C jB; D/ or .A; DjB; C /. (5) If A, B, C are collinear and distinct, then there exists a point D such that .A; BjC; D/. (6) For any five distinct collinear points A, B, C , D and E, if .A; BjD; E/ then .A; BjC; D/ or .A; BjC; E/. The axiom in neutral geometry that says that congruence preserves betweenness is replaced by an axiom saying that congruence preserves separation. To conclude the discussion of basic notions of spherical (or elliptic) geometry, let us now make a few remarks on the derived notions, showing the difference between these notions and those of neutral geometry. In spherical geometry, one cannot talk about the two sides of a point on a line. (Recall that on a great circle on a sphere, the complement of a point consists in one component.) In particular, one cannot define a ray in the same way as in neutral geometry. On the sphere, the complement of a line (great circle) has two connected components. In the projective plane, the complement of a line has one connected component. Talking about a segment in spherical or elliptic geometry has to be done with some care, since a segment is not determined by its two endpoints. The same holds for triangles, which are not determined by their three vertices, as in the case of neutral geometry. Given a straight line l in elliptic space, the family of all lines that are perpendicular to l have a unique common point, namely, the pole of l. The distance from the pole to the line l is equal to half of the length of l. The duality between lines and points can be expressed by the fact that if the pole of a line l lies on a line l 0 , then the pole

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of l 0 lies on l, and the intersection of l and l 0 is the pole of the line joining the poles of l and l 0 . In spherical and in elliptic geometry, there are trirectangular triangles. In such a triangle, the pole of each line containing an edge is the vertex opposite to that edge. Finally, as already mentioned, in spherical and elliptic geometry there is a canonical (or “absolute”) measure for length, that is, a definition of a unit length that is done using only the axioms of spherical (or elliptic) geometry. This follows from the fact that in spherical and elliptic geometry all lines have the same length, and this length is finite. Thus, this common length can be taken as a unit for measuring length, in much the same way as a total angle at any point (whose measure is usually taken to be 2) can be taken as an absolute measure for angles. We also note that there is such a canonical measure for length in hyperbolic geometry, but for completely different reasons; see Section 4.5 and Section 7.6.

2.7 Euclidean trigonometric formulae obtained as limits of hyperbolic and spherical trigonometric formulae It is always instructive to make parallels between results in hyperbolic geometry and results in spherical geometry. We already noted that there are analogies between the trigonometric formulae of the two geometries, and we can see this more explicitly in the table below, which contains some formulae extracted from the survey paper by Alekseevskij, Vinberg and Solodovnikov in [2]. Glancing at the formulae in the Table 1. The table compares trigonometric formulae for a right triangle with angles ˛; ˇ;  and opposite edges a, b, c, and with right angle at  , in the hyperbolic plane (of constant curvature 1), in the Euclidean plane and on the sphere (of radius 1). The formulae in the Euclidean plane are obtained from the corresponding formulae in the other two geometries by taking Taylor expansions, as the edges a, b, c of the triangle tend to 0.

Hyperbolic plane

Euclidean plane

Sphere

cosh c D cosh a cosh b

c Da Cb

cos c D cos a cos b

sinh b D sin c sin ˇ

b D c sin ˇ

sin b D sin c sin ˇ

tanh a D tanh c cos ˇ

a D c cos ˇ

tan a D tan c cos ˇ

cosh c D cot ˛ cot ˇ

1 D cot ˛ cot ˇ

cos c D cot ˛ cot ˇ

cos ˛ D cosh a sin ˇ

cos ˛ D sin ˇ

cos ˛ D cos a sin ˇ

tan a D sinh b tan ˛

a D b tan ˛

tanh a D sin b tan ˛

2

2

2

table, we can see that the spherical formulae are obtained by replacing the hyperbolic functions sinh and cosh of edge lengths by the values of the trigonometric functions sin and cos respectively of these edge lengths. Of course there are explanations for

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this fact. One explanation involves the function E.y/ of de Tilly that we already mentioned above. We shall use this function below, in the proof of the trigonometric formulae. There are other formulae, which transform in the same manner, and which are valid for arbitrary triangles. We mention as an example the “spherical sine law” for a spherical triangle of side lengths a, b, c and opposite angles ˛, ˇ,  , sin b sin c sin a D D sin ˛ sin ˇ sin  (see Section 6.7 below) which becomes, in hyperbolic geometry, the “hyperbolic sine law” which says, using the same notation, that sinh b sinh c sinh a D D : sin ˛ sin ˇ sin  We prove this hyperbolic sine law in Theorem 6.13 below. In Section 9, we shall make precise relations between the trigonometric formulae in the three geometries. Namely, we shall describe a larger space in which the hyperbolic, Euclidean and spherical planes are embedded and in which it is possible to pass continuously from one geometry to another one. The trigonometric formulae can be transformed in a continuous manner from one geometry to another. Another interesting fact is that infinitesimally, the hyperbolic and the spherical trigonometric identities both give rise to the Euclidean ones. Let us work out an example. The first hyperbolic formula in the above table is cosh c D cosh a cosh b: This is the “hyperbolic Pythagorean formula” for a right triangle whose sides have lengths a, b, c, with c being the length of the hypotenuse (the side opposite to the right angle). We shall prove this theorem in Section 6.4 below. Taking order two Taylor expansions, we obtain 1 C c 2 =2 D .1 C a2 =2/.1 C b 2 =2/; which gives, keeping only terms up to order two, c 2 D a2 C b 2 ; which is the Euclidean Pythagorean theorem. We can easily deduce that this limiting Euclidean behavior is true by using trigonometric formulae for more general figures that are decomposable into right triangles. We can also check this fact directly, starting with the trigonometric formulae for arbitrary triangles. For instance, we have the following identity, for a hyperbolic triangle with edges a, b, c and opposite angles ˛, ˇ,  respectively (see Theorem 6.9 below): cosh a D cosh b cosh c  sinh b sinh c cos ˛:

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31

Taking order two Taylor expansions, we get 1 C a2 =2 D .1 C b 2 =2/.1 C c 2 =2/  bc cos ˛; that is (again, up to order two), a2 =2 D b 2 =2 C c 2 =2  bc cos ˛; or, equivalently, a2 D b 2 C c 2  2bc cos ˛; which is the Euclidean cosine rule, which allows one to express the value of an angle in terms of the lengths of its three sides. The explanation for this limiting behavior is that at an infinitesimal level, the hyperbolic plane and the sphere are both Euclidean. Lobachevsky, in his memoirs, stressed on this fact, and he checked it on the trigonometric formulae at each occasion, see e.g. the Pangeometry [59], pp. 31, 34 and 52. In fact, Lobachevsky gave a proof of the fact that at the infinitesimal level, hyperbolic geometry becomes Euclidean, based on the hyperbolic trigonometric formulae, by showing that for a triangle with angles ˛, ˇ,  , taking first-order approximations, the hyperbolic trigonometry formulae give ˛ C ˇ C  D , see [59], p. 31. Finally, we mention that a passage between formulae for volumes of hyperbolic polyhedra and formulae for volumes of spherical polyhedra was noticed by Coxeter in [21], who made a relation between computations of Lobachevsky and formulae discovered by Schläfli on the variation of the volume function.

2.8 Comments on references The standard English (and probably the most useful) reference to Euclid’s Elements is the book by Heath [29], containing a translation with comments, historical remarks and much more. It may be surprising for the neophyte to hear that there is still a huge amount of work to be done by historians of mathematics on Euclid’s Elements. Hilbert’s Grundlagen der Geometrie, first published in 1899, grew out of lectures that Hilbert delivered at the University of Göttingen, during the winter semester of the academic year 1898–1899. After that, the lectures appeared in a book form, in several editions, revised by Hilbert, sometimes with significant changes. For a short review of Hilbert’s Grundlagen der Geometrie, we refer to the paper [13] by Birkhoff and Bennett. Friedrich Schur (1856–1932), in his Ueber die Grundlagen der Geometrie [91] presented another axiom system of geometry. George David Birkhoff (1884–1944) worked out a system of axioms for Euclidean geometry which is different from the ones of Hilbert and Schur, and which is based on the notion of distance; see [12].

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Norbert A’Campo and Athanase Papadopoulos

Hermann Weyl proposed a system of axioms for Euclidean geometry that is based on the notion of vector space;30 see [108]. Andrei Nikolaevich Kolmogorov, who is best known for his axiomatization of probability theory (1933), also gave an axiomatics of Euclidean geometry where distance is an undefined notion. Kolmogorov presented his axioms in a textbook for secondary school teaching. A good reference on axioms and on several other matters discussed in these notes is Efimov’s book [27]. We also note that de Tilly,31 in the nineteenth century, worked out the principles of the three geometries (hyperbolic, Euclidean and spherical) that is based on the notion of distance. He developed an axiomatic approach to these geometries based on metric relations between finite sets of points.

30 It seems that the notion of vector space was discovered and re-discovered several times; the list of people to be credited includes Darboux (1875), Peano (1888), Weyl (1918) and then Banach, Hahn, Wiener and Noether (around 1920); see [73]. 31 Joseph-Marie de Tilly (1837–1906) was an officer in the Belgian army, who taught mathematics at a military school and who did profound work on non-Euclidean geometry. De Tilly did his research in isolation (like Lobachevsky, Bolyai, and to a certain extent, Gauss, regarding this subject). He obtained in 1860 results similar to those of Lobachevsky on hyperbolic geometry, before realizing that his work was already done by the great Russian mathematician. In the introduction to his Etudes de mécanique abstraite (1868) [99], de Tilly writes: “J’ai établi, après plusieurs années de travail, les principes fondamentaux d’une géométrie abstraite, basée sur la négation de l’axiome XI d’Euclide, qui à mes yeux, et en tant que vérité absolue, ne repose sur rien. J’avais tiré de ces principes des démonstrations fort curieuses et je comptais les présenter au jugement des savants losque, il y a à peu près une année, je lus, dans le tome XVII du Journal de Crelle, un mémoire de Lobatschewsky, intitulé: Géométrie imaginaire, dans lequel je retrouvai mes formules fondamentales, sans démonstration, mais suivies de déductions qui, en certains points, étaient poussées bien plus loin que les miennes. Plus tard, je trouvai dans un autre mémoire du même auteur la démonstration de ces formules fondamentales, très différente de la mienne, mais tout aussi exacte, et en même temps l’indication d’autres ouvrages traitant du même sujet. Je perdis ainsi la priorité de mes découvertes et je pus me convaincre que les travaux exécutés avant moi suffisent, et au delà, pour faire présumer que la nouvelle hypothèse qui sert de base à la géométrie abstraite ou imaginaire ne peut conduire à aucune conséquence en opposition avec la logique, ce qui permet de la considérer comme possible aussi bien que celle d’Euclide”. De Tilly became later on a member of the Royal Belgian Academy of Science. He is considered as the main inventor of non-Euclidean mechanics. See [99], [100] and [101]. It seems that de Tilly was among the first mathematicians to give a rigorous definition of a multi-dimensional space, see the report in Mansion [66]. The work of de Tilly has not received the attention that it deserves. His writings are still difficult to find and they need to be thoroughly analyzed.

Notes on non-Euclidean geometry

33

3 The neutral plane 3.1 Introduction The neutral plane is the plane satisfying the axioms of Euclidean geometry except the parallel postulate, and where this axiom is neutralized; that is, this axiom may or may not be satisfied. The first 28 propositions in Euclid’s Book I of the Elements do not make use of the parallel postulate, and therefore they are valid in neutral geometry. This fact has been commented by several authors who used it to argue that Euclid himself thought that the parallel postulate might be a theorem and not a postulate, and that failing to prove it, he delayed its use as much as possible. This is a conjecture that cannot be proved, and it might be that Euclid just decided to start the Elements by highlighting results in neutral geometry, with no secret assumption that the parallel postulate might be deduced from the others. In any case it is remarkable that there is a set of propositions which are grouped together at the beginning of the Elements and that we can use in the setting of neutral geometry. In the next subsection, we record some of these propositions and others on neutral geometry that will be useful in the rest of these notes. The rest of the section essentially deals with triangles and quadrilaterals in the neutral plane.

3.2 Some results in neutral geometry We shall sometimes use the notation AB to denote either a segment ŒA; B or the length of that segment, when no confusion is possible. Likewise, we shall usually write AB D CD (instead of AB  CD) to say that the two segments AB and CD are congruent. All the propositions in this subsection hold for the neutral plane. Proposition 3.1. Assume there exists a line l, a point A and two lines that contain A and that are disjoint from l. Then, there exist infinitely many lines that contain A and that are disjoint from l. Proof. Let l1 and l2 be two distinct lines that contain A and that are disjoint from l. The angle they form at A is nonzero. Take a point B on l1 with B 6D A, and a point D on l such that the segment BD intersects l2 , and call C this intersection point. (To see that this is possible, let A0 be the projection of A on l and take B and D close to A and A0 respectively.) Now let l 0 be a line passing through A and contained in the open angular sector BAC , see Figure 8. The line l 0 enters the triangle ABC and it must exit from it (Pasch’s axiom). It necessarily exits from the edge BC because otherwise we would find two lines that intersect in more than one point. Let E be this intersection point of l 0 with BC . The

1

34

Norbert A’Campo and Athanase Papadopoulos B

A

l1 E

l0

C

l2

A0

D

M 2l

Figure 8

line l 0 does not intersect l. To see this, suppose there is such an intersection point and assume first it is on the ray A0 D. Let M be this point, and consider the triangle EDM . The line AC enters this triangle, it must exit from it (Pasch’s axiom again), and it necessarily exits from the side MD. But AC cannot intersect MD because l2 is parallel to l. Therefore l 0 does not intersect the ray A0 D. The same proof shows that the line l 0 cannot intersect the complement of the ray A0 D of l. Thus, l 0 and l are disjoint. The next three results are from Euclid’s Elements. Proposition 3.2 (Euclid, Book I, Proposition 16). Let ABC be a triangle and let us extend the edge BC at C , producing an exterior angle  at C (Figure 9). Then,  > Ay y and  > B. A

B0

I

B



C Figure 9

Proof. Let I be the midpoint of AC . We extend BI to IB 0 D BI . The two triangles y Since y > B 0 CI , the proof BIA and B 0 IC are congruent, therefore B 0 CI D A. follows.

1

1

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Notes on non-Euclidean geometry

Remarks 3.3. 1) Proposition 3.2 is not correct on the sphere; see Figure 10. The reader can examine what goes wrong if one uses the above proof in the case of the sphere instead of the neutral plane. C > =2

A

B

Figure 10. In spherical geometry, it is not true that in any triangle, an exterior angle is greater than the two other interior angles. (In this figure, the exterior angles at A and B are smaller than the interior angle at C .)

2) Euclid proved later on in the Elements, using the parallel postulate, that the angle sum in a triangle is equal to two right angles, and Proposition 3.2 follows immediately from this fact. It is interesting that Euclid included this special case in the list of results at the beginning of Book I that do not use the parallel postulate. In the next section, we shall give a proof of the result attributed to Saccheri (and sometimes called Legendre’s First Theorem, because the proof that is usually given of this result is due to Legendre) saying that in neutral geometry, the angle sum of any triangle is  . The next proposition is a particular case of that theorem. It says that in any triangle, the sum of two angles is  . This is easier to prove than Saccheri’s theorem, and it is contained in Euclid ([29], Vol. I, p. 281). This proposition will be used below in the proof of Saccheri’s Theorem. Proposition 3.4 (Euclid, Book I, Proposition 17). In any triangle in the neutral plane, the sum of any two angles is  . Proof. Let ABC be an arbitrary triangle and let us prove that By C Cy  . Let I be the midpoint of the edge BC . We take a point A0 on the line AI such that I is the midpoint of AA0 (Figure 11). The two triangles IBA and ICA0 are congruent, therefore ABC  A0 CB. The two angles ACB and BCA0 are adjacent and they are on the same side of the line AC , therefore their sum is  . Thus, ABC C ACB  . This proves the proposition.

1 1

1

1

1 1

Remark 3.5. Proposition 3.4 can be considered as a converse to the parallel postulate. Indeed, one form of this postulate says that if two lines (here, AB and CB) make with

36

Norbert A’Campo and Athanase Papadopoulos B

A0

I

A

C Figure 11

1

1

a third line (AC ) internal angles (here, BAC and B CA) whose sum is less than , then, the first two lines intersect, that is, the triangle ABC exists. Proposition 3.4 says that conversely, if two lines AB and CB intersect, forming a triangle ABC , then these two lines make with AC internal angles whose sum is less than . Proposition 3.6 (Euclid, Book I, Proposition 18). In a triangle ABC , if AC > AB, then By > Cy . In other words, the greater edge subtends the greater angle. Proof. Consider the point D on AC satisfying AD D AB, and join B to D by a segment (Figure 12). From Proposition 3.2, ADB > D CB. But ADB D ABD,

1 1

1 1

A

D

B

C Figure 12

since the triangle ADB is isosceles. (In neutral geometry, a triangle with congruent sides has congruent opposite angles; this can be deduced from the symmetry with respect to the angle bisector of the two congruent edges.) Hence, ABD > D CB and a fortiori ABC > D CB.

1 1

1 1

Proposition 3.7 (Euclid, Book I, Proposition 19). In any triangle ABC , if By > Cy then AC > AB. In other words, the greater angle subtends the greater edge. Proof. This follows easily from Proposition 3.6.

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Notes on non-Euclidean geometry

Proposition 3.8 (Euclid, Book I, Proposition 24.). Let ABC and A0 B 0 C 0 be two triangles satisfying AB D A0 B 0 , AC D A0 C 0 and Ay > Ay0 . Then, BC > B 0 C 0 .

2

Proof. Since Ay > Ay0 , we can consider, at the vertex A0 , an angle C 0 A0 D 0 congruent to CAB, with A0 D 0 D A0 B 0 D AB and situated on the same side than B 0 with respect to the line A0 C 0 (Figure 13).

1

A0

A

B

C

D0

B0

C0

Figure 13

The triangles ABC and A0 D 0 C 0 are congruent, therefore D 0 C 0 D BC . The triangle A0 B 0 D 0 is isosceles, therefore A0 D 0 B 0 D A0 B 0 D 0 . Thus, A0 B 0 D 0 > C 0 D 0 B 0 , which implies C 0 B 0 D 0 > C 0 D 0 B 0 . From Proposition 3.7 applied to the triangle B 0 C 0 D 0 , we obtain C 0 D 0 > B 0 C 0 , which gives BC > B 0 C 0 .

2

2

2 2

2 2

3.3 Saccheri’s Theorem and other results in neutral geometry The next theorem is usually attributed to G. Saccheri, and it is sometimes referred to as Legendre’s First Theorem. Theorem 3.9 (Saccheri). In the neutral plane, the angle sum of any triangle is  . Proof. Let A1 B1 A2 be a triangle with angles ˛, ˇ and  at the respective vertices, and let us show that ˛ C ˇ C   . Consider the infinite ray starting at A1 and containing the segment A1 A2 . We replicate the triangle A1 B1 A2 on that ray and on the same side, taking A2 A3  A1 A2 and so on, as shown in Figure 14. By Proposition 3.4, the sum of two angles in any triangle is  , therefore ˛ C ˇ   and consequently we can indeed do the replication of the triangle by staying on the same side of the ray A1 A2 . We let ı be the angle at A2 in the triangle B1 A2 B2 . Since the total angle at any side of a point on a line is flat, we have, at the point A2 of the line A1 AnC1 ,  C ı C ˛ D . We wish to show  C ˇ C ˛  , or, equivalently, that ˇ  ı, or, equivalently (Proposition 3.8), that jA1 A2 j  jB1 B2 j. The proof is by contradiction. Assume that jA1 A2 j > jB1 B2 j. From the triangle inequality, we have jA1 B1 j C jB1 B2 j C    C jBn1 Bn j C jBn An j  jA1 An j D njA1 A2 j:

38

Norbert A’Campo and Athanase Papadopoulos B1

B2

ˇ

ˇ

˛ A1



ı

˛ A2

B3

Bn1

Bn ˇ



ı

::: ˛

˛

A3

An

 AnC1

Figure 14. Legendre’s proof of Saccheri’s theorem. Note that in the hyperbolic plane, the concatenation B1 B2 B3 : : : is not a straight line.

Therefore, njA1 A2 j  jA1 B1 j C jBn An j C .n  1/jB1 B2 j; which implies .n  1/.jA1 A2 j  jB1 B2 j/  jA1 B1 j C jAn Bn j: Since n can be arbitrarily large, this contradicts the Archimedean axiom. Thus, we have jA1 A2 j  jB1 B2 j. This completes the proof of the theorem. Another proof. Let ABC be an arbitrary triangle and let  C  be its angle sum. We show that   0. Let BC be an edge of shortest length in the triangle ABC , let I be the midpoint of BC and let A0 be a point on the line AI such that I is the midpoint of AA0 (Figure 15). The triangles ABI and A0 IC are congruent, since they have two B I

A0

A C Figure 15

b 1

congruent angles bounded by two congruent edges. Therefore, IAB D I A0 C . This implies that the two triangles ABC and AA0 C have the same angle sum. Thus the angle sum of the triangle AA0 C is  C . Let us denote by ˛ the measure of the angle BAC . This is the smallest angle measure in the triangle ABC , since BC is the shortest edge in that triangle (Proposition 3.6). We have BAC D A0 AC C AA0 C . Therefore, the triangle A0 AC has an angle whose measure is not greater than ˛=2. Iterating some finite number of times this construction, we obtain a triangle whose angle sum is  C  and which has an angle of measure < =2. Since the sum of the measures of the two other angles cannot exceed  (Proposition 3.4), we have   0.

1

1 1 1

Notes on non-Euclidean geometry

39

Remarks 3.10. Theorem 3.9 follows from Theorem XIV in Saccheri’s book [89], saying that the “obtuse-angle hypothesis” for quadrilaterals with two congruent opposite edges making right angles with a common third edge (called “Khayyam–Saccheri quadrilaterals”, on which we give more detail below) cannot hold. Saccheri’s book Euclides ab omni naevo vindicatus : sive conatus geometricus quo stabiliuntur prima ipsa universae geometriae principia [89] was published in 1733, the year of Saccheri’s death. It is unknown whether Saccheri saw it in print or not. The book was soon almost forgotten, and it remained so during more than 150 years, until Angelo Manganotti, who, like Saccheri, was a Jesuit priest, found by accident a copy of that book and communicated it to Beltrami, in 1889.32 Soon after this, Beltrami reported on that book in a paper entitled Un precursore italiano di Legendre e di Lobatschewski [10] (an Italian precursor to Legendre and Lobachevsky, Beltrami’s Collected Works, Vol. IV, p. 348). In that article, Beltrami showed that several results that were attributed to Legendre and Lobachevsky were already contained in Saccheri’s work. Among these is the theorem stating that in neutral geometry, the angle sum of any triangle is  . Saccheri’s Euclides ab omni nœvo vindicatus was translated into German by P. Stäckel (1895), into Italian by G. Boccardini (1904), and into English by G. B. Halsted, first in installments, in the American Mathematical Monthly (1894– 1898), and then as a book in 1920. A proof of the fact that the “obtuse-angle hypothesis” does not hold in Khayyam– Saccheri quadrilaterals was already given by the medieval mathematician Omar Khayyam33 in Book I of his Commentaries on the Difficulties Encountered in Certain Postulates in the Book of Euclid; cf. [81], p. 308 ff.; see also [88], p. 467. Saccheri’s Theorem, with a proof, was also known to Nasir al-Din at-Tusi (1201–1274), see [29], Volume I, p. 209. Rosenfeld and Youschkevitch, in [88], p. 468, report on the fact that al-Tusi, in two treatises, the Treatise that cures doubts in parallel lines and the Exposition of Euclid, studied Khayyam–Saccheri quadrilaterals and analyzed the three possibilities for the remaining two (equal) angles of such a quadrilateral, showing that this angle must necessarily be acute. We shall further comment on this below. 32 Saccheri’s book was not completely forgotten, since it is mentioned in Klügel’s doctoral thesis (1763), in which the author analyzes 28 different existing “proofs” of Euclid’s parallel axiom, highlighting the flaws in these “proofs”. Georg Simon Klügel (1739–1812) studied in Göttingen, and he did his doctorate under the supervision of Abraham Gotthelf Kästner (1719–1800) who was a teacher of Gauss, who compiled encyclopaedias and who wrote mathematics textbooks and books on the history of mathematics and of optics, and on the applications of mathematics to physics and astronomy. It is told that Kästner was an excellent teacher, but that Gauss did not bother to go to his lectures because he found them too elementary. It seems that Kästner had an indirect influence on the three founders of non-Euclidean geometry since, besides being a teacher of Gauss, he taught Farkas Bolyai, the father of Janós Bolyai, and he also taught J. C. M. Bartels, who was the teacher of Lobachevsky. Thus, one can assume that Kästner’s interest in the Parallel Postulate Problem was transmitted in three different ways to the three founders of hyperbolic geometry. After he read Klügel’s thesis, d’Alembert described Euclid’s Parallel Postulate as “the scandal of geometry.” 33 Omar Khayyam (1048–1131 ca.) was a mathematician, philosopher and astronomer. He was born and he died in Nishapur (Persia; actual Iran). He worked in the cities of Samarkand, Bukhara, Ispahan and Marw. A valuable edition of Khayyam’s mathematical texts, with a beautiful introduction, was published by R. Rashed and B. Vahabzadeh [81]. It seems that it cannot be said with certainty that Khayyam the mathematician and Khayyam the poet, author of the celebrated Rubaiyat (quatrains) are the same person, see [81], p. 5.

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Norbert A’Campo and Athanase Papadopoulos

Figure 16. From a manuscript by Nasir al-Din at-Tusi (13th c.) on the parallel postulate. The title of the manuscript is “al-Risala al-shâfiya ’an al-shakk fî al-khutût al-mutawâziyya” (The epistle that cures the doubts concerning parallel lines). [Istanbul, Topkapi Saray, Ahmet III 3342, fol. 197r.]

Gauss also stated Theorem 3.9, with the same proof as the one of Legendre, cf. [35], p. 190. The reason why Theorem 3.9 was attributed to Legendre34 is that, as we already recalled, the work of Saccheri had been almost forgotten for more than 150 years, and was rediscovered in 1889, about eighty years after Legendre published his proof. Another reason for which this theorem is still sometimes attributed to Legendre is that Legendre stated it as a theorem, which he proved using the first 28 propositions of Euclid’s Elements, which as we recalled, concern neutral geometry, whereas the statement was contained in works of his predecessors without the status of a theorem. The first proof presented above was given by Legendre in the third edition of his Éléments de géométrie (1800) (Proposition 19 of Book I) [50]. This proof is also reproduced by Bonola [16], p. 56 of the English translation. The second proof given above is in Lobachevsky’s Geometrische Untersuchungen, [56], §19, and it is a variation on a proof that appeared in the 12th edition (1823) of Legendre’s Éléments (stated there also as Proposition 19 of Book I). Note that this second proof is based on a construction that is contained in the proof of Proposition 16 of Book I of Euclid (Proposition 3.2 above). Finally, we note that the two proofs that are given above of Theorem 3.9 use some form of the Archimedean axiom (either the fact that lines are infinitely extendable, or a continuity property). Max Dehn showed that this result is false for the neutral plane without the Archimedean axiom, see the discussion in Bonola [16], p. 144. Note that the result is false for the geometry of the sphere (where geodesics have finite length). 34 We already mentioned that this theorem is sometimes referred to as “Legendre’s First Theorem”. There is a “Second Theorem of Legendre” which we shall mention below (see Remark 3.23).

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From Saccheri’s Theorem, we deduce a few other results. Theorem 3.11. In the neutral plane, if two distinct lines have a common perpendicular, then they are disjoint. Proof. The proof is by contradiction. Let l and l 0 be two lines with a common perpendicular, intersecting l and l 0 in two points A and B. If l and l 0 intersected at some point C , then the angle sum of the triangle ABC would be > , which contradicts Saccheri’s Theorem. Theorem 3.12. In the neutral plane, if two distinct lines l1 and l2 make equal alternate interior angles (or, equivalently, equal corresponding angles) with a third line l3 , then l1 and l2 are disjoint. Proof. Let A and B be the intersection points of l1 and l2 respectively with l3 (Figure 17), let I be the midpoint of AB and let P and Q be the feet of the perpendiculars from I on l1 and l2 respectively. P

A

l1

I B

l2

Q

l3 Figure 17

b

1 b 1

Since IA  IB and since PAI  QBI , the right-angled triangles AP I and BQI are congruent. Therefore, we have PIA  QIB, which implies that the points P , I and Q are collinear. Thus, l1 and l2 have a common perpendicular, and by Theorem 3.11, they are disjoint. Theorem 3.13. In the neutral plane, for any line l and for any point P not on l, there exists a line containing P and disjoint from l. Proof. From P we draw a perpendicular line to l. Let P 0 be the foot of this perpendicular. Now from P , we draw the perpendicular line l 0 to the segment PP 0 . Then, using again Theorem 3.11, the two lines l and l 0 , having a common perpendicular, are disjoint.

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3.4 Angular deficit in neutral geometry We shall define the notion of angular deficit of a polygon. We start with the definition of a polygon: A polygonal curve in the plane is the homeomorphic image of a circle which is made out of a finite union of segments. A polygonal curve bounds a compact region in the plane, which we call a polygon. The polygonal curve bounding a polygon P is called the boundary of P and is denoted by @P . If P is a polygon, then a point v on @P is said to be a vertex of P (or of @P ) if for every closed disk D centered at p, the intersection D \ @P is not a segment. In the neutral plane, any polygon has at least three vertices. This follows from the fact that two distinct lines intersect in at most one point. An edge of P (or of @P ) is a connected component of the complement of the set of vertices in @P , or the closure of such a component. An edge (or its closure) is a segment. A polygon is completely determined by its set of vertices together with a cyclic ordering on that set. The boundary of the polygon is obtained from these data by joining by segments the consecutive vertices in the given cyclic ordering, and taking the union of these segments. Note that in general a cyclically ordered set of finite points in the plane does not determine a polygon because the curve obtained as the union of the segments joining consecutive vertices might not be embedded. The angle at a vertex of a polygon is the angular sector defined by the two rays starting at that vertex and containing the two edges of the polygon that meet at that vertex, the associated region being the one that contains points of the polygon near the vertex. Such an angle is also called an internal angle of the polygon. The external angle at that vertex is the other angular sector defined using the same rays. The value of an interior or of an exterior angle is in Œ0; 2Œ. An interior or an exterior angle can be either less than or greater than , and at any vertex of a polygon, the sum of the interior and exterior angles is equal to 2. A polygon P (or its boundary @P ) is said to be convex if for any two points in P , the line segment joining them is contained in P . An open half-plane is a connected component of the complement of a line in the plane. A closed half-plane is the union of an open half-plane with the line that bounds it. It is easy to see that a polygon is convex if and only if it is the intersection of (a finite number of) closed half-planes. The interior angle at any vertex of a convex polygon is < . Conversely, if the interior angle at each vertex of a polygon P is < , then P is convex. An n-gon is a polygon with n vertices. Definition 3.14 (Angular deficit of a polygon). The angular deficit of an n-gon P with internal angles 1 C    C n is defined as A.P / D .n  2/  .1 C    C n /: In other words, the angular deficit of a polygon with n sides is equal to the deficit of the angle sum of this polygon with respect to .n  2/.

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Notes on non-Euclidean geometry

In particular, if P is a triangle with angles 1 , 2 , 3 , then its angular deficit is equal to   .1 C 2 C 3 /. From this definition it follows immediately that if we add to a polygon P some extra (fake) vertices which are at the interior of edges, then the angular deficit of P , computed using these extra vertices, is unchanged, since each of the extra angles is equal to . The angular deficit of a polygon P can be computed by taking a triangle decomposition of P and then summing the angular deficits of the triangles that form this triangulation. This result does not depend on the chosen triangulation, as Proposition 3.17 below shows. Before proving this, we start with the following elementary addition formula: Proposition 3.15. Let  be a quadrilateral obtained by gluing along a common edge two triangles 1 and 2 . Then, A./ D A.1 / C A.2 /. Proof. We use the notation of Figure 18: 1 D ABD and 2 D ACD with angles ˛1 ; ˇ1 ; 1 and ˛2 ; ˇ2 ; 2 respectively, and  is obtained by gluing 1 and 2 along the common edge AD. We have A.1 / C A.2 / D   .˛1 C ˇ1 C 1 / C   .˛2 C ˇ2 C 2 /:

A

˛1

˛2

ˇ2 B

C

ˇ1 2

1

D Figure 18

This gives A.1 / C A.2 / D 2  .˛1 C ˛2 C 2 C ˇ2 C 1 C ˇ1 / D A./: The following special case of Proposition 3.15 will be useful below:

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Norbert A’Campo and Athanase Papadopoulos

Corollary 3.16. Given a triangle  which is the union of two triangles 1 and 2 glued along a common edge, we have A./ D A.1 / C A.2 /. Proof. Using the notation of the proof of Proposition 3.15, we have a quadrilateral  D ACDB which is geometrically a triangle (the extra vertex at D is fake, since the interior angle at that vertex is ; see Figure 19). By Proposition 3.15, the angular A ˛1

B

ˇ1

˛2

1 2

ˇ2

C

Figure 19

deficit of this quadrilateral is the sum of the angular deficits of 1 and 2 . We call a triangle decomposition (or a triangulation) of a polygon P a decomposition of P into a union of triangles with disjoint interiors, and such that the intersection of any two of these triangles is either empty or an edge or a vertex. A polygon equipped with a triangle decomposition is called a a triangulated polygon. Proposition 3.17. Let P be a triangulated polygon in the neutral plane, equipped with a triangle decomposition. The angular deficit of P is equal to the sum of the angular deficits of the triangles of this decomposition. Proof. Any triangle decomposition of P can be obtained by successively cutting a polygon into two other polygons, by a broken line segment that joins two vertices (Figure 20). Thus, it suffices to prove that for any decomposition of P into two

Figure 20

Notes on non-Euclidean geometry

45

polygons P1 and P2 obtained by adding a broken line segment that joins two vertices, we have A.P / D A.P1 / C A.P2 /. Consider such a decomposition and let p C 1 be the number of segments in the broken line segment that separates P1 from P2 ; that is, p is the number of vertices of the broken line segment, excluding the endpoints. Let S (respectively S1 , S2 ) be the angle sum of P (respectively P1 , P2 ) and let n (respectively n1 , n2 ) be the number of edges of P (respectively P1 , P2 ). We have S1 C S2 D S C 2p and n1 C n2 D n C 2.p C 1/: Thus, we obtain A.P / D S  .n  2/ D S1 C S2 p    .n1 C n2  2.p C 1/  2/  D .S1  .n1  2// C .S2  .n2  2// D A.P1 / C A.P2 /:



There are more general figures than polygons for which we can compute the angular deficit. For instance, we can consider a union of polygons, glued along edges. More generally one can talk about “triangulated objects”, and define their angular deficit by summing over the angular deficits of triangles. The point is to show that if we change the triangulation of the same object, the sum of the angular deficits is invariant. We shall use the following corollary of Proposition 3.17: Corollary 3.18. If T1 and T2 are two triangles with T1  T2 , then, A.T1 /  A.T2 /. There are various ways of constructing sequences of triangles whose angular deficits tend to 0. One natural way is described in the following lemma which will be useful later on in these notes. This lemma has also the following theoretical significance. In Section 5.2 below, we shall define the area of a hyperbolic triangle to be equal to its angular deficit. Although this is the usual way area is defined in hyperbolic geometry, at first sight, it might seem non-intuitive. Lemma 3.19 can be considered as a step towards showing that this notion of area is reasonable. Theorem 3.17 has also such a significance. The diameter of a subset of the plane is equal to the supremum of the lengths of segments joining pairs of points in that set. Lemma 3.19. If Tn is a sequence of triangles whose diameters tend to 0, then the sequence of angular deficits of Tn tends to 0. Proof. First consider a fixed triangle T , and suppose that its angular deficit is positive. (If there is no such triangle, the angular deficit of any triangle is zero, and there is

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Norbert A’Campo and Athanase Papadopoulos

nothing to prove.) For any integer k  0, we take k disjoint disks in T whose diameters are all bounded by 1=k. Since the diameter of Tn tends to 0 as n ! 1, we can find an integer n0 such that for any integer n  n0 , we can put k disjoint congruent copies of the triangle Tn in T , where k D k.n/ ! 1 as n ! 1. The angular deficits of all such copies of Tn are equal to A.Tn /, the angular deficit of Tn . Thus, by Corollary 3.18, A.Tn / is bounded by k1 A.T /, therefore A.Tn / converges to 0 as n ! 1. Proposition 3.20. In the neutral plane, if the angular deficit of some triangle is positive, then the angular deficit of any triangle is positive. Proof. By Saccheri’s Theorem, the angular deficit of any triangle in the neutral plane is nonnegative. Suppose a triangle A contains a triangle B whose angular deficit is positive. We can take a triangulation of A such that B is part of the triangulation. By the additivity of angular deficit, the angular deficit of A is also positive. Suppose there exists a triangle T with positive angular deficit. Then, given any  > 0, by successively subdividing T into triangles whose diameters tend to 0, and by the additivity of angular deficit, we can find a triangle T of diameter <  and which has positive angular deficit. Now given an arbitrary triangle A, we can take  to be small enough so that A contains a triangle congruent to T . By the discussion at the beginning of the proof, this implies that the angular deficit of A is positive. Remark 3.21. Proposition 3.20 also follows from Theorem 3.22 below, and in this form it is called Legendre’s Second Theorem; see Remark 3.23 below. Two triangles are said to be homothetic if up to some ordering, their side lengths are homothetic. Theorem 3.22. In the neutral plane, the following three properties are equivalent: (1) there exists a triangle whose angle sum is equal to ; (2) there exists a sequence of homothetic triangles with arbitrarily large homothety factors and with angle sum equal to ; (3) the angle sum of any triangle is equal to . Proof. (3) H) (1) is trivial. Let us show that (1) H) (2). Let T be a triangle whose angle sum is equal to . By doubling this triangle along an edge, we obtain a quadrilateral Q whose opposite edges are pairwise congruent and whose angle sum is equal to 4 (Figure 21 (a)). We can tile the entire neutral plane using tiles that are all congruent to Q. Furthermore, from the fact that the angle sum of each triangle used in this tiling is equal to , it follows that the edges of the triangles in the tiling form a network of straight lines (see Figure 21 (b)). By taking unions of adjacent triangles,

47

Notes on non-Euclidean geometry ˇ ˇ



˛

(a)



A

˛

(b)

Figure 21. At the vertex A, we have ˛ C ˇ C  D .

we can find for each integer n > 1 a triangle Tn which is homothetic to T by the homothety factor n. This shows that (1) H) (2). Now we prove that (2) H) (3). Let Tn be a sequence of homothetic triangles with arbitrarily large homothety factors and with angle sum equal to . Let T 0 be an arbitrary triangle. By taking n large enough, T 0 can be covered by a triangle congruent to Tn .35 By Theorem 3.9 and Corollary 3.18, the angle sum of T 0 is equal to . This completes the proof of Theorem 3.22. Remark 3.23. In Theorem 3.22, the implication (1) H) (3) is sometimes called the “Second Theorem of Legendre”, the “First Theorem of Legendre” referring to Theorem 3.9 above (Saccheri’s Theorem). Legendre proved that theorem using the Archimedean axiom and the postulate of continuity. Dehn [23] showed that these axioms are not needed. An elementary proof, without these axioms, is given by Bonola in [16], p. 30. In his New Elements of Geometry (see [54], p. 7 of the French translation), Lobachevsky recalls that Legendre proved this theorem in 1833 (cf. [51]), and that he had given himself a proof of this fact in 1826, in his manuscript Exposition succinte des principes de la géométrie avec une démonstration rigoureuse du théorème des parallèles (A brief Exposition of the Principles of Geometry with a rigorous proof of the theorem on parallels), which unfortunately is lost. The “Second Theorem of Legendre” should be compared with Saccheri’s Proposition V in [89] (and it follows from it) which concerns Khayyam–Saccheri quadrilaterals (which we already mentioned and which we shall study in Section 3.6 below) and which says the following: (Halsted’s translation): “If even in a single case the right-angle hypothesis is true, always in every case it alone is true”. Note also that Propositions VI and VII in [89] say an analogous thing, with right angle replaced by obtuse, respectively acute.

35 Note

that this uses the Archimedean axiom.

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Norbert A’Campo and Athanase Papadopoulos

3.5 Trirectangular quadrilaterals in neutral geometry A trirectangular quadrilateral in the neutral plane is a quadrilateral that has three right angles. We shall refer to the remaining angle as the fourth angle of the quadrilateral. (Of course, the fourth angle might also be right.) We already noted that Lambert, in his Theorie der Parallellinien (1766) [49], thoroughly studied such quadrilaterals, and he discussed the (a priori) three possibilities for the fourth angle: right, acute or obtuse. He thought that the last two cases are excluded. Before Lambert, Ibn al-Haytham already introduced trirectangular quadrilaterals in his Commentaries on the Postulates of the Books of Euclid. Before Lambert, Ibn al-Haytham made an analysis of the three possibilities for the fourth angle, viz. right, acute, or obtuse, and he refuted the second and third possibilities by using a result he considered to be a theorem, namely, that the endpoints of a perpendicular moving on a straight line is a line, a fact which, as we already noted, is equivalent to the parallel postulate. Then, from the existence of quadrilaterals with four angles, Ibn al-Haytham gave a proof of the parallel postulate (see [88], p. 62). Because the use of trirectangular quadrilaterals in neutral geometry was highlighted in the works of Ibn al-Haytham and Lambert, these quadrilaterals are also called Ibn al-Haytham– Lambert quadrilaterals. This terminology is used in Pont [79], cf. also Rosenfeld [87]. Proposition 3.24. In the neutral plane, the fourth angle of a trirectangular quadrilateral is either right or acute. Furthermore, in the neutral plane, the following are equivalent: (1) there exists a trirectangular quadrilateral that has four right angles; (2) any trirectangular quadrilateral has four right angles; (3) there exists a triangle whose angle sum is equal to . Proof. It is easy to see that a trirectangular quadrilateral is convex. Thus, it can be cut by a diagonal (that is, a straight segment joining two non-consecutive vertices) into two triangles, each of them having, by Saccheri’s Theorem (Theorem 3.9), an angle sum  . Since the angle sums of each of the two triangles add up to the angle sum of the trirectangular quadrilateral, this implies that the fourth angle is either right or acute. To prove that (1) H) (2), we start with a trirectangular quadrilateral that has four right angles, we tile the plane with quadrilaterals that are congruent to it, and we use an argument similar to the one of the proof of (1) H) (2) of Theorem 3.22. The proof of (2) H) (3) is done by subdividing a trirectangular quadrilateral into two triangles, and using Saccheri’s Theorem and the additivity of angular deficit. The proof of (3) H) (1) follows by using (3) H) (1) of Theorem 3.22 and doubling a right triangle. A quadrilateral with four right angles is called a rectangle. We reformulate part of Proposition 3.24 as follows:

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Notes on non-Euclidean geometry

Proposition 3.25. In the neutral plane, either all trirectangular quadrilaterals are rectangles, or each of them has an acute angle. Proposition 3.26. Let ABCD be a trirectangular quadrilateral in the neutral plane, with right angles at B; C; D. Then, the angle at A is either right or acute, and we have the following equivalences: (1) Ay D =2 () jADj D jBC j () jABj D jDC j; (2) Ay < =2 () jADj > jBC j () jABj > jDC j. Proof. The first part of the statement has already been proved (Proposition 3.24). We prove the second. Let M be the midpoint of the segment CB and let us erect a perpendicular to CB at M . This perpendicular cannot intersect the edges CD and AB (because if it did, we would have a triangle whose angle sum is > ), therefore it intersects the edge DA. Let N be the intersection of this perpendicular with DA. The reflection about the line MN sends the point C to the point B, and the point D to a point L on the line BA. There are three cases: (1) L D A, and in this case we have jBAj D jCDj, and DAB D =2.

1

(2) L lies strictly between A and B (as in Figure 22 (i)), which is equivalent to jDC j > jABj. In this case, we have DAB < =2, for otherwise the angle sum in the triangle NAL would be > .

1

(3) A lies strictly between B and L (as in Figure 22 (ii)). This case cannot occur because since N LB D =2 , we would have LAN < =2 (since the angle sum in the triangle NLA is  ), which would imply that DAB > , which is excluded (Proposition 3.24).

1

1

1

L D

C

A

N

L

M

N

D

B

C

(i)

M

A

B

(ii) Figure 22

Remarks 3.27. 1) Case (3) in the proof of Proposition 3.26 occurs in the geometry of the sphere.

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Norbert A’Campo and Athanase Papadopoulos

2) Following a certain tradition, if Case (1) (respectively Case (2)) in the statement of Proposition 3.26 holds, we shall say that the right-angle (respectively the acuteangle) hypothesis holds. If Case (3) in the proof of this proposition occurs, then we say that the obtuse-angle hypothesis holds (that is, the geometry is spherical).

3.6 Khayyam–Saccheri quadrilaterals Definition 3.28 (Khayyam–Saccheri quadrilateral). A Khayyam–Saccheri quadrilateral is a quadrilateral having two opposite edges of the same length and making right angles with a third common edge. In the Euclidean plane, a Khayyam–Saccheri quadrilateral is a rectangle. As already noted, Khayyam–Saccheri quadrilaterals were studied by Omar Khayyam in Book I of his Commentaries on the Difficulties Encountered in Certain Postulates in the Book of Euclid; see [81], p. 308 ff., and [88], p. 467. They were also studied by Nasir al-Din al-Tusi (1201–1274) in two treatises, the Treatise that cures doubts in parallel lines and the Exposition of Euclid, see [29], Volume I, p. 209, and [88], p. 468. Several centuries later, Saccheri, in his Euclides ab omni nœvo vindicatus (1733), considered such quadrilaterals in neutral geometry, and he distinguished three cases: the remaining angle of such a quadrilateral is either acute, or right, or obtuse. Saccheri showed that if any one of these hypotheses holds for one quadrilateral, then the same hypothesis holds for any other quadrilateral. He showed that in each case, the angle sum of a triangle in the plane is less than, equal, or larger than a right angle respectively. He then examined thoroughly the three cases. He showed that the obtuse-angle hypothesis is excluded because it leads to the existence of a pair of points joined by two distinct geodesics, which is contrary to one of the axioms of neutral geometry. With a fallacious argument, he also excluded the case of an acute angle. To this end, Saccheri argued that if the hypothesis of the acute angle occurs, then, there would exist two lines with a common perpendicular “situated at infinity” which, he excluded with an argument that is strangely non-rigorous compared to the rest of his work36 . This shows how much Saccheri was close to the discovery of non-Euclidean geometry, and at the same time, how much he was influenced by the dominant idea that Euclid’s parallel postulate follows from the others and, therefore, that there is no geometry other than the Euclidean one. One may reasonably ask why Khayyam–Saccheri quadrilaterals were studied by so many authors. The answer is that these quadrilaterals naturally appear when one studies the parallel problem in terms of equidistance, by conceiving parallel lines as equidistant from each other, that is, if one has in mind the (fallacious) concept that in neutral geometry, equidistance sets to straight lines are straight lines. Indeed, taking 36 In Halsted’s translation, [89], p. 14, “I only say here at length I have disproved the hostile hypothesis of acute angle by a manifest falsity, since it must lead to the recognition of two straight lines which at one and the same point have in the same plane a common perpendicular. That this is contrary to the nature of the straight line is proved by five lemmas [...]”

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Notes on non-Euclidean geometry

two equidistant points to a straight line situated on the same side amounts to taking two perpendicular segments of equal length and on that side, and this gives immediately rise to a Khayyam–Saccheri quadrilateral. In Section 4, we shall study in detail the geometry of Khayyam–Saccheri quadrilaterals in hyperbolic geometry. In particular, we shall prove monotonicity of certain functions of the edges, and this will be an important ingredient in the proof of the trigonometric formulae in the hyperbolic plane. Now we study some properties of Khayyam–Saccheri quadrilaterals in neutral geometry. Figure 23 represents a Khayyam–Saccheri quadrilateral ABCD with two right angles at A and B, and such that AD  BC . Applying the symmetry with respect to the perpendicular bisector of AB (see Figure 23), we immediately obtain the following proposition. D

C

A

B

Figure 23

Proposition 3.29. With the above notation the following holds: (1) the angles at C and D are congruent; (2) the perpendicular bisector of AB is also the perpendicular bisector of CD; (3) the perpendicular bisector of AB cuts the quadrilateral ABCD into two trirectangular quadrilaterals. Proposition 3.30. Let ABCD be a Khayyam–Sacchheri quadrilateral with right angles at A and B. Then, jABj  jDC j. Proof. For n  2, we draw n C 1 congruent Khayyam–Saccheri quadrilaterals, ABCD, BB1 C1 C , B1 B2 C2 C1 , B2 B3 C3 C2 , : : :, Bn1 Bn Cn Cn1 which are adjacent to each other, starting from ABCD, as indicated in Figure 24. We have jABn j D .n C 1/jABj  jADj C .n C 1/jDC j C jCn Bn j which gives .n C 1/.jABj  jDC j/  2jADj:

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Norbert A’Campo and Athanase Papadopoulos D

C

C1

C2

C3

Cn1

Cn

Bn1

Bn

::: A

B

B1

B2

B3

Figure 24

Since n can take arbitrarily large values, the Archimedean axiom implies jABj  jDC j. We also record the following corollary: Corollary 3.31. Let ABCD be a Khayyam–Sacchheri quadrilateral in the neutral plane with right angles at A and B, and let E and F be the midpoints of AB and CD. Then, we have the following equivalences: (1) Cy D =2 () jEF j D jADj D jBC j; (2) Cy < =2 () jEF j < jADj D jBC j. Proof. This follows from Proposition 3.26. Theorem 3.32. In neutral geometry, the following properties are equivalent: (1) there exists a Khayyam–Saccheri quadrilateral with two acute angles; (2) there exists a trirectangular quadrilateral with an acute angle; (3) there exists a triangle with angle sum < . Proof. Let ABCD be a Khayyam–Sacchheri quadrilateral, with two right angles at A and B and whose other angles are acute. Dividing the Khayyam–Saccheri quadrilateral by the perpendicular bisector of AB, we obtain two trirectangular quadrilaterals with acute angles. This shows that (1) H) (2). We already noted that a trirectangular quadrilateral is convex. Cutting such a quadrilateral with an acute angle by one of its diagonals, we obtain two triangles whose total angle sum is < 4. Therefore, the angle sum of one of them is < . This proves (2) H) (3). Suppose now that there exists a triangle with angle sum < . Then, by Theorem 3.22 the angle sum of any triangle is < . A trirectangular quadrilateral is the union of two triangles, therefore its angle sum is < 4. Thus, it has an acute angle. This proves (3) H) (2). Finally, (2) H) (1) is obtained by doubling a trirectangular quadrilateral. Proposition 3.33. Let ABCD be a Khayyam–Saccheri quadrilateral in the neutral plane, having two right angles at A and B, and with AD D BC (Figure 25). Then the following occurs:

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Notes on non-Euclidean geometry

y and Cy are right angles; (1) if the right-angle hypothesis holds, then D y and Cy are acute angles. (2) if the acute-angle hypothesis holds, then D D

F

C

B

A

E Figure 25

Proof. By Proposition 3.29, EF cuts the Khayyam–Saccheri into two trirectangular quadrilaterals. The proposition then follows from Proposition 3.24. If we deform a Khayyam–Saccheri quadrilateral, keeping the two right angles and the edge (called the base) joining them and making one of the two perpendiculars to the base larger than the other one, then the largest perpendicular is adjacent to a smallest angle. This is expressed in the following: Proposition 3.34. Let ABCD be a quadrilateral with right angles at A and B and with AD  BC . Then, ADC  B CD.

1 1

Proof. Extending BC to a segment BE congruent to AD, we obtain a Khayyam– Saccheri quadrilateral ABED (Figure 26). D

E C

A

B Figure 26

1 1 1 1 1 1 B CD  B ED D A DE  A DC :

By Proposition 3.33, we have ADE D BED. Using Proposition 3.2, we have

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Norbert A’Campo and Athanase Papadopoulos

3.7 Projection in neutral geometry Projection, perpendicularity and shortest distance are three related notions in neutral geometry. We shall need these notions in the sequel. We start with a discussion of the notion of perpendicularity. The notion of perpendicularity is introduced by Euclid in Book I, Definition 10 of his Elements at the same time as the notion of right angle. This definition says the following (Heath’s translation [29], p. 181): “When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands”. Proposition 3.35 (Euclid, Book I, Proposition 12). Let l be a line and let P be a point not on l. Then, we can draw a perpendicular from P to l. Proof. Take a point Q which is not on l and which is situated on the side of l that does not contain P . Draw the circle of center P and containing Q. This circle cuts the line l in two points A and B. By symmetry, the perpendicular bisector of the segment AB is the required perpendicular (see Figure 27).

P A

B

l

Q Figure 27

Let l be a line and let P be a point not on l. A point P 0 on l is said to be a projection of P on l if the line PP 0 is perpendicular to l. We recall that it is a result of neutral geometry that in a triangle, an edge opposite to a smaller angle is smaller than an edge opposite to a larger angle (Euclid, Book I, Proposition 19, Proposition 3.7 above). From this, it follows that if P 0 is the projection of a point P on a line l, then the length of the segment PP 0 is not larger than the length of any segment joining the point P to a point on l. In the neutral plane, the projection of a point P on a line l is unique, for otherwise we could construct a triangle whose angle sum is > =2, which contradicts Saccheri’s Theorem (Theorem 3.9). Proposition 3.36. In neutral geometry, projection on a line cannot increase distances between points.

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Proof. Let l be a line, let P and Q be two points in the plane and let A and B be the projections of P and Q respectively on l. We wish to show that jPQj  jABj. There are two cases, depending on whether P and Q are or are not on the same side of l. We first consider the first case (Figure 28 (i)). We may assume that jAP j  jBQj. Let D P

P

Q

D

C A

B

B

A

l

(i)

(ii) O

Figure 28

be the point on the segment jAP j satisfying jADj D jBQj. In the Khayyam–Saccheri quadrilateral ABQD, we have jDQj  jABj. In the triangle PDQ, the angle at D is obtuse, therefore we have jPQj  jDQj (see Proposition 3.7). Combining the two inequalities gives the desired result. Now we consider the case where P and Q are on different sides of the line l. Here, the segment PQ intersects the line l, and we get two right triangles CAP and CBQ (Figure 28 (ii)). Since in a right triangle the hypotenuse is larger than any of the other edges (we can use again Proposition 3.7) we have jP C j > jAC j and jQC j > jBC j. By addition, this gives jPQj > jABj.

4 The hyperbolic plane 4.1 Some basic properties in hyperbolic geometry We saw (Theorem 3.22) that in the neutral plane, one of the two following cases occurs. (1) The angle sum of every triangle is equal to . (2) The angle sum of every triangle is < . The hyperbolic plane is the neutral plane in which (2) occurs. Equivalently, the hyperbolic plane is obtained from the neutral plane by adding the axiom saying that there exists a triangle whose angle sum is less than two right angles (or by an equivalent statement, see e.g. Theorem 2.6). Proposition 4.1. In the hyperbolic plane, the angle sum of any quadrilateral is < 2.

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Proof. The idea is to introduce a diagonal that cuts the quadrilateral into two triangles, and to use the fact that the angle sum in triangles is < . The only complication is that a quadrilateral is not necessarily convex, that is, a segment joining two opposite edges is not necessarily contained in the quadrilateral (that is, this segment is not necessarily a diagonal). We show that a quadrilateral has at least one diagonal. To see this, let ABCD be a quadrilateral and consider an arbitrary edge AB of that quadrilateral. We distinguish several cases, depending on the interior angles at A and B. 1) The two interior angles Ay and By cannot both be >  because otherwise the vertices C and D would be on the same side of the line AB, and the edge CD would not intersect the other side of AB, therefore the angles we started with cannot be interior angles of the quadrilateral (Figure 29 (i)), which is absurd.

A

B

(i)

A

B

A

(ii)

B

(iii)

A

B

(iv)

Figure 29

2) We cannot have an angle >  at A and an angle  =2 and <  at B because in this case C and D would be on different sides of the line AB, the edge CD would cut the line AB, and the intersection point would lead to a triangle with angle sum >  (Figure 29 (ii)), which is absurd. 3) We could have an angle >  at A and an angle < =2 at B (Figure 29 (iii)) In this case, the diagonal that joins A to the vertex opposite to it is entirely contained in the quadrilateral. 4) The remaining case to consider is when A and B are both less than  (Figure 29 (iv)). In this case, the quadrilateral is convex. Thus, a quadrilateral has at least one diagonal. Therefore, we can subdivide a quadrilateral by a diagonal into a union of two triangles. The angle sum of each of these triangles is < . By Proposition 3.15, the angle sums of the two triangles add up to the angle sum of the quadrilateral we started with. Therefore, the angle sum of the quadrilateral is < 2. Proposition 4.2. In the hyperbolic plane, any two lines have at most one common perpendicular. Proof. Assume there exist two lines with two distinct common perpendiculars. In the case where the two perpendiculars do not intersect, we obtain a quadrilateral whose angle sum is 2, which is excluded by Proposition 4.1. In the case where the two perpendiculars intersect, we obtain a triangle whose angle sum is greater than , see Figure 30. Both cases are excluded.

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Notes on non-Euclidean geometry

Figure 30

Remark 4.3. In the hyperbolic plane, there exist disjoint lines that have no common perpendicular (these lines are said to be parallel, see Section 7). If two lines in the hyperbolic plane have a common perpendicular, then they are called ultra-parallel, and sometimes hyper-parallel. In particular, the two lines that contain any pair of opposite edges of a Khayyam–Saccheri quadrilateral are ultra-parallel. Theorem 4.4. In the hyperbolic plane, if two triangles have their three angles pairwise congruent, then the two triangles are congruent. Proof. Let ABC and A0 B 0 C 0 be two triangles satisfying Ay D Ay0 , By D By0 and Cy D Cy 0 . We show that jABj D jA0 B 0 j. Assume without loss of generality that jABj > jA0 B 0 j. We take a plane motion that sends the angle B 0 A0 C 0 to the angle BAC , the ray 0 0 A B being sent to AB and the ray A0 C 0 being sent to AC . The image of A0 by this motion is the point A. Let B 00 and C 00 be respectively the images of B 0 and C 0 . The point B 00 is on the edge AB of the triangle ABC and it satisfies jAB 00 j D jA0 B 0 j. The point C 00 is on the line AC . We claim that C 00 lies between A and C . The proof is by contradiction. Assume that C 00 is beyond C (as represented in Figure 31 (ii)). Let I be the intersection point between BC and B 00 C 00 . Since the two triangles ABC and A0 B 0 C 0 have all their angles congruent, C C 00 I D ACB. Since ACB C IC C 00 D , the angle sum of the triangle C C 00 I is greater than , which is a contradiction. Thus, C 00 lies between A and C (as in Figure 31 (iii)). But then, the angle sum of the quadrilateral C C 00 B 00 B is equal to 2 (this angle sum being the sum of the two flat angles at C 00 and B 00 ), contradicting Proposition 4.1. Thus, we have jABj D jA0 B 0 j, and the two triangles ABC and A0 B 0 C 0 are congruent.

2

1 1

1

1 1

Remark 4.5. Theorem 4.4 implies that the lengths of the edges of a triangle are determined by the angles of that triangle. This shows that there is a canonical notion of length in the hyperbolic plane. The existence of a canonical measure of length in neutral geometry is equivalent to the negation of Euclid’s parallel postulate. In a letter to Gerling, dated 11 April 1816 (see [35], Vol. VIII, p. 168), Gauss talked, in the following terms, about the existence of the canonical measure of length: “It would be desirable that Euclidean geometry were not true, for we would then have a

58

Norbert A’Campo and Athanase Papadopoulos C 00 C

C I

C0

C 00

A0

B0 (i)

B 00

A

B

A

(ii)

B 00

B

(iii)

Figure 31

universal measure a priori. One could use the side of an equilateral triangle with angle D 59o 590 5900 :9999 ..... as a unit of length.”37 There is no canonical measure for lengths of segments in Euclidean geometry; that is, one has to choose a unit of length before talking about actual length. The canonical measure for length in hyperbolic geometry can also be deduced from the canonical measure for angles, via Lobachevsky’s parallelism function and we shall talk about this in Section 7.6 below. J. H. Lambert, in his Theorie der Parallellinien (1766) [93] already noticed (using other arguments) that in (the still hypothetical) hyperbolic geometry, there exists a canonical measure for length. An immediate corollary of Theorem 4.4 is the following Corollary 4.6. In hyperbolic geometry, there are no homothetic non-congruent triangles.

4.2 On quadrilaterals in hyperbolic geometry There are no “rectangles” (quadrilaterals with four right angles) in the hyperbolic plane, nor on the sphere. There are two classes of figures that play, at several places, the role of rectangles in hyperbolic geometry, and we already studied them in our review of neutral geometry (Sections 3.5 and 3.6 above). These are the Khayyam–Saccheri quadrilaterals (sometimes also called two-right-angled isosceles quadrilaterals) and the trirectangular quadrilaterals (also called Ibn al-Haytham–Lambert quadrilaterals). We recall that the following two statements are equivalent to Euclid’s parallel postulate: 37 [Es wäre sogar wünschenswerth, dass die Geometrie Euklids nicht wahr wäre, weil wir dann ein allgemeines Mass a priori hätten, z.B. könnte man als Raumeinheit die Seite desjenigen gleichseitigen Dreiecks annehmen, dessen Winkel = 59o 590 5900 ; 9999..... ]

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59

(1) the four angles of some (or, equivalently, of any) Khayyam–Saccheri quadrilateral are right angles; (2) the fourth angle of some (or, equivalently, of any) trirectangular quadrilateral is a right angle. We also recall the following: (1) in hyperbolic geometry, the value of the fourth angle of some (or, equivalently, of any) trirectangular quadrilateral is < =2; (2) in hyperbolic geometry, the common value of the two non-right angles of some (or, equivalently, of any) Khayyam–Saccheri quadrilateral is < =2. We note by passing that trirectangular quadrilaterals and Khayyam–Saccheri quadrilaterals can be analogously defined on the sphere, and that we have the following: (1) in spherical geometry, the value of the fourth angle of some (or, equivalently, of any) trirectangular quadrilateral is > =2; (2) in spherical geometry, the common value of the two non-right angles of some (or, equivalently, of any) Khayyam–Saccheri quadrilateral is > =2. Trirectangular quadrilaterals appear naturally as a central object of study in building coordinates for the hyperbolic plane. To see this, consider two oriented lines x 0 Ox and y 0 Oy in the hyperbolic plane, intersecting orthogonally at a point O. We can use these lines to define a coordinate system in the plane. In analogy with what is usually done in Euclidean plane geometry, given any point P in the hyperbolic plane, we consider its projections Px and Py respectively on x 0 Ox and y 0 Oy. We obtain a quadrilateral OPx PPy , which has right angles at O, Px and Py . Unlike the case of Euclidean geometry, the angle at P of this quadrilateral is not right. The quadrilateral OPx PPy is a trirectangular quadrilateral. We shall study how the shape of this quadrilateral varies in terms of the position of P , e.g. in terms of the distance from Px to the origin, and in terms of the angle between the rays Ox and OP . Such a study will be useful in the derivation of the trigonometric formulae and for other important results in hyperbolic geometry.

4.3 Trirectangular quadrilaterals in hyperbolic geometry We start with a few basic properties of trirectangular quadrilaterals. Proposition 4.7. For any trirectangular quadrilateral in the hyperbolic plane, any edge adjacent to the acute angle is strictly shorter than the edge opposite to it. Proof. Consider the trirectangular quadrilateral ABCD of Figure 32, where the three right angles are at B, C , D. Let M be the midpoint of BC . The perpendicular bisector MN of BC intersects the segment DA at a point N . Consider the symmetry with respect to this perpendicular bisector. This symmetry sends the line CD to the line

60

Norbert A’Campo and Athanase Papadopoulos N

D

A L

C

B

M Figure 32

BA. The image L of D by this symmetry cannot be the point A, since otherwise we would have a quadrilateral with four right angles, which is excluded. We claim that L lies between A and B, as shown in Figure 32. Indeed, otherwise, A would lie between L and B, and the angle sum of the triangle NAL would be > , which is excluded. From this the proposition follows. Proposition 4.8. A trirectangular quadrilateral is determined up to congruence by the value of its acute angle and the length of any of the two edges adjacent to the acute angle. Proof. Let ABCD be a trirectangular quadrilateral with acute angle at A, let A0 B 0 C 0 D 0 be another trirectangular quadrilateral with acute angle at A0 , and assume that Ay  Ay0 and AB  A0 B 0 . There exists a plane congruence that sends the segment AB onto the segment A0 B 0 , and the angles at A and B respectively of the quadrilateral ABCD on the angles at A0 and B 0 of the quadrilateral A0 B 0 C 0 D 0 . After transporting the quadrilateral A0 B 0 C 0 D 0 by this congruence, we may assume that the segment AB coincides with the segment A0 B 0 , that the ray r.A; D/ coincides with the ray r.A0 ; D 0 /, and that the ray r.B; C / coincides with the ray r.B 0 ; C 0 / (see Figure 33 in which we have represented two a priori possible relative positions of the pairs .D; D 0 / and .C; C 0 /). D

C

0

C

D0

A D A0

B D B0

D0

C

0

D

C

A D A0

B D B0

Figure 33

Now the two segments DC and D 0 C 0 are perpendicular to the two lines AB and DC , and since two lines in the hyperbolic plane can have at most one common

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perpendicular (Corollary 4.2), the segments DC and D 0 C 0 coincide. Thus, ABC is congruent to A0 B 0 C 0 D 0 . As an application of the geometry of trirectangular quadrilaterals, we prove the following important property of pairs of lines in hyperbolic geometry: Proposition 4.9. Let l and l 0 be two lines having a common perpendicular AB, with A 2 l and B 2 l 0 . Then, jABj is the shortest distance from a point on l to a point on l 0 . Furthermore, if P is a point on l varying on one of the two rays contained in l with origin A and if x denotes the distance jAP j, then the distance from P to l 0 is a strictly increasing function of x. Proof. The first property follows from the second. Thus, it suffices to prove the second property. We refer to Figure 34. l

A

P

Q E

l0 B

P0

Q0

Figure 34

Let P and Q be two points on l which are on the same ray on l with origin A, and such that jAP j < jAQj, and let P 0 and Q0 be respectively the projections of P and Q on l 0 . The three segments AB, PP 0 and QQ0 are pairwise disjoint because otherwise we would get a triangle whose angle sum is > . The quadrilateral APP 0 B is trirectangular, and its angle at P is therefore acute. Thus, in the quadrilateral PQQ0 P 0 , the angle at P is obtuse. For similar reasons, the angle at Q in this quadrilateral is acute. Now from P we draw a perpendicular line to PP 0 . Since the angle P 0 PQ is obtuse, one side of P on this perpendicular line is strictly contained in the angular sector P 0 PQ. For obvious reasons, this perpendicular cannot intersect the sides PQ, PP 0 and P 0 Q0 of the quadrilateral PQQ0 P 0 . Therefore, it intersects the segment QQ0 in some point E between Q and Q0 . The quadrilateral PEQ0 P 0 is trirectangular and by Proposition 6.1, jEQ0 j > jPP 0 j, which implies that jQQ0 j > jPP 0 j, which completes the proof of the proposition.

2

2

We deduce the following:

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Corollary 4.10. Let l1 and l2 be two lines in the hyperbolic plane having a common perpendicular. Then, the distance from a point on l1 to the line l2 cannot take the same value at more than two different points. Corollary 4.10 should be contrasted with the corresponding situation in Euclidean geometry, where disjointedness of two lines is equivalent to equidistance. We already recalled that adding to the axioms of neutral geometry an axiom saying that given any two lines l and l 0 the distance from a point on l to the line l 0 is constant when the given point moves on l is equivalent to adding Euclid’s parallel axiom. We study equidistance in the next section.

4.4 Equidistance The study of equidistant sets is interesting, in particular because it highlights some important differences between hyperbolic and Euclidean geometry. An equidistance set to a point (that is, the set of points that are at some fixed positive distance from that point) is, by definition, a circle. Circles have the same qualitative properties in Euclidean and in non-Euclidean geometries. The difference in equidistance sets starts in the various geometries with lines. In the hyperbolic plane, the set of points which are at a fixed distance from a line l has two connected components, but unlike the situation in the Euclidean plane, these components are not lines. We shall see below that each of these components is a convex curve. More precisely, there is one connected component of the complement of such a curve that is a convex set. This convex component is the one that contains the line l (Proposition 4.13 below). It can easily be seen from the triangle inequality that in the neutral plane, any component of an equidistance set to a line is unbounded in both directions. In the hyperbolic plane, a connected component of an equidistance set to a line is called a hypercycle.38 and the line itself is called an axis of the hypercycle.39 It also follows easily from the definitions that a hypercycle is symmetric with respect to any line perpendicular to its axis. Using such symmetries, we see that if we take two points A and B on different connected components of an equidistance set to a line l, then the intersection of the segment AB with l is the midpoint of AB. We shall make below a few remarks on equidistance and on hypercycles on the sphere. 38 The terminology is probably due to Gauss; cf. his Letter to F. Bolyai on 6 March 1832, in which he comments on J. Bolyai’s Appendix: “On may call hypercycle the collection of all points in a plane that are at equal distance from a straight line; similarly for hypersphere.” [Hypercykel könnte der Complexus aller Punkte heissen, die von einer Geraden, mit der sie in einer Ebene liegen, gleiche Distanz haben; eben so Hypersphäre.] 39 We use this common terminology although, strictly speaking, a figure with an “axis” should be symmetric with respect to that axis, and therefore, a line is an axis of the union of the two components of an equidistance set, and not of a single component of that set. Rather, one should call axis of an equidistance set to a line l (or of a component of such a set) a perpendicular to the line l.

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We note by the way that an equidistance set to a line is drawn using a set square. We recall that in Euclid’s Elements, all figures are constructible (in principle by ruler and compass), and hence the question of whether an equidistance set is constructible or not is quire relevant. The following is a consequence of Corollary 4.10. We now provide another proof. Theorem 4.11. Let l be a line in the neutral plane and suppose that there exist three points that are equidistant from l and that are situated on some line l 0 . Then, the plane is Euclidean (and therefore, all points on l 0 are equidistant from l). Proof. Let A, B, C be three distinct points in that order on l 0 that are at the same distance ı from l. Let P and Q be the midpoints of AB and BC respectively and let P 0 and Q0 be the feet of the perpendiculars from P and Q respectively on l (Figure 35). Since l is perpendicular to PP 0 , the symmetry with respect to the line PP 0 preserves A

P

B

Q

P0

C

Q0 Figure 35

the line l. Therefore, this symmetry also preserves the ı-equidistant set to l. Thus, it preserves the points A and B, and therefore it also preserves the line l 0 that joins them. Thus, the line l 0 is perpendicular to the line PP 0 at P . By the same proof, l 0 is perpendicular to the line QQ0 at Q. Thus, the distinct lines l and l 0 have two distinct perpendiculars, which implies that the plane is Euclidean (Proposition 4.2). Corollary 4.12. Let l be a line in the hyperbolic plane, let m be a connected component of a hypercycle with axis l, let A be a point on m and let A0 be the projection of A on l. Then, AA0 is perpendicular to the curve m (Figure 36). A m

l A0 Figure 36

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The last sentence means that the line l 0 which is perpendicular to AA0 at the point A is a support line to m, that is, l 0 intersects the curve AA0 in a unique point, and the curve is contained in one side of l 0 . Corollary 4.12 follows from Theorem 4.11 by passing to the limit. Proposition 4.13. Let l be a line in the hyperbolic plane and let h be a hypercycle with axis l. Then, h is a curve that is strictly concave when viewed from l. Proof. Let A and B be two points on h, let P be the midpoint of the segment AB. It suffices to show that the point P is in the interior of the connected component of the complement of h containing l. Let A0 , P 0 , B 0 be respectively the feet of the perpendiculars from A, P , B on l (Figure 37). A

B P

A0

P0

B0

Figure 37

The symmetry with respect to the line PP 0 preserves the segment AB, therefore PP 0 is perpendicular to AB. Thus, PP 0 is the common perpendicular to the lines l and AB. In the trirectangular quadrilateral PP 0 B 0 B, we have jPP 0 j < jBB 0 j (Proposition 4.7). This proves the proposition. Remark 4.14 (Spherical geometry). Recall that in the geometry of the sphere (or of the projective plane), the lines are the great circles, and the great circles are also geometric circles, that is, they are equidistant sets to points. These circles are centered at any of the two points that are farthest away from them. Connected components of equidistant curves to these great circles are also circles centered at the same points, and at the limit, when the distance to the great circle becomes equal to half of the diameter of the sphere (or to the diameter of the projective plane), the equidistant curves degenerate to their centers. One can see this the other way round: in spherical geometry the family of equidistant curves to a point (that is, the family of circles centered at that point) contains a line which is the circle farthest away from this center. The equidistance set to a line on the sphere is a union of two circles (and in the projective plane, it is one circle). In spherical geometry, there is an analogue of Proposition 4.13 except that in this geometry the convexity with respect to a line is in the direction opposite to the one in the hyperbolic geometry.

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65

Some of the properties used in the proofs of the preceding propositions hold in absolute geometry (that is, they are properties common to hyperbolic, Euclidean and elliptic geometry), and we record them in the following two propositions. Proposition 4.15. In absolute geometry, given any two distinct points A and B on a hypercycle of axis l, the perpendicular bisector of A and B is orthogonal to the line l. Proof. Let l 0 be the perpendicular bisector of the segment AB. Let A0 and B 0 be the feet of the perpendiculars from A and B respectively on l (use Figure 37 again). The symmetry with respect to l 0 preserves the pair .A0 ; B 0 /, and therefore it preserves the line l. Thus, l is orthogonal to l 0 . Proposition 4.16. In absolute geometry, given any two distinct points on a hypercycle, their perpendicular bisector is orthogonal to this hypercycle. Proof. Let A and B be the two points on a given hypercycle, and let l 0 be their perpendicular bisector. By definition, l 0 is orthogonal to the segment AB. Making A and B come closer and closer to each other in a symmetric way, their perpendicular bisector remains the line l 0 , and we see at the limit that l 0 is perpendicular to the hypercycle. Now we return to the hyperbolic plane. Proposition 4.17. In the hyperbolic plane, any hypercycle intersects a line in at most two points. Proof. Let h be a hypercycle, let l be its axis, met m be a line and assume there exist three points A; B; C in h \ m, such that B lies between A and C . (See Figure 38.) Let A0 ; B 0 ; C 0 be respectively the projections of A, B, C on l. A

B

C

A0

B0

C0

Figure 38

The quadrilaterals ABB 0 A0 and BC C 0 B 0 are Khayyam–Saccheri quadrilaterals, therefore, the angles ABB 0 and B 0 BC are both acute, which is a contradiction.

1

1

The following is just another formulation of Proposition 4.17: Proposition 4.18. In the hyperbolic plane, no three points on the same hypercycle are collinear.

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Since three distinct points on a hypercycle are not collinear, they determine a triangle. We have the following Proposition 4.19. In the hyperbolic plane, the three perpendicular bisectors of any triangle whose vertices are on a given hypercycle are pairwise disjoint and have a common perpendicular. Proof. From Proposition 4.15, the perpendicular bisectors are orthogonal to the axis of the hypercycle. Being orthogonal to a common line, the perpendicular bisectors are disjoint. Conversely, we have the following. Proposition 4.20. In any triangle in the hyperbolic plane, if two perpendicular bisectors are ultra-parallel, then the three vertices of the triangle are on a hypercycle. Furthermore, the three perpendicular bisectors of that triangle have a common perpendicular. (In particular, they are pairwise ultra-parallel.) Proof. Let A, B, C be the three vertices of a triangle, let M and N be the midpoints of AB and AC respectively and suppose they have a common perpendicular PQ, with P being on the perpendicular bisector of AB and Q being on the perpendicular bisector of AC . Let B 0 , A0 , C 0 be respectively the projections of B, A, C on the line PQ (Figure 39). From the symmetry with respect to the line MN , we have A N

M B

B0

C

P

A0

Q

C0

Figure 39

AA0 D BB 0 . From the symmetry with respect to the line NQ, we have AA0 D C C 0 . This implies that the three points A, B, C are at the same distance from the line B 0 C 0 . Furthermore, BB 0 D C C 0 implies that BB 0 C 0 C is a Khayyam–Saccheri quadrilateral. Thus, the perpendicular bisector of B 0 C 0 is also the perpendicular bisector of BC (Proposition 3.29). This proves the second part of the statement.

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4.5 Geometric relations in quadrilaterals In this section, we start by proving a theorem which concerns the geometry of a quadrilateral with two right angles and a third angle which is obtuse or right. Note that since we are working in the hyperbolic plane, the fourth angle of this quadrilateral is necessarily acute. We shall use this theorem to obtain results about two special figures: (1) right triangles, obtained by making an edge of the above quadrilateral shrink to a point; (2) trirectangular quadrilaterals, obtained by taking the third angle in the above description equal to a right angle. A

P

t  2

l

ˇ

C˛

h

l0 A0

x

P0

Figure 40

To fix notation, we consider the situation represented in Figure 40, in which l and l 0 are two lines, A a point on l and A0 the projection of A on l 0 , P D P .t/ a point on l at distance t from A and P 0 D P 0 .t/ the projection of P on l 0 . We suppose that the point P varies on the side of A on l for which the angle A0 AP is at least equal to a right angle. We set the value of this angle to be equal to 2 C ˛, where ˛  0. We also use the following notation: x D x.t/ D jA0 P 0 j; h D h.t / D jPP 0 j; ˇ D ˇ.t / is the value of the angle APP 0 .

1

1

Theorem 4.21. With the above notation, when P varies on l as above, we have the following: (1) x.t / < t, and, more generally, x.t / is a 1-Lipschitz function of t ; that is, for t1 and t2 satisfying 0  t1 < t2 , we have jx.t1 /  x.t2 /j < jt1  t2 j; (2) x.t/ is an increasing function of t ; (3) ˇ.t / is a decreasing function of t ;

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(4) h.t / is an increasing function of t ; (5) h.t / is a 2-Lipschitz function of t ; (6) x.t/=t is a decreasing function of t ; (7) h.t /=t is an increasing function of t . (In particular, h.t / ! 1 as t ! 1.) Before proving the theorem, we make a few remarks. Remarks 4.22. 1) Property (4) can be deduced from Property (7), but we state it separately because it is useful as such and because it is useful in the proof of Property (7). 2) In this theorem, we allow the case where A D A0 , that is, the case where the two lines l and l 0 intersect at a point A. In that case, we assume that l and l 0 are distinct, that is, we assume that ˛ > 0. The proof below also works in this case. For future reference, we shall state separately the corresponding statements for the case where A D A0 , as Theorem 4.23 below. Note that in that special case, the quantities x.t /=t and h.t /=t are reminiscent of trigonometric functions of Euclidean geometry, namely, x.t/=t D cos ˛ and h.t/=t D sin ˛. In the hyperbolic plane, these two quantities are not constant (they depend on t and not only on ˛). We shall comment on that later on. 3) Note that x.t / 6! 1 as t ! 1. We shall see in Theorem 4.26 that x.t / tends to a finite value as t ! 1. Proof. (1) x.t / is a 1-Lipschitz function of t : Given two real numbers t1 and t2 satisfying 0 < t1 < t2 , the segment P 0 .t1 /P 0 .t2 / is the projection of the segment P .t1 /P .t2 / on the line l 0 . From Proposition 3.36 (or Proposition 4.7), we have jP 0 .t1 /P 0 .t2 /j < jP .t1 /P .t2 /j, that is, jx.t1 /  x.t2 /j < jt1  t2 j. This proves Property (1). (2) x.t/ is an increasing function of t: Let t1 and t2 be two real numbers satisfying 0 < t1 < t2 . If we had x.t1 /  x.t2 /, then the two segments P .t1 /P 0 .t1 / and P .t2 /P 0 .t2 / would intersect at some point I , giving rise to a triangle IP .t1 /P .t2 / which has two right angles, which is a contradiction. Thus, x.t1 / < x.t2 /. (3) ˇ.t / is a decreasing function of t : Let 0 < t1 < t2 . From Property (2), the quadrilateral AP .t1 /P 0 .t1 /A0 is strictly contained in the quadrilateral AP .t2 /P 0 .t2 /A0 . Therefore, the angular deficit of the quadrilateral AP .t1 /P 0 .t1 /A0 is strictly less than the angular deficit of the quadrilateral AP .t2 /P 0 .t2 /A0 . This implies that ˇ.t2 / < ˇ.t1 /. (4) h.t / is an increasing function of t : Again, let 0 < t1 < t2 . By the continuity of x.t/ (which follows from Property (1)), there exists a real number t3 satisfying t1 < t3 < t2 such that P 0 .t3 / is the midpoint of the segment P 0 .t1 /P 0 .t2 / (see Figure 41). Let s be the orthogonal symmetry with respect to the line P .t3 /P 0 .t3 /. This symmetry preserves the line l 0 , and it sends P 0 .t1 / to P 0 .t2 /, therefore it sends the line P .t1 /P 0 .t1 / to the line P .t2 /P 0 .t2 /. Let R be the point s.P .t1 //. This point is on the line P .t2 /P 0 .t2 /. The quadrilateral P .t3 /P 0 .t3 /P 0 .t1 /P .t1 / is sent by s to the congruent quadrilateral P .t3 /P 0 .t3 /P 0 .t2 /R. Note that the angle P 0 .t3 /P .t3 /R, which is

6

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A



P .t1 /

P .t2 /

P .t3 / 

ˇ.t1 /

R

A0 P 0 .t1 /

P 0 .t3 /

P 0 .t2 /

Figure 41

equal to ˇ.t3 /, is acute (considering the angle sum in the quadrilateral AP .t /P 0 .t /A0 , we see that ˇ.t / is acute for any t  0). Since the angle P 0 .t3 /P .t3 /P .t2 / is obtuse, the ray P .t3 /R is strictly contained in the angular sector P 0 .t3 /P .t3 /P .t2 /, and the point R is therefore contained in the interior of the segment P .t2 /P 0 .t2 /. Thus, we have jP 0 .t2 /Rj < jP 0 .t2 /P .t2 /j, that is, h.t2 / > h.t1 /. (5) h.t / is a 2-Lipschitz function of t: We take two real numbers t1 and t2 satisfying 0 < t1 < t2 and we use the same notation as in the proof of Property (4). We have

8 8

jP .t1 /P .t2 /j D t2  t1 and h.t2 /  h.t1 / D jRP .t2 /j < jRP .t3 /j C jP .t3 /P .t2 /:j

5

We already noted that the angle P .t3 /RP .t2 / is congruent to the angle ˇ.t1 /, which implies, using Property (3),

5

P .t3 /RP .t2 / > ˇ.t2 /: Thus, in the triangle P .t3 /RP .t2 /, the value of the angle opposite to the side RP .t3 / is less than the value of the angle opposite to the side P .t3 /P .t2 /. From Theorem 3.6, we have a corresponding inequality between the edge lengths of this triangle: jRP .t3 /j < jP .t3 /P .t2 /j: We therefore obtain jh.t1 /  h.t2 /j < 2jP .t3 /P .t2 /j < 2jP .t1 /P .t2 /j D 2jt1  t2 j: This proves Property (5). (6) x.t/=t is a decreasing function of t : Let us fix a position P0 of P on the line l and let x0 D x0 .t/ be the distance from A0 to the projection P00 of P0 on l 0 . We take an integer n  2, we let a D x0 =n and we consider the sequence a; 2a; : : : ; na D x0 of real numbers and the sequence P 0 .t1 /; P 0 .t2 /; : : : ; P 0 .tn /

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of successive points on l 0 , such that P 0 .t0 / D A0 , jP 0 .ti /P 0 .tiC1 /j D a for all i D 1; : : : ; n  1 and P 0 .tn / D P00 . For each i D 1; : : : ; n  1, P 0 .ti / is the projection of a point on l which we denote by P .ti /. Finally, let b1 ; b2 ; : : : ; bn be the sequence of real numbers defined by t1 D b1 , t2 D b1 C b2 ; : : : ; tn D b1 C b2 C    C bn . From the proof of Property (5) (more precisely, from the fact that jP .t3 /Rj < jP .t3 /P .t2 /j), we have b1 > b2 >    > bn : The corresponding values of x.t /=t satisfy 2a na a < <  < : b1 b1 C b2 b 1 C    C bn Thus, x.t/=t , restricted to the discrete set of points t1 ; t2 ; : : : ; tn is a decreasing function of t . By making n tend to infinity and using the fact that x.t / is continuous, we obtain that x.t/=t is a decreasing function of t , for all t between 0 and x0 . Since x0 is arbitrary, this implies that x.t /=t is a decreasing function of t . (7) h.t /=t is an increasing function of t : We use the notation of Figure 42. We take three real numbers, t1 , t2 , t3 satisfying 0 < t1 < t1 < t3 and t2  t1 D t3  t2 . Thus, P .t2 / is now the midpoint of the segment P .t1 /P .t3 /. As before, for i D 1; 2; 3, P 0 .ti / is the projection of P .ti / on the line l 0 . M N P .t1 /

P 0 .t1 /

P .t3 /

P .t2 / L

P 0 .t2 /

P 0 .t3 /

Figure 42

We let L and M be points on the line P .t2 /P 0 .t2 / satisfying jP .t1 /P 0 .t1 /j D jP 0 .t2 /Lj and jP .t3 /P 0 .t3 /j D jP 0 .t2 /M j: Since the function h.t/ is strictly increasing, the point L is in the interior of the segment P .t2 /P 0 .t2 / and M is outside this segment.

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71

Finally, let N be a point on the line P .t2 /P 0 .t2 / such that P .t2 / is the midpoint of the segment LN . We claim that N lies between the points P .t2 / and M . The two triangles P .t2 /LP .t1 / and P .t2 /NP .t3 / are congruent, since they have congruent angles bounded by congruent sides. Thus, we have

5 5

P .t2 /LP .t1 /  P .t2 /NP .t3 /: The quadrilateral P .t1 /P 0 .t1 /P 0 .t2 /L being a Khayyam–Saccheri quadrilateral, the angle P .t1 /LP 0 .t2 / is acute, therefore the angle P .t1 /LP .t2 / is obtuse. Likewise, the quadrilateral MP 0 .t2 /P 0 .t3 /P .t3 / being a Khayyam–Saccheri quadrilateral, the angle P .t2 /MP .t3 / is acute. These facts imply that N lies strictly between P .t2 / and M . Thus, we have jP .t2 /N j < jP .t2 /M j. Since jP .t2 /N j D jP .t2 /Lj D h.t2 /  h.t1 /

6 6

5

and jP .t2 /M j D h.t3 /  h.t2 /; we obtain h.t3 /  h.t2 / > h.t2 /  h.t1 /: Given a real number t > 0, we consider the following sequence of points on l: P .t0 /, P .2t0 /, P .3t0 /; : : : . We can write the values taken by the function h.t / at these points as h0 D h.t0 /, h0 C h1 , h0 C h1 C h2 ; : : : . We write h0 < h0 C h1 < h0 C h1 C h2 <    : The values taken by the function h.t/=t at these points are h0 h0 C h 1 h0 C h1 C h2 < < <  : t0 2t0 3t0 Thus, the restriction of the function h.t/=t to the sequence of points t0 ; 2t0 ; 3t0 : : : is an increasing function of t . Since t0 can be taken as small as we wish and since the function h.t/ is continuous, we obtain the desired result, that is, that h.t /=t is increasing. Now we apply Theorem 4.21 to the special case where the angle at ˛ is a right angle. The quadrilateral APP 0 A0 becomes trirectangular. The situation is represented in Figure 43, in which the notation is slightly different from the one we used in Theorem 4.21, and it is as follows: The two rays Ox and Oy intersect at right angles at O; P .t/ is a point on a ray starting at O, contained in the triangular sector Ox; Oy and making an angle with the ray Ox; t is the distance from O to P .t/; Px .t / is the foot of the perpendicular from P .t / on the ray Ox ; Py .t/ is the foot of the perpendicular from P .t/ on Oy;

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x.t/ is the distance between O and Px .t/; y.t / D jOPy .t /j; h.t / D jP .t/Px .t /j; e.t/ D jP .t /Py .t/j; .t / is the angle OP .t /Px .t /.

5

y e.t/

Py .t/

P .t/

.t/ t

y.t/

h.t/

O

x.t/

Px .t/

x

Figure 43

Theorem 4.23 (Monotonicity properties in trirectangular quadrilaterals). With the above notation, we have the following: (1) e.t / > x.t/ and h.t/ > y.t/. (2) As t increases on the ray that makes an angle with Ox, we have the following: • x.t / and y.t / increase; • .t / decreases; • e.t / and h.t/ increase; • x.t /=t and y.t /=t decrease; • h.t/=t and e.t /=t increase. Proof. The results follow from Theorem 4.21 (see Remark 4.22 that follows the statement of that theorem), and they can also be proved using the same techniques. Recall that the distance function from a point P to a line l is, by definition, the distance from P to the foot of the perpendicular from P to l. We note the following useful result, which follows from the last property stated in Item (2) of Theorem 4.23. Corollary 4.24. Let l1 and l2 be two intersecting lines in the hyperbolic plane. For any sequence xn of points on l1 that tends to infinity as n ! 1, the sequence of distances from xn to its projection xn0 on l2 tends to infinity as n ! 1. Remark 4.25. The property stated in Corollary 4.24 is commented by Greenberg in his paper [40], in the context of the foundations of geometry. In that paper, Greenberg

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shows that this statement can replace the Archimedean axiom in the collection of axioms of the hyperbolic plane. The axiom is called Aristotle’s axiom, and it is stated as follows by Greenberg: “Given any acute angle formed by rays r, s having a common vertex O, and any segment AB, there exists a point P on r such that if Q is the foot of the perpendicular from P to s, then PQ > AB (Figure 44).” r P

O Q

s

Figure 44

Stated in this manner, Aristotle’s axiom has the advantage of using only geometric variables instead of the integer variables that are used in the Archimedean axiom. We extract in the following theorem a property that concerns right triangles in hyperbolic geometry, and we shall comment on its significance in Remark 4.27 that follows the theorem. Theorem 4.26 (Monotonicity properties in right triangles). Consider a right triangle with an acute angle , let t be the length of the hypotenuse, x the length of the edge adjacent to and h the length of the edge opposite to . Then, we have the following: (1) as t decreases to 0, x=t increases to a finite and nonzero limit; (2) as t decreases to 0, h=t decreases to a finite and nonzero limit. Proof. Using the notation of Theorem 4.23, we have x.t / < e.t / and therefore x.t /=t < e.t/=t . Since x.t /=t increases and e.t /=t decreases as t decreases to 0, the two quantities converge and have finite and non-zero limits. Likewise, from Theorem 4.23, we have y.t/ < h.t/, and therefore y.t /=t < h.t /=t . Since y.t /=t increases and h.t/=t decreases as t decreases to 0, the two quantities converge and have finite and non-zero limits. Remark 4.27 (The sine and cosine functions in hyperbolic geometry). The quantities h.t /=t and x.t /=t that appear in Theorem 4.26 play the role of the sine and cosine of the angle in the right triangle considered, since these functions are the quotients of the lengths of the opposite (respectively adjacent) angle to the length of the hypotenuse in a right triangle. But these “hyperbolic sines and cosines”, for a given angle, are variable quantities; unlike the situation in Euclidean geometry, they depend on the length of the hypotenuse. Nevertheless, in hyperbolic geometry, Theorem 4.26 says that these

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quantities have finite limits as the length of the hypotenuse tends to 0. Actually, we can define these limits to be sin and cos respectively, and their quotient to be tan . Thus, the “sine”, “cosine” and “tangent” functions that are defined in Euclidean geometry have a definition within hyperbolic geometry. One can make a relation between this definition and the fact that when the edges of a triangle become infinitely small, the geometry of the triangle becomes that of a Euclidean one. See also Corollaries 6.16 and 6.14 below, for the appearance of the sine and cosine functions in hyperbolic geometry. For later use, we state this fact in the following:

1

Theorem 4.28. Let ABC be a right angle in the hyperbolic plane, with vertex at B. Let .Cn /, n  1 be a sequence of points on the edge C that converges to B as n ! 1, and for each n  1 let An be the foot of the perpendicular from Cn to BA (Figure 45). Then, we have jCn An j sin jCABj D lim ; n!1 jCn Bj

2

2

jCn An j ; n!1 jCn Bj

cos jCABj D lim and

2

jCn An j : n!1 jCn An j

tan jCABj D lim

C

C1 C2 C4 B

C3

A4 A3 A2

A1

A

Figure 45

5 Area 5.1 Introduction In the Euclidean plane, we usually define the area of a rectangle to be the product l  h of its width by its height. The area of a triangle is then taken to be equal to half of the product of a base by the corresponding altitude. This is justified by the fact that such

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a triangle can be decomposed into three pieces that can be rearranged so as to obtain a rectangle of area 12 l  h; see Figure 46, in which I is the midpoint of the altitude from A to the edge BC . See also Proposition 5.6 below. A

A0

1

2

B0

I

B

C Figure 46

After this, one can define areas of polygons by subdividing them into triangles. To define the area of more complicated figures, one has to use some infinite processes and continuity arguments and in doing so, one is led to develop a measure theory. As is well known, one then encounters in some way or another delicate notions, like the one of non-measurable subset, and paradoxes like the Banach–Tarski paradox (see below) 40 and so on. Thus, the least we can say is that the notion of area in Euclidean geometry is an intricate one. Euclid, in his Elements, did not define area as a function on figures, but he defined the notion of “two figures having the same area”. In fact, Euclid used more simply the terminology of two figures being equal. This notion of area is discussed in Books I to IV of the Elements. It starts at Proposition 35 of Book I which says (Heath’s translation [29], p. 326) that “parallelograms which are on the same base and in the same parallels are equal to one another”. Euclid then proved in Book II of the Elements that any triangle can be cut into a finite number of pieces that can be reassembled in such a way that the result is a square. The square so obtained has the same area as the original triangle. This operation of cutting into a finite number of pieces and reassembling is usually called “dissection”. In fact it is this cut-and-paste operation that allows one to define two figures to have the same area. Thus, two figures are said to have the same area precisely when they are congruent up to dissection. Euclid also talked about ratios of areas. For instance, we can find in the Elements that the ratios of two spheres is equal to the square of the ratios of their radii. It may also be useful to recall that the development of area in the setting of a metric geometry in which one computes distances, lengths of curves, area and volume, is attributed to Archimedes, who flourished a few decades after Euclid. Let us note by the way that in Euclid, the notion of area of a rectangle as the product of its height by its width is important from another theoretical point of view. In this respect, we recall that Book II of the Elements starts with ten propositions that concern 40A

paradox ia a theorem that look absurd because it is counter-intuitive.

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algebraic identities like .a C b/2 D a2 C b 2 C 2ab (and there are more involved ones) that are all expressed and proved in terms of figure dissection. From this point of view, The Elements contain the foundations of the theory of geometrical algebra based on area comparison. We shall describe a construction in hyperbolic geometry which is the analogue in the hyperbolic plane of Euclid’s dissection. Khayyam–Saccheri quadrilaterals in the hyperbolic theory of dissection play the role that squares play in Euclidean plane dissection. We use the definition of area as angular deficit, that is, Area D   (angle sum):

(5.1)

This notion of area in the hyperbolic plane is simple though highly non-intuitive. The main result, presented in Section 5.5, is that two triangles have the same area if and only if they can be obtained from each other by dissection. This is a way of justifying the definition of area as angular deficit, and we shall see others. We saw in Section 4.1 (as a consequence of Theorem 4.4, saying that the edge lengths of a triangle are determined by its three angles) that in the hyperbolic plane, there is a canonical notion of length. We shall see in the next section that there is a notion of area which is canonical up to multiplication by a positive constant. More precisely, we shall see that any “area function”  7! Area./ defined on the set of triangles in the hyperbolic plane and satisfying a certain number of reasonable properties is of the form Area./ D kA./, where A./ denotes angular deficit, and where k is a positive constant that depends only on the choice of a particular triangle to be a triangle of unit area. The existence of canonical length and of canonical area (up to a constant factor) were both pointed out by Lambert in his Theorie der Parallellinien [49]. In particular, Lambert showed that if the parallel postulate fails, then the area of a triangle can be defined as its angular deficit. The theory of area as angular deficit was developed independently by the three founders of Euclidean geometry. In Lobachevsky’s works, this is done for instance in §23 of his treatise [56]. In what follows, after defining area by angular deficit in Section 5.2, we shall prove in Section 5.3 that in hyperbolic geometry, the angular deficit is, up to a constant, the unique area function that satisfies some small set of very reasonable conditions. After that, we shall review dissection in Euclidean geometry in Section 5.4. Then, we shall consider dissection in hyperbolic geometry in Section 5.5.

5.2 Angular deficit in hyperbolic geometry We recall that the angular deficit of a triangle ABC is defined as A.ABC / D   .Ay C By C Cy /; and that angular deficit is strictly positive (respectively zero, negative) in hyperbolic (respectively Euclidean, spherical) geometry. In all this section, we work in the hyperbolic plane.

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In Lemma 3.19, we described a construction of a sequence of triangles whose angular deficits tend to 0. We now describe another way of obtaining a sequence of triangles having this property, which will be useful to us later on. Lemma 5.1. Let ABC be a triangle and let An , n D 1; 2; 3; : : : be a sequence of points on the edge AC converging to the point C . Then, A.An BC / ! 0 as n ! 1. Proof. We first prove the lemma in the particular case where ABC is a right triangle, with right angle at C , and An being a sequence of points on the edge AC converging to C . We start by finding a sequence of triangles Mn BC with Mn on AB and converging to C , such that the angular deficit of Mn BC tends to 0 as n ! 1. Consider, in the triangle ABC , the angle bisector at B. It intersects the edge AC at an interior point M . Let H be the foot of the perpendicular drawn from M on AB (Figure 47). Since the angle at A is acute, H is in the interior of the edge AB. The A

H

M B

C Figure 47

two triangles BMH and BM C , having two congruent sides bounded by pairwise congruent angles, are congruent; therefore they have the same angular deficit. By additivity, this angular deficit is less than the angular deficit of the triangle ABC . We therefore have 1 A.BM C /  A.ABC /: 2 Taking now the angle bisector of the triangle MBC at B and repeating the process, we find the sequence of points Mn on AC defining the required sequence of triangles. Now if An is the sequence of points satisfying the hypothesis of the lemma, each triangle An BC is contained in a triangle Mkn BC , for some sequence kn of integers tending to infinity as n ! 1. Again, by the additivity of angular deficit, and since A.Mkn BC / ! 0, we have A.An BC / ! 0 as n ! 1. This completes the proof of the lemma in the case where the triangle ABC has a right angle at C . In the case where ABC is not right, we can extend the line BAn so that the triangle BAn C is contained in a triangle with a right angle at C (see Figures 48 (i) and (ii), the cases depending on whether the angle at C of ABC is obtuse or acute). In this way we can conclude using the case where the angle at C is a right angle and by the additivity of angular deficit. Note that in the case described in Figure 48 (ii),

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A An An C

B

C

(i)

B

(ii) Figure 48

the line BAn necessarily intersects the perpendicular at C for provided An is close enough to C , that is, if n is large enough.

5.3 The area function Let P be the set of polygons in the hyperbolic plane. We already defined the angular deficit function on P (see Section 3.4). We now show that in hyperbolic geometry this function is, up to a multiplicative constant, the unique function that satisfies a small set of axioms that can reasonably be required from an area function. Theorem 5.2. Up to a multiplicative factor, the angular deficit function P ! RC , P 7! A.P / is the unique function satisfying the following properties: (1) A.T / > 0 for any triangle T ; (2) if P1 and P2 are congruent polygons, then A.P1 / D A.P2 /; (3) if a polygon P is the union of two polygons P1 and P2 with disjoint interiors, then A.P / D A.P1 / C A.P2 /. Proof. From the properties of angle sum of triangles in the hyperbolic plane, we already know that the angular deficit function satisfies the three stated properties. To prove the theorem, it suffices to show that for any function f W P ! RC satisfying these properties, we have, for any triangle T , A.T 0 / f .T 0 / D : A.T / f .T / Let T be an arbitrary triangle and let p be a positive integer. We can subdivide T into a union of p triangles with disjoint interiors and all having the same area. To do this, we use Lemma 5.1, starting with a sub-triangle of T that has two vertices in common with T and whose angular deficit is almost zero, and vary continuously the

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Figure 49. Finding a subdivision of T into p triangles whose area is A.T /=.p/, by using Lemma 5.1.

third vertex on an edge of T until the area of that triangle is A.T /=p (see Figure 49). Repeating this, we obtain the required subdivision of T into p triangles. Now let T 0 be any triangle distinct from T . By the same construction, we can find a nonnegative integer p 0 that depends on p such that the triangle T 0 is subdivided into p 0 C 1 sub-triangles, with p 0 of them having area A.T /=p and the last one having area  A.T /=p. From this construction, we get A.T 0 / A.T / A.T 0 /   ; p0 C 1 p p0 which gives A.T 0 / p0 C 1 p0   ; p A.T / p from which we deduce

A.T 0 / p0 D : p!1 p A.T / lim

(5.2)

Now consider a function f satisfying Properties (1)–(3) stated in the theorem, and let us take the same decomposition of the triangle T and T 0 into p and p 0 C 1 triangles respectively than the one we took for the area function, with the area function replaced now by the function f . From the fact that f satisfies Properties (2) and (3), we have p0 f .T 0 / p0 C 1   : p f .T / p Thus, we also have f .T 0 / p0 D : p!1 p f .T / lim

(5.3)

Equations (5.2) and (5.3) give the desired result. The three properties in the statement of Theorem 5.2 are properties that one naturally expects from an area function defined on the set of figures in a plane. We can

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express this fact in the following theorem, which was known to Lobachevsky, Bolyai and Gauss, and before them, to Lambert.41 Theorem 5.3. In the hyperbolic plane, the angular deficit function is, up to a scalar factor, the unique area function. One consequence of the above definition of area is that the area of triangles in the hyperbolic plane is bounded (by ). Thus, adding to the axioms of neutral geometry an axiom saying that there exist triangles of arbitrary large area amounts to adding the parallel postulate.42 We already mentioned that in Euclidean geometry, the definition of the area function is more complicated than the one in hyperbolic geometry, and that there is no uniqueness property analogous to that of an area function in the non-Euclidean plane. In the Euclidean plane, area is usually defined locally as a product measure of two onedimensional measures. There is no canonical one-dimensional measure in Euclidean geometry, and no canonical area function. Besides areas of polygons, one can naturally talk about areas of singular figures obtained as unions of polygons with disjoint interiors, glued together along boundary edges. One can also talk about areas of more general figures. For instance, the area enclosed by a circle can be defined using a limiting procedure. There is a theory of conics in hyperbolic geometry which has been developed at the end of the nineteenth century (see e.g. Story [96] and Barbarin [5]), and it is possible to compute areas of ellipses and other figures. Remark 5.4 (The area function in spherical geometry). There is a similar definition of area in spherical (or elliptic) geometry, where angular deficit is replaced by angular excess (angular deficit on the sphere is negative);43 that is, the area of a triangle ABC 41 Lambert wrote : “If it were possible under the third hypothesis [that is, the hypothesis of the acute angle] to cover a large triangle with two equal triangles, then we could prove in a simple manner that for every triangle the excess from 180o of the sum of its three angles is proportional to the area of the surface.” [Wenn es bey der dritten Hypothese möglich wäre, mit gleichen und ähnlichen Triangeln einer grössern Triangel zu bedecken: so würde es sich auch leichte darthun lassen, dass bey jedem Triangel der Ueberschuss von 180 Gr. über die Summe seiner drey Winkel dem Flächenraume des Triangels proportional wäre.] ([93], p. 202). Gauss also wrote on that topic, in his notes and in his correspondence. For instance, in a letter to Gerling dated 16 March 1819, [35], p. 182, Gauss wrote: “The angular deficit from 180o in the plane triangle, for example, does not only get greater as the area gets greater, but it is exactly proportional to it”. [Der Defect der Winkelsumme im ebenen Dreieck gegen 180o ist z.B., nicht bloss desto grösser, je grösser der Flächeninhalt ist, sondern ihm genau proportional.] In a letter to Farkas Bolyai, dated 6 March 1832, [35], p. 220, Gauss wrote a proof of the fact that the area of a triangle is equal to its angular deficit. 42 This sentence has to be taken with some care because the definition we gave of area in the hyperbolic plane and in the Euclidean plane are different. Note however that area in hyperbolic geometry can also be defined, as in the Euclidean plane, by integrating an infinitesimal area element (and there is no need for that to work in a model, that is, there is no need for a particular coordinate system). Lobachevsky developed such a theory since his early writings, see [53]. This was also developed in several later memoirs of Lobachevsky, in particular his last one, the Pangeometry, see [59]. 43 The fact that the area of a spherical triangle, or of a triangle in the projective plane, is equal to its angular excess, is usually called Girard’s Theorem, named after the Flemish mathematician Albert Girard (1595–1632),

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is defined by the formula A.ABC / D Ay C By C Cy  . The analogue of uniqueness (Theorem 5.3) holds in spherical geometry. There is no analogous definition in Euclidean geometry, since angular deficit is zero. Remark 5.5 (Volume). Let us conclude this section on area in the hyperbolic plane by a few words on volume in hyperbolic 3-space. In 3-dimensional hyperbolic geometry, the volume of a tetrahedron is a function of the dihedral angles. But the theory of volumes of polyhedra is much more difficult than the one of areas of polygons. In particular, there is no simple formula for the volume of a hyperbolic tetrahedron in terms of its dihedral angles. In this respect, it is significant that after Farkas Bolyai, in 1832, sent to Gauss a copy of his son’s achievements on non-Euclidean geometry, Gauss, in his response to the older Bolyai, asked him to transmit to his son the question of studying volumes of tetrahedra. In his work [60], Lobachevsky obtained a formula for the volume of certain tetrahedra in hyperbolic 3-space, called birectangular tetrahedra. Such a tetrahedron is characterized by the property that its vertices can be denoted by A, B, C , D in such a way that the edge AB is perpendicular to the plane BCD and the edge DC is perpendicular to the plane CBA. Any 3-dimensional tetrahedron is the union of two birectangular tetrahedra. Lobachevsky’s formula expresses the volume of a birectangular tetrahedron in terms of its dihedral angles. The formula involves the function Z  L./ D  log j2 sin ujdu 0

which is known today as the Lobachevsky function. (This is the form used in Thurston’s notes [102], Chapter 7.) The development of the theory of volume in hyperbolic geometry is surveyed in Milnor [69] and Vinberg [105]; see also the survey by Alekseevskij, Vinberg & A.S. Solodovnikov [2]. Thurston’s 1976 Princeton Notes [102] contain a chapter on volumes of 3-dimensional tetrahedra. There is a nice formula for the volume of an ideal tetrahedron , that is, a tetrahedron having its four points at infinity. Denoting by ˛; ˇ;  the three dihedral angles made by the faces meeting at some ideal vertex, the volume of the ideal tetrahedron is given by Vol./ D L.˛/ C L.ˇ/ C L. /; and this does not depend on the choice of the ideal vertex. One has always ˛ C ˇ C  D , and it is also known that an ideal tetrahedron has maximal volume (among all ideal tetrahedra) if and only if it is regular, that is, if and only if all dihedral angles are equal (and therefore equal to =3). and it appeared in print in his De la mesure de la superficie des triangles et polygones sphériques, nouvellement inventée (On the newly invented measure of area for triangles and polygons). Another proof was given by Thomas Harriot (1560–162) and by Bonaventura Cavalieri (1598–1647). Euler, in his De mensura angulorum solidorum (On the measure of solid angles) gave a simple proof of this fact, see [31] and the exposition and the history in Rosenfeld [87], p. 27 ff.

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In 1858, L. Schläfli established a very elegant formula which gives a relation between the volume of a spherical tetrahedron and its dihedral angles, in the form of a system of partial differential equation [90]. Coxeter made the relation between this formula and Lobachevsky’s formula in hyperbolic geometry. This gave rise to interesting developments by J. Milnor, I. Rivin, M. Brägger and F. Luo and others (see for instance the recent paper [64] and the references there). Walter Brägger used this formula for proving Andreev’s Theorem on planar Euclidean disk packings; see [18] and [17]. Physicists used Schläfli’s formula in variational problems related to general relativity.

5.4 Dissection in Euclidean geometry In the Euclidean plane, two figures are said to be scissors equivalent if they can be dissected into a common figure. Equivalently, two figures are scissors equivalent if one of them can be cut into a certain number of pieces that can be re-arranged so as to give the second figure. Hilbert treated this question in Chapter IV of his Grundlagen der Geometrie [43]. Proposition 14 of Book II of Euclid’s Elements treats the problem of dissecting a rectangle into a square. In some sense, this is equivalent to solving geometrically the quadratic equation x 2 D ab. In fact, Book II of the Elements can be considered as entirely devoted to geometrical algebra, that is, to the question of solving geometrically algebraic equations; see Heath’s presentation and comments in [29], p. 372. Crelle’s book Sammlung mathematischer Aufsätze und Bemerkungen (1821) [22] contains a wealth of formulae of this sort, for figures in three-dimensional Euclidean space. For instance, Crelle proves the following formula for a simplex, d 2 D a2 C b 2  2ab cos.ab/  2ac cos.ac/  2bc cos.bc/; where a, b, c, d denote the areas of the faces, and where a term .ab/ under the cosine denotes the dihedral angle made by the planes whose names are a and b; we refer the reader to [22], p. 108, for the details. Another result proved by Crelle, in the same book, is the following: Consider the triple obtained by multiplying the lengths for each of the three pairs of opposite edges of a Euclidean tetrahedron. Then, this triple satisfies the triangle inequality. Construct a Euclidean triangle whose side lengths are these three numbers. Then Crelle gives a formula for the area of this triangle in terms of the volume of the tetrahedron and the radius of the circumscribed sphere; see [22], p. 114 ff. We propose, as a good research project, to find hyperbolic-geometry analogues of these results by Crelle. We now return to Euclid. Euclid’s proof of Pythagoras’ theorem consists in showing that in a right triangle, the area of the square drawn on the hypotenuse is equal to the sum of the areas of the two squares drawn on the other sides (see Figures 50 and 51).

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Figure 50. A Greek manuscript with Euclid’s proof of Pythagoras’ theorem. The proof is based on the notion of area. One may compare the figure in this page with Figure 59 below that is used in the proof of the hyperbolic Pythagorean theorem.

Proposition 5.6 (Euclid, Book II). In the Euclidean plane, any triangle can be dissected into a square. Proof. The proof is done in two steps. Step 1: Any triangle can be dissected into a rectangle. Let ABC be a triangle, and assume that BC is its longest edge. The altitude from A to BC is then entirely contained in the triangle. Let I be the midpoint of this altitude. The parallel to BC through I cuts AB in B 0 and AC in C 0 . Let D and E be respectively the intersection points of B 0 C 0 with the perpendiculars to BC at B and C respectively (see Figure 52). Then, the triangles AIB 0 and BDB 0 are congruent, and the triangles AIC 0 and CEC 0 are congruent. This shows that the triangle ABC can be dissected into the rectangle BCED. Step 2: Any rectangle can be dissected into a square. To see this, we start by recalling the construction of the geometric mean of two line segments b and h, that is, the construction of a segment m satisfying m2 D bh. This is done in Figure 53. Now let ABCD be a rectangle with base AB D b and height AD D h. We may assume that b  2h, because if this is not the case, we can cut the rectangle into two congruent rectangles along the perpendicular bisector of the base and superpose the two smaller rectangles along their base, repeating this operation several times if needed, until we get the desired property. Now we take a segment AH D AD on AB and we consider the altitude to AB at H . It cuts the circle of diameter AB at a point M satisfying AM 2 D AH  AB (Figure 54 (a)). Thus, AM is the geometric mean of AH and AB and it is equal to the edge of the desired square.

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Figure 51. An Arabic manuscript of Nasir al-Din al-Tusi, containing Euclid’s proof of Pythagoras’ theorem

To perform the dissection of the rectangle into a square, we take on AB a point I satisfying AI D AM , and a point J on CD satisfying CJ D AM . Translating the two pieces IFB and BCJ along the edge BJ , we obtain the square X Y IA with basis AI D AM (Figure 54 (b)). Proposition 5.6 implies that two triangles have the same area if and only if they are scissors equivalent. From that proposition, the following follows easily:

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Notes on non-Euclidean geometry A D

B0

C0

E

I

B

C Figure 52

m

h b Figure 53 J

X

C

D

Y

M D

J

F H

A

I

B A

I (b)

(a) Figure 54

Proposition 5.7 (Bolyai–Gerwien). Two polygons in the Euclidean plane are scissors equivalent if and only if they have the same area. We note that the name Bolyai attached to Proposition 5.7 is the one of Farkas Bolyai, the father of Janós Bolyai, the latter being, together with Lobachevsky and Gauss, one of the three founders of hyperbolic geometry. This proposition says that equality of area between figures (at least between polygons) in the Euclidean plane can be defined using the notion of scissors equivalence. A natural question that immediately arises is whether this holds in dimension greater than two. The next remark is about this.

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Remark 5.8 (Dimension three). The question of dissection in the 3-dimensional Euclidean space is much more complicated than in dimension two. The theory starts as in dimension two: the volume of a rectangular parallelepiped is defined as the product of three mutually perpendicular edge lengths, and the volume of a tetrahedron is one third of the product of the area of a base times the corresponding altitude. Euclid started the discussion of 3-dimensional geometry in Book XI of the Elements, and there he defined equality of volume between solid figures using dissection. He started with volume equality of parallelepipeds having the same base and congruent heights (Proposition 29 of Book XI). The complication starts in Book XII, where Euclid discussed volumes of pyramids. He used a method called “exhaustion”, which involves removing from a solid figure an infinite number of sub-figures.44 Sometimes, the limiting process that is used in the calculation of the volume of a pyramid is called “devil’s staircase”. The third of Hilbert’s 23 Problems that he presented in 1900 at the second International Congress of Mathematicians (Paris) deals with the question of Euclidean volume. More precisely, Hilbert asked whether the method of exhaustion is really necessary for 3-dimensional volume theory, that is, whether there exist two solid figures that have the same volume and that are not equivalent by a finite sequence of scissors cuts. The problem was solved affirmatively, in the same year, by Max Dehn. Dehn proved that there exist two solid figures, namely, a regular tetrahedron and a cube which have the same volume and which are not equivalent by dissection (that is, by a finite sequence of scissors cuts). In particular, Dehn showed that Euclid was right in assuming that an infinite-process is needed for the volume theory of figures in dimension three, unlike the case of dimension two. It seems that the question was asked, before Hilbert, by Gauss. Dissection in Euclidean geometry is discussed in Heinz Hopf’s lectures [46]. It follows from Dehn’s work that there is associated to each 3-dimensional polyhedron a real number, which is now called its Dehn invariant, such that two figures equivalent by dissection have the same Dehn invariant. The definition of this invariant involves the angle measures of the polyhedron. The Dehn invariant of a tetrahedron is nonzero, and the Dehn invariant of a cube is zero (see [24], [25]). Polyhedra with the same volume may have different Dehn invariants, and a theorem by J.-P. Sydler (1965) [97] says that two polyhedra are equivalent by dissection if and only if they have the same volume and the same Dehn invariant. In 1990, J. Dupont & C.-H. Sah gave a homological interpretation of that result [26]. The theory is still active. In dimension 3, one can also mention the Banach–Tarski paradox, stating that one can dissect the sphere into six pieces and reassemble these pieces to form two spheres, the volume of each one of them being equal to the volume of the original sphere. Let us conclude this section by quoting Hilbert’s Problem III verbatim. (The translation is from the article in the 1902 Bulletin of the AMS.) 44 The exhaustion method is developed in Book XII of the Elements, and this theory is attributed to Eudoxus of Cnidus, the student of Plato who was a mathematician and an astronomer and which we already mentioned in these notes in relation to spherical trigonometry.

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Problem 3: The equality of the volumes of two tetrahedra of equal bases and equal altitudes. In two letters to Gerling, Gauss45 expressed his regret that certain theorems of solid geometry depend upon the method of exhaustion (or upon the axiom of Archimedes). Gauss mentions in particular the theorem of Euclid that triangular pyramids of equal altitudes are to each other as their bases. Now the analogous problem in the plane has been solved.46 Gerling also succeeded in proving the equality of volume of symmetrical polyhedra by dividing them into congruent parts. Nevertheless, it seems to me probable that a general proof of this kind for the theorem of Euclid just mentioned is impossible, and it should be our task to give a rigorous proof of its impossibility. This would be obtained, as soon as we succeed in specifying two tetrahedra of equal bases and equal altitudes which can in no way be split up into congruent tetrahedra, and which cannot be combined with congruent tetrahedra to form two polyhedra which themselves could be split up into congruent tetrahedra.47

5.5 Dissection in non-Euclidean geometry The following result is a hyperbolic analogue of Theorem 5.6. Theorem 5.9. Let ABC be a triangle in the hyperbolic plane, let B 0 and C 0 be the midpoints of AC and AB respectively, and let F , G, H be the feet of the altitudes from A, B, C respectively on the line C 0 B 0 . Then, we have the following: (1) AF  BG  CH ; (2) the quadrilateral BGH C is a Khayyam–Saccheri quadrilateral; (3) the triangle ABC is scissors equivalent to the Khayyam–Saccheri quadrilateral BGH C . Proof. In Figure 55 (i) and (ii), we have represented the cases where the angle at A is acute and obtuse respectively. To prove (1), we first note that the two right triangles AF C 0 and BGC 0 have congruent hypotenuses, and two congruent acute angles. Therefore, the two triangles are congruent, and we have BG  AF . In the same manner, we see that the two right triangles AFB 0 and CHB 0 are congruent, which gives CH  AF . This proves (1). 45 Werke, Vol.

8, pp. 241 and 244. (This note, and the next two, are Hilbert’s.) beside earlier literature, Hilbert, Grundlagen der Geometrie, Leipzig, 1899, ch. 1 [Translated by Townsend, Chicago, 1902.] 47 Since this was written Herr Dehn has succeeded in proving this impossibility. See his note: “Ueber raumgleiche Polyeder,” in Nachrichten d. K. Gesellsch. d. Wiss. zu Göttingen, 1900, and a paper soon to appear in the Math. Annalen [Vol. 55, pp. 465–478]. 46 Cf.,

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Norbert A’Campo and Athanase Papadopoulos A

A

C0 G

B0 F

B

H

E

A0 (i)

E

G

C

C0

H B0

B

A0 (ii)

F

C

Figure 55

Property (2) follows from the definitions, since the quadrilateral CHBG has right angles at G and H , and BG  CH . Now we can dissect the triangle AB 0 C 0 into the two smaller triangles AF C 0 and AFB 0 , and glue them back at BC 0 G and CB 0 H . We obtain by dissection the Khayyam–Saccheri quadrilateral BGH C . From this, we can see that any triangle can be dissected into a Khayyam–Saccheri quadrilateral having the same angle sum. We record this as the following. Corollary 5.10. In neutral geometry, any triangle can be dissected into a Khayyam– Saccheri quadrilateral, and two triangles have the same angular deficit if and only if they can be dissected into Khayyam–Saccheri quadrilaterals having the same angular deficit. Theorem 5.11 (Triangle dissection and hypercycles). Let ABC be a triangle, let B 0 and C 0 be the midpoints of AC and AB respectively, let F be the projection of A on the line B 0 C 0 and let h be the hypercycle at distance jAF j from the line B 0 C 0 and containing A. Then, h is the locus of points A0 characterized by the following two properties: (1) the triangles A0 BC and ABC have the same area; (2) A0 is situated from the same side than A with respect to the line BC . Proof. We can use again Figure 55. Let G and H be as before the projections of B and C respectively on the line A0 C 0 . Take any point A0 on the hypercycle h and let F 0 be its projection on the line A0 B 0 . Since jA0 F 0 j D jAF j, we have jA0 F 0 j D jBGj D jCH j. Therefore, by Theorem 5.9, the two triangles A0 BC and ABC are scissors equivalent, and therefore they have the same area. This proves that every point on h satisfies Properties (1) and (2). The converse follows from a monotonicity property of Khayyam–Saccheri quadrilaterals having the same base: We use the same notation as before. Consider the

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triangle ABC , and let A00 be a point on the same side as A with respect to the line BC , which has the same projection F on the line B 0 C 0 , and such that jA0 F j > jAF j (respectively jA0 F j < jAF j). By Theorem 5.9, the triangle A00 BC is equivalent to a Khayyam–Saccheri quadrilateral whose area is larger (respectively smaller) than the area of the triangle ABC . Therefore, the triangles A00 BC and ABC do not have the same area. Theorem 5.12. In the hyperbolic plane, any two triangles of the same area are scissors equivalent. Proof. From Theorem 5.9, any two triangles that have the same area and that share a common basis BC , are scissors equivalent. Furthermore (using for instance Theorem 5.9), we can increase continuously the length of the side AB of the triangle ABC , until this length takes an arbitrarily large value, and in such a way that the resulting triangle is scissors equivalent to the original triangle. We can apply this operation to two arbitrary triangles of the same area, so that they have an edge of the same length. Applying a plane motion, we can assume that the two triangles have a common edge. The result then follows again from Theorem 5.9. Remark 5.13 (Spherical geometry). The results of this section are valid in spherical geometry, with the same methods. An analogue of Theorem 5.11 stating that the locus of the third vertex of a triangle having a fixed basis and a given area is a hypercycle on the sphere was proved by Euler in his paper [33] and by Lexell48 in his paper [52] . Euler declares in [33] that the idea of studying this problem was brought to him by Lexell.

6 Trigonometry 6.1 Introduction The trigonometric formulae in a geometry give expressions of distances and angles, in a given figure (most often, a triangle), in terms of other distances and angles. We already saw that in hyperbolic geometry two triangles are congruent if and only if they have congruent angles (Theorem 4.4). It is therefore not surprising that in hyperbolic geometry there are formulae that express the lengths of the edges of a triangle in terms of its angles. In hyperbolic geometry, the Euclidean trigonometric functions (sin, cos, etc.) are not useful as such, because there are no similar triangles, and the ratio, in a right 48A. I. Lexell (1740–1784) is considered as a student of Euler. Although Euler did not have any official student, Lexell is mentioned by Fuss in Euler’s Eulogy [34], together with seven other scientists including Fuss himself, as having had the privilege of being taught by Euler at the Saint-Petersburg Academy of Science.

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rectilinear triangle, between the adjacent (respectively opposite) edge to an angle and the hypotenuse is not determined by the angle made by the two edges. Hyperbolic geometry is infinitesimally Euclidean. This can be expressed by the fact that trigonometric formulae in non-Euclidean geometry give, taking first order Taylor series approximations, the Euclidean formulae. Like in Euclidean geometry, the building blocks for the system of trigonometric formulae in hyperbolic (and in elliptic) geometry are those for a right triangle. Such a formula is called a “Pythagorean theorem”. To prove the Pythagorean theorem in hyperbolic geometry, one starts with a right triangle ABC with right angle at A, with side lengths a, b, c opposite to the vertices A, B, C respectively. Two operations are performed on this triangle. We first slide slightly the triangle along one side of the right angle, in the direction of the right angle, and then along the hypotenuse of the new triangle (see Figure 59 below). The proof of the Pythagorean trigonometric formula is then based on the variation of a certain function defined using trirectangular quadrilaterals. We now consider more closely this function. We consider the trirectangular quadrilateral of Figure 56, and we define the function E.y/ as the limit e.x/ : x!0 x

E.y/ D lim

e

P

x O

x Figure 56

This function was already defined in the Introduction. By reasoning on the small triangles thus obtained (the triangles D1 B1 B2 , DBB2 , etc. in Figure 59) and then taking limits, we shall prove the relation E.a/ D E.b/E.c/:

(6.1)

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Now E is a continuous function satisfying the functional equation E.x  y/ C E.x C y/ D 2E.x/E.y/:

(6.2)

For any constant k, the functions E.x/ D cos.kx/ and E.x/ D cosh.kx/ are solutions of Equation 6.2. The solution to Equation 6.2, for the case of the hyperbolic plane, is E.x/ D cosh.kx/, and in that case Equation 6.1 becomes cosh.ka/ D cosh.kb/ cosh.kc/; which is Pythagoras’ theorem for the hyperbolic plane. We shall work out the details in Section 6.4 below. The constant k is not determined from the axioms of hyperbolic geometry; it depends on the curvature of the space. This constant is equal to 1 in the case of the hyperbolic plane of constant curvature 1. In the case of spherical geometry, the constant is equal to 1 for the sphere of radius 1 (which is also the case of constant curvature 1).49 We note that the construction works as well in spherical geometry. The function E.x/, in the case of the sphere, satisfies E.x/  1 for all x, and, for the hyperbolic plane, it satisfies E.x/  1. We shall not go into the details of the case of spherical geometry in these notes, but we mention that the solution to Equation 6.2 in that case is E.x/ D cos.kx/, and Equation 6.1 becomes cos.ka/ D cos.kb/ cos.kc/; which is the Pythagorean theorem for the sphere. In the rest of this section, we work in the hyperbolic plane.

49All three founders of hyperbolic geometry noticed that there is an undetermined constant when they worked out their trigonometric formulae, and none of them did make the relation between this constant and the curvature of the space. Lobachevsky noticed that when the value infinity is attributed to the constant k, the formulae of hyperbolic geometry become those of Euclidean geometry. Of course Lobachevsky and Bolyai did not have the notion of curvature. But Gauss had it (he discovered it), but there is no evidence at all that he made the relation between curvature and the constant k that appears in the hyperbolic trigonometry formulae. In a letter to F. A. Taurinus, on 8 November 1824 Gauss wrote: “The assumption that in a triangle the sum of three angles is less than 180o leads to a curious geometry, quite different from ours, but thoroughly consistent, which I have developed to my entire satisfaction, so that I can solve every problem in it with the exception of the determination of a constant, which cannot be designated a priori. The greater one takes this constant, the nearer one comes to Euclidean geometry, and when it is chosen infinitely large, the two coincide.” (Greenberg’s translation.) [Die Annahme, dass die Summe der 3 Winkel kleiner sei als 180o , führt auf eine eigene, von der unsrigen (Euklidischen) ganz verschiedene Geometrie, die in sich selbst durchaus consequent ist, und die ich für mich selbst ganz befriedigend ausgebildet habe, so dass ich jede Aufgabe in derselben auflösen kann mit Ausnahme der Bestimmung einer Constante, die sich apriori nicht ausmitteln lässt.]

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6.2 The function E.y/ Let P be a point in a rectangular sector .Ox; Oy/. Let Px and Py be respectively the feet of the perpendiculars from P on Ox and Oy, and let x D jOPx j and e D e.x/ D jPy P j (Figure 57). We note that the value e.x/ also depends on the ordinate y of Py , y

e D e.x/

Py

P

y x O

Px

x

Figure 57

but in the next proposition, we consider it as a function of x, with y fixed. Consider the following function of x: e.x/ r.x/ D : (6.3) x Proposition 6.1. With the above notation, we have the following. (1) For x > 0, we have r.x/ > 1. (We note that r.x/ is not defined if x is large. When P tends to infinity on the perpendicular at Py to the y-axis, its projection on the x-axis has a finite limiting value; see the remark before Proposition 8.1) (2) The function r.x/ is an increasing function of x; that is, r.x1 / < r.x2 / for 0 < x1 < x 2 . (3) The limit limx!0 r.x/ exists and is  1. (We shall see later on that this limit is > 1.) Proof. Property (1) follows from Proposition 3.26 (also from Proposition 4.7). Let us prove Property (2). Let us fix a value x0 of x and let n be an integer  2. We divide the segment OPx into n segments of equal length. The length of each such segment is therefore a D x0 =n. From the proof of Property (5) of Theorem 4.21, we know that if we successively increase the value of x by some equal quantity, the corresponding values e.x/ increase by a quantity that is strictly smaller. Thus, to the cutting points of the segment ŒOPx , whose values are a; 2a; : : : ; na, the n corresponding values of e,

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namely, en D e.na/ for n D 1; 2; : : : ; n, increase by addition of a sequence of strictly decreasing positive numbers. Thus, we can write e1 D b1 ; e2 D b1 C b2 ; e3 D b1 C b2 C b3 ; : : : ; en D b1 C    C bn ; with b1 < b2 < b3 <    < bn : With this notation, the sequence of values of r.x/ D e.x/=x at the points a; 2a; : : : ; na is the following increasing sequence b1 C b2 b1 C b2 C b3 b1 C b2 C    C bn b1 < < <  < : a 2a 3a na This shows that the restriction of the function x 7! e.x/=x to the set of values a, 2a, 3a; : : :, na D x0 is an increasing function of x. Since we can take n as large as we wish and since e is a continuous function of x, this shows that e.x/=x is an increasing function of x, for x varying on the segment OPx0 . Since x0 is arbitrary, this proves Property (2). Property (3) follows from Properties (1) and (2), since r.x/ decreases as r decreases, and it is bounded below by 1. Remark 6.2. In spherical geometry, the function r.x/ can be defined similarly, and it has monotonicity properties in the opposite direction. We already noted that the values of the functions e.x/ and r.x/ that appear in Proposition 6.1 also depend on the ordinate y of the point Py . In the next proposition, we need to take this fact into account, and for that purpose we now denote the function e.x/ by e.y; x/. We let E.y/ be the function defined, for y  0, by e.y; x/ : (6.4) x!0 x The limit exists by Proposition 6.1. It also follows from the same proposition that E.y/ is always  1, and that in any trirectangular quadrilateral with edges denoted like in Figure 56, we have E.y/ < e=x. E.y/ D lim

Proposition 6.3. The function E satisfies the following functional equation: E.y C z/ C E.y  z/ D 2E.y/E.z/ for all y; z  0: Proof. We first prove the inequality E.y  z/ E.y C z/ C  2E.y/: E.z/ E.z/

(6.5)

We use the construction represented in Figure 58, in which we assume 0 < z < y.

94

Norbert A’Campo and Athanase Papadopoulos y e.y C z; x/ yCz b

y

q

p

e.y; x/

a e.y  z; x/

yz

O

x

x Figure 58

We start with the relation e.y; x/  p 2e.y; x/ C q  p e.y; x/ C q C D x x x which we transform into the following: e.y  z; x/ e.y; x/  p 2e.y; x/ C q  p e.y C z; x/ e.y; x/ C q C D : x e.y C z; x/ x e.y  z; x/ x Taking limits as x ! 0 while y and z are kept fixed, we obtain E.y C z/

1 qp 1 C E.y  z/ D 2E.y/ C lim : x!0 E.z/ E.z/ x

From the proof of property (2) of Proposition 6.1, we have q  p, and q  p is a exists and is non-positive. monotonous function of t . This implies that limx!0 qp x From this we deduce Inequality (6.5). Now we prove E.y C z/ E.y  z/ C  2E.y/: E.z/ E.z/ We start with the relation .e.y; x/ C q/ C

b b b .e.y; x/  p/ D e.y; x/ C e.y; x/ C q  p; a a a

(6.6)

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which gives e.y; x/ C ab e.y; x/ C q  ab p b e.y; x/  p e.y; x/ C q C D ; x a x x therefore b e.y  z; x/ e.y; x/  p e.y C z; x/ e.y; x/ C q C x e.y C z; x/ a x e.y  z; x/ e.y; x/ C ab e.y; x/ C q  ab p : x Taking limits as x ! 0 while y and z are kept fixed, we obtain D

E.y C z/

q  ab p 1 1 C E.y  z/ D 2E.y/ C lim ; x!0 E.z/ E.z/ x

since as x ! 0, ab ! 1. For y and z fixed, q=x and p=x are monotonic functions of x. We already saw that p  q. From Theorem 4.23, we have pa > qb . Therefore, b q a p x

> 0. This quantity is also monotonous as a function of x, and the limit

limx!0

b p q a x

exists and is nonnegative. Thus, we obtain Inequality (6.6).

Theorem 6.4. In the hyperbolic plane, for all x  0, we have x E.x/ D cosh k where k is some constant that is independent of x. Proof. The proof follows from Proposition 6.6 which we prove in the next section. Remark 6.5. The constant k in Theorem 6.4 is given by E.1/ D cosh k1 , that is, k D arccosh1 E.1/ . This constant is related to the constant of curvature of the space. Taking k D 1 corresponds to the fact that the space is Euclidean (the angular deficit of triangles is 0). Indeed, in the Euclidean plane, the function x 7! E.x/ is still defined, and its is a constant equal to 1, which is equal to the value cosh 0. In the case where k 6D 1, choosing k amounts to choosing a length unit. In the sequel, we shall take k D 1, which corresponds to a space of constant curvature 1.

6.3 The functional equation .x C y/ C .x  y/ D 2.x/.y/ We study the solution of the equation .x C y/ C .x  y/ D 2.x/.y/; where x and y vary in R.

(6.7)

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This is a well-known functional equation, and it is studied for instance by Picard in [74]. We first note the following: (1) Taking y D 0, we obtain .x/ C .x/ D 2.x/.0/, which implies .0/ D 1. (2) Taking x D 0 and using the fact that .0/ D 1, we obtain .y/C.y/ D 2.y/ which gives .y/ D .y/ for all y in R, that is,  is an even function. (3) The constant function .x/ D 1 is a solution of Equation (6.7), which becomes 1 C 1 D 2. (4) For all 2 R, the functions x 7! cos x and x 7! cosh x are solutions of (6.7). To find the general solution of Equation (6.7), we start by fixing a real number ˛. Taking x D ˛ and y D k˛, we obtain  ..k C 1/˛/ C  ..k  1/˛/ D 2.˛/.k˛/ Now taking x D y D  



˛ , 2k

for all k D 1; 2; 3; : : :.

(6.8)

we obtain  





2 1 ˛ ˛ D  k1 C 1 for all k D 1; 2; 3; : : :. (6.9) k 2 2 2 From well-known properties of the functions cos x and cosh x, or using the fact that these functions satisfy Equation (6.7), we have the following formulae:



cos ..k C 1/˛/ C cos ..k  1/˛/ D 2 cos.˛/ cos.k˛/

for all k D 1; 2; 3; : : :, (6.10) cosh ..k C 1/˛/ C cosh ..k  1/˛/ D 2 cosh.˛/ cosh.k˛/ for all k D 1; 2; 3; : : :, (6.11)   2     1 ˛ ˛ cos k D cos k1 C 1 for all k D 1; 2; 3; : : :. (6.12) 2 2 2 and   2     1 ˛ ˛ cosh k D cosh k1 C 1 for all k D 1; 2; 3; : : :. (6.13) 2 2 2 Now since .0/ D 1 and assuming that  is continuous, we can find a positive real number ˛ such that .x/ is positive for all x between 0 and ˛. Given such an ˛, we have either 0 < .˛/ < 1 or .˛/ > 1. With such an ˛ in hand, we have the following: Proposition 6.6. Let  be a continuous function satisfying the functional equation (6.7). Then, (1) if for some positive ˛ we have .˛/ < 1, there exists 2 R such that .x/ D cos x for all x in R; (2) if for some positive ˛ we have .˛/ > 1, there exists 2 R such that .x/ D cosh x for all x in R;

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(3) if for some positive ˛ we have .˛/ D 1, the solution to Equation (6.7) is the constant function .x/ D 1. Proof. Assume there exists ˛ > 0 satisfying .˛/ > 1. Then, let  be a positive real number such that .˛/ D cos . Since the function  satisfies Equation (6.8) and using Equation (6.10), we have, by induction, for all k D 1; 2; : : :, .k˛/ D cos k: In the same way, since  satisfies Equation (6.9), we have by induction, and using Equation (6.12), for all k D 1; 2; : : :, 



˛ 2k



D cos





 : 2k

We conclude that for all positive integers k and l we have 









k k ˛ D cos l  : l 2 2

Since the function  is continuous, we obtain, for all x  0, .x˛/ D cos.x /. Since  is even, the last equality holds for all x in R. Setting y D x˛, we obtain, for all y in R, .y/ D cos y, with D =˛. This proves Statement (1) of Proposition 6.6. Statement (2) follows in the same way by starting with a real number ˛ satisfying .˛/ D cosh  and using Equations (6.10) and (6.13) instead of (6.10) and (6.12).

6.4 Pythagoras’ theorem In Euclidean geometry, the familiar Pythagoras theorem gives a relation between the three edges of a right triangle. In non-Euclidean geometries (spherical and hyperbolic), the term “Pythagoras’ theorem” is also used for a theorem which gives such a relation. In this section, we prove such a theorem. From such a theorem, one can deduce formulae that hold for arbitrary triangles by subdividing them into right triangles. We shall also see that from the trigonometric formulae in hyperbolic or in spherical geometry, one can deduce trigonometric formulae for Euclidean triangles, by taking limits. This is a consequence of the fact that non-Euclidean geometries are infinitesimally Euclidean. To prove Pythagoras’ theorem, we start with a right triangle ABC and we slide it two times along one of its edges, as in Figure 59. This gives rise to trirectangular quadrilaterals drawn on the sides of the right triangle, and we apply to them the limiting formulae that we found for such quadrilaterals in Section 6.2. This construction has the flavor of the figure that one draws for proving Pythagoras’ theorem in Euclidean geometry (see Figures 50 and 51). In this hyperbolic proof, the trirectangular quadrilaterals play the role of the squares that one draws in Euclidean geometry.

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Proposition 6.7. The function E defined by Equation (6.4) satisfies: E.a/ D E.b/E.c/: Proof. The proof we give is a variation on a proof given in [6], p. 54. We start with a right triangle ABC , with right angle at A. We slightly slide this triangle along the side AC , in the direction of the vertex A. The new position of the triangle is A1 B1 C1 . We let B2 be the intersection point AB \ B1 C1 (see Figure 59). We next slide the triangle A1 B1 C1 along the side B1 C1 , in the direction of the vertex C1 , until the vertex B1 reaches the position B2 . The new triangle is called A2 B2 C2 . B B1

D1

D B2

E

I

H

K

C1

C

A

J

A1

E1

C2 H2

A2

L

Figure 59

The edge A2 B2 intersects the line AC in a point K which lies strictly between A and C . To see this, let us assume for contradiction that A lies between C1 and K, and consider the two triangles B2 C1 A and B2 C2 A2 . Our hypothesis implies that the latter contains the former, which implies that the angular deficit of B2 C2 A2 is greater than the angular deficit of B2 C1 A, which is a contradiction since (by our assumption) the angle at B2 of the triangle B2 C1 A is less than the angle at B2 of B2 C2 A2 and the other two angles of these triangles are congruent.

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From the construction, we have jAA1 j D jC C1 j

(6.14)

jB1 B2 j D jC1 C2 j:

(6.15)

and We let s D jAA1 j. Let D1 be the foot of the perpendicular from B1 on the line AB. The quadrilateral D1 AA1 B1 is a trirectangular quadrilateral, therefore the angle at B1 is acute, and from this we can see that the foot D1 of B1 on the line AB falls on the segment AB. Indeed, the quadrilateral AA1 B1 B being a Khayyam–Saccheri quadrilateral, its angle at B is acute, and in order to get a right angle at B while keeping the vertices B1 , A1 , A of this quadrilateral fixed, one has to move the vertex B towards the interior of the edge BA. Now let D be the foot of the perpendicular from B on the line B1 C1 . As we already noted, the angle at B1 in the Khayyam–Saccheri quadrilateral D1 AA1 B1 is acute, therefore the angle BB1 B2 is also acute, which implies that the point D falls on the segment B1 C1 , since BB2 B1 is also acute. Let L be the intersection point of B1 A and C2 A2 . Let J be the midpoint of C1 C2 and let H be the foot of the perpendicular from J to the line C C1 . Since J C1 C and J C C1 are acute, H is on the segment C C1 . Let H2 be the intersection of the lines HJ and C2 A2 . It is easy to see that H2 is on the segment HJ . The two triangles H C1 J and H2 C2 J are congruent, since C1 JH D C2 JH2 , C C1 J D B1 C1 A1 D J C2 H2 and since jCJ j D jJ C2 j. Therefore, JH2 cuts C2 A2 perpendicularly. Let I be the midpoint of C C1 . Let IE be the perpendicular from I to BC and let E1 be the point where the line IE cuts B2 C2 . The two triangles ICE and IC1 E1 are congruent, having all their angles equal, and one congruent side. Therefore, IE1 is perpendicular to C2 B2 . We have jBDj jB1 D1 j lim D lim ; (6.16) s!0 jBB2 j s!0 jB1 B2 j

2 2

1

1

1

1 3 2

2

1

since as s ! 0 the two quantities tend to sin ABC (Theorem 4.28). Likewise, we have jEE1 j jHH2 j lim D lim s!0 jC C1 j s!0 jC1 C2 j

1 1

(6.17)

because as s ! 0 the two limits are equal to sin ACB (Theorem 4.28). Now as s ! 0, the triangle BB1 B2 tends to a right Euclidean triangle, D1 ! B, the angles BB2 B1 and DD1 B1 both tend to ˇ D ABC , and we have (see Figure 60):

2

2

100

Norbert A’Campo and Athanase Papadopoulos B D D1

B1

ˇ D ˇ

B2 Figure 60

lim

s!0

jBB2 j jBDj D lim D cos ˇ: s!0 jB1 B1 j jB1 D1 j

(6.18)

We claim the jBDj jB1 D1 j jB1 B2 j D lim  lim : s!0 jEE1 j s!0 jAA1 j s!0 jHH2 j lim

(6.19)

To prove the claim, we set jB1 D1 j jBDj D jEE1 j jAA1 j jB1 D1 j D jAA1 j

jAA1 j jEE1 j jC C1 j  jEE1 j 

jBDj jB1 D1 j jBDj  jB1 D1 j 

(using (6.14)):

Thus, (6.19) is equivalent to jB1 B2 j jC C1 j jBDj D lim  lim : s!0 jHH2 j s!0 jEE1 j s!0 jB1 D1 j

(6.20)

lim

Using (6.17), (6.20) is equivalent to jB1 B2 j jC1 C2 j jBDj D lim  lim ; s!0 jHH2 j s!0 jHH2 j s!0 jB1 D1 j lim

which is equivalent to

(6.21)

jB1 B2 j jC1 C2 j D lim ; s!0 jBDj s!0 jB1 D1 j

(6.22)

lim

which follows from (6.18). This shows (6.19). The line A2 B2 cuts the segment AC in an interior point, which we call K. To prove this claim, we first note that

3 2

C2 B2 A2  C1 B2 A:

2

3

This follows from Theorem 4.23 (2), applied to the angles C1 B2 A and C1 B1 A1 , the latter being equal to C2 B2 A2 . Let K D A2 B2 \ AC and L D BA \ C2 A2 .

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We claim that jALj  jBB2 j  jA2 Kj:

(6.23)

To prove the first inequality in (6.23), we note that since B2 A2 is perpendicular to A2 C2 , we have jB2 A2 j  jB2 Lj; or, equivalently, jB2 Kj C jKA2 j  jB2 Aj C jALj; or, equivalently, jABj  jB2 Aj C jALj; or, equivalently, jBB2 j C jB2 Aj  jB2 Aj C jALj; or, equivalently, jB2 Aj  jALj: For the second inequality in (6.23), we note that since B2 A is perpendicular to AC , we have jB2 Aj  jB2 Kj: Since we have jBB2 j C jB2 Aj  jB2 Kj C jKA2 j; we obtain jBB2 j  jKA2 j: This completes the proof of (6.23). Now lim

jBDj D E.a/ jEE1 j

(6.24)

jB1 D1 j D E.c/: jAA1 j

(6.25)

s!0

and lim

s!0

From (6.23) we obtain

jALj jBB2 j jA2 Kj   : jHH2 j jHH2 jHH2

We also have lim

s!0

Thus, we obtain

jALj jKA2 j D lim D E.AC /: s!0 jHH2 j jHH2 j

jBB2 j D E.AC / D E.b/: s!0 jHH2 j lim

The result then follows from (6.19), (6.24), (6.25) and (6.26)

(6.26)

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Theorem 6.8 (The Pythagorean theorem in hyperbolic geometry). Let ABC is a right triangle, with sides a, b, c, with a being the hypotenuse. Then, we have cosh a D cosh b cosh c: Proof. The proof follows from Proposition 6.7 and Theorem 6.4.

6.5 Trigonometry in an arbitrary triangle From the Pythagorean theorem valid for a right triangle, we can deduce trigonometric formulae that hold for an arbitrary triangle by considering such a triangle as the union of two right triangles. We produce one of these formulae, known as the “sine law”. Theorem 6.9. Consider an arbitrary triangle of side lengths a, b, c, and let ˛ be the angle opposite to a side of length a. Then, we have cosh a D cosh b cosh c  sinh b sinh c cosh ˛; where cosh ˛ is a quantity that only depends on ˛. (Secretly, we know that cosh is the familiar Euclidean cosine function.) Proof. Let A, B, C be the vertices of the triangle that are opposite to edges of lengths a, b, c respectively. We suppose that the angle at C is obtuse. The proof for the case where this angle is acute can be done with minor modifications. Let C 0 be the foot of the altitude from C on AB, and B 0 the foot of the altitude from B on AC . We use the notation indicated in Figure 61, where u, v, k etc. denote the lengths of the corresponding sides. B0 v k

C

b A

h

a

˛ u

C0

cu Figure 61

B

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Pythagoras’ theorem applied to the right triangle C C 0 B gives cosh a D cosh h cosh.c  u/ D cosh h.cosh c cosh u  sinh c sinh u/:

(6.27)

From Pythagoras’ theorem applied to the right triangle AC C 0 , we obtain cosh b D cosh h cosh u:

(6.28)

From (6.27) and (6.28), we obtain cosh a D cosh b cosh c  sinh c sinh b

tanh u : tanh b

(6.29)

We now use Pythagoras’ theorem in the triangles AB 0 B and CBB 0 . We obtain cosh a D cosh k cosh v

(6.30)

cosh c D cosh.b C v/ cosh k:

(6.31)

and From (6.30) and (6.31), we obtain cosh v cosh c D cosh.b C v/ cosh a:

(6.32)

We claim that cosh b cosh c cosh.b C v/  sinh b cosh c sinh.b C v/ D cosh v cosh c:

(6.33)

Indeed, from the usual laws for sinh and cosh, we have cosh b cosh c cosh.b C v/  sinh b cosh c sinh.b C v/ D cosh b cosh c.cosh b cosh v C sinh b sinh v/  sinh b cosh c.sinh b cosh v C cosh b sinh v/ D cosh b cosh c cosh v cosh b  sinh b cosh c sinh v cosh c D cosh v cosh c.cosh2 b  sinh2 b/ D cosh v cosh c; which proves (6.33). Now (6.32) and (6.33) imply: cosh.b C v/ cosh a D cosh b cosh c cosh.b C v/  sinh b cosh c sinh.b C v/: (6.34) Dividing by cosh.b C v/, we obtain cosh a D cosh b cosh c  sinh b cosh c tanh.b C v/ and finally cosh a D cosh b cosh c  sinh b sinh c

tanh.b C v/ : tanh c

(6.35)

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Remark 6.10. The formula proved in Theorem 6.9 should be compared to the “cosine law” of Euclidean geometry, that is, to the formula c 2 D a2 C b 2  2ab cos : The Euclidean formula can also be obtained from the hyperbolic formula by taking power-series approximations. Using the notation of Figure 61, we have, from (6.34) and (6.35): tanh u tanh.b C v/ D : tanh b tanh c Proposition 6.11. The quantity

tanh u tanh b

(6.36)

in Equation (6.36) depends only on ˛.

Proof. Given an angle whose value is ˛, if we fix the point B on one edge of that angle, as in Figure 61, then, the projection B 0 of B on the other edge is also fixed, and if we vary the point C on the segment AB 0 , then the value tan.bCv/ is constant. tan c u Now if we fix the point C and vary B, then the point C 0 is fixed, and the value tan tan b is constant. We denote this constant by cosh ˛. In any right triangle in which ˛ is an acute angle with y being the hypotenuse and x the other side of ˛, this quantity is tanh x equal to tanh . y We call this quantity cosh ˛ (the “cosine function defined in the hyperbolic plane”), or simply cos ˛ because it coincides with the usual cosine function. Indeed, making x tends to 0, we have tanh x x D lim D cos ˛: lim x!0 tanh y x!0 y From this, one can also define the sine function for angles between 0 et  and so on. As a corollary, we have the following result (which was already proved using a different method, cf. Theorem 4.4). Corollary 6.12. There are no similar (except congruent) triangles in hyperbolic geometry; that is, triangles with congruent angles are congruent. As a geometric consequence of Theorem 6.9, we have the following. Consider an isosceles triangle of vertices A, B, C with opposite edge lengths a, b, c satisfying b D c (Figure 62 (a)), and let ˛ be the angle opposite to a. We have, from Theorem 6.9, cosh a D cosh2 b  sinh2 b cos ˛: From the identity cosh2 b  sinh2 b D 1, we obtain cosh a D 1 C . cos ˛/ sinh2 b

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Notes on non-Euclidean geometry B

B

b

A

˛

A

c C

(a)

(b)

C

Figure 62

which implies that if b is very large, then a is also very large, therefore cosh a

ea : 2

Taking logarithms, we obtain a  2b  K˛ where K˛ is a quantity that depends only on ˛. This tells us that the edge joining B to C passes close to A, at a distance of the order  K˛ (as represented in Figure 62 (a)). We also record the following “hyperbolic sine law”: Theorem 6.13. Consider a triangle with sides a, b, c and opposite angles respectively ˛, ˇ,  respectively. Then, we have sin ˇ sin  sin ˛ D D : sinh a sinh b sinh c Proof. It suffices to find an expression for This is done as follows. From Theorem 6.9, we have cos ˛ D

sin ˛ sinh a

which is symmetric in a, b and c.

cosh b cosh c  cosh a : sinh b sinh c

This gives cos2 ˛ D

cosh2 b cosh2 c C cosh2 a  2 cosh a cosh b cosh c : sinh2 b sinh2 c

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Using cos2 ˛  1 D  sin2 ˛, we obtain  sin2 ˛ D

cosh2 b cosh2 c C cosh2 a  2 cosh a cosh b cosh c  sinh2 b sinh2 c : sinh2 b sinh2 c

Using the identity cosh2 x  sinh2 x D 1, we obtain 1  .cosh2 a C cosh2 b C cosh2 c/ C 2 cosh a cosh b cosh c sin2 ˛ D ; sinh2 a sinh2 a sinh2 b sinh2 c which is the relation needed. We shall use the following particular case of Theorem 6.13 Corollary 6.14. Consider a right triangle with hypotenuse c, with ˛ being an angle adjacent to the hypotenuse, and a the side opposite to ˛. Then, sinh a D sinh c sin ˛: Proof. From Theorem 6.13, we have sin  D 1. This gives the result.

sin ˛ sinh a

D

sin  . sinh c

Since  D =2, we have

We can draw the following consequence, which we already proved using a different method (Corollary 4.24). Corollary 6.15. If two straight lines intersect making a nonzero angle, then the distance from a point on one line to the other line tends to infinity as the point moves to infinity. Proof. Let O be the intersection point of the two lines and let ˛ > 0 be the angle at this intersection point. If ˛ is a right angle then the result follows easily. Thus, we can assume that ˛ is acute. We then have a triangle AOB with right angle at B (Figure 63). A

O

˛ B Figure 63

From Corollary 6.14 we have sin ˛ sinh OA D sinh AB: Thus, when the distance OA tends to infinity, the distance AB also tends to infinity.

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From Proposition 6.11, we also have the following Corollary 6.16. Consider a right triangle with hypotenuse c, with ˛ being an angle adjacent to the hypotenuse, and b the other side adjacent to ˛. Then, tanh b tanh c

cos ˛ D

We can use this formula to prove the following result on the intersection of the three altitudes in small hyperbolic triangles. Theorem 6.17. In the neutral plane, in any triangle of sufficiently small diameter, the three altitudes are concurrent. Proof. We treat the problem in an indirect way, starting with three lines l1 , l2 , l3 that have a common point O. We take a point A1 on l1 , and from A1 we draw a perpendicular to the line l2 . If A1 is close enough to O, then this perpendicular intersects the line l3 . Let A2 and A3 be the intersection points of this perpendicular with l2 and l3 respectively (Figure 64). From the point A3 we continue in the same cyclic order; drawing a perpendicular to l1 , we get the points A4 et A5 on l1 and l2 respectively (assuming the construction is done in a small enough neighborhood of O), and from A5 we draw again a perpendicular to l3 and get the points A6 and A7 (again, assuming the construction is done in a small enough neighborhood of O); see Figure 64. l2 l3

l1 A1

A2

A3

A4

A7

ˇ ˛  O A5

A6

Figure 64. The figure used in the proof of the fact that the altitudes in small triangles are concurrent.

It suffices to prove that A7 D A1 .

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2 2 2

To prove this, let us first do the reasoning in the more familiar Euclidean plane. Let ˛, ˇ,  denote respectively the angles A1 OA2 , A2 OA3 , A3 OA4 , as in Figure 64. We have OA2 D OA1 cos ˛, and we view this equality as saying that by the first step of the above the construction (the step that produces the point A2 out of the point A1 ), the segment of length a D OA1 has been contracted by cos ˛. Likewise, by the second step of the construction (the one that produces the point A3 ), the segment OA2 is dilated by cos1 ˇ , and so forth. The overall effect of the construction on the segment of length a is to contract it by the product cos ˛  cos ˇ  cos  and to expand it by the product cos1 ˛  cos1 ˇ  cos1  . Thus, the segment of length a is unchanged by the whole construction, and we have A7 D A1 . 2 , we use the Now in the hyperbolic plane. Instead of the formula cos ˛ D OA OA1 OA2 formula cos ˛ D tanh (Corollary 6.16). We consider in this formula that instead of tanh OA1 measuring distances from O on the lines l1 , l2 , l3 by arclength, we measure them using the scale tan.arclength/. The rest of the proof is then as in the Euclidean case.

Further applications of Corollary 6.16 are given below (see Proposition 7.33.)

6.6 Geometric relations in trirectangular quadrilaterals We use the notation of Figure 65 which represents a trirectangular quadrilateral in which P is the acute angle, with edges a; q; b; p, and with diagonals c and d . The diagonal d has one vertex at the acute angle. b

b

P

˛1

1 2

ˇ1 p

P

c

q

d

p

˛2

q

˛

ˇ2

ˇ

a

a Figure 65

Theorem 6.18. With the above notation, we have the following: (1) tanh2 d D tanh2 p C tanh2 a; sinh b sinh q (2) D cosh q; D cosh b; sinh a sinh p

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(3)

cosh q 1 cosh b D D ; cosh a cosh p sin P

(4)

tanh b D cosh p D cosh q sin P ; tanh a

(5) sinh a sinh p D tanh b tanh q D cos P . 2

2

tanh p tanh a 2 Proof. Let us prove (1). From Corollary 6.16, we have tanh 2 d D cos ˛ and tanh2 d D cos2 ˇ. Since ˛ C ˇ D =2, we have cos2 ˛ D sin2 ˇ. Using the formula cos2 ˛ C sin2 ˛ D 1, we obtain tanh2 p tanh2 a 1D C ; 2 tanh d tanh2 d which gives (1). Now we show the first equality in (2). From Theorem 6.13 applied to the triangle whose sides are p, d , b, we have

sinh d sinh b D D sinh d: sin ˛ sinh =2 Since ˛ C ˇ D =2, we have sin ˛ D cos ˇ, which gives sinh b D sinh d cos ˇ: Using Corollary 6.16, we obtain sinh a cosh d tanh a D : tanh d cosh a Using now Pythagoras’ Theorem, we get sinh b D sinh d

sinh b D sinh a cosh q; which gives the desired result. The proof of the second inequality in (2) is identical. Now we prove (3). By Pythagoras’ Theorem, we have cosh b cosh p D cosh d D cosh a cosh q: The first inequality in (3) follows. Next, we prove the second inequality. We have, using Theorem 6.13 and the Corollary 6.16, sinh c sinh b sinh b tanh c D : D D sinh b sin P sin ˛2 cos ˇ2 tanh a Therefore, sinh b sinh b sinh b 1 D D D : sin P tanh a cosh c .sinh a= cosh a/ cosh a cosh p sinh a cosh p Using

sinh b sinh a

D cosh q (proved in (2)), we obtain the result.

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Now we prove the first equality of (4). Applying twice the hyperbolic Pythagorean theorem, we have sinh b cosh a cosh a cosh d tanh b D D cosh q D D cosh p: tanh a cosh b sinh a cosh b cosh b The second equality of (4) is the second equality of (3). Now we prove (5). For the first equality, we use (2), and we obtain: tanh b tanh q D

sinh a cosh q sinh p cosh b sinh b sinh q D : cosh b cosh q cosh b cosh q

which gives tanh b tanh q D sinh a sinh p

(6.37)

For the second equality, we cut the angle at P in two parts by the line d (the right hand side picture in Figure 65), and we use the cosine addition formula, which gives: sinh p tanh b tanh q  tanh d tanh d sinh d tanh b tanh q  sinh p sinh a : D tanh b tanh q C sinh2 d From Formula (6.37), the second term in the last line above is equal to 0. We obtain the desired result. cos P D

6.7 Some spherical trigonometry We can take as a model of spherical geometry the unit sphere in the three-dimensional Euclidean space R3 . In this model, the lines are the great circles, that is, the intersection of the sphere with the planes passing through the origin. The angle made by two lines at any one of their two intersection points is the dihedral angle in R3 made by the corresponding planes. The distance between two points on the sphere is the Euclidean angle made at the origin of R3 by the rays starting at this point and passing through the two points. The spherical trigonometric formulae can then be deduced from Euclidean considerations. The proof of the two next theorems is an illustration of how this works. Theorem 6.19 (The sine law in spherical geometry). Let ABC be a spherical triangle, and let a, b, c be the lengths of the angles opposite to A, B, C respectively. Then we have sin A sin b sin B sin c sin C sin a D I D I D : sin c sin C sin a sin A sin a sin A Proof. It suffices to prove the first formula. We work on the unit sphere in Euclidean 3-space, with center O. We draw the Euclidean segments OA, OB, OC . The angles A, B and C are the dihedral angles made by the pairs of planes .OAC; OAB/, .OAB; OBC /, .OBC; OCA/.

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Let H be the projection of the point B on the plane OAC , let A0 be the projection of H on OA and let C 0 be the projection of H on OC (see Figure 66). The line OA B

C0

O

C

H A0

A

Figure 66

is orthogonal to the lines BH and A0 H , therefore it is orthogonal to the plane A0 BH . Similarly, the line OC is orthogonal to the plane C 0 BH . The Euclidean triangle OA0 B has a right angle at A0 , the Euclidean triangle OC 0 B has a right angle at C 0 , and both Euclidean triangles A0 HB and C 0 HB have right angles at H . The angle BA0 H is equal to the dihedral angle between the pair of hyperplanes .OBA; OAC /. Therefore, we have A D BA0 H . Likewise, we have B D B C 0 H . Thus, in the Euclidean triangle A0 BH , we have

1

1

1

sin A D sin BA0 H D

2

BH ; BA0

and in the Euclidean triangle C 0 BH , we have

2

sin C D sin B C 0 H D Hence, we obtain

BH : BC 0

BC 0 sin A BH BC 0  : D D sin C BA0 BH BA0

4

4

On the other hand, since a D .OB; OC / and since c D .OA; OB/, we have

1

sin a D sin BOC 0 D and

1

sin c D sin BOA0 D

BC 0 OB BA0 : OB

(6.38)

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This gives BC 0 OB sin a BC 0 D  D : sin c OB BA0 BA0 From (6.38) and (6.39), we get

(6.39)

sin A sin a D : sin c sin C Remark 6.20. From Theorem 6.19, we deduce, using the same notation, that sin b sin c sin a D D : sin A sin B sin C In this form, the formulae can be compared to the hyperbolic sine law of Theorem 6.13. We now prove the Pythagorean theorem in spherical geometry. Theorem 6.21 (The Pythagorean theorem in spherical geometry). Consider a right spherical triangle with edge lengths a, b, c, with c being the length of the hypotenuse. Then, we have: cos c D cos a cos b: Proof. As in the proof of Theorem 6.19, we work on the unit sphere in Euclidean 3-space, with center O. Let ABC be the right triangle with opposite edge lengths a, b, c respectively. The right angle is at C . Let A0 be the orthogonal projection in R3 of the point A on the line OC , and let 00 A be the orthogonal projection of the point A0 on the line OB (see Figure 67). We claim that A00 is also the projection of A on the line OB. Indeed, AA0 is perpendicular to the plane OBC . (Recall that the dihedral angle made by the two planes OAC and OCB is equal to =2.) Thus, the line OB is perpendicular to the two lines A00 A and A0 C , and therefore OB is also perpendicular to A00 A. In the Euclidean right triangle OAA00 , we have OA00 D OA cos c D cos c: In the Euclidean triangle OA0 A00 we have OA00 D OA0 cos a: In the Euclidean triangle OAA0 , we have OA0 D OA cos b D cos b: Putting everything together, we obtain cos c D cos a cos b:

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Notes on non-Euclidean geometry A

O A00

B

A0

C Figure 67

Note that the Euclidean reasonings that we made in the proofs of the two preceding theorems do not have any parallel in hyperbolic geometry because unlike the spherical case, the geometry of the hyperbolic plane does not extend to a Euclidean geometry of 3-space. Remark 6.22. Euler worked out trigonometric formulae for the sphere in several of his memoirs, see e.g. [30] and [32]. Euler’s approach to the trigonometric formulae was completely new, because to prove them he used the calculus of variations that he had invented (with Lagrange as a co-inventor). In some sense, Euler worked out the formulae using the intrinsic geometry of the sphere.

6.8 The function E.y/ in spherical geometry We already recalled the spherical Pythagorean theorem (Theorem 6.21), stating that for a right triangle of sides a, b, c where c is the hypotenuse, we have: cos c D cos a cos b: We also noted that we can define, in spherical geometry, the function E.y/ that we already encountered at several places in hyperbolic geometry (see Sections 1 and 6.2). Using the Pythagorean theorem, we prove the following result Theorem 6.23. On the sphere, we have E.y/ D cos y.

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Proof. For a given segment of length y, we compute the quantity E.y/ by using a trirectangular quadrilateral, as we did in the case of the hyperbolic plane (see Figure 1 and Figure 56 above). On the sphere, a trirectangular quadrilateral gives rise to a larger figure on which we can read various informations; see the trirectangular quadrilateral of edge lengths x, y, e, t in Figure 68 (the obtuse angle is made by e and t ). In the notation of

˛

 2

x  2

A

e

t

x

 2

t

e B

ˇ  2

y

y

Figure 68

Figure 68, and using the above remarks on angles and distances on the sphere, the length y is equal to the angle ˛, and the length x is equal to the angle ˇ. The spherical Pythagorean theorem in the triangle B gives:     cos 2  y D cos 2  t  cos e: The Pythagorean theorem in the triangle A gives     cos 2  e D cos 2  x  cos t: From this we get, taking limits as x ! 0 while keeping y constant (or, equivalently, keeping ˛ constant): E.y/ D lim

cos. 2  e/ e D cos.y/: D lim cos. 2  x/ x

Notes on non-Euclidean geometry

115

Recall also that the cosine function that appears here in spherical geometry satisfies, like the function cosh that appears in hyperbolic geometry, the functional equation discussed in Section 6.3 and that is satisfied by the function E.y/ in hyperbolic geometry, as we already noted in Section 6.2. The properties of the function E.y/, in the three geometries, in relation to the trigonometric formulae in these three geometries, were discovered by the Belgian mathematician de Tilly, whom we already mentioned in these notes (Section 2.8), cf. [99] and [100].

7 Parallelism 7.1 Introduction The notion of parallelism is one of the most intricate notions in geometry, and it led to many mistakes. One reason for this might be the fact that unlike other basic notions such as betweenness and perpendicularity, parallelism has to do with the fact that lines are infinitely extendable, and thus, a statement about parallel lines usually involves in its essence the notion of infinity. We already recalled at several occasions that, for a period that lasted more than 20 centuries (from the times of Aristotle until the times of Gauss), a huge effort by many great geometers was spent in trying to prove a statement about parallels that was erroneous, namely, that Euclid’s parallel postulate follows from his other postulates. A section on parallelism could have been included at a very early stage in these notes, but we preferred to put it at this place, in order to make use of results obtained in previous sections. We note that Bolyai and Lobachevsky discuss parallelism right at the beginning of their memoirs; see in particular the first page of Lobachevsky’s Pangeometry ([59], p. 3), and [14] and [56] in the appendix to [16]. We shall give precise definitions below, but let us first make a short summary. First of all, we have to agree on the definition of parallelism. In Euclidean geometry, two lines are said to be parallel if they do not intersect. In hyperbolic geometry, such a definition turns out to be a poor one because among the pairs of non-intersecting curves there are two different classes, which have very different behaviors. Thus, there is a more useful notion of parallelism in hyperbolic geometry, and this notion can be formulated so that it also holds in neutral geometry, and generalizes the Euclidean one. This fact has been noticed independently by the three founders of non-Euclidean geometry, Lobachevsky, Bolyai and Gauss. The definition is as follows. Given a line l and a point P not on l, a parallel line l 0 from P to l is the limiting position of a sequence of lines joining P to a sequence of points on l that tend to infinity in a given direction.

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Such a definition, made in neutral geometry, is equivalent, when restricted to Euclidean geometry, to the fact that l and l 0 are disjoint. But in hyperbolic geometry, this is not the case. Disjointness is clearly a symmetric relation. The notion of parallelism in hyperbolic (and in neutral) geometry is also symmetric, but this will need a proof. Since parallelism involves the choice of an orientation on a line, it is sometimes more convenient to deal with parallel rays instead of parallel lines, and this is what we shall do now. In fact, one can start with the definition of parallel rays. Then, using the fact that any ray can be extended in a unique way to a line, one can define two oriented lines to be parallel if they contain parallel rays, with orientations respected regarding the sequences of points that tend to infinity that are involved in the definition of parallelism. Two lines that are parallel with respect to a given orientation on each of them are not necessarily parallel with respect to the other orientations. In the neutral plane, the existence of a line that is parallel to a second one with respect to different choices of orientations is equivalent to the fact that the plane is Euclidean. To say it in different terms, in the neutral plane, for any line l and for any point A which is not on l, we can draw a unique parallel from A to l, for each choice of an orientation on l and these two parallels coincide if and only if the plane is Euclidean. In the hyperbolic plane, the two parallels are distinct. Let us note by the way that there is no interesting notion of line parallelism in spherical geometry. This is because in that geometry any two lines have nonempty intersection. Thus, on the sphere, given any line l and any point A not on l, there is no line parallel to l and passing by A, in any sense of the word. In the rest of this section, after introducing parallels and their main properties, we discuss the notions of horocycles, parabolic motions and the angle of parallelism function. All these notions are proper to hyperbolic geometry.

7.2 Parallelism in hyperbolic geometry We define parallelism and show that it is symmetric and transitive. The results in this section are contained in Lobachevsky’s and Bolyai’s works, e.g. [56] and [14]. We recall that when we denote a ray by r.A; M /, then A is the origin of the ray and M a point on that ray. Let us also make the following convention: We shall assume that on a ray r.A; M /, the point M is as much far away from A as is needed for the discussion in progress. For instance, when we say that X is a point on a ray r.A; M /, then we shall implicitly assume that X lies between A and M . Definition 7.1 (Parallel rays). Given two rays r.A; M / and r.B; N /, we say that r.A; M / is a parallel to r.B; N / starting at A, and we write r.A; M / k r.B; N /, if the following two conditions are satisfied: (1) r.A; M / \ r.B; N / D ;;

1

(2) Any ray starting at A and situated in the open angular sector BAM intersects r.B; N / (see Figure 69).

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Notes on non-Euclidean geometry

B

N

M

A Figure 69

It is not completely trivial from this definition that parallelism is a symmetric relation. This will be proved below (Proposition 7.11). Proposition 7.2. Given a ray r.B; N / and a point A … r.B; N /, there exists one and only one ray r.A; M / parallel to r.B; N /. Proof. The proof uses a continuity and a limiting argument. Let C be a point on r.B; N / which is distinct from B (Figure 70). As C varies on the ray r.B; N / in the B

C

N

M

A Figure 70

1

positive direction, the value of the angle BAC increases. Since this value is bounded (by the value of two right angles), the angle has a limit as C tends to infinity. The ray starting at A and making this limiting angle with r.A; B/ is parallel to r.B; N /. Uniqueness is shown by contradiction. Assume there exist two rays, r.A; M / and r.A; M 0 / that are parallel to r.B; N /, and suppose that BAM 6D BAM 0 . Without loss of generality, we suppose that BAM 0 < BAM . Since r.A; M / k r.B; N /, r.A; M 0 / intersects r.B; N /, which is absurd. Therefore, the parallel to r.B; N / starting at A is unique.

2 1

1 2

Proposition 7.3. Ray parallelism is a congruence invariant. Proof. This follows from the fact that a plane congruence sends intersecting rays to intersecting rays, and non-intersecting rays to non-intersecting rays.

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Later on, we shall define two oriented lines to be parallel if they contain parallel rays, with the orientations on the lines being induced by those of the rays they contain. From now on, when we say that an oriented line l contains a ray r, we shall implicitly assume that the orientations of l and r coincide. Clearly, if two rays r10 and r20 are contained in rays r1 and r2 respectively, then the natural orientations of r10 and r20 are induced from those of r1 and r2 . We shall prove the following: Let r1 be a ray parallel to a ray r2 , and let r10 and r20 be two rays such that – either ri0 is contained in ri for i D 1; 2, – or ri is contained in ri0 for i D 1; 2. Then r10 is parallel to r20 (Propositions 7.6 and 7.7 below). We shall prove below that parallelism is a symmetric relation, that is, if r1 k r2 , then r2 k r1 . (Proposition 7.11). After we prove this, we can safely use the terminology “the two rays r1 and r2 are parallel”. But until we prove Proposition 7.11, let us agree that the sentence “the two rays r1 and r2 are parallel” means that r1 is parallel to r2 . Let us now recall the following property of the Euclidean plane. Let l be a line, let A be a point that is not on l, let B be the foot of the perpendicular from A on l and let l 0 be the perpendicular to AB through A (Figure 71). Then, l 0 is the unique line A

B

l0

l

Figure 71

passing through A which is disjoint from l. Indeed, any line passing through A which is not perpendicular to AB makes with AB an angle that is not equal to a right angle. Now one form of Euclid’s parallel postulate says that when two lines make with a transversal interior angles whose sum from one side is less than two right angles, then the first two lines intersect in that side. Thus, the line l 0 is disjoint from the line l, and l 0 is the unique line disjoint from l containing A. We shall see that this property does not hold in the hyperbolic plane. That is, the line l 0 making a right angle with AB is not parallel to l, with the appropriate definition of parallelism (Definition 7.10 and Proposition 7.26 below). We start with the following proposition, which will be strengthened in the sequel (Corollary 7.28). Proposition 7.4. In the hyperbolic plane, if two oriented rays are parallel, then their distance decreases in the direction of parallelism. More precisely, if a ray r 0 is parallel to a ray r, then the distance from a point on r 0 to the ray r decreases in the sense of parallelism.

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Proof. Consider a ray r, two distinct points A and B on r, and two segments AC and BD, perpendicular to r, of the same length, and situated on the same side of r (see Figure 72). The quadrilateral ABDC is a Khayyam–Saccheri quadrilateral, C

D G

A

B

r0 r

Figure 72

therefore its angles at C and D are congruent (Proposition 3.29) and these angles are acute (Theorem 3.32). The reflection about the perpendicular bisector of AB shows that this perpendicular bisector is also perpendicular to DC . Therefore, the lines AB and CD have a common perpendicular, which implies that they cannot be parallel. Therefore, the parallel to r passing by C cuts the segment BD in a point G satisfying jBGj < jAC j. From this, the proposition follows.

1 1

Proposition 7.5. If r.A; M / k r.B; N /, then ABN C BAM  . Proof. Take a sequence of points Cn (n D 1; 2; : : :) on r.B; N / tending to infinity (Figure 73). From the proof of Proposition 7.2, the angle BAM is the limit of the

1

Cn

B

N

M

A Figure 73

1

sequence of angles BAC n as n ! 1. For each n  1, in the triangle BACn , we have ABN C BAC n  . Passing to the limit, we obtain ABN C BAM  .

1 1

1 1

Proposition 7.6. If r.A; M / k r.B; N / and A0 2 r.A; M /, then r.A0 ; M / k r.B; N /. Proof. The rays r.A0 ; M / and r.B; N / are clearly disjoint. We need to show that every ray r.A0 ; P / contained in the interior of the angular sector BA0 M has nonempty intersection with the ray r.B; N / (Figure 74). Let P 0 be a point on r.A0 ; P / which is close to A0 . Since the ray r.A; P 0 / is in the interior of the angular sector BAM , it intersects r.B; N /. Let F be this intersection

2

1

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Norbert A’Campo and Athanase Papadopoulos F

B

N

P P A

0

M

A0 Figure 74

point. The ray r.A0 ; P / enters in the triangle ABF . It therefore leaves this triangle from another edge (Pasch’s axiom). Since this edge cannot be AB, r.A0 ; P / intersects BF . Therefore, r.A0 ; P / intersects r.B; N /. Proposition 7.7. If r.A; M / k r.B; N / and A 2 r.A0 ; M /, then r.A0 ; M / k r.B; N /. Proof. We first show that r.A0 ; M / \ r.B; N / D ;. Indeed, if not, then let I be the intersection point of these two rays (Figure 75). For any point I 0 close enough to the

I0 A0

A

I B Figure 75

point I , the ray r.A; I 0 / intersects the ray BN . Therefore, r.A; M / cannot be parallel to r.B; N /. It remains to show that for every point P such that the ray r.A0 ; P / is contained in the interior of the angular sector BA0 M , we have r.A0 ; P / \ r.B; N / 6D ;. Extend the ray r.A0 ; P / in the direction of A0 to a point Q outside that ray (Figure 76). Then, the ray r.Q; A/, after it passes the point A, enters in the interior of the angular sector MAB, and therefore it intersects r.B; N / in some point we call R. Since the ray r.A0 ; P / is in the angular sector BA0 A, it enters the triangle BAR from the side BA. By Pasch’s axiom, it must get out of this triangle. Since it cannot intersect the side AR, it gets out from side BR.

2

1

1

Proposition 7.8. If r.A; M / k r.B; N / and B 0 2 r.B; N /, then r.A; M / k r.B 0 ; N /. Proof. We must show that any ray r.A; P / starting at A and contained in the open angular sector B 0 AM intersects the ray r.B 0 ; N / (see Figure 77). Now r.A; P /

2

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R

B P A0

M

A

Q

Figure 76 B

B0

P

N M

A Figure 77

1

intersects r.B; N / because it is contained in the angular sector BAN , and since r.A; M / k r.B; N /, and furthermore, r.A; P / cannot intersect the segment BB 0 because it does not intersect the rays r.A; B 0 / and r.A; B/ except in A. Proposition 7.9. If r.A; M / k r.B; N / and B 2 r.B 0 ; N /, then r.A; M / k r.B 0 ; N /.

2

Proof. We must prove any ray r.A; P / in the open angular sector B 0 AM intersects the ray r.B 0 ; N / (Figure 78). Indeed, either this ray is in the open angular sector BAM , and in this case it intersects r.B; N / because r.A; M / k r.B; N /, therefore it intersects r.B 0 ; N / or it is in the angular sector B 0 AB, and in this case it intersects the line BB 0 of the triangle ABB 0 , by Pasch’s axiom.

1

1

B0

B

N

M A Figure 78

The last four propositions allow us to define parallel oriented lines: Definition 7.10 (Parallel lines). Let l1 and l2 be two oriented lines. We say that l1 is parallel to l2 , and we write l1 k l2 , if l1 contains a ray r1 and l2 contains a ray r2

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such that r1 k r2 . (As already mentioned, when we say that an oriented line contains a ray, it is implicitly assumed that the orientation on the line agrees with the natural orientation of the ray.) By Propositions 7.6, 7.7, 7.8 and 7.9, if l1 k l2 , then for any two rays r1 and r2 contained respectively in l1 and l2 with the induced orientations, we have r1 k r2 . From Proposition 7.2 we deduce that given an oriented line l and a point A that is not on l, there is a unique oriented line passing through A that is parallel to l with the given orientations. From Proposition 7.3 we deduce that line parallelism is a congruence invariant. Propositions 7.11 and 7.14 below show that the parallelism relation between rays is almost an equivalence relation. It fails to be an equivalence relation, just because it is not reflexive. (In the definition of being parallel, a ray is not parallel to itself). The next proposition says that the relation is symmetric: Proposition 7.11. If r.A; M / k r.B; N /, then r.B; N / k r.A; M /. Proof. Suppose that r.A; M / k r.B; N /. To show that the converse is true, it suffices to show that there is a plane congruence that transforms some ray containing r.A; M / or contained in r.A; M / into a ray containing r.B; N / or contained in r.B; N /. We start by finding a point P equidistant from r.A; M / and r.B; N /. For this, we let P be the intersection point of the angle bisectors of BAM and ABN , and we let H1 , H2 and H3 be the feet of the perpendiculars from P on r.B; N /, BA and r.A; M / respectively (Figure 79). From the definition of the angle bisectors,

1

H1

B

1

N P

K

H2 M A

H3

1

Figure 79

we have PH1 D PH2 D PH3 . Let r.P; K/ be the geodesic ray that bisects the angle .r.P; H1 /; r.P; H3 //. The symmetry with respect to the line PK exchanges the rays r.H1 ; N / and r.H3 ; M /. As already noted, this implies the equivalence r.H1 ; N / k r.H3 ; M / () r.H3 ; M / k r.H1 ; N /. Thus, from now on, the sentence “two lines or rays r1 and r2 are parallel” means that either r1 is parallel to r2 or r2 parallel to r1 , and the two statements are equivalent. From the proof of Proposition 7.11, we extract the following useful property: Proposition 7.12. Given any two lines, there is a plane congruence that interchanges them.

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Proof. Let l1 and l2 be two lines. If these lines intersect and if O is their intersection point, then the symmetry with respect to the angle bisector exchanges these two lines. If the lines l1 and l2 do not intersect, then the proof of Proposition 7.11 provides a line l3 such that the symmetry with respect to l3 exchanges l1 and l2 . We already noted that in the neutral plane, the complement of any line l has two connected components, and we called them the two (open) half-planes defined by the line l. If H1 and H2 are these components, then if we take two points A 2 H1 and B 2 H2 , any line joining A and B intersects l. Given three pairwise non-intersecting lines l1 , l2 , l3 , we shall say l1 lies between l2 and l3 if l2 and l3 lie in different half-spaces defined by l1 . Note that in the hyperbolic plane, unlike in the Euclidean, given three pairwise non-intersecting lines, it is possible that neither of them lies between the other two (see Figure 80).

Figure 80

We shall use the following lemma in the proof of the proposition that follows it. Lemma 7.13. Let l1 , l2 , l3 be three oriented lines such that l1 k l3 and l2 k l3 , with l1 6D l2 . Then, there exists a line that intersects l1 , l2 and l3 , and one of the three lines lies between the other two. Proof. Any two parallel lines are disjoint, thus, l1 and l3 are disjoint, and l2 and l3 are disjoint. Finally, l1 and l2 are disjoint because otherwise there would be two distinct oriented lines that are parallel to l3 containing the point l1 \ l3 . Now we prove that there exists a line that intersects the three lines l1 , l2 and l3 . Take a point A on l3 , and let B and C be the feet of the perpendiculars from A on l1 and l2 respectively. There are two cases, represented in Figure 81 (i) and (ii), depending on whether B and C are on the same side of l3 or not. Case (i) occurs when B and C are on different sides of l3 . In that case, the line joining B to C intersects l3 , and we are done. In Case (ii), there are two sub-cases: either the three points A, B, C are collinear (and here also the claim follows), or not. Let us consider the latter case.

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Norbert A’Campo and Athanase Papadopoulos l1

l3

l2

B

A

A

l1

C B

C (i)

l3

l2

(ii) Figure 81

5 5

Either r.A; B/ is in the interior of the angular sector .r.A; C /; l3 /, or r.A; C / is in the interior of the angular sector .r.A; B/; l3 /. (In the notation .r.A; C /; l3 /, etc. we consider on l3 the orientation referred two in the parallelism statement). We can assume without loss of generality that the ray r.A; B/ is in the interior of the angular sector .r.A; C /; l3 /, as in Figure 81 (ii). In that case, the ray r.A; B/ intersects the line l1 since the sub-ray of the oriented line l3 starting at A is parallel to l1 . Thus, there is a line l that intersects the three lines l1 , l2 and l3 . Now let K; L; M be the three points on l that are on the three lines l1 , l2 , l3 , with L lying between K and M . The line among the three lines l1 , l2 , l3 that passes through K is completely contained in one of the half-planes defined by the line that passes through L, and the line through M is completely contained in the other half-plane. This completes the proof of the proposition.

5

5

Next, we show the transitivity of the relation of ray parallelism, up to a minor condition (disjointedness). Proposition 7.14. Assume that r1 , r2 , r3 are three rays such that r1 k r2 and r1 k r3 , and suppose that r2 and r3 are disjoint. Then, r2 k r3 . Proof. We extend each ray ri to a line li . The three lines are disjoint (Propositions 7.7 and 7.9) and they are equipped with the orientations induced by the rays they contain. By Lemma 7.13, there is a line intersecting l1 , l2 and l3 , and one of the three lines lies between the other two. We shall say that the ray ri is contained between the other two rays rj and rk if the line li is contained between the two lines lj and lk . We distinguished two cases, represented in Figure 82. Case (i): The ray r1 lies between the rays r2 and r3 (Figure 82 (i)). We take a line intersecting r1 , r2 and r3 in three points (Lemma 7.13), which we call respectively B, A and C . Let r.A; D/ be a geodesic ray in the angle sector .r.A; B/; r2 /. Since r2 is parallel to r1 and r3 , the r.A; D/ intersects r1 and then r3 . Thus, r2 is parallel to r3 .

5

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Notes on non-Euclidean geometry A

r1

r2

D

r2

r1

B

r3

r3

C

(ii)

(i) Figure 82

Case (ii): The line r1 is not between the rays r2 and r3 (Figure 82 (ii)). We can assume that r2 lies between r1 and r3 ; the case where r3 lies between r1 and r2 is treated similarly. If r2 is not parallel to r3 , then we take an arbitrary point on r3 , and from that point a ray parallel to r2 . By Case (i), this ray is also parallel to r1 , which is absurd. This completes the proof. We note that Proposition 30 of Book I of Euclid’s Elements asserts the Euclidean counterpart of Proposition 7.14, that is, if two distinct lines are parallel (in the Euclidean sense) to a third one, then the first two lines are parallel. An equivalence class of parallel rays is called a point at infinity, or an ideal point of the hyperbolic plane. (Note that equivalence classes are well-defined, even though, as we already noted, the relation of parallelism between rays is not, strictly speaking, an equivalence relation because it is not reflexive.)

7.3 Parabolic motions and horocycles In this section we study parabolic motions, horocycles, and related matters in the hyperbolic plane. Parabolic motions are transformations that exists only in hyperbolic geometry. Likewise, horocycles are curves that are proper to hyperbolic geometry. They can be defined as limits of circles whose centers converge to a point at infinity along a line, and that pass through a fixed point on that line.50 In spherical geometry horocycles do not exist, and in Euclidean geometry they are straight lines. There are several ways in which the hyperbolic plane can be considered as being more symmetric than the Euclidean plane. One way is to notice that there are motions 50 The term “horocycle” was coined by Lobachevsky (in the form horicycle), cf. his Geometrische Untersuchungen §31. The root of the word are the two Greek words “hori” (boundary) and “cyclos” (circle). The word “horizon” has the same root. The word “horicycle” was later on transformed in the English and French literature into “horocycle”. In his Pangeometry, Lobachevsky describes a horocycle, which he also calls a limit circle, as a circle whose radius is infinite, cf. [59], p. 7.

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of the hyperbolic plane that have no analogue in the Euclidean plane. These motions can be considered as rotations about points at infinity. They are called parabolic transformations, and we shall describe them below. Each parabolic transformation f is an element of a one-dimensional non-compact subgroup of motions whose elements are the parabolic transformations that fix a certain point at infinity which is associated to f . To describe a parabolic transformation f of the hyperbolic plane, we consider two parallel lines l and l 0 and we call ! their common point at infinity. This point will be called the center of f , and f will send l to l 0 . To specify the motion f , we start by specifying which point on l 0 is the image by f of which point of l. From the proofs of Propositions 7.11 and 7.12, we deduce the following useful fact. Proposition 7.15. Given two parallel lines l and l 0 with common point at infinity !, there exists a unique line l 00 that lies between l and l 0 , with the property that the reflection along l 00 interchanges the two lines l and l 0 . The restriction of the desired parabolic transformation f to l will be the restriction to l of the reflection provided by Proposition 7.15. Now that we have specified the action of f on l, we have a unique way of extending f to the entire hyperbolic plane. Indeed, each of the lines l and l 0 divides the plane into two closed half-planes, which we call the sides of these lines. The map f sends the side of l that does not contain l 0 to the side of l 0 that contains l. It is easy to see that since f is a motion, its action on the boundary of the first half-plane completely determines the action of f on that half-plane. Likewise, the map f sends the side of l that contains l 0 to the side of l 0 that does not contains l, and the action of f on the boundary of these first half-plane completely determines the action of f on that half-plane. This defines the parabolic transformation f of center !. A pencil of parallel lines is a maximal collection of parallel lines. The lines are equipped with the orientation defined by the parallelism relation. A pencil of lines defines a foliation of the hyperbolic plane; that is, each point of the plane passes through one and only one line of that pencil. This follows from the existence and uniqueness of parallel lines to a given line that pass through a given point. We shall use this fact below. There is a natural one-to-one correspondence between the set of pencils of lines and the set of points at infinity, or ideal points, of the hyperbolic plane. (In fact, we can identify each pencil with its point at infinity.) Given two parallel lines l and l 0 as above, with the associated parabolic transformation f that sends l to l 0 , the pencil P of lines parallel to l (and l 0 ) is preserved by f . Indeed, the parallelism relation is preserved by motions of the plane, and since f sends the line l to a parallel line l 0 , it sends any line parallel to l to a line parallel to l 0 . In particular, the parabolic motion f preserves the point at infinity ! of the pencil of lines P.

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Notice that from the definition of the transformation f restricted to l that we gave above (as the symmetry with respect to the intermediate line l 00 ), the line segment joining A and its image A0 makes equal angles with l and l 0 (Figure 83). A ˛ ˛ A

00

l

l0

A0 Figure 83

The point A0 is called the point on l 0 corresponding to A. Note that this correspondence is canonical. The parabolic transformation f is also called a rotation about the point at infinity ! because of the following property which is analogous to a property that holds for rotations about points in the plane: Take two points A and A0 in the plane, join them by two rays to the point at infinity !. Then the resulting infinite triangle A!A0 has congruent angles at A and A0 . Now we introduce horocycles. The following point by point construction defines the horocycle passing through a point A in the hyperbolic plane and centered at a given point at infinity !. The construction is analogous to the construction of a circle. We consider the pencil P of lines having ! as point at infinity and we let l be the line in P that contains A. We choose an angle ˛ whose value is in 0; =2 and we consider a ray starting at A and making an angle ˛ with l (the direction on l being the direction of parallelism). There are two such rays, one on each side of l. On each of these rays there is a unique point where this ray makes an angle ˛ with the line in P that contains it (with the appropriate orientations). These two points belong to the required horocycle. The points are denoted by H and H 0 in Figure 84. From the symmetry in the above construction, the horocycle through A is perpendicular to the line l in the pencil P. We record this in the following proposition, which can be compared to Proposition 4.12 that concerns hypercycles. Proposition 7.16. Given a pencil P of parallel lines and a point A in the plane, then the horocycle passing through A and whose point at infinity is defined by P is the orthogonal curve to the pencil P that contains the point A. Lobachevsky in [56], §31, gave an equivalent definition of a horocycle, as a plane curve which has the property that all the perpendicular bisectors of pairs of points on that curve are parallel. We also single out the following proposition:

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Norbert A’Campo and Athanase Papadopoulos H ˛ ˛ A

l

˛ ˛ H0

Figure 84. The point by point construction of a horocycle.

Proposition 7.17. Given two parallel lines l and l 0 with common point at infinity !, there is a canonical bijection between them. This bijection sends a point A on l to a point A0 on l 0 which is the unique point on l 0 contained in the horocycle passing through A and centered at !. Proof. We already know that for any point A on l, there exists a well-defined point A0 on l 0 corresponding to A. To prove uniqueness, notice that if we move the point A0 on l 0 to a point A00 on that line and disjoint from A (Figure 83), then the angle made with l and l 0 by the geodesic joining A and A00 varies continuously in a strictly monotonic way. Therefore, the two points cannot be equal. Thus, for any two parallel lines l and l 0 , there is a correspondence from l to l 0 which is given by the action of the unique parabolic transformation which sends l to l 0 . From the above discussion, we record the following useful fact. Proposition 7.18. Let l and l 0 be two parallel lines with common point at infinity !, let A be a point on l and let A0 be its image on l 0 by the parabolic transformation of center ! that sends l to l 0 . Then the line AA0 makes equal angles with l and l 0 . We deduce the following proposition concerning horocycles, which can be compared to Proposition 4.17 that concerns hypercycles. Proposition 7.19. In the hyperbolic plane, a horocycle has at most two points of intersection with a line. We also point out the following proposition, which gives another characterization of horocycles, and which follows from the definition we gave of these curves and Proposition 7.18. This proposition makes the relation between horocycles and parabolic transformations.

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Proposition 7.20. Let l be a line, let ! be a point at infinity of l and let A be a point on l. Then the horocycle of center ! and axis l and passing through A is the set of images of A by the one-parameter parabolic transformation group of center !. As a complement to Proposition 7.17, we note that there is an analogous fact for any two intersecting lines l and l 0 : there is a unique orientation-preserving congruence sending l to l 0 and preserving the intersection point, namely, a rotation about l \ l 0 . We also note for completeness that for any two ultra-parallel lines l and l 0 in the hyperbolic plane, there is also a well-defined correspondence between them. Indeed, the unique common perpendicular to these lines associates to some point on l a welldefined point on l 0 , and then, there is a unique orientation-preserving motion of the plane that sends l to l 0 preserving these points. In conclusion, we have the following three kinds of orientation-preserving motions of the hyperbolic plane. (1) A rotation of center O, preserving setwise each circle of center O. This transformation is called an elliptic motion of the hyperbolic plane. (2) A translation along a line l, preserving setwise each hypercycle of axis l. This transformation is called a hyperbolic motion of the hyperbolic plane. (3) A parabolic motion of center !, preserving setwise each horocycle of center !. From the definition of the correspondence between two parallel lines l and l 0 given by the action of the reflection about the intermediate line l 00 that we gave above (proofs of Propositions 7.15 and 7.11), we have the following: Proposition 7.21. Given any two points on a horocycle relative to a certain pencil of curves, their perpendicular bisector is a line of the same pencil of curves. We deduce the following: Proposition 7.22. The three perpendicular bisectors of a triangle whose vertices are on a given horocycle are parallel with the same point at infinity. Proposition 7.22 is an analogue for horocycles of Proposition 4.19 that concerns hypercycles. We also have the following converse of Proposition 7.22, which has to be compared with Proposition 4.20 concerning triangles whose three vertices are on a hypercycle. Proposition 7.23. In the hyperbolic plane, if two perpendicular bisectors of a triangle are parallel with point at infinity !, then the three vertices of this triangle are on a horocycle with point at infinity !, and the three perpendicular bisectors are parallel. Proof. Let ABC be a triangle, and assume that the perpendicular bisectors of AB and AC are parallel, with point at infinity !, cf. Figure 85. The symmetry with respect

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Norbert A’Campo and Athanase Papadopoulos A

C0 B0

B

!

A0 C Figure 85

to the perpendicular bisector of AB sends A to B. Therefore, in the infinite triangle AB! the angles at A and B are congruent. Likewise, in the infinite triangle BC ! the angles at B and C are congruent. The definition of the horocycle that we gave above using a point by point construction shows that A and C are on the horocycle containing B and having ! as point at infinity. Using any of the characterization of horocycles that we mentioned, it is easy now to prove the following: Proposition 7.24 (Horocycles as limits of circles). Let l be a line in the hyperbolic plane, A a point on l and rn a sequence of points converging to infinity on one side of A. Let ! denote the corresponding point at infinity. Then the horocycle containing A and having ! as point at infinity is the limit as n ! 1 of the sequence of circles centered at rn and containing A. The convergence here is in the sense that each point on the horocycle is a limit of a sequence of points xn 2 rn . In the Euclidean plane, the limit of such a sequence of circles is a line. Like circles and hypercyles, horocycles are equidistant sets; they are equidistant sets to the point at infinity to which they are associated. (One can conceive this as an expression of the fact that horocycles are limits of circles.) The following is a property of horocycles that distinguishes them from circles and from hypercycles: Proposition 7.25. In the hyperbolic plane, any two horocycles are congruent. Proof. A horocycle is completely determined by a line l (the axis of the horocycle), the choice of a point at infinity ! on that axis, and a point A on that line through which the horocycle passes. Therefore it suffices to prove that for any pair of triples .l; !; A/ and .l 0 ; ! 0 ; A0 / where l and l 0 are two lines, ! and ! 0 are points at infinity on l and

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l 0 respectively, and A and A0 are points on l and l 0 , we can find a plane congruence taking .l; !; A/ to .l 0 ! 0 ; A0 /. We already noticed that for any two lines l and l 0 , there is a plane congruence sending l to l 0 . For a given line l, the symmetry about a perpendicular to l exchanges the two points at infinity defined by the two sides of l. This shows that for any two lines with a specified point at infinity on each of them, there is a congruence exchanging the two lines and the given points at infinity. Finally, given a line l with two points A and A0 on l, there is a plane congruence taking l to itself and A to A0 (use a hyperbolic motion preserving l). This completes the proof of the proposition. For circles, there is no statement analogous to Proposition 7.25, since two circles are congruent if and only if they have the same radius. Likewise, there is no analogous statement for hypercycles because two hypercycles are congruent if and only if they are at the same distance from their respective axes. The next result should be contrasted with the result in Euclidean geometry stating that if two lines are parallel, then any perpendicular to one of them is also perpendicular to the other one. Proposition 7.26. In the hyperbolic plane, if two distinct lines have a common perpendicular, then they are not parallel for any choice of an orientation on them. Proof. From Theorem 4.21, if two lines l1 and l2 have a common perpendicular, the distance from a point P on l1 to l2 tends to infinity as P tends to infinity on l1 in any given direction. By Proposition 7.4, l1 and l2 cannot be parallel.

7.4 Horocycle contraction and applications The following theorem is proved by Lobachevsky in [56], §33, and in [58], p. 7. It describes in a precise way the distance contraction property between parallel geodesics in the hyperbolic plane. Theorem 7.27. Let l1 and l2 be two parallel lines, and let ! D l1 .1/ D l2 .1/ be their common point at infinity. Let AB and A0 B 0 be two horocyclic arcs centered at !, with A and A0 on l1 , B and B 0 on l2 , such that the point A seen from A0 is in the direction of !, and let x D jAA0 j. Let s and s 0 be the lengths of the horocyclic arcs AB and A0 B 0 respectively. Then, we have s 0 D se x with being a universal constant.51 51 The constant  is equal to 1 in the hyperbolic plane of constant curvature 1. Lobachevsky had no normalization for the curvature of the space, since he did not have the notion of curvature of a space. Thus, for him, there was this constant  pending because there was no preferred choice for the curvature of space. Indeed, any simply connected complete space of constant negative curvature satisfies the axioms of hyperbolic geometry, and is therefore a model of hyperbolic space.

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Proof. The proof will use the properties of parabolic and of hyperbolic transformations of the plane. We refer to Figure 86. B0

B

!

M0

M

A

l2 l3

A0

l1

Figure 86

From the existence of a parabolic transformation centered at ! and sending l1 to l2 , we have jBB 0 j D jAA0 j D x. Through each point of the segment AB passes a line that is parallel to l1 and l2 . This line lies between l1 and l2 , and it intersects orthogonally the horocyclic arc AB. Let M be the midpoint of the horocyclic arc AB, let l3 be the parallel to l1 and l2 that passes through M and let M 0 be the intersection point of l3 with the horocyclic arc A0 B 0 . The parabolic transformation of center ! that sends A to M preserves the pencil of lines that have ! as point at infinity. Thus, this transformation sends l1 to l3 and l3 to l2 . Therefore, it sends the horocyclic arc A0 M 0 to the horocyclic arc M 0 B 0 . Thus, M 0 is the midpoint of the horocyclic arc A0 B 0 . By the same reasoning, for any given positive integer q, if we subdivide the horocyclic arc AB into q sub-arcs of equal length, then, the lines parallel to l1 and l2 passing through these subdivision points intersect the horocyclic segment A0 B 0 into q equally spaced points. The same argument shows that if C is any point on the arc AB such that the length t of AC satisfies t D pq s, with p and q being positive integers, if C 0 is the intersection point of A0 B 0 with the line parallel to l1 and l2 and passing through C , and if t 0 is the length of the horocyclic arc A0 C 0 , then we have p t 0 D s0: q Finally, by continuity, we conclude that for any point C on AB, if C 0 is the intersection of A0 B 0 with the line parallel to l1 and l2 and passing through C , and if t is the length of the horocyclic segment AC and t 0 the length of the horocyclic segment A0 C 0 , we have t0 t D 0: (7.1) s s Now for y > 0, let A00 be the point on l1 at distance y from A and on the same side as A0 (Figure 87). Let  be the hyperbolic motion that translates along l1 in

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the direction from A to A0 , with translation distance y. We have  .A/ D A00 . Let A000 D  .A0 /. Since  preserves lengths we have 







length A0 B 0 length .A0 /.B 0 /  00  D   ; length A .B/ length AB where “length” means here the length of the horocyclic arc joining the two points indicated. Let B 00 and B 000 2 l2 be the endpoints of the horocyclic segments starting at A00 and  .A0 / respectively and contained between the two lines l1 and l2 (see Figure 87). By the result stated in Equation (7.1), we have 







length .A0 /.B 0 / length  .A0 /B 000  00  D   : length A .B/ length A00 B 00 B 00 B !

B0

A

.C 0 /

.C / A

0

A

l2 .B 0 /

.B/

C0

C

B 000

00

0

l1

.A /

Figure 87

Therefore, given the two lines l1 and l2 and a fixed point A on l1 , if we denote by f the function   length A0 B 0  ; x 7! length AB where as above B is the point at distance x from A in the direction opposite to that of !, and where A0 and B 0 are the points on l2 that are also on the horocycle passing by A and B respectively and centered at !, then, from the equation       length A00 B 00 length A000 B 000 length A000 B 000  D   D    length AB length A00 B 00 length AB we obtain f .x/f .y/ D f .x C y/. The solution of this functional equation is recalled in the next section, and it is of the form f .x/ D e x for some > 0. This completes the proof of Theorem 7.27. Let us draw some consequences from Theorem 7.27. The first result improves Proposition 7.4: Corollary 7.28. In the hyperbolic plane, if two lines are parallel, then their distance tends to 0 in the direction of parallelism.

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Proof. From Proposition 7.4, the distance between a point on one line to the other line decreases as the point moves in the direction of parallelism. This implies that the constant in Theorem 7.27 is negative. Thus, the distance tends to 0. Proposition 7.29. Let l be a line in the hyperbolic plane, let A be a point on l and let AB be a perpendicular segment to l at A. Any parallel ray to l through B makes with BA an angle that is < =2. Proof. Let l 0 be the perpendicular line to BA through B (Figure 88). We recall that l 0 is disjoint from l (this is a result of neutral geometry; cf. Theorem 3.11). Assume B

A

l0

l

Figure 88

that a ray in l 0 makes with BA an angle > =2. Then, by Theorem 4.21 the distance to l from a point on l 0 tends to infinity as the point moves to infinity. This contradicts Corollary 7.28. Thus, a parallel ray to l starting at B makes an angle < =2 with AB. Corollary 7.30. In the hyperbolic plane, two lines cannot be parallel in both directions. We formulate the result of Corollary 7.30 in a different manner: Corollary 7.31. Under the axioms of neutral geometry, if the two parallel rays to a line starting at a given point are contained in a single line, then the plane is Euclidean.

7.5 The functional equation f .x/f .y/ D f .x C y/ In this section, we prove the result on the solution of the functional equation that was used in Theorem 7.27. We first study the following functional equation f .x/ C f .y/ D f .x C y/;

(7.2)

where x and y vary over the real numbers. This equation was already considered by d’Alembert (see [74]). We assume that f is continuous.

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135

Taking x D y D 0, we obtain 2f .0/ D f .0/, that is, f .0/ D 0. Taking x D y and applying the equation n times, we obtain f .nx/ D nf .x/ for every integer n  0. By the same reasoning, we have nf . px / D f . pn x/ and pf . px / D f .x/ for all integers n  0 and p > 1. This gives   n n f .x/ D f x p p and

n n f .x/ D f .1/: p p

for all integers n  0 and p > 1. Since f is continuous, we obtain f .x/ D C x, with C D f .1/, for every x  0. For x < 0, we have, from (7.2), f .x/ C f .x/ D f .0/ D 0, which gives f .x/ D f .x/. Thus, the solution of (7.2) is f .x/ D C for all x 2 R. Now we consider the functional equation f .x/f .y/ D f .x C y/:

(7.3)

We have .f .x//2 D f .2x/, which implies f .x/  0 for all x 2 R. If f .x/ D 0 for some x, then, Equation (7.3) implies that f .x/ D 0 for all x. Thus, we assume now that f .x/ > 0 for all x in R. Taking logarithms, we obtain the functional equation log.f .x// C log .f .y// D log.f .x C y//: Thus, the function log f satisfies Equation (7.2), which implies that log.f .x// D C x for all x 2 R and for some constant C . Thus, the solution of Equation (7.3) is f .x/ D e C x:

7.6 Lobachevsky’s angle of parallelism function The trigonometric formulae that Lobachevsky established in his memoirs are all expressed in terms of the angle of parallelism function. Lobachevsky introduced this function in his first memoir that survived, the Elements of geometry (1829) [53], where it is denoted by F .p/. This function was used later on by several authors, with a reference to Lobachevsky, e.g. by Beltrami in his Saggio di Interpretazione della geometria non-Euclidea [8] and by Klein in his Über die sogenannte Nicht-Euklidische Geometrie [48]. There are several ways of defining the angle of parallelism function in the hyperbolic plane, and we record here two of them: (1) Consider and arbitrary line l and a point P that is not on l, and let P 0 be the projection of P on l. The angle of parallelism is a function of the segment PP 0 defined as the value of the angle made by a line parallel to l through P with the

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segment PP 0 . This function depends only on the distance from P to l, that is, it is a function of the length of the segment PP 0 . (2) Consider a right triangle  which has exactly one vertex at infinity (Figure 89). This triangle has exactly one edge of finite length, and this edge is adjacent to

p

Figure 89

the right angle. Let p be the length of that edge. The angle of parallelism ˛, as a function of p, is the value of the acute angle of . Lobachevsky’s angle of parallelism function is a function that is proper to hyperbolic geometry. (In Euclidean geometry, this function is also defined but it is constant and equal to =2. In spherical geometry, the angle of parallelism function is not defined.) It is a function that expresses angles in terms of lengths of segments. The existence of a function with this property, that is, the fact than one can express angles in terms of length in such an absolute manner, is an expression of the fact that there exists an absolute measure of length in the hyperbolic plane. (Recall that we already deduced this fact from Theorem 4.4 which implies that the lengths of the edges of a triangle are determined by its angles.) To see this, recall that there is an absolute notion of angle measure in the three geometries. This follows from the fact that at any point, the total angle is the same. We usually call this total angle 2, and we can measure all angles in terms of this total angle. We can do this measurement entirely within our geometry, without any extra notion, by first dividing a total angle into congruent angles, and hence defining the value of any integral fraction of 2, and then using again congruence to define the value of a rational multiple of a total angle, and finally, by continuity, we can define the value of any angle. There is no such absolute measure of length in Euclidean geometry. In this geometry, in order to define length, we have to agree beforehand on a length unit, and such a choice of a unit cannot be done intrinsically, that is, using the axioms and nothing else. Such an agreement is necessarily artificial, that is, it has to be done using a device that is extrinsic to the elements of Euclidean geometry. In hyperbolic geometry, there is no need of making a choice of a unit length. Using the angle of parallelism function that gives a canonical one-to-one correspondence between length in 0; 1Œ and angles in 0; =2Œ. We mention by the way that in spherical (or in elliptic) geometry, even though the angle of parallelism function is not defined, there is a canonical measure for length that follows from the fact that all lines in this geometry are of finite length and congruent, and one can choose this common length as a length unit. We shall also comment on this fact in Section 9.2 below.

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Given a line l and a point A … l, there are two parallel rays from A to l, one from each side of the perpendicular AD from A on l. Let r.A; H / and r.A; K/ be these two rays (Figure 90). Let D be the foot of the perpendicular from A to l. The orthogonal symmetry with respect to the line AD exchanges the two angles ˛ and ˇ that are made respectively by r.A; H / and r.A; K/ with r.A; D/. In other words, ˛ D ˇ. A ˇ

K

˛

D

H

l

Figure 90

The angle ˛ is called the angle of parallelism at the point A with respect to the line l. Thus, the angle of parallelism of a point A with respect to a line l is half of the angle made by the two parallels drawn from A to l. It is easy to see from the congruence-invariance properties that the angle of parallelism of A with respect to l depends only on the distance jADj from A to l. Since this is an important property, we state it as a theorem: Theorem 7.32. Given a line l and a point A not on l, the angle of parallelism from A to l depends only on the distance from A to l. We denote the angle of parallelism function by …. (This notation is Lobachevsky’s.) Thus, setting jADj D p, we can write ˛ D ….p/: We have the following precise formula for the angle of parallelism ˛ D ….p/ in terms of the distance p D jADj. Proposition 7.33. In the hyperbolic plane, for every positive real p, we have cos ….p/ D tanh p: Proof. Consider the right triangle ADH of Figure 90, whose vertex H being at infinity, and with angle ˛ at A. Applying Corollary 6.16 to this limiting case, we have cos ˛ D cos ….AD/ D

tanh AD AD D D tanh AD: AH 1

Lobachevsky proved Proposition 7.33 with a different reasoning, see [59], p. 25. We shall give another formula for the angle of parallelism in Proposition 8.5.

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Proposition 7.34. In the Euclidean plane, Lobachevsky’s angle of parallelism function is constant and equal to =2. Proof. This follows from the fact that the parallel l 0 to l drawn from a point A makes an angle of =2 with the perpendicular AD from A to l (Figure 91). A

l0

l

D Figure 91

Besides its theoretical importance, the angle of parallelism function was used by Lobachevsky in the expression of the trigonometric formulae of hyperbolic geometry. Let us give a few examples. For a right triangle, with edges a, b, c and angle 2 opposite to c, Lobachevsky obtained the formula: sin ….a/ sin ….b/ D sin ….c/ (see [59], p. 25). For a trirectangular quadrilateral with edges c, y, x, a in that order and with an acute angle ' at the vertex common to c, a and right angles everywhere else, Lobachevsky gave the formulae cos ….y/ D sin ….x/ cos ….a/ and tan ' D

tan ….a/ cos ….x/

(see [59], p. 41). In the so-called .x; y/-coordinates52 , Lobachevsky gave the following equation for a circle of radius r centered at the origin: sin ….x/ sin ….y/ D sin ….r/ (see [59], p. 40).

52 These coordinates are defined as follows. One first chooses an x-axis with origin O. Then the x-coordinate of a point P in the plane is defined as the distance (with sign) of the projection of P on the x-axis to the origin. The y-coordinate is the distance from P to its projection (with the appropriate sign). Note that such a y-coordinate is not the distance to O measured on some y-axis of the projection of P on that axis.

Notes on non-Euclidean geometry

139

8 Models 8.1 Introduction We have a good intuition of Euclidean geometry that we gain from our experience in the physical world that surrounds us. We have also an intuition of spherical geometry because it is the geometry of a sphere embedded in 3-dimensional Euclidean space. For hyperbolic geometry, we are used to work in models. The first model of the hyperbolic plane, as the open unit disk in the Euclidean plane in which the hyperbolic lines are represented by the Euclidean chords, was discovered by Eugenio Beltrami in 1869 in his paper [8]. There are several other models available, which are more or less intuitive and we shall present two of them in this section. Models are important, each of them being useful for doing computations, and some models play a significant role in various field of mathematics. For instance, the Poincaré disk and upper half-plane models have applications in complex analysis, in number theory, and in the study of differential equations. The Beltrami–Klein model is intimately related to projective geometry, and it has been at the origin of the definition of the so-called Hilbert geometries that are defined on general open convex sets. The Minkowski model is useful in the study of quadratic forms and in general relativity. Furthermore, the existence of models has a theoretical significance. It implies the relative consistency of hyperbolic geometry with respect to Euclidean geometry. Indeed, in a Euclidean model of hyperbolic geometry, the undefined notions of this geometry are interpreted as (not necessarily undefined) notions of Euclidean geometry. For instance, in each of the two Poincaré models, lines are interpreted as circles orthogonal to the boundary, and so on. In these models, the hyperbolic notions and relations that are stated in the axioms of hyperbolic geometry are expressed in terms of Euclidean notions and relations. In this way, (the hypothetical existence of) an absurd statement involving the undefined notions of hyperbolic geometry would lead to an absurd statement of Euclidean geometry. The relative consistency of hyperbolic geometry with respect to the Euclidean one, as a consequence of the existence of such Euclidean models, was most clearly stated by Hilbert.

8.2 A Euclidean model of the hyperbolic plane In this section, we describe a classical model of the hyperbolic plane. The underlying space is the Euclidean unit disk D D f.x; y/ 2 R2 j x 2 C y 2 < 1g. The congruence group of the hyperbolic plane becomes in this model a group of projective transformations, that is, transformations that preserve collineation, or, equivalently, transformations that preserve the cross ratio of aligned quadruples of points. We start by constructing this model. We denote by H the hyperbolic plane. In H, we choose a point O which we call the origin, and two oriented lines which we call the axes, X 0 OX and Y 0 OY , that meet perpendicularly at O (see Figure 92).

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Norbert A’Campo and Athanase Papadopoulos Y

.; C/

P

.C; C/

p

X0

X

a

O .; /

.C; /

Y0 Figure 92

Using the axes, we assign coordinates to points in H. To any point P 2 H, we consider its projections on the lines X 0 OX and Y 0 OY and we denote by a  0 and p  0 respectively the distances from the origin to these projections. We consider the map B W H ! R2 ; P .a; p/ 7! .˙ tanh a; ˙ tanh p/;

(8.1)

where the sign, depending on the quadrant to which the point P belongs, is indicated in Figure 92. This sign convention is the same as the one used when we deal with Cartesian coordinates in the Euclidean plane. A chord of D is the intersection of a Euclidean line in R2 with D. A diameter of D is a chord that passes through the center of that disk. We shall prove that the image of the hyperbolic plane by the map B is the disk D, and that the map B has the following properties: (1) The image by B of any hyperbolic line in H is a chord of D. (2) Angles at the origin are preserved. In other words, the image by B of any two lines l and l 0 that pass through O are diameters of D that make at the origin an angle equal to the angle made by l and l 0 . (3) Given any two chords in D that are perpendicular (in the Euclidean sense), if one of them is a diameter, then they are images of perpendicular lines in the hyperbolic plane.

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(4) Perpendicularity in the hyperbolic plane can be expressed in terms of polarity in projective geometry. For instance, two lines in the hyperbolic plane are perpendicular if and only if their images in the Euclidean disk are polar conjugate. In other words, if M and N are the endpoints on the circle S1 D @D of a line D in D and if P is the intersection point of the tangents to S1 at M and N (P might be at infinity), then a line in D is perpendicular to D if and only if its extension in the Euclidean plane contains P . As a consequence, the construction of a perpendicular common to two ultra-parallel lines in the hyperbolic plane is obtained through the construction of the polar dual in the complement of the Euclidean disk (see Figure 93). M D1 P D2 N Figure 93. In this Euclidean model of the hyperbolic plane, MN is the common perpendicular to the two lines D1 and D2 . The point P is the polar dual of the line MN . The pencil of Euclidean lines that contains D1 and D2 induces on the disk D a family of lines that are perpendicular (in the hyperbolic sense) to the line MN .

(5) If two hyperbolic lines in H are parallel, then their images in D are chords that meet at a point on the boundary circle. (6) If two hyperbolic lines are ultra-parallel, then their images in D are chords which, when extended, either are parallel lines in the Euclidean sense, or they meet at a x This point is the polar dual of the point in the exterior of the closed unit disk D. common perpendicular to these two lines. Now we study this model in some detail. In what follows, we shall use the notation P .a; p/ to say that .a; p/ are the coordinates of a point P in H; that is to say, the projections of the point P on the two axes X 0 OX and Y 0 OY are respectively at distance a and p from the origin. We shall talk about signs when this is necessary. We consider the map B W H ! D defined in (8.1). A first consequence of Formula (5) of Theorem 6.18 is the following: If a and p become very large, then sinh a sinhp is > 1. Since cos P < 1, this implies that the angle at P does not exist for a and P large, which shows that the two lines perpendicular to the axes and at distances a and p from the origin cannot intersect. Thus, we see immediately that the map B from H to the plane R2 is not surjective.

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Proposition 8.1. The map B is a bijection from H to the Euclidean open unit disk D. Proof. Using the inequality 1 < tanh x D Theorem 6.18 implies that

sinh x cosh x

< 1 valid for all x, Formula (1) of

tanh2 p C tanh2 a D tanh2 d < 1; which shows that the image of H by B is contained in D. Using the fact that the function tanh is injective and taking into account the signs of the coordinates given in Equation (8.1), we can see that if P .a; p/ and P 0 .a0 ; p 0 / are two points in H that have the same image by B, then a D a0 and p D p 0 . Now if two distinct lines in the hyperbolic plane that are perpendicular to the axes 0 X OX and Y 0 OY respectively and at distance a and p respectively from the origin meet, then they meet at a unique point. This shows that B is injective. The fact that B is surjective onto D also follows from Theorem 6.18. Indeed, consider a point in D of coordinates .x; y/, and let us construct a point in H such that B.P / D .x; y/. For this it suffices to know one of the angles ˛ or ˇ that the line OP makes with the axes X 0 OX and Y 0 OY respectively. Formula (1) of Theorem 6.18 gives tanh2 d D x 2 C y 2 , and the Euclidean definitions of the sine and cosine give cos ˇ D .x 2 Cyx2 /1=2 , and cos ˛ D sin ˇ D .x 2 Cyy2 /1=2 . Thus, ˛ and ˇ are determined. From now on, we consider B as a homeomorphism B W H ! D: Abusing notation, we shall denote by the same letter O the image of the point O 2 H by B. Thus, O also denotes the center of the disk D. We let x 0 Ox and y 0 Oy be the images in D of the axes X 0 OX and X 0 OY by the map B. Proposition 8.2. The map B satisfies the following properties: (1) The image by B of any line in H is a Euclidean chord of D. (2) The image by B of any line in H containing O and making an angle ˛ with X 0 OX is a diameter in D making the same angle ˛ with the axis x 0 Ox. (3) If l is a line in H that contains the origin and if l 0 is a line that is perpendicular to l, then the image chord B.l 0 / is perpendicular to the image chord B.l/. Proof. We first consider a special case of (1). From the definition of the map B, it easily follows that x 0 Ox and y 0 Oy are Euclidean chords in D. Now consider the special case of a line l in H that is perpendicular to X 0 OX (respectively Y 0 OY ). Then, all the points on that line have the same projection on X 0 OX (respectively Y 0 OY ), and the formula for B shows that the image of l by the map B is a Euclidean chord in D which is perpendicular to the chord x 0 Ox (respectively y 0 Oy) at a point at distance tanh a (respectively tanh p) from the origin (Figure 94).

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Notes on non-Euclidean geometry B.l 0 / Y

l0 tanh b

b X0

O

X

a

Y0

O tanh a B.l/

l Figure 94

Let us now prove (2). Consider a line l in H passing through the origin and making an angle ˛ with the axis X 0 OX (Figure 95). P p

d

˛

˛

B.l/

a l

Figure 95

We claim that the image of l by B is a Euclidean chord in D that passes through the origin. Indeed, we can write an equation of the image B.l/ as follows. Using the notation of Figure 95, we have, from Corollary 6.16, cos ˛ D

tanh a tanh d





and sin ˛ D cos

tanh p  ˛ D ; 2 tanh d

which gives y D .tan ˛/x: This is the equation of a diameter in D, and this proves (2).

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Now if l is a line in H that does not pass through the origin O, then we choose new coordinate axes U 0 OU and V 0 OV in H such that the new “horizontal” axis U 0 OU intersects perpendicularly the line l. Let ˛ be the angle between the lines X 0 OX and U 0 OU . From (2), the image of U 0 OU by the map B is a diameter u0 Ou in the Euclidean disk D that makes the same angle ˛ with the horizontal axis x 0 Ox (Figure 96). V

l

v

u

U ˛

B.l/

˛

O

O u0

U0

v0 V0 Figure 96

We can take this diameter u0 Ou as the horizontal axis of a new coordinate system in D. The construction of the line B.l/ now follows from the particular case considered at the beginning of the proof: it is a chord in D perpendicular to u0 Ou (Figure 96). This proves at the same time (1) and (3). It is easy to visualise in this Euclidean model the two parallels drawn to a line l from a point A that is not on that line (see Figure 97).

A l

Figure 97. The two parallels drawn from a point A to a line l in the Euclidean (or projective) disk model.

We recall the definition of the cross ratio.

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If .a; b; P; Q/ is an ordered quadruple of pairwise distinct points in the Euclidean plane which are on the same line, their cross ratio is defined by the formula

.a; b; P; Q/ D

ja  Qj jb  P j ; jb  Qj ja  P j

where j j denotes Euclidean distance. We shall use the cross ratio for a quadruple as in Figure 98, that is, a quadruple

P

a

b

Q

Figure 98

.a; b; P; Q/ in which a and b are two distinct points in the unit disk D and P and Q are the intersection with the boundary of D of the Euclidean line joining a and b. Remark 8.3. The cross ratio is defined in a much more general situation than the one we consider here, but we do not need this fact. Figure 99 provides a ruler and compass construction of the cross ratio of a quadruple of points that are on the same line. The construction depends on the choice of a point C outside the closed disk. In that figure, using the invariance of the cross ratios by projectivities, we have X D Y , hence the construction of . We denote by dH the distance in H, and dD D dH B B 1 the induced distance on the unit disk D. Thus, for x and y in D, we let 



dD .x; y/ D dH B B 1 .x/; B 1 .y/ : A congruence of the disk D that we obtain by transporting a congruence of H by B sends chords to chords. Therefore, by the so-called fundamental theorem of projective geometry, this map of D is a projective transformations of the disk. Thus, the image by B of the isometries of H is a subgroup of the projective transformation group of the unit disk. In particular, such a transformation preserves the cross ratio. Since the action is transitive on segments of the same length, the transformation group is the full group of projective transformations of the unit disk. In fact, the distance function dD in D can be expressed in terms of the cross ratio, as follows:

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Norbert A’Campo and Athanase Papadopoulos Q

b parallel a

P X

Y

C

parallel Figure 99. The construction of the cross ratio of .P; a; b; Q/: we have X D Y .

Proposition 8.4. For a and b in D, we have dD .a; b/ D

1 j . .a; b; P; Q// j; 2

where P and Q are the intersection points of the line ab with the boundary of the disk D, and where the four points P , a, b, Q are aligned in that order. Proof. For a point A in H, let Ax D B.A/ 2 D. Assume first that Ax is on the X-axis, and let x be its coordinate on that axis. We have, from the definition,   x D dH B 1 .O/; B 1 .A/ x D dH .O; A/: dD .O; A/ Since x D tanh dH .O; A/, we obtain

  1Cx 1 x : dD .O; A/ D arctanh x D log 2 1x   is the cross ratio of the quadruple .0; x; 1; 1/, that is, The quantity 12 log 1Cx 1x we have 1 dD .O; x/ D j log. .0; x; 1; 1/j: 2

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The general case follows from this special case. Indeed, given two arbitrary distinct points in H, there is a congruence of H that sends them to O and a point A on the X -axis respectively. Now since the group of projective transformations of the disk preserves the cross ratio, the result follows. The disk D with the distance dD is a model of the hyperbolic plane. Note that this model is not conformal; in general, hyperbolic angles are distorted, that is, they are not transformed into Euclidean angles having the same value. We can see this on the situation represented in Figure 100. The lines X 0 OX, Y 0 OY , l and l 0 Y

l0 X0

A

C

O

B

P

X

l

Y0 Figure 100. The hyperbolic angle at C of the quadrilateral OACB is less than a right angle.

form a quadrilateral OACB that has three right angles, at O, A and B. The angle at C is a right angle when measured with the Euclidean metric, and we know it cannot be a right angle for the hyperbolic metric. (Note that we can construct the perpendicular P C to l at C using the pole P of l; thus we can see on the picture that the angle made by this perpendicular and l is different from the angle ACB.) Note that Properties (2) and (3) in the list of properties that are at the beginning of this section say however that certain specific angles are preserved. As an application of this Euclidean model, let us find a formula for the relation between the angle of parallelism ˛ of a point A with respect to line l in terms of the distance p D jADj from A to D.

1

Proposition 8.5. In the hyperbolic plane, for every positive number p, we have 1 e p D cot ….p/: 2

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Proof. We compute distances in the projective model D using the cross ratio. We take a point A at the entre of the disk, and we take a line l such that the distance from A to l is p > 0. The Euclidean angle between two lines intersecting at the center of the disk is equal to the hyperbolic angle between these lines. (We already proved this, but this also follows simply by symmetry.) We use the notation of Figure 101. P

A ˛ l

D

Q Figure 101. Computing the angle of parallelism in the projective model.

We have

  jPDj jQAj ˇˇ 1 1 C cos ˛ ˛ 1 ˇˇ D log D log cot p D log 2 jPAj jQDj 2 1  cos ˛ 2

which gives

˛ : 2 This formula is given by Lobachevsky, with another proof, in [59], p. 24. We already gave a formula for the angle of parallelism function in Proposition 7.33 above. It is not difficult to deduce each of these formulae from the other one. As an example of the use of hyperbolic geometry in Euclidean geometry, we now give a proof of Brianchon’s Theorem.53 This is a theorem of Euclidean geometry54 stating the following: e p D cot

Theorem 8.6 (Brianchon). In the Euclidean plane, if a hexagon is circumscribed about a circle, then the three large diagonals are concurrent. The large diagonals of a hexagon are the segments that join vertices n and n C 3 (using cyclic order notation). These diagonals are represented in continuous lines in Figure 102. 53 Charles 54 More

Julien Brianchon (1783–1864). correctly, it is a theorem of projective geometry.

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Figure 102. The large diagonals of the hexagon are those represented in continuous lines. The dashed lines are the large diagonals of the hexagon whose vertices are the tangency points.

Note that since the statement of Theorem 8.6 is invariant under projective maps, we can replace in the statement the word “circle” by “conic”. Proof. Consider a large diagonal AB and the edges of the hexagon at the endpoints of this diagonal. We use the notation of Figure 103, where C1 , D1 , C2 , D2 are the points of tangency of these edged with the circle. We equip the interior of the circle with the hyperbolic metric, as in the model described above. The hyperbolic symmetry with respect to the line AB sends C1 to D1 and C2 to D2 . A

D1

C1

D2 C2 Figure 103

B

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Consider now the three large diagonals of the inner hexagon formed by the six tangency points. These diagonals are drawn in dashed lines in Figure 102, and two of them are represented in Figure 103. Using again the above symmetry, each two such consecutive dashed diagonals intersect at a point on the large diagonal of the initial hexagon that lies between them. Now the three dashed large diagonals define a small triangle whose three vertices lie on the three large diagonals (drawn in continuous lines) of the initial hexagon, and the three large diagonals are the angle bisectors of this small triangle, therefore they are concurrent. (The fact that the three angle bisectors in an arbitrary hyperbolic triangle are concurrent follows easily from the definition of angle bisector, based on an equidistance argument from the three edges.) Remark 8.7. Brianchon’s Theorem is equivalent to Pascal’s Theorem stating that if a hexagon is inscribed in a circle (or, more generally, in a conic), then the intersection points of the three pairs of opposite edges of that hexagon form an aligned triple of points. The equivalence of the two theorems is a consequence of a duality (“polarity theory”) in projective geometry. Brianchon proved his theorem in 1810, about 170 years after Pascal proved his theorem (at the age of 16). Duality theory (and hence the equivalence of the two theorems) was discovered at the end of the 1820s, that is, a few years after Brianchon proved his theorem.

8.3 A model arising from algebra The aim of this section is to describe a model of the hyperbolic plane whose points are prime ideals of the ring RŒX of polynomials with one real variable. Eventually, we shall see that this model is the Poincaré upper half-plane model, equipped with its Riemannian metric. This is an instance where the upper-half model arises in a non-expected manner in algebra.55 The congruence group of hyperbolic geometry contains the group of automorphisms of the ring of polynomials.56 We shall also give the classical description of the action of the congruence group of hyperbolic geometry as a group of inversions, and then as a group of linear fractional transformations. The fact that the automorphisms of the upper half-plane can be represented by matrices allows one to use the tools of linear algebra to study these automorphisms, both individually and in subgroups. Anticipating on the theory we shall develop below, we note that the set of prime ideals (that is, the “spectrum”) of the ring RŒX  splits into two parts, one part which is identified with the hyperbolic plane H, and another part which is identified with the line R, equipped with its affine geometry, which is seen as sitting on the boundary of the hyperbolic plane. Thus, the boundary R of the Poincaré model also admits a description in terms of prime ideals of RŒX . 55 We

note that there are many other places in which this model is hidden in mathematics. congruence group of the hyperbolic plane is strictly larger than the automorphisms of the ring of polynomials for reasons that will become clear below. 56 The

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It is natural that an “invariant” subset of a space has more symmetries than the whole space itself, and this explains the fact that the symmetry group of the H-part of the spectrum (that is, the hyperbolic plane congruence group) is larger than the symmetry group of the spectrum of RŒX. On the other hand, it is remarkable that the symmetry group of the R-part of the spectrum coincides with the symmetry group of the spectrum. Let us now start from the beginning. We recall that the polynomial ring RŒX in one variable X with coefficients in R is the set of expressions of the form pm X m C pm1 X m1 C    C p1 X C p0 ; where p0 ; : : : ; pm are real numbers, and where the ring operations are the usual operations on polynomials. We also recall the following facts: • RŒX  is an integral domain; that is, it is a commutative ring with unity, in which a product of two non-zero elements is also nonzero. • There is a factorization theory in RŒX , which is analogous to the factorization theory of positive integers into products of primes. • Every nontrivial ideal I of RŒX which is proper (i.e. I 6D RŒX ) consists of the set of multiples of a single non-zero polynomial, which is the “greatest common divisor” of all polynomials in I . This property is expressed by the fact that the polynomial ring RŒX  is a principal ideal domain. An ideal P of a ring R is said to be prime if it is proper and if whenever the product AB of two ideals is contained in P , then A or B (or both) is contained in P . The spectrum of a ring is the set of its prime ideals. The terminology for the factorization theory in the ring RŒX  is strongly motivated by that of the ring Z of integers, whose set of prime ideals is in natural one-to-one correspondence with the set of prime numbers (up to sign). We let H be the upper half-plane, that is, H D fz 2 C j Im.z/ > 0g: We first describe the following set-theoretic equality: Proposition 8.8. We have Spec.RŒX/ D R [ H: Proof. The ring RŒX has two kinds of prime ideals: (1) For r 2 R, the ideal .X  r/RŒX is a prime ideal of the first kind. (2) For b; c 2 R with b 2  4c < 0, the ideal .X 2 C bX C c/RŒX  is a prime ideal of the second kind. We shall use the notation Ir D .X r/  RŒX  and Ib;c D .X 2 CbX Cc/  RŒX  for prime ideals of the first and second kind respectively.

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Observe that the ideals Ir and Ib;c are indeed maximal ideals in RŒX , and that the quotient fields RŒX=Ir and RŒX=Ib;c are isomorphic to the field R and to the field C respectively. A polynomial X 2 C bX C c, with b; c 2 R satisfying b 2  4c < 0 has exactly one root z in H, and this root satisfies b and b 2  4c D 4Im.z/2 : 2 For this root z, we also denote the corresponding ideal Ib;c by Iz , and we set Re.z/ D 

b.z/ D 2Re.z/

and c.z/ D Re.z/2 C Im.z/2 :

Putting together notation, we have the following: Spec.RŒX/ D fIr j r 2 Rg [ fIz j z 2 Hg D R [ H: Remark 8.9. The description of the spectrum of the polynomial ring RŒX; Y  in two variables is much more complicated. We wish to have more than the set-theoretic equality Spec.RŒX / D R [ H. We shall recover, from the theory of polynomials, the set H equipped with its hyperbolic geometry, and the set R equipped with its affine geometry. We begin by studying the natural symmetries of the spectrum. We recall that the automorphism group of a ring naturally acts on the spectrum. The symmetries of Spec.RŒX / are those that originate in the symmetries of the ring RŒX . As already noted, it will turn out that the symmetry group of the H-component of Spec.RŒX / is much larger than the symmetry group of its R-component. We shall then define a metric Dist on the H-component of the spectrum, that is, on the set of prime ideals of the second kind. This metric will be natural with respect to the action of the automorphism group Aut.RŒX / on RŒX . This means that the automorphism group Aut.RŒX/ will act on the metric space .H; Dist/ by isometries. The action of the automorphism group Aut.RŒX / of RŒX  is the group action on RŒX  induced by the variable substitution X ! X C T , for 2 R and T 2 R. Given an element g in Aut.RŒX/ represented by a substitution X ! X C T , and given a polynomial X in RŒX, we shall denote by P g .X / the image polynomial, that is, the polynomial P . X C T /. The action of Aut.RŒX/ in terms of variable substitution can also be described in terms of matrices:     

T X

X C T D : 0 1 1 1   We note by the way that the group of matrices of the form 0 T1 with . ; T / 2 .R  R/ is in some sense the smallest non-commutative Lie group. We shall restrict our attention to the subgroup Aut C.RŒX/ of substitutions X !

X C T with > 0. This corresponds to the matrices 0 T1 with > 0. We now study the induced action of Aut C .RŒX / on the spectrum of RŒX .

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Note that a substitution X ! X C T does not change the set of values taken by a polynomial, and for each value, it do not change its multiplicity. Proposition 8.10. The natural action of Aut C .RŒX / on the spectrum R [ H is given for p in R [ H, where g is the substitution X ! X C T . by g  p D pT  Proof. First, note that the action of the group Aut C .RŒX / on the spectrum preserves the type of elements (first or second kind). A real polynomial X 2 C bX C c represents a prime ideal of the second kind if and only if it has negative discriminant, which amounts to saying that the polynomial does not take the value zero. If this holds, then this is also true for the polynomial . X C T /2 C b. X C T / C c, obtained after the substitution of X by X C T . A polynomial X  r representing an element of the first kind of the spectrum is encoded by the real number r. The substitution of X by X C T transforms this , which is encoded by the real number rT . polynomial into the polynomial X  rT   Next, we represent a prime ideal of the second kind by the corresponding element in H, and we study the action of Aut C .RŒX / on H. Recall that p is the complex root of the polynomial P .X/ D pX 2 C bX C c that has 2 . The element positive imaginary value and that Re.p/ D  b2 and Im.p/ D 4cb 2 p  P .X / is the polynomial T 2 C bT C c 2T C b XC :

2 A simple computation then shows that the complex root of the above polynomial p 2 and imaginary value 4cb . that has positive imaginary value has real value  bT 2 2 This shows that the action of the automorphism group on the element of the spectrum is indeed given by g  p D pT .  X2 C

Remarks 8.11 (The harmonic ratio and the cross ratio). The induced action of AutC .RŒX / on R is that of the affine group. We recall that the affine group of the line R is the group of bijections of R that preserve the harmonic ratio57 of triples, that is, 3 the function on distinct triples of real numbers defined by .x1 ; x2 ; x3 / 7! xx12 x . The x3 action on R of the affine group is simply transitive on pairs of points. The reader can search, in this setting of polynomial theory, for a relation between the harmonic ratio and the following function of ordered triples .x1 ; x2 ; x3 / of distinct real numbers: take the degree-two polynomial that takes the value 0 at x1 , 1 at x2 , and evaluate that polynomial at the point x3 . We note by the way that the cross ratio is a symmetrization of the harmonic ratio, in a sense that can easily be made precise. 57 The harmonic ratio is sometimes called the affine ratio, or the section ratio. We also note that the harmonic ratio can be defined more generally (with the same formula) for distinct triples of complex numbers, and that the harmonic ratio of a triple z1 , z2 , z3 in C is real if and only if the three points lie on the same Euclidean line. Likewise, the cross ratio can be defined more generally for distinct quadruples of complex numbers, and the cross ratio of a quadruple z1 , z2 , z3 , z4 in C is real if and only if the four points are on the same circle (a Euclidean line being here a particular case of a circle).

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The restriction of the Euclidean metric dC .p; q/ D jp  qj of the complex plane to the upper half-plane H is not natural with respect to the action of Aut.RŒX / on H. For instance, the substitution of X by 2X maps the ideals .X 2 C 1/ and .X 2 C 9/ to the ideals .X 2 C 14 / and .X 2 C 94 / respectively. Solving the equations X 2 C 1 D 0 and X 2 C 9 D 0, we find that the points in H corresponding to the ideals .X 2 C 1/ and .X 2 C 9/ are respectively i and 3i . Likewise, the points in H corresponding to the ideals .X 2 C 14 / and .X 2 C 94 / are respectively 2i and 3i2 . But we have ji 3i j 6D j 2i  3i2 j. Now we shall see that the H-component of Spec.RŒX / is equipped with a Riemannian metric which is natural (that is, invariant by the action of Aut C .RŒX /). As already noted, it will turn out that the isometry group of this metric is much larger than the group Aut C .RŒX/. On the other hand, the symmetry group of the R-part of the spectrum (that is, the affine group) coincides with the Aut C .RŒX /. In fact, we can make the symmetries of H act transitively on the set of distinct triples of R, and this property is not true for the symmetries of R, as we noted in Remark 8.11 above. We consider the natural lines in RŒX considered as an R-vector space, that is, the lines ftp C .1  t /qg for p; q 2 RŒX and for t varying in R, and we study the corresponding lines in the spectrum. Let p; q 2 H, p 6D q. In the vector space RŒX, we consider the parametrized line t 2 R 7! w t .X/ ´ .1  t/.X 2 C b.p/X C c.p// C t .X 2 C b.q/X C c.q// D X 2 C .b.p/ C t .b.q/  b.p///X C c.p/ C t .c.q/  c.p// 2 RŒX : The ideal .w t .X// is a prime ideal of the second kind for all t such that the discriminant p;q .t/ of w t .X/ is negative. Let us compute this discriminant.   p;q .t / D b.p/ C t .b.q/  b.p///2  4.c.p/ C t .c.q/  c.p// D ..b.q/  b.p//2 t 2 C 4.c.q/  c.p//t C 2b.p/b.q/  b.p/2  4c.p/: The polynomial function p;q W t 2 R 7! p;q .t / 2 R has leading term t 2 .b.q/  b.p//2 . We distinguish two cases: Case 1. b.p/ 6D b.q/. We may assume b.p/ D 2Re.p/ < 2Re.q/ D b.q/. The coefficient of t 2 in p;q is strictly positive. Since p;q .0/ D b.p/2  4c.p/ and p;q .1/ D c.q/2  4c.q/ and since p and q represent prime ideals of the second kind, p;q .0/ and p;q .1/ are both negative. Hence the equation p;q .t / D 0 has a root  C < 0 and a root tp;q > 1. tp;q  C For t 2 tp;q ; tp;q Œ, the discriminant p;q .t / is negative, therefore .w t .X // is a prime ideal of the second kind and the polynomial w t .X / has a root in H. Let zp;q .t / 2 H be that root.

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We now consider the following path in H:  C ; tp;q Π7! zp;q .t / 2 H: t 2 tp;q

We have p D zp;q .0/ and q D zp;q .1/.  C ; tp;q Π7! zp;q .t / 2 H is the intersection Lemma 8.12. The image of the path t 2 tp;q with H of the Euclidean circle in C that passes through p and q with center on R.

Proof. Let S 2 R be the point at equal distance from p and q. Thus, we have the equation jp  Sj2 D jq  S j2 : This equation is in fact linear in S : .Re.p/  S/2 C Im.p/2 D .Re.q/  S /2 C Im.q/2 : The last equation leads to 2S.Re.p/  Re.q// D Re.p/2  Re.q/2 C Im.p/2  Im.q/2 : Now we recall that 2b.p/ D Re.p/ and c.p/ D jpj2 . From this, we obtain c.p/  c.q/  S.b.p/  b.q// D 0:  C 2 tp;q ; tp;q Œ,

For t polynomial

we have zp;q .t/ 2 H, hence, since zp;q .t / is a root of the

wp;q .t/ WD X 2 C .b.p/ C t .b.q/  b.p///X C c.p/ C t .c.q/  c.p//; we have jzp;q .t /  Sj2 D .zp;q .t/  S/.Nzp;q .t/  S / D S 2  2S.zp;q .t/ C zNp;q .t // C zp;q .t /zNp;q .t / D S 2  S.b.p/ C t .b.q/  b.p/// C c.p/ C t .c.q/  c.p// D S 2 C Sb.p/t.Sb.q/  b.p// C c.q/  c.p// D S 2  Sb.p/ C c.p/ D jp  Sj2 :  C Thus, the Euclidean distance jpSj does not depend on t . Hence, for t 2 tp;q ; tp;q Œ, the point zp;q .t/ is on the circle of center S and radius jp  Sj D jq  Sj.

Remark 8.13. The lines considered in the preceding construction can be used to define the Hilbert metric on the convex set in R2 defined by the parabola of equation b 2  4c D 0 in the space of polynomials of the form X 2 C bX C c. This would give a projective model of the hyperbolic plane, defined algebraically using the polynomial ring RŒX .

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We now define a natural distance on H. For p and q in H, we define the number ˇ

Dist.p; q/ WD

ˇ

 C /.tp;q  0/ ˇˇ .1  tp;q 1 ˇˇ log ˇ ˇ: C  /.t 2 .0  tp;q p;q  1/

(8.2)

Observe that log

 C /.tp;q  0/ .1  tp;q  /.t C  1/ .0  tp;q p;q

D  log

 C /.tp;q  0/ .1  tq;p  /.t C  1/ .0  tp;q q;p

and that the absolute value in Formula (8.2) for Dist.p; q/ makes this function symmetric in p, q. Case 2. b.p/ D b.q/; that is, p and q have the same real part. We may assume that c.p/ < c.q/. We consider again the line t 2 R 7! w t .X/ D .1  t/.X 2 C b.p/X C c.p// C t .X 2 C b.q/X C c.q// 2 RŒX  which becomes in this case w t .X/ D X 2 C b.p/X C c.p/ C t .c.q/  c.p//: Now p;q .t/ D b.p/2  4.c.p/ C t .c.q/  c.p///:  C Again, we have p;q .t/ < 0 for t 2 tp;q ; tp;q Œ with  D tp;q

b.p/2  4c.p/ 0

jhj 1 Dist.p; p C sh/ D : s Im.p/

Proof. Let us set p D x C iy, h D u C iv. For p 2 H, q WD p C sh, we consider the line t 7! w t .X/ in RŒX defined above. We have w t .X / D .1  t/.X  x  iy/.X  x C iy/ C t .X  x  su  i.y C vs//.X  x  su C i.y C vs// D X 2 C .2t us  2x/X C 2tyvs C 2txsu C x 2 C y 2 C t v 2 s 2 C t s 2 u2 : We assume that we are in Case 1, i.e. u 6D 0. We compute p;pCsh .t/=4 D t 2 u2 s 2 C .s 2 u2  2yvs  v 2 s 2 /t  y 2 : C  From this we get the (lengthy) formulae for tp;pCsh and tp;pCsh and we obtain, after simplifications and computations: p jhj Dist.p; p C sh/ u2 C v 2 D D : lim s!0;s>0 s y Im.p/

In Case 2 we have u D 0.     1 y2 y2 1  ln 1 C ln 2 2ysv C s 2 v 2 2 2ysv C s 2 v 2   2 2 1 2ysv C s v D log C1 : 2 y2

Dist.p; p C sh/ D

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Hence we have, as in Case 1, lim

s!0;s>0

v jhj Dist.p; p C sh/ D D : s y Im.p/

For p 2 H fixed, consider the function Distp W q 2 H 7! Dist.p; q/2 2 R: Proposition 8.18. The function Distp W H 7! R is twice differentiable, and it is a Morse function with exactly one critical point, which is at p. The Hessian Bp of Distp at p is positive definite. The map p 2 H 7! Bp is a Riemannian metric on H. Moreover, at any point p 2 H, identifying two tangent vectors at p with two complex numbers u; v 2 C, we can write the scalar product at the tangent space at p induced by this Riemannian metric as Bp .u; v/ D

Re.uv/ N : Im.p/2

Proof. The function Distp is at q 2 H, q 6D p, infinitely differentiable. The differential .DDistp /q at q 2 H, q 6D p is nonzero. In fact, .DDistp /q .h/ 6D 0, where h is the Re.uv/ N speed vector zPp;q .1/. The point p 2 H is critical with Hessian Bp .u; v/ D Im.p/ 2, since this is the scalar product that is associated to the quadratic form provided by Proposition 8.17. We now use the infinitesimal form of the metric to show that the function .p; q/ 2 H  H 7! Dist.p; q/ 2 R is a distance on H. The method is the usual one, in which one recovers a distance function from a norm defined infinitesimally, that is, by a family of norms on tangent spaces (in the present situation, we have a Riemannian metric). We proceed in several steps. Using the notation of differential forms, the infinitesimal distance Distpassociated dx 2 Cdy 2

. to the Riemannian metric B at a point x Ciy 2 H is also usually written as y Each of the above formulae for the infinitesimal metric shows that the norm of a tangent vector at a point p 2 H does not depend on the real part of x but only on the imaginary part y. This implies that the Euclidean translations that preserve H, that is, the translations whose vector is parallel to the x-axis, preserve this infinitesimal metric. Likewise, Euclidean reflections with respect to lines that are perpendicular to the x-axis also preserve the infinitesimal line element.  C We note that the parametrized curves t 2 tp;q ; tp;q Π7! zp;q .t / 2 H, p; q 2 H, p 6D q, are not geodesics. For instance, if we take p D i , q D 2i , then the map ; C1Π7! zp;q .t/ D i C 3ti 2 H. It is not a geodesic, because of its is t 2 1 3 parametrization. The curve s 2 R 7! .s/ D i e s 2 H has the same image, and it is a parametrized geodesic. This is because the image .R/ lies on a geodesic line, as we already noticed that the intersection with the upper half-plane of vertical Euclidean

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lines are hyperbolic geodesic lines, and furthermore the parametrization of the arc  R et is by arclength, since for all s  t 2 R, the length of  jŒs;t  is equal to es dy D t  s. y An easy computation also shows that a homothety centered at the origin also preserves this infinitesimal metric. Indeed, if .x C iy/ 7! .u C iv/ D .x C iy/, with

> 0, is such a homothety, then we have p p p

du2 C dv 2

du2 C dv 2 dx 2 C dy 2 D D ; v v y which shows that the norms of tangent vectors are preserved. By composing the above homothety with translations directed by vectors parallel to the x-axis, we obtain the following: Proposition 8.19. Any homothety centered at any point on the x-axis preserves the infinitesimal metric Dist. Inversive geometry now enters into the story. An inversion is a transformation of the Euclidean plane that is associated to a circle or to a line. It is defined as follows. • Given a circle C of center and radius k, the inversion with respect to C is a map that assigns to every point P in the plane the point P 0 situated on the ray P satisfying j P 0 j  j P j D k 2 . • The inversion with respect to a Euclidean line in the plane is the Euclidean symmetry with respect to that line. (In inversive geometry, one can consider Euclidean lines as circles of infinite radii, and one can make a common definition of inversions with respect to circles and lines, but this is not important for our purpose here.) It is clear that inversions are involutive, that any involution interchanges the two sides of the circle (or line) to which it is associated, and that each point of that circle (respectively line) is fixed by the inversion. We prove the following: Proposition 8.20. Inversions about circles centered on the x-axis and about lines perpendicular to the x-axis preserve the infinitesimal metric Dist on H. Proof. It is clear that such inversions and reflections preserve the upper half-plane H. We already noted that reflections with respect to lines perpendicular to the x-axis preserve the infinitesimal metric. To see that inversions preserve the infinitesimal metric, it suffices, by composing with translations, to consider inversions with respect to circles centered at the origin. Furthermore, by composing with a homothety centered at the origin, it suffices to consider the inversion with respect to the unit circle (that is, the circle of radius one centered at the origin). The proof of this fact can easily be done by writing the explicit

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formulae: If u C iv is the image of x C iy by this transformation, we have y x and v D 2 : uD 2 2 x Cy x C y2 This gives du D

y2  x2 2xy dx C 2 dy .x 2 C y 2 /2 .x C y 2 /2

dv D

2xy x2  y2 dx C dy: .x 2 C y 2 /2 .x 2 C y 2 /2

and

A computation gives

du2 C dv 2 dx 2 C dy 2 D : v2 y2

Thus, the inversions along circles centered at the origin preserve the infinitesimal metric. Corollary 8.21. The intersection of H with vertical Euclidean lines and with circles centered on the real axis are geodesic lines of the infinitesimal metric. Proof. The circles centered on the real axis and the vertical lines, being fixed loci of involutions that preserve the infinitesimal metric Dist, are geodesic lines of this infinitesimal metric. Proposition 8.22. Inversions with respect to circles centered on the real axis or lines perpendicular to the real axis preserve the class of circles centered on the real axis and lines perpendicular to that axis. This Proposition would be an immediate consequence of Corollary 8.21 if we had shown there that vertical Euclidean lines and with circles centered on the real axis are the only geodesic lines of the infinitesimal metric Dist. Proposition 8.22 can be proved in several ways, for instance using the fact that inversions preserve orthogonality and preserve the class of Euclidean circles and lines. (The last fact can be proved analytically, writing the equation of a circle/line as A.x 2 C y 2 / C Bx C Cy C D D 0 and checking that an inversion preserves the class of objects defined by such an equation.) The inversions along circles centered on the x-axis and along lines perpendicular to the x-axis constitute all the isometries of the infinitesimal metric Dist. The Riemannian manifold .H; Dist/ is an “inversive model” of the hyperbolic plane. In this inversive model, many constructions can be done using (Euclidean) ruler and compass. For instance, let us consider a hyperbolic line segment AB in H, contained in a Euclidean circle centered on the x-axis, and let us describe a construction of its midpoint I and of its perpendicular bisector l. We first assume that this Euclidean line

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AB is not parallel to the x-axis, and we let S be the intersection point of the Euclidean line with the x-axis. We draw the tangent Euclidean line from S to the circle arc AB and we let I be the intersection point of this tangent with this arc. From Euclidean geometry, we have jSI j2 D jSAj  jSBj (distances measured in the Euclidean metric). Furthermore, the arc AB being tangent to SI is orthogonal to the Euclidean circle C of center S and radius SI . This implies that A and B are images of each other by the inversion with respect to the Euclidean circle C , that I is the hyperbolic midpoint of A and B and that the trace l of this circle with the upper half-plane is the perpendicular bisector of the hyperbolic segment AB (see Figure 104).

I B A S Figure 104. Finding the midpoint and the perpendicular bisector of a segment in the upper half-plane model.

In the case where the Euclidean line AB is parallel to the x-axis, it is easy to see that the perpendicular bisector of AB is the intersection with the upper half-plane of the vertical line that bisects A and B into two Euclidean congruent parts. We continue studying the hyperbolic plane.   A real two-by-two matrix ac db with a; b; c; d 2 R and ad  bc > 0 acts on H by the formula az C b ; z 7! cz C d where z is a point in the upper half-plane expressed as a complex number. This action is called the action of GL.2; R/C on H by linear fractional transformations. In fact, since homothetic matrices have the same action on H, we can restrict ourselves to the group SL.2; R/ of 2  2 matrices with real coefficients and with determinant one, called the special linear group.58 We can also use this action to determine the geodesics of H. We shall show that this action is by isometries, and that it is transitive. Note that the symmetries along vertical lines and inversions along circles that are perpendicular to the x-axis, which as we know preserve the infinitesimal metric, do not belong to the SL.2; R/ group action because they reverse the plane orientation. The SL.2; R/ 58 In fact, we could also restrict to the quotient PSL.2; R/ of SL.2; R/ by the action of the group fId; Idg. However, this quotient group, called the projective special linear group, is less practical to work with because it is not really a group of 2  2 matrices.

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action gives the subgroup of orientation-preserving isometries. This is a subgroup of index two. A linear fractional transformation is a product of inversions. This is easily seen by noting that a linear fractional transformation is a product of maps of the form z 7! az C b and z 7! z 1 . The following lemmas describe some basic properties of the action of SL.2; R/ on H by linear fractional transformations.   Lemma 8.23. The action of any real two-by-two matrix ac db with ad  bc > 0 by azCb z 7! f .z/ D czCd takes bijectively the upper half-plane H to itself. Proof. A complex number z in H has positive imaginary part. We need to check that the imaginary part of f .z/ is also positive. A computation shows that the imaginary part is .ad  bc/Im.z/ ; Im.f .z// D jcz C d j2   which is positive. That f is bijective follows from the fact that the matrix ac db is invertible and has positive determinant. Lemma 8.24. The group SL.2; R/ acts on H by isometries. Proof. Given an element f of SL.2; R/, it acts on H by diffeomorphisms (in fact, it acts biholomorphically). We show that it preserves the infinitesimal metric Dist.   We consider the differential of a diffeomorphism f represented by a matrix ac db . Since f is holomorphic, its differential at a point p 2 H is multiplication by its complex derivative. This derivative is f 0 .p/ D

.cp C d /a  .ap C b/c ad  bc D : 2 .cp C d / .cp C d /2

Thus, given two tangent vectors at p represented by two complex numbers u and v, we have, from Proposition 8.18,  0  1 jf 0 .p/j2 0 Re f .p/uf .p/ v N D Re.uv/ N .Imf .p//2 .Imf .p//2 1 D Re.uv/ N Im.p/2 D Bp .u; v/

f  Bp .u; v/ D

Thus, f is an isometry. Proposition 8.25. The action of PSL.2; R/ on H is transitive. Proof. It suffices to prove that any point z D x C iy, with y > 0, is the image of the point i by a map represented by an element in PSL.2; R/. One can start by taking

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p 1= y 0



p which takes i to the point yi , and then apply the matrix the matrix 0 y which takes iy to the point x C iy.

163 1 x 0 1

Remarks 8.26. In this section, we described one way in which the hyperbolic plane H appears in algebra. As already said, there are several ways in which this space appears in mathematics. Without entering into details, we mention the following instances: 1) In the theory of moduli of complex structures: The hyperbolic plane H is the space of complex structures on R2 that are translation-invariant. More precisely, we have H D fJ W R2 ! R2 j J linear; J 2 D Id for all u 2 R2 ; .u; J u/ positively orientedg: 2) In probability theory: The upper half-plane H is the space of Gaussian distributions on the real line. Here, the real line R is seen as the boundary at infinity of H, and the Gaussian measure is defined as follows. Given a point z 2 H and an interval I  R, the measure of I is taken to be =.2/, where  is the angle between the two rays starting at z and joining this point to the boundary points of the interval I . 3) In Teichmüller theory : The hyperbolic plane is the Teichmüller space of the torus (parametrising the flat structures), and that of the once-punctured torus (parametrising the complete finite volume hyperbolic structures), each of these spaces being equipped with its classical Teichmüller metric. 4) The hyperbolic plane is the moduli space of positive real quadratic forms with determinant one. The isomorphism can be reached through matrix representation, 2 2 where  a b  a quadratic Q.x; y/ D ax C 2by C cy form is identified with the matrix : b c 5) The hyperbolic plane is the space of elliptic curves 6) The upper-half space H D fx C iy; x 2 R; y > 0g is a basic example of a symmetric space.

9 Transitional geometries 9.1 Introduction In this section, we describe a space which contains the hyperbolic plane and the spherical one, and in which we can make a continuous transition between these two planes, passing through the Euclidean plane. During this transition, we can keep track of geometrical properties of figures, that is, we can follow in a continuous manner, in terms of a real parameter t , points, pairs of points, triangles, lines, and so on. More precisely, we want a space containing the three geometries equipped with a projection onto the real numbers and such that the three geometries, hyperbolic, Euclidean and spherical, correspond respectively to the values t D 1, 0 and 1 of the parameter.

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We would like that properties of the following sort hold: (1) The distance functions between pairs of points can be followed up in a continuous manner in the three geometries. In particular, Euclidean distances, obtained for t D 0, can be seen as limits of spherical (for t > 0/ and of hyperbolic (for t < 0) distances. (2) Angle measure can be followed up in a continuous manner in the three geometries. (3) The monotonicity properties of edge lengths of quadrilaterals, in hyperbolic and spherical geometries transit through an equality formula in Euclidean geometry. (4) The trigonometric formulae for the three geometries can be globally described in a continuous manner. This is related to the fact that the functional equation .x C y/ C .x  y/ D 2.x/.y/ that arises in the proof of the Pythagorean theorem that we gave in Section 6 has three families of solutions that describes the three geometries. (5) The definition of area can be made in a coherent manner. The area of a triangle is defined in hyperbolic (respectively elliptic) geometry as angular deficit (respectively angular excess), and the passage between the two definitions is made through Heron’s formula for the area of Euclidean triangles. (6) The constructions made for scissors equivalence in the three geometries can be described globally in a coherent manner. Although we shall not dwell about this fact here, let us also note that there is a 3dimensional analogue of this theory, in which we can vary tetrahedra, from Euclidean to spherical, and study Schläfli-type variation formulae for their volume, transiting from the Schläfli formulae for hyperbolic and for spherical geometry through Cayley’s determinant formulae for the Euclidean ones. Let us say a few more words on this space of transitional geometries, which we call the “transitional model”. The description of the geometries and of the passage between them is given in terms of transformation groups. Points, lines and the other notions of geometry are described in terms of groups. More precisely, the transitional model is based on the fact that each of the three plane geometries is a homogeneous space in the sense of Lie group theory; that is, it is a space of cosets G=H0 where G is a Lie group (which for each value of the parameter t will be the congruence group of the given geometry) and H0 the stabilizer group of a point. We call this model a coherent model for the three geometries for reasons that will become clear below. We consider the 3-dimensional vector space R3 , equipped with a basis, and we denote the vector coordinates by .x; y; z/. The congruence groups of the hyperbolic plane and of the sphere are the orthogonal groups of the quadratic forms .x; y; z/ 7! x 2  y 2 C z 2

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and .x; y; z/ 7! x 2 C y 2 C z 2 respectively. We can recover the hyperbolic plane (respectively the sphere) as a homogeneous space, by quotienting its congruence group by the subgroup of automorphisms that fix a point. Introducing a nonzero real parameter t does not make a difference at the level of the axioms of the geometries (although it affects the curvature). For any t < 0, the congruence group of hyperbolic geometry is the orthogonal group of the quadratic form .x; y; z/ 7! tx 2 C ty 2 C z 2 and for any t > 0, the congruence groups of spherical geometry is the orthogonal group of the quadratic form .x; y; z/ 7! tx 2 C ty 2 C z 2 : But the spaces, equipped with their metrics, are different. For t < 0, we can consider the orthogonal group of the quadratic form tx 2 C ty 2 C z 2 as the congruence group of the hyperbolic plane of constant curvature 1=t 2 , and for t > 0, we can consider the orthogonal group of the quadratic form tx 2 C ty 2 C z 2 as the congruence group of the sphere of constant curvature 1=t 2 . It is tempting to try to include the Euclidean plane in this picture, by considering the real number t as varying between 1 and C1 and the Euclidean plane as sitting between the family hyperbolic planes and spheres, corresponding to the value 0 of the parameter t . (In fact, taking t in Œ1; 1 will suffice for our purpose.) Surprisingly, it turns out that the hyperbolic and the spherical planes are more easily described than the Euclidean one. The Euclidean plane will look like the mysterious object which is hidden in between the two others. For instance, making t D 0 in the above picture does not give the Euclidean congruence group, but rather the orthogonal group of the quadratic form .x; y; z/ 7! z 2 ; which is the group of matrices that preserve the hyperplane z D 0 in R3 , and which therefore is larger than the Euclidean plane isometry group. We shall introduce a device, the notion of a “coherent element”, in such a way that the Euclidean plane automorphism group (and the Euclidean plane itself) appears in a continuous way between hyperbolic and spherical geometries. This will give a global transition picture between the three geometries.

9.2 A coherent model for the three geometries We consider the vector space R3 equipped with a basis, and we denote the coordinates of a point p in this space by .x.p/; y.p/; z.p//.

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We then consider on R3 the family .q t / t 2Œ1;1 of quadratic forms defined in coordinates by q t .p/ D tx.p/2 C ty.p/2 C z.p/2 ; p 2 R2 : For any t 2 Œ0; 1, the matrix of q t is 0 1 t 0 0 @0 t 0A 0 0 1 and its associated bilinear form is  1 Q t .p; q/ D q t .p C q/  q t .p/  q t .q/ 2 D tx.p/x.q/ C tx.p/y.q/ C z.p/z.q/: For each t 2 Œ1; 1, let I t be the connected component of the identity in the orthogonal group of .R3 ; q t /. (Thus, we consider only orientation-preserving elements in this orthogonal group.) For any t > 0, I t is isomorphic to the topological group of orientation-preserving motions of the sphere. For any t < 0, I t is isomorphic to the topological group of orientation-preserving motions of the hyperbolic plane. The group I0 is the matrix group I0 D fA 2 GL.3; R/ j A3;1 D A3;2 D 0; A3;3 D 1; det.A/ > 0g: This is the group of matrices of the form 0 a b @c d 0 0

1 e fA 1

with ad  bc > 0. The fact that the determinant is positive is a consequence of the orientation-preserving assumption we made. Note that each such matrix preserves the plane fz D 1g. The group I0 is the usual matrix representation group of the orientation-preserving affine group of R2 as a semi-direct product of R2 acted upon by R2 . Thus, the group I0 is the group of matrices preserving the .x; y/-plane in R3 , and it is isomorphic to the topological group of orientation-preserving automorphisms of the real affine plane. In particular, I0 is not isomorphic to the group of orientation-preserving Euclidean motions of the plane (the dimension of I0 is larger). We shall reduce the size of I0 , by restricting the type of matrices that we consider. functions ai;j .t /, We call an element A 2 I0 coherent if there exist analytic   1  i; j  3, such that for all t 2 Œ1; 1 the matrix A t D ai;j .t / represents an element of I t , and A0 D A. In other words, we say that an element  A 2 Is , s 2 Œ1; 1 is coherent if there exists an analytic family of matrices A t D Ai;j .t / with A t 2 I t

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for all t 6D 0 and A0 D A. This analytic setting is convenient for using of the implicit function theorem. The coherent elements of I0 have interesting particular features. For instance, if A 2 I0 is coherent, then det.A/ D 1, since for t 6D 0, we have det.A/ D 1 and since the map t 7! det.A t / is continuous. Form the definition, we have the following: Proposition 9.1. Every A 2 I t with t 6D 0 is coherent. Proposition 9.2. An element A 2 I0 is coherent if and only if A is an orientationpreserving motion of the Euclidean plane. a b e  Proof. We already noticed that the matrix of A is of the form c d f with ad bc > 0 0 1

0. From the fact that the determinant of a coherent element is 1, we have the additional condition ad  bc D 1. For t 6D 0, we have q t .u/ D t 2 q1 .u/ for any vector u in the .x; y/-plane (restricting the quadratic forms to the .x; y/-plane). Thus, for any nonzero vectors u and v in the .x; y/-plane, the quotient qqtt .u/ is constant. Therefore, .v/ a b  by the coherence property, the block c d in the above matrix must be conformal with respect to the metric q1 restricted to the .x; y/-plane, which is the standard Euclidean metric. Thus, this block is the product of a linear orientation-preserving Euclidean isometry (that is, a rotation) by some factor . Since the determinant is equal to 1, we  necessarily have D 1. Thus, the block ac db corresponds to a Euclidean rotation. The entries e; f in the third column correspond to a translation. Therefore, the set of coherent elements of I0 coincides with the set of orientation-preserving isometries of the Euclidean plane. From now on, we denote by I0 the subgroup of coherent elements of the previously defined group I0 . In this way, for each t 2 Œ1; 1, each element of I t is coherent. Two groups Is and I t are isomorphic if and only if t and s are nonzero and have the same sign or, equivalently, if and only if t s > 0. So far, the space of all coherent elements is a space of matrices, and equipped with a map onto Œ0; 1. This is a topological subspace of the product fibration GL.3; R/Œ0; 1, but it is not a fibre bundle, since the topological type of the fibres above points of Œ0; 1, which are the groups Is , is not constant. But by construction, this space has continuous (and even analytic) sections. For t > 0 (respectively t < 0), the homogeneity of the sphere (respectively of the hyperbolic plane) implies that the sphere (respectively the hyperbolic plane) is the quotient of I t by the stabilizer subgroup of a point. Thus, we can recover the sphere (respectively the hyperbolic plane) from a group I t for t > 0 (respectively t < 0). We shall show below how to recover the Euclidean plane from the topological group I0 . For every t 2 Œ1; 1, we define a space E t . The points in E t are the maximal abelian subgroups of I t . For each p 2 E t , we shall sometimes denote by K t  I t the maximal subgroup that corresponds to p. For each p 2 E t , there exists a unique element sp 2 Kp which is strictly of order 2.

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For t 6D 0, we consider the set E t of all maximal abelian compact subgroups of I t . Each such group is isomorphic to the circle group SO.2/. (Recall that we consider only orientation-preserving elements in the orthogonal group, and this is why we get SO.2/ instead of O.2/.) Using this description, a coherent family of points becomes a coherent family of maximal abelian compact subgroups of E t depending analytically on the parameter t . We can also consider coherent families of pairs (respectively triples) and study the corresponding distance (respectively the area) function, in algebraic terms. For each t 6D 0 and for p 2 E t , considered as a subgroup of I t , there exists a unique element sp 2 I t of order two in this subgroup. We call sp the reflection, or involution in I t , of center p. In this way, any point in E t can be encoded by an involution, that is, a map of the space (the sphere or the hyperbolic plane) that fixes this point, whose square is the identity, and whose differential at the point is Id. This algebraic description of points as involutions has certain advantages. In particular, we can define composition of involutions and use this operation to describe algebraically lines and other geometric objects. In this setting, a line in E t is defined as a maximal subset L of E t such that the subgroup of I t generated by the products sp sp0 2 I t , for p; p 0 2 L, is abelian. In other words, each time we take four points (represented by involutions) sp1 , sp2 , sp3 , sp4 in E t , then sp1 sp2 commutes with sp3 sp4 . The group I t acts by conjugation on E t and by reflections along lines. Let us consider in more detail the case t > 0. We work in the projective plane (elliptic space) rather than in the sphere. In this way, as already noted, the set of points of the geometry E t , t > 0, can be considered as the set of unordered antipodal points on the sphere in the 3-dimensional Euclidean space .R3 ; q t /. To each line in E t is associated a well-defined point called its “pole”. In algebraic terms (that is, in the description where elements are replaced by involutions), the pole is the unique involution sp which, as an element of I t , fixes globally the line and is not an element of that line. Conversely, given a point N (which we think of as a pole), we can associate to it a line of which N is the pole (and we think of that line as the “equator” of N ). There are several equivalent algebraic characterizations of that line. For instance, it is the unique line L such that for any point q on L, the involution sq fixes the point N . (Note that this correspondence between poles and lines holds because we work in the elliptic plane, and not on the sphere. In the latter case, there would be two “poles”, which are exchanged by sq .) Now note that if two points p and q in E t are distinct, the product of the corresponding involutions sp and sq is a translation along the line joining these points. Concretely, sp sq is a rotation along the line which is perpendicular to the plane of the great circle determined by p and q. This line passes through the pole sN of the great circle, and therefore the product sp sq commutes with the pole sN (seen as an involution).

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Given p 2 E t , the equator of p is the set of all q 2 E t for which the path from p under q (for the action of I t on E t ) is a straight line. We shall measure lengths in the geometry E t for t > 0. We already mentioned that in such a geometry, there is a natural length unit. This length unit can be taken to be the diameter of a line (which is homeomorphic to a circle), or as the diameter of the whole space (which is compact). We can also use the correspondence between lines and poles and define a normalized distance on E t , by fixing once and for all the distance from a point to its equator. We can normalize this distance by setting it equal  to 2p . t After defining the distance on E t , we can check that the line which joins a pole to a point on its equator is orthogonal to the equator. Note that in spherical (or elliptic) geometry, a symmetry with respect to a point is at the same time a symmetry with respect to a line. This can be seen using the above description of points as involutions. From the definition, the involution corresponding to the pole of a line (the equator associated to that pole) fixes the pole, but it also fixes pointwise the equator. The pole can be characterized in this setting as being the unique isolated fixed point of its involution. The set of other fixed points is the equator. Thus, to a line in elliptic geometry is naturally associated an involution. Now we consider the case t D 0. We define a line L in E0 as a maximal subgroup satisfying the following three properties: (1) L is non-empty and proper (that is, not equal to E0 ); (2) L is connected; (3) the subgroup of I t generated by the products sp sp0 , for p; p 0 2 L is abelian. Note that the set of products sp sp0 , with p and p 0 in E0 , is precisely the set of translations of E0 . Also note that Property (2) excludes situations where L is the set of integer points of a line, or a lattice, or the set of rational points, and so on. Any two maximal abelian compact subgroups p and q in E t are conjugate, and they are all isomorphic to the circle group SO.2/. Therefore, by choosing an orientation and an angle measure on one of these subgroups and transporting it by conjugation, we obtain an orientation and an angle measure on E t . After normalization we obtain a distance. Our next goal is to formulate for t 2 Œ0; 1 a coherent law which will be a “Pythagorean theorem” in positive curvature and which for t D 0 corresponds to the familiar Euclidean Pythagorean theorem. We first discuss coherent families of distances and of triangles. We want the distance between two points in the geometry E0 to be the limit (after normalization) of the distance in E t between corresponding points, for t > 0 and for t < 0. To be more precise, for t 2 Œ1; 1, let K t be the stabilizer subgroup of the point a D .0; 0; 1/ 2 R3 under the action of the group I t . The group K t is a maximal abelian

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subgroup of I t . There is a coherent family of points A t 2 E t ; t 2 Œ1; 1 represented by a. For all x 2 R, let B tx 2 E t , t 2 Œ0; 1 and C tx 2 E t , t 2 Œ0; 1, be the coherent families of points represented by the stabilizers I t of the vectors b x D .x; 0; 1/ and c x D .0; x; 1/ of R3 . The family  t .x; y/ D .A t ; B tx ; C ty /, t 2 Œ0; 1 is a coherent family of triangles. The triangle 0 .x; y/ is a right triangle with catheti ratio x=y. For this family of points A t , B tx and C tx , the distance in E0 between A0 and B0x x is equal to the limit as t ! 0; p t > 0, of the distance from A t to B t in E t (t > 0), normalized by dividing it by t . We have q t .a/ D 1; q t .b x / D q t .c x / D tx 2 C 1 and Q t .a; b x / D Q t .a; c x / D 1: Therefore the angular distance from A t to B tx , measured with q t , t > 0, is 



Q t .a; b x / D arccos p q t .a/q t .b x /   1 D arccos p tx 2 C 1 p D j arctan. tx/j ˇp ˇ 1p 3 3 1p 5 5 ˇ ˇ D ˇ tx  t x C t x  ˇ 3 5 and therefore, after our choice of a unit length for the distance on E t , the distance between the two points is   ˇ ˇ 1 1 1 1 ˇ ˇ x D ˇx  x 3 C t 2 x 5     ˇ: D t .A t ; B t / D p arccos p 2 3 5 t tx C 1 w t .A t ; B tx /

The limit as t ! 0, t > 0, in D t .A t ; B tx / gives simply jxj. Therefore, in the geometry E0 , the distance from A0 to B0x is equal to jxj. A similar computation gives that the distance in E0 from A0 to C0y is equal to jyj. We shall also need to know the distance in E0 between the points B0x and C0y . The angular distance from B tx to C ty , measured with q t , t > 0, is 

w t .B tx ; C ty / D arccos p

Q t .b x ; c y /



q t .b x /; q t .c y /  1 D arccos p p tx 2 C 1 ty 2 C 1   1 D arccos p t 2 x 4 y 4 C tx 2 C ty 2 C 1 

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and up to higher order terms, this expression is equal to p t 2 x 4 y 4 C tx 2 C ty 2 : p After our normalization (division by t ), the same computation as above gives that the distance between the two points is given by p D t .B tx ; C ty / D x 2 C y 2 : We collect this information in the following Lemma 9.3. In the coherent geometry E0 , we have D0 .A0 ; B0x / D jxj; D0 .A0 ; C0y / D jyj and

D0 .B0x ; C0y / D

p

x2 C y2:

We use this lemma to show that we obtain the Euclidean Pythagorean formula in E0 as a limit of the Pythagorean formula in E t for t > 0. in the triangle  t .x; y/, the three angles are p For fixed xx; yp 2 R and ty > 0, p t D .A ; B /, t D .A ; C / and t D t .Cy t ; B tx /. (Note that we multiplied by t t t t t t p t to recover the angle, since in our normalization of the distance we divided by this quantity.) Therefore, we have the following law (Pythagorean theorem in spherical geometry): p p 1 cos. t D t .A t ; B tx // cos. t D t .A t ; C ty // D p p 2 tx C 1 ty 2 C 1 p D cos. t D t .C ty ; B tx //: Taking the limit, as t ! 0, we obtain the well-known formula 1  1 D 1: We can obtain a more useful result by taking another limit. We transform the spherical Pythagorean theorem into the following one: p p p  t 1  cos. t D t .A t ; B tx // cos. t D t .A t ; C tx // p p  D t 1  cos. t D t .C ty ; B tx // : We use the definition of the distance in E0 . For this, we write this equation at the first order and we take the limit as t ! 0, with t > 0. This provides, after normalization, D0 .A0 ; B0x /2 C D0 .A0 ; C0y /2 D D0 .B0x ; C0y /2 ; which is the Pythagorean theorem in the geometry E0 , and this indeed is the familiar Pythagorean law in Euclidean geometry.

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Now we analyze what happens for t < 0. The stabilizer of .x; 0; 1/ in I t , t 2 Œ1; 0, is abelian, and therefore it is also compact as soon as tx 2 C 1 > 0, that is, if jt j < x12 . To see this, we compute the norm of the vector .x; 0; 1/ with respect to the quadratic form on E t . This norm must be < 1, in order for the point to be inside the disk, so that its stabilizer is compact. (Recall that the stabilizer of a point in the interior of the hyperbolic plane is compact, whereas the stabilizer of a point on the boundary is not compact (it is a parabolic subgroup) and the stabilizer of a point outside the unit disk is also non-compact.) Thus, we have as before a coherent family of triangles  t .x; y/ D .A t ; B tx ; C ty / with t 2 Œ1; 1 if tx 2 C 1 > 0 and ty 2 C 1 > 0. We can therefore take t 2 Œ 2x1 2 ; 0Œ \Œ 2y1 2 ; 0Œ. Then a miracle happens. The equation 

1 1 D t .A t ; B tx / D p arccos p t tx 2 C 1



gives for t 2 Œ 2x1 2 ; 0Œ a positive real number, which we interpret as a distance in the geometry E t between the points A t and B t . Let us look at this in more detail. Since p 12 > 1, arccos. p 12 / is pure tx C1 tx C1 p and D t .A t ; B tx / is real and positive, if we take imaginary, t is also pure imaginary, p the appropriate branches of t and arccos. p 12 /. We could have taken, instead of tx C1 p arccos. p 12 /, the expression arccosh . p 12 / and instead of t , the expression tx C1 tx C1 p t . Recall that cos.iv/ D cosh.v/ D 12 .e v C e v / and that p arccosh W u 2 Œ1; C1 7! v D log.u C u2  1/ 2 Œ0; C1 is the inverse function of u D cosh.v/. We obtain, for the geometry E t , t < 0, a Pythagorean theorem: For fixed x; y 2 R with tx 2 C 1 > 0, ty 2 C 1 > 0 and 1  t < 0, in the triangle  t .x; y/, we have p p 1 cosh. t D t .A t ; B tx // cosh. t D t .A t ; C ty // D p p tx 2 C 1 ty 2 C 1 p D cosh. t D t .C ty ; B tx //: To summarize: In a right triangle with catheti x, y and hypotenuse z, we have, in the hyperbolic geometry E1 , the relation cosh z D cosh x cosh y; in the Euclidean geometry E0 the relation z2 D x2 C y2;

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and in the spherical geometry E1 the relation cos z D cos x cos y: If the edges of lengths x, y bound a sector of angle  , then applying the relation of the geometries E1 , E0 , E1 , we obtain E1 W cosh z D cosh x cosh y  sinh x sinh y cos ; E0 W z 2 D x 2 C y 2  2xy cos ; E1 W cos z D cos x cos y  sin x sin y cos : Since angles are in Œ0; , and the function cos is injective on Œ0; , it follows that in the geometries E1 , E0 , E1 the three edges of a triangle determine the three angles. We already noted that, conversely, in the geometries E1 , E1 , the three angles determine the edge lengths. Remark 9.4. Regarding transitional geometries, there has been a recent activity in dimension 3, on moving continuously between the eight Thurston geometries, and also on varying continuously between Riemannian and Lorentzian geometries on orbifolds. We mention in this respect the works of Porti [75] and [76] and Porti & Weiss [77] and Cooper, Hodgson & Kerckhoff [20]. We also mention work on combinatorial transitions, by Kerckhoff and Storm [47].

References [1]

N. A’Campo, Géométrie hyperbolique. Handwritten notes by Dupeyrat (Orsay 1978), 50 p., available at the libraries of Orsay, Strasbourg and some other French universities.

[2]

D. V. Alekseevskij, E. B. Vinberg and A. S. Solodovnikov, Geometry of spaces of constant curvature. In Geometry. II: Spaces of constant curvature, Encyclopaedia of Mathematical Sciences 29, Springer-Verlag, Berlin, 1993, 1–138. Russian original published by Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Fundamental’nye Napravleniya 29 (1988), 5–146.

[3]

Aristotle, Metaphysics. Vol. I. Translation by H. Tredennick, Loeb Classical Library 271, Harvard University Press, Cambridge, Mass., first printed 1933, several later editions.

[4]

Aristotle, Posterior analytics. Translation by H. Tredennick and E. S. Forster, Loeb Classical Library 391, Harvard University Press, Cambridge, Mass., first printed 1960, several later editions.

[5]

P. Barbarin, Etudes de géométrie analytique non Euclidienne. Mémoires Acad. Royale de Belgique 60 (1901), 1–168.

[6]

P. Barbarin, La géométrie non euclidienne (suivie de notes sur la géométrie non euclidienne dans ses rapports avec la physique mathématique par A. Buhl). Third edition, Gauthier-Villars, Paris 1928 (first edition 1902, second edition 1907).

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[7]

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N. I. Lobachevsky, New elements of geometry, with a complete theory of parallels (Russian). Uchenye Zapiski Kazanskogo Imperatorskogo Universiteta (1835), no. 3, 3– 48; (1836), no. 2, 3–98, (1836), no. 3, 3–50; (1837), no. 1, 3–97; (1838), no. 1, 3–124, (1838), no. 3, 3–65. Publlished in one volume by the Kharkov Mathematical Library, Kharkov, 1912, with a biography and notes by D. M. Sintsov. Complete works [62], Vol. II, 147–454, with an introduction and a commentary by B. L. Laptev, A. P. Norden and A. N. Khovansky. German translation of Chapters I to XI by F. Engel in [28], 67– 236. French translation by F. Mailleux, Nouveaux principes de le géométrie avec une théorie complète des parallèles. Mémoires de la Société Royale des Sciences de Liège, 3e Série, tome 2, no. 5, 101; 3–101, tome 3, no. 2, 1–32 (1899). English translation of the introduction by G. B. Halsted in [42]. An extract is contained in the Selected works [63], Greek–Russian edition, Greek translation by K. Philippides and K. Philippidou, 232–250 even pages (Greek) and 233–251 odd pages (Russian).

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N. I. Lobachevsky, Imaginary geometry (Russian) Uchenye Zapiski Kazanskogo Imperatorskogo Universiteta (1835), no. 1, 3–88. Reprinted in Lobachevsky’s Collected geometric works [61], Vol. 1, 71–120, and in Lobachevsky’s Complete works [62], Vol. III, 16–70, with an introduction and a commentary byA. P. Norden andA. N. Khovansky. German translation by H. Liebmann, Imaginäre Geometrie, Abhandlungen zur Geschichte der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, Vol. 19, 1–50, Leipzig, Druck und Verlag von B. G. Teubner, 1904. An extract (Part I) is contained in the Selected works [63], Greek–Russian edition, Greek translation by K. Philippides and K. Philippidou, 208–230 even pages (Greek) and 209–231 odd pages (Russian).

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N. I. Lobachevsky, Pangeometry (Russian). Uchenye Zapiski Kazanskogo Imperatorskogo Universiteta I (1855), 1–56. Reproduced in Lobachevsky’s Collected geometrical works [61], Vol. I, 489–550, and in his Complete works [62], Vol. III, 435–524, with an introduction and commentary by V. F. Kagan.

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A. Tarski, What is elementary geometry? In The axiomatic method, L. Henkin, P. Suppes, and A. Tarski, eds., with special reference to geometry and physics, Stud. Logic Found. Math., North Holland, Amsterdam 1959.

[99]

J.-M. de Tilly, Etudes de mécanique abstraite. Mémoires couronnée et autres textes autres mémoires Académie Royale de belgique XXI (1870).

[100] J.-M. de Tilly, Essai sur les principes fondamentaux de la géométrie et de la mécanique. Mémoires de la Société des Sciences Physiques et Naturelles de Bordeaux Série 2, III (1879), 1–190. [101] J.-M. de Tilly, Essai de géométrie analytique générale. Mémoires de l’Académie Royale de Belgique 47, Bruxelles 1892. [102] W. P. Thurston, The geometry and topology of three-manifolds. Mimeographed notes, Princeton University, Princeton, NJ, 1976; available on the web at http://www.msri.org/ publications/books/gt3m/ [103] W. P. Thurston, Three-dimensional geometry and topology. Volume 1, Princeton University Press, Princeton, N.J., 1997. [104] A. Turc, Introduction élémentaire à la géométrie lobatschewskienne. Kündig, Genève 1914. Reprinted by Editions P. Blanchard, Paris 1967. [105] E. B. Vinberg, Volumes of non-Euclidean polyhedra. Russian Math. Surv. 48 (1993), no. 2, 15–45; translation from Uspekhi Mat. Nauk 48 (1993), no. 2 (290), 17–46. [106] J. Wallis, Demostratio Postulati Quinti Euclidis, in Operum Mathematicorum, t. II, Oxford, 1693. [107] S. Walter, La vérité en géométrie : sur le rejet de la doctrine conventionnaliste. Philosophia Scientiae 2 (1997), 103–135.

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[108] H. Weyl, Raum, Zeit, Materie, 8. Auflage, Julius Springer, Berlin, first edition 1918, several later revised editions. English translation: Space, Time, Matter, reprinted by Dover, 1952 and later editions. [109] H. Weyl, David Hilbert and his mathematical work. Bull. Am. Math. Soc. 50 (1944), 612–654. [110] I. M. Yaglom, Felix Klein and Sophus Lie. Evolution of the idea of symmetry in the nineteenth century. Translated from the Russian by S. Sossinsky, edited by H. Grant and A. Shenitzer, Birkhäuser, Boston, Mass., 1988. [111] A. P. Youschkevitch, Les mathématiques arabes (VIIIe –XVe siecles). Translated from the Russian by M. Cazenave and K. Jaouiche, Collection L’Histoire des Sciences, Textes et Études, Librairie Philosophique J. Vrin, Paris 1976.

Crossroads between hyperbolic geometry and number theory Françoise Dal’Bo IRMAR, Campus de Beaulieu, Université Rennes 1 35042 Rennes Cedex, France email: [email protected]

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Geometry on the Poincaré half-plane . . . . . . . . . . . . . . . 1.1 Basic concepts in Riemannian geometry . . . . . . . . . . . 1.2 Poincaré half-plane and Möbius transformations . . . . . . 1.3 Area, length and distance . . . . . . . . . . . . . . . . . . . 1.4 Circles, neighborhoods and perpendicular bisectors . . . . . 1.5 Boundary at infinity and projective action . . . . . . . . . . 1.6 Classification of elements of G . . . . . . . . . . . . . . . 2 Vector approach to horocycles . . . . . . . . . . . . . . . . . . . 2.1 Horocycles . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Vector approach . . . . . . . . . . . . . . . . . . . . . . . 3 Actions of the modular group . . . . . . . . . . . . . . . . . . . 3.1 The action of  on H.1/ . . . . . . . . . . . . . . . . . . 3.2 A tiling of H . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Hyperbolic characterization of rational numbers . . . . . . 3.4 An application to the linear action of SL.2; Z/ . . . . . . . 4 Topology of the horocyclic trajectories . . . . . . . . . . . . . . 4.1 Basic concepts in topological dynamics . . . . . . . . . . . 4.2 Dynamics of U on SL.2; Z/n SL.2; R/ . . . . . . . . . . . 4.3 Dynamics of the horocyclic flow . . . . . . . . . . . . . . . 5 Bounded geodesic trajectories . . . . . . . . . . . . . . . . . . . 5.1 Introduction to the geodesic flow . . . . . . . . . . . . . . 5.2 Characterization of some trajectories . . . . . . . . . . . . 5.3 Bounded non-periodic trajectories . . . . . . . . . . . . . . 6 Geodesic trajectories and Diophantine approximation . . . . . . . 6.1 Classical results in number theory . . . . . . . . . . . . . . 6.2 Dynamical characterization of badly approximated numbers 6.3 Application to small values of binary quadratic forms . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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184 187 187 189 191 194 195 196 197 197 200 202 203 204 206 208 211 211 214 215 218 219 220 223 223 224 225 227 230

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Introduction Number theory and geometry are closely related since a long time. In the first part of the 20th century, E. Artin [1] introduced a fundamental new idea of linking the topology of geodesics in the hyperbolic modular surface SL.2; R/= SL.2; Z/ to continued fraction expansion of real numbers. Sixty years later, G. Margulis used this geometrical approach to solve the following long-standing conjecture, known as the Oppenheim–Davenport conjecture, [22], [23], [5] Theorem 0.1. If Q is an irrational, non-degenerate and indefinite quadratic form on Rn with n > 2, then Q.Zn / is dense in R. The proof of Margulis ([6], [9], [20]) consists in connecting this conjecture to the dynamics of some group action on SL.3; R/= SL.3; Z/. This stream of ideas has continuously stimulated research up to the present times [3], [21]. The motivation of our text is to explain some of these links between number theory and dynamics in an elementary context. Our goal is not to prove new results but to highlight the existence of gateways between different mathematical fields and the advantage of using them. Many results can be extended to the general context where the modular group PSL.2; Z/ is replaced by a non-elementary discrete subgroup of finite type of PSL.2; R/. Patterson’s thesis [24] is one of the first important papers in this area. In our text, we chose proofs which can be adapted in this more general context. Such proofs are not always the simplest ones for SL.2; Z/. We begin with an introduction to the geometry of the Poincaré half-plane H. In Section 2, we construct a dictionary between the horocycles of H and the space E D R2  f0g=f˙ Idg. Using this dictionary, we translate the linear action of the G D PSL.2; R/ on E in terms of hyperbolic geometry. In Section 3, we focus on the modular  D PSL.2; Z/  G and prove the following theorem: Theorem 0.2. Let v 2 R2 . If v is not collinear to a vector in Q2 , then SL.2; Z/.v/ is dense in R2 . A similar result holds when SL.2; Z/ is replaced by an arbitrary non-elementary discrete subgroup H  G of finite type. Such group acts on the projective line and admits a minimal non-empty closed H -invariant subset L.H /. In this setting, the condition that v is not collinear to a vector in Q2 has to be replaced by the fact that v ¤ 0 and 0 2 H.v/. Under this assumption, the conclusion is that H.v/ is the set of all vectors whose direction belongs to L.H / [7]. Our proof of Theorem 0.2 is purely geometric in the sense that it is based on the hyperbolic interpretation of the linear action introduced in Section 2 and on a tiling of H by . As applications of Theorem 0.2, we describe in Section 4 the dynamics of the horocyclic flow hR on the quotient by  of the unit tangent bundle T 1 H of H, and we prove Hedlund’s Theorem [13]:

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Theorem 0.3. An orbit of hR on =T 1 H is either dense or periodic. In the general case where  is replaced by a non-elementary discrete subgroup H  G of finite type, the behavior of the horocyclic flow on H=T 1 H is also regular, in the sense that the orbits of hR restricted to its non-wandering set are dense or periodic [7]. In Section 5, we show that this regularity disappears when we replace the horocyclic flow hR by the geodesic flow gR . Theorem 0.4. There exist bounded orbits for the geodesic flow on =T 1 H which are not periodic. The idea of a continued fraction expansion of a real number x can be traced back to the 17th century. It it is defined ([18]) by 1

Œn0 I n1 ; : : : ; nk  D n0 C

;

1

n1 C

n2 C : : :

C

1 nk1 C

1 nk

where the sequence .ni /i>0 is given by the following algorithm: x0 D x, n0 D E.x0 /, where E.x/ denotes the integral part of x, and for any i > 1, xi D 1=.xi1  ni1 /

ni D E.xi /:

and

The sequence .Œn0 I n1 ; : : : ; nk / sequence converges to x. One branch of the theory of Diophantine approximations consists of constructing a one-to-one correspondence between some algebraic properties of an irrational number and those of the sequence of integers .ni /i>0 associated with its continued fraction expansion. For example, it is know that quadratic real numbers correspond to almost periodic sequences ([18], [7]). Another branch of the same theory focuses on the speed of convergence of the sequence of rationals associated with a continued fraction expansion. For example, one of the problems considered is to find the “best” (in the sense of asymptotic behavior) function ‰ W N ! RC decreasing to 0 such that for all x 2 R  Q, there exists a sequence of rationals .pn =qn /n>1 satisfying jx  pn =qn j 6

.jqn j/

and

lim jqn j D C1:

n!C1

Using continued fraction expansion, G. Lejeune-Dirichlet proved that for any irrational number x, there exists a sequence .pn =qn /n>0 of rational numbers with qn > 0 and limn!C1 qn D C1 such that ˇ ˇ ˇ p ˇ 1 ˇx  n ˇ 6 : ˇ ˇ qn 2qn2

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Can one find a function ‰.n/ converging to zero faster than 1=n2 ? The answer is “No”. For example, one can check that for all p 2 Z and q in N , the following inequality is satisfied: p j 2  p=qj > 1=.4q 2 /: The function ‰ that we are looking for therefore satisfies 1=4 6 n2 ‰.n/ 6 1=2: This naturally leads us to define for each irrational number x the quantity .x/ D inffc > 0 j there exists .pn =qn /n>1 2 Q such that jx  pn =qn j 6 c=qn2 and

lim jqn j D C1g:

n!C1

Clearly, this quantity is less than 1=2 for all x. More precisely, for every irrational x, one has ([18]) p .x/ 6 1= 5: p Furthermore .x/ D 1= 5 if and only if there exist a, b, c, d in Z such that ac  bd D 1 and

xD

aN C b ; cN C d

p where N D .1 C 5/=2 is the golden ratio. Among the irrational numbers, the badly approximated real numbers x for which p .x/ is strictly positive are of special interest. For example, this is the case for 2. Following the path of Artin, we give in Section 6 examples of some relations between the topology of the projection of a geodesic ray Œz; x/ with z 2 H and x 2 R, on the modular surface S D =H and the continued fraction expansion of the real number x. Theorem 0.5. Let x 2 R=Q and z 2 H. The projection on S of the geodesic ray Œz; x/ is bounded if and only if x is badly approximated. This geometric approach to Diophantine approximations allows one to rediscover classical theorems, like the Lejeune-Dirichlet Theorem [7], but also to generalize them [24]. The theory of Diophantine approximations is closely related to the study of quadratic forms. For example, a real number x is badly approximated if and only if 0 belongs to Qx .Z2 /  f0g; where Qx .X; Y / D XY  xY 2 . In the second part of Section 5, we associate to a quadratic form Q an element uQ of the unit tangent bundle of H and connect the orbit by the geodesic flow gR of its projection  1 .uQ / into =T 1 H with the set Q.Z2 /. Theorem 0.6. If Q.X; Y / D .aX C bY /.cX C d Y / is a non-semirational, nondegenerate and indefinite quadratic form, then Q.Z2 /  f0g contains 0 if and only if the trajectory gR . 1 .uQ // is not bounded.

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Other relations between Q and uQ are obtained in [10], [5] and [26]. Theorem 0.6 allows us to understand why the hypothesis n > 2 is necessary in Theorem 6.12. More precisely, at the end of Section 6, we deduce from Theorems 0.4 and 0.6 the following corollary. Corollary 0.7. There exist non-degenerate and indefinite non-semirational and irrational quadratic forms Q such that Q.Z2 /  f0g ¤ R. One of the open problems in number theory which presently motivates many mathematicians is the Hardy–Littlewood conjecture. This conjecture states that for any pair .x; y/ in R2 , there exist sequences of integers .qn /n>0 , .pn /n>0 and .rn /n>0 with qn > 0 such that    pn rn y D 0: lim qn3 x  n!C1 qn qn This conjecture is directly related to the orbits of the diagonal subgroup of SL.3; R/ acting on SL.3; R/= SL.3; Z/ (for an introduction to this conjecture, see the text of M. Queffelec in [3], and also [25]). These notes grew out of a course which I gave in Strasbourg during the “Semaine spéciale de géométrie et théorie des groupes”, organized in May 2008 by Athanase Papadopoulos. I am indebted to Athanase for the organization of this special week and for reading in detail this text. I thank the referee for useful comments.

1 Geometry on the Poincaré half-plane In this section, we recall some basic properties of planar hyperbolic geometry. A reader who is not familiar with this geometry will find more details in [16], [17], [4]. We begin with a short introduction to Riemannian geometry.

1.1 Basic concepts in Riemannian geometry This section outlines some results proved in [12]. Let M be a connected smooth manifold. A Riemannian metric on M is a family of scalar products .gm /m2M defined on each tangent space Tm M and depending smoothly on m. For example, the Euclidean space Rn is canonically equipped with a Riemannian structure .gm /m2M , where gm is the ambiant scalar product. More generally, if M is a submanifold of Rn , then the restriction of gm to each tangent space Tm M induces a Riemannian metric on M . This is the case for example for the torus T 2 viewed as a revolution surface in R3 induced by the map W R2 ! R3 defined by .; / D ..2 C cos / cos ; .2 C cos  / sin ; sin  /:

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Given a Riemannian metric .gm /m2M on M , we are led to define a canonical measure vg on M . More precisely, let .Uk ; k / be a chart and consider the local expression of the metric in this chart: X gijk dxi dxj : 16i;j 6Dim.M /

q The volume of the parallelotope generated by the vectors @=@xi is det .gijk /. We define the measure vg as corresponding to the density which is given in the atlas .Uk ; k / by q det .gijk /L;

where L is the Lebesgue measure on Rn . By definition, the volume of a subset B  M is given by Z vol.B/ D vg ; B

when this integral exists. For the torus T 2 viewed as revolution surface in R3 , the area of a subset B D .A/  T 2 associated with the induced metric is given by Z s @d @d ^ jdd; j vol.B/ D d d A where ^ denotes the vector product in R3 . The notion of length of a piecewise C 1 curve c W Œ0; a ! M is also well defined and is given by Z aq length.c/ D gc.t / .c 0 .t /; c 0 .t // dt: 0

This notion does not depend on the choice of a regular parametrization. Using the notion of length, we define a distance on M associated to the Riemannian metric .gm /m2M . The following proposition is proved in [12] (Proposition 2.91). Proposition 1.1. Let d W M  M ! RC be the map defined for m and m0 as the infimum of the lengths of all piecewise C 1 curves from m to m0 . This map is a distance on M , whose associated topology is the topology of M . In Euclidean space, straight lines are length minimizing. The curves which (locally) minimize length in a Riemannian manifold are called the geodesics. Namely we have ([12], Corollary 2.94) Definition 1.2. A curve c W I  R ! M , parametrized proportionally to the arc length, is a geodesic if and only if for any t 2 I there exists " > 0 such that d.c.t /; c.t C"// D length.cjŒt;t C" /.

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For the metric on T 2 viewed as revolution surface in R3 meridian lines and parallels ( D constant) parametrized proportionally to length are geodesics (see Exercise 2.83 in [12]). A diffeomorphism f between a Riemannian manifold .M; .gm /m2M / and a smooth manifold M 0 induces a metric on M 0 defined for m0 2 M 0 and uÅ0 ; vÅ0 2 Tm0 M 0 by 0 1 Å0 Å0 Å0 .u /; Tm0 f 1 .vÅ0 //: gm 0 .u ; v / D gf 1 .m0 / .Tm0 f 0 The Riemannian manifolds .M; .gm /m2M / and .M 0 ; .gm 0 /m0 2M 0 / are isometric in the following sense. 0 Definition 1.3. Let .M; .gm /m2M / and .M 0 ; .gm 0 /m0 2M 0 / be two Riemannian mani0 folds. A map f W M ! M is an isometry (resp. local isometry) if f is a diffeomorphism (resp. local diffeomorphism), satisfying the following relation for any m 2 M and u Å; vÅ 2 Tm M : u/; Tm f .Å v // D gm .Å u; vÅ/: gf0 .m/ .Tm f .Å

When M 0 D M , the set of isometries f W M ! M is a group. Let  be a discrete group of isometries of M . We suppose that  acts on M freely (i.e. for m 2 M and g 2   fIdg, g.m/ ¤ m), and properly (i.e. for any m; m0 2 M , if m0 …  m, then there exist two neighborhoods V .m/ and V .m0 / such that gV \ V 0 D ¿, for any g 2 ). Under these conditions, there exists a unique Riemannian metric on nM such that the canonical projection of M onto nM is a smooth covering map and a local isometry ([12], Proposition 2.20). For example, if  is a group of translations associated to a basis of R2 , we obtain a Riemannian metric on the torus T 2 , which is said to be flat ([12], Exercise 2.25). For a flat Riemannian metric on T 2 , the geodesics are the projections of the straight lines of R2 parametrized proportionally to length. More generally, we have ([12], Proposition 2.81) Proposition 1.4. If  is a discrete group of isometries of .M; .gm /m2M / acting freely and properly on M , then the geodesics of nM are the projections of the geodesics of M , and the geodesics of M are the lifts of those of nM .

1.2 Poincaré half-plane and Möbius transformations We consider the Poincaré half-plane H D fz 2 C j Im z > 0g; equipped with the Riemannian metric g defined for any point z 2 H and for any two vectors u Å and vÅ in the tangent plane Tz H by u; vÅ/ D gz .Å

1 hÅ u; vÅi; .Im z/2

where h ; i is the usual scalar product on R2 .

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By definition, the unit tangent space at z is the set u 2 Tz H j gz .Å u; u Å/ D 1g: Tz1 H D fÅ

  Let us introduce the group G D PSL.2; R/ of elements ˙M D ˙ ac db with a, b, c, d in R satisfying the identity ad  bc D 1. To each ˙M 2 G we associate the Möbius transformation hM acting on the extended complex plane C [ f1g defined as follows. • If c D 0, then hM .z/ D .az C b/=d if z ¤ 1, otherwise h.1/ D 1. • If c ¤ 0, then hM .z/ D .az C b/=.cz C d / if z … f1; d=cg, otherwise h.1/ D a=c, h.d=c/ D 1. Since for any z 2 C  R we have Im z ; jcz C d j2

Im hM .z/ D

the group G acts on H. This action can be extended to the tangent bundle T H: Å/ D .hM .z/; Tz hM .Å u//: hM .z; u Making this expression explicit, one obtains Tz hM .Å u/ D

u Å ; .cz C d /2

where multiplying u Å by a complex number means applying the linear map associated to this number. Clearly, hM preserves the orientation induced on each tangent plane by R2 . Moreover hM satisfies the relation ghM .z/ .Tz hM .Å u/; Tz hM .Å v // D gz .Å u; vÅ/: It follows that the group G is included in the group of orientation-preserving isometries of H (actually both groups coincide [7]). In particular, the group G acts on the unit tangent bundle T 1 H. Proposition 1.5. Let u Å 2 Ti1 H. For any z 2 H and vÅ 2 Tz1 H, there exists a unique ˙M 2 G such that Å/ D .z; vÅ/: hM .i; u Before we prove this proposition, let us introduce the following subgroups of SL.2; R/:  ˚   U D u.t/ D 10 1t I t 2 R ; ˚  0   D D d.s/ D 0s 1=s Is>0 ; and

˚  s0 K D k.s 0 / D cos sin s 0

 sin s 0 cos s 0



 I s0 2 R :

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The actions of u D u.t/, d D d.s/ and k D k.s 0 / on T 1 H are given by Å/ D .z C t; u Å/; hu .z; u hd .z; u Å/ D .s 2 z; s 2 u Å/; and 0

hk .i; u Å/ D .i; e 2is u Å/:

p Proof of Proposition 1.5. Write z D x C iy. Consider the matrices d D d. y/ 2 D and u D u.x/ 2 U , and set h D hu B hd . We have h.i / D z and Ti h.Å u/ D yÅ u. Let s 0 be the measure of the oriented angle between u Å and vÅ, and k be the matrix of K defined by k D k.s 0 =2/. We have h B hk .i / D z and Ti h B hk .Å u/ D vÅ. This proves that the action of G on T 1 H is transitive. Now suppose that there is h 2 G satisfying h.i; u Å/ D .i; u Å/: Since h.i / D i and K is the stabilizer of i in G, there exists k D k.s 0 / 2 K such that h D hk . We have 0 hk .i; u Å/ D .i; e 2is u Å/. Moreover Ti hk .Å u/ D u Å. It follows that the real number 2s 0 is a multiple of 2, and hence that h D Id. We equip the group G with the topology induced by R4 , and H and T 1 H with the topology induced by R2 and R2 R2 . Clearly, we deduce directly from Proposition 1.5 the following result. Corollary 1.6. (i) The map f W G ! H defined by f .h/ D h.i / induces a homeomorphism between the quotient space G=K, equipped with the quotient topology, and H. (ii) The map F W G ! T 1 H defined by F .h/ D h.i; u Å/ is a homeomorphism. Moreover we deduce from the proof of Proposition 1.5 the following decomposition, called Iwasawa decomposition: Corollary 1.7. For any ˙M 2 G, there exist u 2 U , d 2 D and k 2 K such that hM D hu hd hk : One can check that u, d and k in the Iwasawa decomposition are unique.

1.3 Area, length and distance By recalling some facts from Euclidean geometry, we observe that the hyperbolic metric .gz /z2H on H allows us to define new notions of length and area on each tangent plane Tz H. Namely, the area of a parallelogrampof T 1 H is its Euclidean area divided by Im z 2 , and if u Å is in Tz H, then its length is gz .Å u; u Å/. These notions give rise to the following global definitions: the hyperbolic area of a domain B  H is defined by “ dxdy ; A.B/ D 2 B y

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when this integral exists, and the hyperbolic length of a parametric piecewise-smooth curve c W Œa; b ! H with c.t / D x.t / C iy.t / is defined by Z bp 0 2 x .t / C y 0 .t /2 dt: length.c/ D y.t / a One can check that all these definitions do not depend on a particular parametrization of the domain B and of the curve c. Thus the notion of hyperbolic length is well defined for piecewise-smooth geometric curves (by geometric curve, we mean the image – sometimes also called the trace – of the curve which is a set of points in H.) Notice that the hyperbolic length of the segment Œi b; a C i b with b > 0 is jaj=b; likewise, for the segment Œi; ib with b > 0, it is j ln bj. It follows that the length of the segment Œib; a C ib goes to 0 if b goes to C1, while a is kept fixed, and that the length of Œi; ib goes to C1 if b goes to 0 or to C1. Clearly, the group G preserves all these notions. Namely, for any g 2 G we have length.g.c// D length.c/

and

A.g.B// D A.B/:

As a subgroup of the Möbius transformations, the group G acts on the extended complex plane C [ f1g, and preserves the family of circles (we regard straight lines in C as being circles in C [ f1g that pass through 1). Moreover the circle R [ f1g is globally invariant by G and G preserves the angles. It follows that G preserves the subfamily of vertical half straight lines and half-circles orthogonal to the real axis included in H. We will show that this family of curves plays a specific role in H. For any z and z 0 in H define d.z; z 0 / D inf length.c/; S

where S is the set of piecewise smooth curves c W Œa; b ! H with c.a/ D z and c.b/ D z 0 . Proposition 1.8. For every z and z 0 in H, there exists a unique (up to the parametrization) piecewise smooth curve c W Œa; b ! H, with c.a/ D z and c.b/ D z 0 satisfying length.c/ D d.z; z 0 /: Moreover, • if Re.z/ D Re.z 0 /, then c.Œa; b/ is the Euclidean segment Œz; z 0 ; • otherwise, c.Œa; b/ is the arc with endpoints z and z 0 , included in the half-circle passing through these two points, with center on the real axis. Proof. First suppose that z D is and z 0 D i s 0 with s > 0 and s 0 > 0. Let c W Œa; b ! H be a piecewise smooth curve with c.a/ D z and c.b/ D z 0 . Define c.t / D x.t/ C iy.t /. We have ˇZ b 0 ˇ ˇ ˇ y .t / length.c/ > ˇˇ dt ˇˇ; a y.t /

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with equality if and only if x.t / D 0 for all t 2 Œa; b and y 0 does not change sign. Thus, length.c/ > j ln.s=s 0 /j with equality if and only if c.Œa; b/ is the Euclidean segment Œi s; is 0 . Now let z and z 0 be any two points. Using Proposition 1.5, one obtains h 2 G such that h.z/ D i . Replacing h with hhk , for some k 2 K, we can suppose that h.z 0 / also belongs to the positive imaginary axis. Applying the previous argument, we obtain that the Euclidean segment Œh.z/; h.z 0 / is the unique piecewise smooth curve c W Œa; b ! H, with c.a/ D h.z/ and c.b/ D h.z 0 / and satisfying length.c/ D d.h.z/; h.z 0 //: We finish the proof using the fact that the action of G is by isometries and preserves the family of vertical half-lines and half-circles with center on the real axis. In the Euclidean space, straight lines are length minimizing. In hyperbolic geometry we have: Definition 1.9. Vertical lines and Euclidean half-circles included in H, with center on the real axis, are the geodesics for the metric .gz /z2H (Figure 1).

H

Figure 1

With this characterisation of geodesics, we can immediately see that Euclid’s parallel postulate fails in H; given for example the point i and the vertical geodesic C D fz 2 H; Re.z/ D 2g, it is clear that there are infinitely many geodesics passing through i which do not intersect C . We deduce directly from Proposition 1.8 (but also from Proposition 1.1) that: Corollary 1.10. The map d W H  H ! RC defined by d.z; z 0 / D inf length.c/ S

is a distance function. Moreover, for any h 2 G we have d.h.z/; h.z 0 // D d.z; z 0 /: The following formula gives a relation between the hyperbolic distance and the Euclidean one.

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Proposition 1.11. For any z and z 0 in H we have sinh. 12 d.z; z 0 // D where sinh.x/ D

jz  z 0 j ; 2.Im z Im z 0 /1=2

e x e x . 2

Proof. For z D it and z 0 D it 0 , this formula is easy to check since d.i t; i t 0 / D j ln.t =t 0 /j. In the general case, we introduce h 2 G such that Re h.z/ D 0 and Re h.z 0 / D 0. Since we have jh.z/  h.z 0 /j jz  z 0 j D ; 2.Im z Im z 0 /1=2 2.Im h.z/ Im h.z 0 //1=2 this case reduces to the previous one.

1.4 Circles, neighborhoods and perpendicular bisectors We define the hyperbolic circle (resp. hyperbolic disk) of radius r > 0 centered at z 2 H as the set of those z 0 in H such that d.z; z 0 / D r (resp. d.z; z 0 / 6 r). The family of hyperbolic circles is the same as the family of Euclidean ones. Property 1.12. The hyperbolic circle of radius r > 0 centered at z D a C ib is the Euclidean circle having the segment Œa C ibe r ; a C i be r  as a diameter. Proof. It follows from Proposition 1.11 that the hyperbolic circle of radius r > 0 centered at z D a C ib is the set of z 0 2 H such that jz  z 0 j D sinh. 2r /: 2.Im z Im z 0 /1=2 If z D i , clearly this circle is the set of z 0 D x 0 C iy 0 such that 







e r C e r 2 e r  e r 2 D : 2 2 This implies that this hyperbolic circle is the Euclidean circle having the segment Œi e r ; i e r  as a diameter. Let z D a C ib. The hyperbolic circle of radius r > 0 centered at z is the image by the isometry h.z 0 / D bz 0 C a of the hyperbolic circle centered at i with the same radius. On the other hand, the image by h of a Euclidean circle with Œz1 ; z2  as a diameter is the Euclidean circle with Œh.z1 /; h.z2 / as a diameter. It follows that the hyperbolic circle of radius r > 0 centered at z D aCi b is the Euclidean circle having the segment Œa C ibe r ; a C i be r  as a diameter. x 02 C y 0 

Let c W R ! H be a geodesic. This geodesic determines two points in R [ 1, called the endpoints of c, defined by lim t!1 c.t / D x and lim t !C1 c.t / D y. We write c.R/ D .xy/.

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Fix " > 0 and let us describe the "-neighborhood of .xy/. First suppose that x D 0 and y D 1. Since the hyperbolic disk with center i and radius " is the Euclidean disk with diameter Œe "i ; e "i , the "-neighborhood of .01/, denoted V , is the image of this disk by the one parameter family of Möbius transformations h t .z/ D t z with t > 0. This implies that this neighborhood is given by the set of z 2 H satisfying the inequality Re z 6 "0 Im z for some "0 > 0. We reduce the general case to the previous situation using a transformation h 2 G such that h.0/ D x and h.1/ D y. We thus obtain that the "-neighborhood of .xy/ is equal to h.V /. In particular we obtain the following property: Property 1.13. Let Œz; y/ be a geodesic ray in H. For any " > 0 and any z 0 2 H there exists z 00 2 Œz 0 ; y/ such that Œz 00 ; y/ is included in the "-neighborhood of Œz; y/. For z ¤ z 0 in H, we define the perpendicular bisector of these points as the set of z 00 in H such that d.z; z 00 / D d.z 0 ; z 00 /. Property 1.14. The perpendicular bisector of z and z 0 is the geodesic passing through the midpoint of the geodesic arc with endpoints z and z 0 , orthogonal to this arc. Proof. Suppose that Im z D Im z 0 . Using Proposition 1.11, we obtain that the set of z 00 in H satisfying d.z; z 00 / D d.z 0 ; z 00 / is the vertical half-line passing through the point z0 D .Re z C Re z 0 /=2 C i Im z. In the general case, consider h 2 G such that Im h.z/ D Im h.z 0 /. The set of z 00 in H satisfying d.z; z 00 / D d.z 0 ; z 00 / is the image by h1 of the vertical half-line passing through the point z1 D .Re h.z/ C Re h.z 0 //=2 C i Im h.z/. Therefore, it is the geodesic passing through the midpoint of the geodesic arc with endpoints z and z 0 , orthogonal to this arc.

1.5 Boundary at infinity and projective action Clearly, the topology induced by d on H is the one induced by the Euclidean distance. For this topology, the space H is not compact. We compactify it by taking its closure in the extended complex C [ f1g. The set H.1/ D R [ f1g is called the boundary at infinity of H. The restriction to H of the topology on H [ H.1/ is the topology induced by d . More precisely, an open set of H [ H.1/ is either an open set of H [ R (relative to the topology induced by the Euclidean distance on R2 ) or the union of the point 1 and the complement of a compact set in H [ R. As we have already seen, the group G acts on H.1/.

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Proposition 1.15. The action of G on H.1/ is conjugate to the projective action of PSL.2; R/ on the real projective line RP 1 . Proof. Let W H.1/ ! RP 1 be the map defined by   • .x/ D R x1 if x 2 R, and   • .1/ D R 10 . This map is a homeomorphism satisfying .hM .y// D ˙M .y/ for any ˙M 2 G and any y 2 H.1/.

1.6 Classification of elements of G Since the action of G on H.1/ is conjugate to the projective action of G on RP 1 , the eigenlines of a matrix M 2 SL.2; R/ correspond to the fixed points of hM acting on H.1/. Hence if M ¤ Id, then hM fixes 0, 1 or 2 points in H.1/. Definition 1.16. Let h 2 G be different from Id. If h fixes exactly two points of H.1/, then h is called hyperbolic. If h fixes exactly one point on H.1/, then h is called parabolic. Otherwise, h is called elliptic. The transformation h.z/ D sz with s > 1 is hyperbolic. More precisely h fixes the points 1 and 0, and for any z 2 H we have 1 D lim hn .z/ n!C1

and

0 D lim hn .z/: n!1

It follows that the geodesic .01/ is invariant by h. The translation h.z/ D z C1 is parabolic. It fixes 0 and any horizontal line (i.e. any horocycle centered at 1, see Definition 2.2) is invariant by h. The transformation h.z/ D .cos z C sin  /=. sin  z C cos  / with  ¤ k is elliptic and it fixes i . Any circle centered at i is invariant by h. Recall that the subgroups U , D, K of SL.2; R/ are defined by  ˚   U D u.t/ D 10 1t I t 2 R ; ˚  0   D D d.s/ D 0s 1=s Is>0 ; and

˚  s0 K D k.s 0 / D cos sin s 0

 sin s 0 cos s 0



 I s0 2 R :

Let M 2 SL.2; R/. Clearly there exists M 0 2 SL.2; R/ such that M 0 MM 01 or M 0 MM 01 belongs to U [ D [ K. Using this property, one easily obtains the following result: Proposition 1.17. Let h 2 G  fIdg.

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– h is hyperbolic if and only if there exist h0 2 G and s > 1 such that h0 hh01 .z/ D sz. – h is parabolic if and only if there exists h0 2 G such that h0 hh01 .z/ D z C 1. – h is elliptic if and only if there exist h0 2 G and  ¤ k such that h0 hh01 .z/ D .cos z C sin /=. sin z C cos /. Corollary 1.18. If h 2 G is hyperbolic, then there exists a unique geodesic, called the axis of h, invariant by h. The endpoints of this geodesic are the attractive fixed point hC D limn!C1 hn .z/ and the repulsive one h D limn!1 hn .z/, where z is any point in H.

2 Vector approach to horocycles Proposition 1.15 gives an interpretation in terms of hyperbolic geometry of the action of the group PSL.2; R/ on the real projective line RP 1 . Following this way, we construct in this section a dictionary between a family of curves in H [ H.1/ invariant by G, called horocycles, and the space E D R2  f0g=f˙ Idg. Using this dictionary, we translate the linear action of the group PSL.2; R/ on E in terms of hyperbolic geometry.

2.1 Horocycles The family of extended horizontal lines (i.e. with f1g added) and of circles tangent to the real line is clearly a family of curves in H invariant by G. There are different approaches to these curves. One approach is related to the geodesics of H. Obviously, a horizontal line is orthogonal to the pencil of all vertical geodesics. Replacing this line by a circle tangent to the real line at some point x 2 R, and using a transformation g 2 G such that g.1/ D x, we obtain that this circle (without x) is orthogonal to the pencil of all geodesics .x  x C /, whith x C D x. Such curves can also be viewed as limit circles. Namely, one checks that an extended horizontal line is the limit in H [ H.1/ of hyperbolic circles passing through a fixed point z in H, with center converging to 1 along the geodesic ray Œz; 1/. The same property holds for a circle tangent to the real line, replacing the point 1 by the point of tangency. For this reason, a horizontal line or a circle tangent to the real line (without its point of tangency) is usually called a horocycle, and its boundary at infinity is called its center. We now give a metric approach to these curves. The idea is to sit at a point x on the boundary at infinity H.1/ and to observe the points of H from x. To do this, we associate to each pair of points z and z 0 in H an algebraic quantity reflecting the

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relative position of these two points as seen from x. Denote by .r.t // t >0 the geodesic ray Œz; x/ parametrized by arc length and set B.t / D d.z; r.t //  d.z 0 ; r.t //. Proposition 2.1. The function B has a limit as t goes to C1. This limit is called the Busemann cocycle centered at x, evaluated at z, z 0 and it is denoted Bx .z; z 0 / (Figure 2). Moreover we have (i) for any h 2 G, Bh.x/ .h.z/; h.z 0 // D Bx .z; z 0 /; (ii) d.z; z 0 / 6 Bx .z; z 0 / 6 d.z; z 0 /; (iii) Bx .z; z 0 / D d.z; z 0 / (resp. d.z; z 0 /) if and only if z 0 belongs to the geodesic ray Œz; x/ (resp. z 2 Œz 0 ; x/); (iv) for any z 00 2 H, Bx .z; z 0 / D Bx .z; z 00 / C Bx .z 00 ; z 0 /. z z

B x (z; z  ) < 0

B ∞ (z; z  ) > 0 z z H

x

Figure 2

Proof. It follows directly from the definition of B.t / that if Bx .z; z 0 / exists, then Properties (i) and (ii) are satisfied. First suppose that x D 1 and z D ib. Set z 0 D a0 C ib 0 and denote by .s.t / D 0 a C i be t / t >0 the geodesic ray Œa0 C ib; 1/ parametrized by arc length. For t large, we have d.s.t /; z 0 / D ln.b=b 0 / C t; thus

B.t / D d.s.t /; z 0 /  d.z 0 ; r.t // C ln.b 0 =b/:

In addition d.s.t/; r.t// 6 ja0 j=be t , hence lim t !C1 B.t / D ln.b 0 =b/. Note that the limit does not depend on a0 . Furthermore, this limit is equal to d.z; z 0 / (resp. d.z; z 0 /) if and only if a0 D 0 and b 0 > b (resp. b 0 6 b) (Property (iii)). Now let z 00 D a00 Cib 00 in H. Since B1 .z; z 00 / D ln.b 00 =b/, B1 .z; z 0 / D ln.b 0 =b/ and B1 .z 0 ; z 00 / D ln.b 00 =b 0 /, we have B1 .z; z 00 / D B1 .z; z 0 / C B1 .z 0 ; z 00 /; hence Property (iv). If x D 1 and z is arbitrary, using a translation, we obtain the previous case.

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If now x ¤ 1, using the Möbius transformation h.z/ D

xz  x 2  1 ; zx

we are again in the first situation. In conclusion, the limit of B.t / as t goes to C1 exists for all x 2 H.1/ and all z, z 0 in H, and the Properties (i), (ii), (iii), (iv) are satisfied. For all t > 0, define H t .x/ D fz 2 H j Bx .i; z/ D ln tg

H tC .x/ D fz 2 H j Bx .i; z/ > ln t g:

and

If x D 1, the set H t .1/ is the horizontal line defined by Im z D t , and H tC .1/ is the closed half-plane included in H and bounded by this line. If x ¤ 1, consider h 2 G such that h.1/ D x. Using Proposition 2.1, we obtain H t .x/ D g.H t 0 .1//

with t 0 D t e Bx .i;g.i // :

It follows that H t .x/ is a Euclidean circle tangent to the real axis at x, and that H tC .x/ is the closed Euclidean disk bounded by this circle (Figure 3). Ht+ (∞)

Ht (∞)

it

Ht+ (x)

Ht (x)

H x

Figure 3

Definition 2.2. The set H t .x/ (resp. H tC .x/) is called a horocycle (resp. horodisk) centered at x. Let h 2 G such that h.1/ ¤ 1. Write h.z/ D .az C b/=.cz C d / with ad  bc D 1. Note that c ¤ 0 and hence that the horocycle h.H tC .1// is a Euclidean circle. The following proposition gives an expression of its diameter. Proposition 2.3. The Euclidean diameter of the Euclidean circle h.H tC .1// is equal to 1=c 2 t . Proof. The image by h of H tC .1/ is the disk passing through h.i t / and tangent to the real axis at the point a=c D h.1/ (Figure 4).

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it + a=c

it

iı + a=c g(Ht (∞)) H a=c

0

Figure 4

Let ı be its Euclidean diameter. We have Bh.1/ .it C a=c; a=c C iı/ D ln ı=t; therefore B1 .h1 .it C a=c/; h1 .a=c C i ı// D ln ı=t . Since the point a=c C iı belongs to h.H t .1// and the point a=c C i t belongs to H t .1/, we have B1 .h1 .it C a=c/; h1 .a=c C i ı// D B1 .h1 .i t C a=c/; i t /. It follows that t : ln t =ı D ln 1 Im h .a=c C it / In conclusion we have ı D 1=c 2 t . Corollary 2.4. For any horocycle H t .x/, there exists h 2 G such that h.H1 .1// D H t .x/. Moreover h.H1 .1// D H1 .1/ if and only if h D hu with u 2 U . Proof. If x D 1 then h.H1 .1// D H t .1/ with h.z/ D t z. Otherwise, H t .x/ is a Euclidean circle. Denote by D its diameter. Consider h 2 G such that h.1/ D x. Write h.z/ D .az C b/=.cz C d / with ad  bc D 1. Applying Proposition 2.3, we obtain that for any t 0 > 0, the diameter of the Euclidean circle h.H t 0 .1// is the real number 1=c 2 t 0 . It follows that for t 0 D 1=c 2 D, we have h.H t 0 .1// D H t .x/, and hence that there exists h0 2 G such that h0 .H1 .1// D H t .x/. Now we have: h.H1 .1// D H1 .1/ if and only if h.1/ D 1 and Im h.i / D 1. It follows that h.H1 .1// D H1 .1/ if and only if h is a translation.

2.2 Vector approach We introduce the set H of horocycles: H D fH t .x/ j t > 0 and x 2 H.1/g: It follows from Corollary 2.4 that the map p W SL.2; R/ ! H

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defined by p.M / D hM .H1 .1// induces a bijection P W SL.2; R/= ˙ U ! H : The group SL.2; R/ acts by left translation on the quotient SL.2; R/= ˙ U . On H , this action corresponds to the “natural” action of G. More precisely, let M 2 SL.2; R/, set ŒM U D M ˙ U 2 SL.2; R/= ˙ U and H t .x/ D P .M /. For any M 0 2 SL.2; R/, we have P .M 0 .ŒM U // D hM 0 .H t .x//: Consider now the linear action of the group SL.2; R/ on R2 . This action induces a transitive action of G D SL.2; R/=f˙ Idg on the space E D .R2  f0g/=f˙ Idg: Moreover the subgroup consisting of the ˙M 2 G such that ˙M.˙e1 / D ˙e1 ,  where e1 D 10 , is precisely ˙U . Thus the map p0 W G ! E defined by p 0 .˙M / D ˙M.e1 / induces a bijection P 0 W SL.2; R/= ˙ U ! E: By construction, for any M and M 0 in SL.2; R/, we have P 0 .M 0 .ŒM U // D ˙M 0 .M.e1 //: Using P and P 0 , we obtain a bijection Vect D P 0 B P 1 W H ! E which allows us to conjugate the action of G on H to the “linear” action of PSL.2; R/ on E. More precisely for any H 2 H and any M 2 SL.2; R/; we have Vect.hM .H // D ˙M.Vect.H //: The following proposition gives an explicit expression of Vect.H t .x// in terms of the parameters t > 0 and x 2 H.1/. Theorem 2.5. Let H t .x/ 2 H . We have ´ p p   ˙ t = 1 C x 2 x1 if x ¤ 1; p   Vect.H t .x// D ˙ t 10 if x D 1: Proof. By construction, Vect.H1 .1// D ˙e1 and Theorem 2.5 is true in this case. Consider now H t .x/ 2 H and M D ac db 2 SL.2; R/ such that hM .H1 .1// D H t .x/. We have hM .1/ D x, therefore a=c D x if x ¤ 1, and c D 0 otherwise. 1 .i /; i / D ln t . Moreover Bx .i; hM .i// D ln t hence B1 .hM

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On the other hand, since Vect.hM .H1 .1// D ˙M.Vect.H1 .1// , we have Vect.H t .x// D ˙M.˙e1 /: Let us first analyze the direction   of the vector Vect.H t .x//. Since Vect.H t .x// D ˙M.˙e1 /, thevector w D ac is collinear to M.e1 /. It follows that M.e1/ is collinear to x1 if x ¤ 1 because c ¤ 0 and a=c D x, and is collinear to 10 otherwise. In conclusion, regarding the directions of vectors, Theorem 2.5 is true. Let us now analyze the Euclidean norm of the vector Vect.H t .x//. By Corol1 lary 1.7, there exist u 2 U , d 2 D and k 2 K such that hM D hu hd hk (hence 1 M D ˙ud k). Since B1 .hu hd hk .i/; i / D ln t , we have B1 .hd .i /; i / D ln t , hence hd .z/ D .1=t/z. This implies that d is the diagonal matrix d.s/ with s > 0, and s 2 D 1=t . On the other hand the norm of the vectors ˙M.˙e1 / is equal to the norm of d.1=s/.e1 / p D 1=se1 . In conclusion we have proved that the norm of Vect.H t .x// is equal to t . Using Theorem 2.5, we obtain the following equivalent properties: Corollary 2.6. Let H t .x/ 2 H and v 2 R2 such that ˙v D Vect.H t .x//. Consider a sequence .Mn /n>0 in SL.2; R/. The following properties are equivalent. (i) The sequence .Mn .v//n>0 converges to 0. 1 (ii) The sequence .Bx .i; hM .i///n>0 goes to C1. n

Proof. Set hn D hMn and H tn .xn / D hn .H t .x//. For z 2 H t .x/, we have ln tn D Bhn .x/ .i; hn .z//, hence ln tn D Bx .h1 .i /; z/. Applying Theorem 2.5, we obtain pn that the norm of Mn .v/ is equal to tn D e . Bx .h1 n .i /; z/=2/. It follows that this sequence of norms goes to 0 if and only if the sequence .Bx .h1 n .i /; z/=2/n>0 goes to 1. We finish the proof using the relation Bx .i; h1 Bx .h1 n .i/; z/ n .i // C Bx .i; z/ D : 2 2

3 Actions of the modular group We restrict our attention to the modular group  D PSL.2; Z/  G. The goal of this section is to use hyperbolic geometry to prove the following theorem: Theorem 3.1. Let v 2 R2 . If v is not collinear to a vector in Q2 , then SL.2; Z/.v/ is dense in R2 . Our proof is not the simplest one. Actually, our motivation is to give a proof which can be generalized by replacing the group PSL.2; Z/ by any non-elementary discrete subgroup H  PSL.2; R/ of finite type. In this general context, there exists a smallest

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(for the inclusion), non-empty closed subset of RP 1 which is invariant by H [7]. This set, denoted L.H /, is called the limit set of H . It can also be seen as a subset of H.1/ (Proposition 1.15). We associate to this set the subset R2 .H /  R2 containing 0 and all vectors whose direction belongs to L.H /. Clearly, R2 .H / is a closed subset on R2 invariant by the linear group p 1 .H /, where p is the projection of SL.2; R/ onto PSL.2; R/. The following theorem, proved for example in [7], generalizes Theorem 3.1 Theorem 3.2. Let H  PSL.2; R/ be a non-elementary discrete subgroup of finite type. For any non-zero vector v 2 R2 .H / whose direction is not fixed by a non-trivial parabolic transformation of H , we have p 1 .H /v D R2 .H /.

3.1 The action of  on H.1/ Let us first analyze the action of the group  on H.1/. The following result says in particular that this action is minimal, which means that H.1/ is the limit set of . Proposition 3.3. For any x 2 H.1/, the orbit .x/ is dense in H.1/. Moreover, .1/ D Q [ f1g: Proof. We begin with the last assertion. The inclusion .1/  Q [ f1g is clear. Now, let p=q 2 Q with gcd.p; q/ D 1. Consider p 0 , q 0 in Z such that pq 0  qp 0 D 1 and set h.z/ D .pz C p 0 /=.qz C q 0 /. This Möbius transformation is associated to an element of  and h.1/ D p=q, thus p=q 2 .1/: Now take x 2 H.1/ and consider the translation t .z/ D z C1. We have t D hu.1/ and u.1/ 2 SL.2; Z/. Since the sequence .t n .x//n>0 converges to 1, the closure of .x/ contains this point. Therefore Q [ f1g is included in .x/. In particular we obtain that .x/ is dense in H.1/. In general the limit set of a non-elementary discrete subgroup H  PSL.2; R/ may be different from H.1/ ([4], [7], [16]). The group  contains parabolic transformations since for example t .z/ D z C 1 2 . The fixed point of t is 1. It follows from Proposition 3.3 that any rational number is fixed by a parabolic transformation of . One can check that any parabolic transformation of  is conjugate to some t k , and hence the set Q [ f1g is the set of all fixed points of parabolic transformations of  (see for example [7] for a proof). On the contrary, the group  contains infinitely many non-conjugate hyperbolic transformations having no fixed points in common. Let A be the subset of H.1/ defined by A D fhC I h is a hyperbolic transformation of g: This set is invariant by the group  because h0 .hC / is the attractive fixed point of h0 hh01 . Since H.1/ is minimal for the action of , we have

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Corollary 3.4. The closure of A is equal to H.1/. If  is replaced by a non-elementary discrete subgroup H  PSL.2; R/, then obviously Corollary 3.4 holds, replacing H.1/ by the limit set of H .

3.2 A tiling of H Let us analyze the action of  on H. This action is properly discontinuous, in the sense that no -orbit accumulates in H. This property holds for any discrete subgroup of G. Property 3.5. Let z 2 H and .n /n>0 be a sequence in . If the set fn .z/ I n > 0g is infinite, then it is not bounded. Proof. Let z 2 H. Suppose that the set fn .z/ I n > 0g is bounded. Since the n are isometries we can replace z by i . Write n D hMn with Mn 2 SL.2; R/ and   an bn : Mn D cn dn We have Im n .i/ D 1=.cn2 C dn2 / and Re n .i / D .an cn C bn dn /=.cn2 C dn2 /. By hypothesis there exist A > 0, B > 0 and C > 0 such that, for any n > 0, we have A 6 Im n .i/ 6 B and C 6 Re n .i/ 6 C . Moreover an dn  bn cn D 1 and  is discrete in G, hence the set fn I n > 0g is finite, a contradiction. Let us now construct a tiling of H associated to . For each  2 , consider the subset H2i ./  H defined by H2i ./ D fz 2 H I d.z; 2i / 6 d.z; .2i //g: It is easy to check that .2i/ ¤ 2i for any  2  different from Id. It follows that the set fz 2 H I d.z; 2i/ D d.z; .2i//g is the perpendicular bisector between 2i and .2i /. This set is the geodesic passing through the midpoint of the geodesic arc with endpoints 2i and .2i/, orthogonal to this arc (Property 1.14). Hence H2i . / is the closed subset of H containing 2i, bounded by this geodesic. For example for t .z/ D z C 1 and s.z/ D 1=z, we have H2i .t/ D fz 2 H j Re z 6 1=2g; H2i .t 1 / D fz 2 H j Re > 1=2g and H2i .s/ D fz 2 H j jzj > 1g: We introduce now the set D2i ./, called the Dirichlet domain of  at the point 2i, defined by \ D2i ./ D H2i . /: 2

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B

The interior of the set D2i ./  H is denoted D2i ./. The following theorem says that D2i ./ is a fundamental domain for . This theorem holds for any discrete subgroup H  G, replacing 2i by any z 2 H such that h.z/ ¤ z for any non-trivial h 2 H ([16]). Theorem 3.6. The following properties are satisfied: S (i) 2  D2i ./ D H; B

B

(ii) D2i ./ \  D2i ./ D ¿, for all  2   fIdg. Proof. (i) SupposeS that there exists z 2 H such that for any  2 , the point .z/ does not belong to 2 D2i ./. It follows that for any  2 , there exists  0 2  such that d.2i; .z// > d.2i;  0 .z//: Therefore there exists an infinite sequence fn .z/ I n > 0g with n 2  satisfying d.2i; 0 .z// > d.2i; 1 .z// >    > d.2i; n .z// > d.2i; nC1 .z// : : : : We obtain a contradiction to Property 3.5. It follows that for any z 2 H, there exists 0 2  such that d.2i; 0 .z// 6 d.2i; .z//; for any  2 , and hence that 0 .z/ 2 D2i ./. (ii) If there exists z 2 D2i ./ \ D2i ./ for some  2   fIdg, then d.z; 2i / D B

d.z; .2i//, hence z does not belong to D2i ./: The transformations t and s will be useful to describe the domain D2i ./. Property 3.7. The Dirichlet domain D2i ./ is equal to the set defined by D fz 2 H j jzj > 1 and  1=2 6 Re z 6 1=2g: t E

s

H −1

− 12

1 2

Figure 5

1

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Proof. The domain D2i ./ is clearly included in the set H2i .t /\H2i .t 1 /\H2i .s/. Hence it is included in .B Let z be in the interior of . Suppose that there exists .z/ D .az C b/=.cz C d / with  2   fIdg such that .z/ 2 . Then c ¤ 0 because j Re.z C b/j > 1=2 for B all b 2 Z . Moreover we have Im.z/ D Im z=jcz C d j2 and, since z 2 , jcz C d j2 > .jcj  jd j/2 C jcjjd j: It follows that Im .z/ > Im z. B If .z/ 2 , applying the same argument, we obtain Im z > Im .z/, which contradicts the previous inequality. B B B To summarize, for all  2   fIdg, we have  \ D ¿. This implies that is included in D2i ./. In conclusion D2i ./ D . Notice that the hyperbolic area of is finite. By definition, a discrete subgroup of G admitting a Dirichlet domain with finite area is called a lattice. The following result proved in [4] shows that the Dirichlet domain D2i ./ allows us to visualize the modular surface S D nH (Figure 6) associated with . More cusp

Figure 6

precisely, let nD2i ./ be the set of elements of D2i ./ modulo , and let the function  W nD2i ./ ! S be defined by .z 0 \ D2i .// D z 0 : Then we have: Proposition 3.8 ([4], Theorem 9.2.4). The map  W nD2i ./ ! S is a homeomorphism.

3.3 Hyperbolic characterization of rational numbers We introduce the open horodisk H C D fz 2 H I Im z > 2g. For  2 , write .z/ D .az C b/=.cz C d / with ad  bc D 1 and a, b, c, d integers. If c D 0, then .H C / D H C . Otherwise, this set is a Euclidean disk with diameter 1=2c 2

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207

(Proposition 2.3). Since c 2 Z, observe that .H C / D H C

or

.H C / \ H C D ¿:

Consider now the unbounded set B and the compact set C defined by B D HC \

and

C D  B:

By construction, is the disjoint union of B and C . Since is a fundamental domain for , any geodesic ray Œi; x/ with x 2 H.1/ is covered by some sets .K/[.C / with  2 . The following result gives a geometric characterization of the irrationality of x. Corollary 3.9. For x 2 H.1/, the set of  2  such that .C / \ Œi; x/ ¤ ¿ is infinite if and only if x 2 R  Q. Proof. Suppose that the set of  2  such that .C / \ Œi; x/ ¤ ¿ is finite. Since .H C / D H C or .H C / \ H C D ¿; there exist z 2 Œi; x/ and  2  such that Œz; x/  .H C /. Hence x D .1/ and thus x 2 Q [ f1g. Suppose now that x 2 Q and consider 0 2  such that x D 0 .1/ (Proposition 3.3). Because the ray Œ01 .i/; 1/ is vertical, there exists z on this ray such that Œz; 1/  H C . Consider the set S of  2  such that .C / \ Œ01 .i /; 1/ ¤ ¿: It is equal to the set of  2  such that .C / \ Œ01 .i /; z ¤ ¿: Then there exists A > 0 such that if  2 S , then d.i; .i // < A. Applying Property 3.5, we obtain that the set f.i/ I  2 S g is finite. Since S  , this set, hence also 0 .S /, is finite. Let x 2 H.1/ be an irrational number and .r.t // t >0 be the arc length parametrization (i.e. d.r.t /; r.t 0 // D t 0  t for t 0 > t ) of the geodesic ray Œi; x/. It follows from Corollary 3.9 that there exist A > 0, a sequence .n /n>0   and a sequence of positive numbers .tn /n>0 with limn!C1 tn D C1 such that d.n .i/; r.tn // < A: This remark implies the following property for the values of the Busemann cocycle centered at x. Property 3.10. If x 2 H.1/ is irrational, then there exists a sequence .n /n>0   such that lim Bx .i; n .i // D C1: n!C1

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Proof. Let us prove that this property holds for the sequence .n /n>0   satisfying d.n .i/; r.tn // < A: Since G acts transitively on H.1/, there exists h 2 G such that h.x/ D 1. Moreover one can choose h 2 G such that h.i / D i . We have h.Œi; x// D Œi; 1/. By construction of h, the point hn .i/ belongs to the hyperbolic disk centered at the point h.r.tn // D ie tn with radius A. This disk is the Euclidean disk with diameter Œi e .tn A/ ; i e .tn CA/  (Property 1.12). Since tn goes to C1, we obtain lim Im.hn .i // D C1:

n!C1

It follows from the definition of the Busemann cocycle centered at 1 that lim B1 .i; hn .i // D C1:

n!C1

Using Proposition 2.1, we obtain B1 .i; hn .i // D Bh1 .1/ .h1 i; n .i //, hence limn!C1 Bx .i; n .i// D C1: Property 3.10 can be generalized by replacing  with a discrete subgroup H  G of finite type. In this general context we obtain that if x is in the limit set of H and is not fixed by a parabolic transformation of H , then there exists a sequence .hn /n>0 in H such that limn!C1 Bx .i; hn .i// D C1 ([7]). Such a point x is called a horocyclic point with respect to H .

3.4 An application to the linear action of SL.2 ; Z/ Recall from Theorem 2.5 that E D .R2 f0g/=f˙ Idg and that the mapping Vect W H ! E defined for the horocycle H t .x/ D fz 2 H I Im z D t g, t > 0 and x 2 H.1/, by ´ p p   if x ¤ 1; ˙ t = 1 C x 2 x1 p 1 Vect.H t .x// D if x D 1 ˙ t 0 is a bijection satisfying hM .H t .x// D ˙M.Vect.H t .x//: for any M 2 SL.2; R/ Using this bijection, we deduce the following from Corollary 2.6 and Property 3.10: Property 3.11. Let v 2 R2 . If Rv \ Q2 D f0g, then the closure of SL.2; Z/.v/ in R2 contains 0. This property can be generalized replacing  by a discrete subgroup H  G of finite type. In this general context, we obtain that if v is a non-zero vector in R2 .H / which is not fixed by a parabolic transformation of H , then the closure of H.v/ contains 0.

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We now want to show that the existence of small vectors in the closure of .v/ implies the density of this orbit. Let M 2 SL.2; R/ be a hyperbolic matrix, which means that hM is hyperbolic, M is diagonalizable with different eigenvalues. Denote by the eigenvalue of M with absolute value strictly greater than 1. An eigenvector of M associated to (resp. 1= ) is called an attractive (resp. repulsive) eigenvector of M . The following corollary holds also for a discrete subgroup H  G of finite type, replacing v by a vector whose direction is in L.H / and which is not fixed by a parabolic transformation of H . Corollary 3.12. Let M be a hyperbolic matrix of SL.2; Z/ and v be a vector in R2 . If Rv \ Q2 D f0g, then the closure of SL.2; Z/.v/ contains an attractive eigenvector of M . Proof. Since Rv \ Q2 D f0g, by Property 3.11 there exists a sequence .Mn /n>0 in  such that limn!C1 Mn .v/ D 0. Let v C and v  be respectively an attractive and a repulsive eigenvector of M . Decompose Mn .v/ as follows: Mn .v/ D an v C C bn v  : The sequences .an /n>0 and .bn /n>0 converge to 0, and for example .an /n>0 is not stationary. Consider kn 2 Z such that .an kn /n>0 converges to a real number ˛ ¤ 0. We have lim Mn M kn .v/ D ˛v C : n!C1

We are now ready to prove the following theorem. Theorem 3.13. Let v 2 R2 . If Rv \ Q2 D f0g, then SL.2; Z/.v/ is dense in R2 . We first prove the existence of at least one dense orbit. Proposition 3.14. There exists v 2 R2 such that the closure of SL.2; Z/.v/ is dense in R2 . Let us begin with the following lemma. Lemma 3.15. Let D1 and D2 be open disks included in R2  f0g. There exists M 2 SL.2; Z/ such that M.D1 / \ D2 ¤ ¿. Proof. For i D 1; 2 denote by Ci the set of points M such that for some > 0, !

OM belongs to Di . Using Proposition 2.3, we obtain an attractive eigenvector v1C ! of a hyperbolic matrix M1 2 SL.2; Z/ such that v1C D OB with B 2 D1 . We can suppose that the eigenvalue associated to v1C is positive. Choose a hyperbolic matrix M 2 SL.2; Z/ having no common eigenvectors with M1 , positive eigenvalues, and

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let uC and u be respectively an attractive and a repulsive eigenvector of M . If we replace M1 by M1n , and uC , u by vectors in R uC , R u , we can suppose that ! ! M1 .uC / D OB 0 and M1 .u / D OB 00 with B 0 and B 00 in D1 . Consider now a hyperbolic matrix M2 2 SL.2; Z/ having no common eigenvectors with M1 such that its attractive eigenline meets C2 . If we replace M2 by ˙M2n , we can suppose ! that the half-line D RC M2 .uC / is included in C2 . Let M2 M n .u / D OAn ! and M2 M n .uC / D OBn . The sequence of sets .ŒAn ; Bn /n>0 converges to the open half-line , when n goes to C1. On the other hand we have ŒAn ; Bn  D M2 M n M11 .ŒB 00 ; B 0 /. Since the half-line is included in C2 and ŒB 00 ; B 0  is included ! ! in D1 , there exist some w D OP , P 2 D1 , and n such that M2 M n M11 .w/ D OP 0 with P 0 2 D2 . Proof of Proposition 3.14. Consider a sequence .Dn /n>1 of open disks in R2  f0g such that for any open set O  R2  f0g there exists n > 1 satisfying O \ Dn ¤ ¿: Choose an open set D0  R2  f0g. Applying Lemma 3.15, one obtains M1 2 SL.2; Z/ such that M1 .D0 / \ D1 ¤ ¿. Let K1 be relatively compact open set included in D0 , satisfying M1 .K1 /  D1 . Applying the same argument to the open sets K1 and D2 , we obtain M2 2 SL.2; Z/ and a relatively compact open set K2  K1 such that M2 .K2 /  D2 . Following this process, we construct a sequence of matrices .Mn /n>1 in SL.2; Z/ and a sequence of relatively compact open sets .Kn /n>1 such T that KnC1  Kn and Mn .Kn /  Dn . Let B 2 C1 nD1 Kn . For any n > 1, the point ! ! Bn satisfying Mn .OB/ D OBn belongs to Dn . ! ! Let us prove that SL.2; Z/.OB/ D R2 . Consider v 0 D OB 0 in R2  f0g and a sequence .Dn0 /n>1 of open disks centered at B 0 with radius converging to 0. Each Dn0 contains a disk Din , hence Min .B/ belongs to Dn0 . It follows that v 0 belongs to SL.2; Z/.v/. Proof of Theorem 3.13. According to Corollary 3.12, the set SL.2; Z/.v/ contains an attractive eigenvector for a hyperbolic matrix of SL.2; Z/. Let us prove Theorem 3.13 in the case where M v D v with M 2 SL.2; Z/ and > 1. Consider v0 2 R2 such that SL.2; Z/.v0 / is dense in R2 . Such a vector exists according to Proposition 3.14. Denote by p.v/ the image of v in the projective line. Applying Propositions 3.3 and 1.15, there exists a sequence .Mn /n>0 in SL.2; Z/ such that the sequence .p.Mn .v///n>0 converges to p.v0 /. Since M v D v, we can find a sequence of integers .kn /n>0 such that .Mn M kn .v///n>0 converges to a vector in R v0 . It follows that SL.2; Z/.v/ D R2 .

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211

4 Topology of the horocyclic trajectories We introduce two equivalent systems: the action by right translation of   1 tdynamical the group U D fu.t/ D 0 1 I t 2 Rg on the space SL.2; Z/n SL.2; R/ and the horocyclic flow hR on the quotient space nT 1 H. Our goal is to describe the topology of the orbits of U and hR using Theorem 3.13. Let us first give a short introduction to abstract topological dynamics, which will be also used in Section 5.

4.1 Basic concepts in topological dynamics This introduction is inspired by Recurrence and topology by M. Alongi and G. S. Nelson ([2]). This book contains the solutions of the exercises of this section. We encourage the reader who wants to know more on this field to read A. Katok and B. Hasselblatt’s Introduction to the modern theory of dynamical systems ([19]), and W. H. Gottschalk and G. A. Hedlund’s Topological dynamics ([11]). Let Y be a topological space. By definition, a flow on Y is a map W RY ! Y satisfying the following conditions: (i)  is continuous; (ii) .t; :/ W Y ! Y is a homeomorphism for each t 2 R; (iii) .s; .t; y// D .s C t; y/ for all y in Y and any real numbers s, t . For each real number t, we denote by  t W Y ! Y the map defined for all y 2 Y by  t .y/ D .t; y/. Exercise 4.1. Prove that 0 D Id, and that  t D  t 1 for each t 2 R. Many examples arise from smooth vector fields f on smooth manifolds, and are determined by a differential equation of the form dy ; dt where dy=dt denotes the derivative of a function y with respect to a single independent variable. In most cases, there exists a unique smooth function  W R  Y ! Y satisfying f .y/ D

d.t; y/ .0/ D f .y/ dt such that (i) .t; :/ W Y ! Y is a diffeomorphism for each t 2 R; (ii) .s; .t; y// D .s C t; y/ for all y in Y and any real numbers s, t .

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Examples. (i) If Y D R2 and f .y/ is the constant vector field vÅ ¤ Å 0, then .t; y/ D y C t vÅ. (ii) Notice that the flow .t; y/ D y C t vÅ induces a flow ˆ on the torus T 2 D R2 =Z2 given by ˆ.t; y mod Z2 / D y C t vÅ mod Z2 : More generally, when Y is a compact smooth manifold, classical theorems for ordinary differential equations guarantee the existence (and uniqueness) of a flow associated to a smooth vector field on Y . Definition 4.2. If  W R  Y ! Y is a flow, then the trajectory (respectively the positive or negative semi-trajectory) from a point y in Y is the set of points  t .y/, where t is in R (respectively RC or R ). In Example (i), the trajectory from y 2 R2 is the straight line passing through y with direction vÅ. In Example (ii), the trajectories of ˆ on T 2 are the projections on T 2 of the trajectories of Example (i). Proposition 4.3. Let  W R  Y ! Y be a flow. If two trajectories have a non-empty intersection, then they are equal. Exercise 4.4. Prove Proposition 4.3. It follows from Proposition 4.3 that the family of all trajectories is a partition of the space Y . Definition 4.5. Let  W R  Y ! Y be a flow. A point y is a periodic point if there exits T > 0 such that T .y/ D y. The period of y is the infimum of such T . A flow  associated to a non-zero constant vector field on R2 does not admit periodic points. On the contrary, if vÅ 2 Q2  f0g , then all points in the torus T 2 are periodic for the flow induced by . Exercise 4.6. Prove that a flow on T 2 induced by a constant vector field vÅ ¤ o Å on R2 has periodic points if and only if vÅ 2 Q2  f0g. Proposition 4.7. Let  W R  Y ! Y be a flow. If y is a periodic point, then its trajectory is compact. Exercise 4.8. Prove Proposition 4.7. Exercise 4.9. Let ˆ be a flow on T 2 induced by a constant vector field vÅ ¤ o Å. Prove that, if vÅ … Q2  f0g, then each trajectory is dense in T 2 .

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213

Notice that if y is a periodic point for a flow  or if its trajectory is dense then there exists an unbounded sequence of real numbers .tn /n>0 such that lim tn !1  tn .y/ D y. More generally we introduce the following notion. Definition 4.10. Let  W R  Y ! Y be a flow. A point y is non-wandering if for any neighborhood V of y there exists an unbounded sequence of real numbers .tn /n>0 such that  tn V \ V ¤ ¿: We denote by  .Y / the set of non-wandering points of . Note that in Examples (i) and (ii) above we have  .R2 / D ¿ and ˆ .T 2 / D T 2 . In general the situation is more complicated. x D fz 2 C; jzj 6 1g Exercise 4.11. Let  be the flow on the closed unit disk D associated to the vector field defined in polar coordinates .r;  / by dr D r.r  1/ dt

and

d D : dt

x D S1 [ f0g. Prove that the set of periodic points is S1 [ f0g and that  .D/ (Hint: see [2], Exercises 2.3.8 and 2.5.12.) Proposition 4.12. Let  W R  Y ! Y be a flow. The non-wandering set  .Y / is a closed set which is invariant with respect to the flow. Exercise 4.13. Prove Proposition 4.12. In Example (i), no trajectory has accumulation points. More generally we define the notion of divergent points. Definition 4.14. Let  W R  Y ! Y be a flow. A point y is said to be divergent (respectively positively or negatively divergent) if for all unbounded sequences .tn /n>1 in R (respectively RC or R ), the sequence of points . tn .y//n>1 diverges. Notice that the notion of divergent point makes sense only for non-compact manifolds. Exercise 4.15. Prove that a point y is positively divergent (respectively negatively divergent) if and only if for some T 2 R the function from ŒT; C1/ (respectively .1; T ) into Y , which sends t to  t .y/, is a homeomorphism onto its image (i.e. a topological embedding). There is no general relation between divergent points and wandering points. Exercise 4.11 gives an example without divergent points, where  .Y / ¤ Y .

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Definition 4.16. A set M  Y is minimal with respect to the flow  if M is a nonempty closed subset in Y such that for each m 2 M its trajectory R .m/ is dense in M . Equivalently, a non-empty subset in Y is minimal if it does not contain a proper non-empty closed subset which is invariant with respect to the flow . For example, if y is periodic or is positively and negatively divergent, then its trajectory is minimal.

4.2 Dynamics of U on SL.2 ; Z/n SL.2 ; R/ Consider the linear action of SL.2; R/ on R2  f0g. The orbit of e1 D .1; 0/ is equal to R2  f0g and its stabilizer is the unipotent subgroup  ˚   U D u.t/ D 10 1t I t 2 R : It follows that the map

f 0 W SL.2; R/ ! R2  f0g

defined by f 0 .M / D Me1 induces a homeomorphism F 0 W SL.2; R/=U ! R2  f0g; where SL.2; R/=U is equipped with the quotient topology. Denote by ŒM U the projection of M 2 SL.2; R/ in SL.2; R/=U . For any M 0 and M in SL.2; R/ we have F 0 .ŒM 0 M U / D M 0 .F 0 .ŒM U //: Using this remark, we can translate Theorem 3.13 as follows: Theorem 4.17. Let M 2 SL.2; R/ such that R.M e1 / \ Q2 D f0g. We have SL.2; Z/ŒM U D SL.2; R/=U: Consider now the dual action of U by right translation on the quotient space SL.2; Z/n SL.2; R/. If M 2 SL.2; R/, denote by SL.2;Z/ ŒM  the projection of M on SL.2; Z/n SL.2; R/. It follows directly from the definition of the quotient topology that the orbit of ŒM U under SL.2; Z/ is dense in SL.2; R/=U if and only if the orbit of SL.2;Z/ ŒM  under U is dense in SL.2; Z/n SL.2; R/. The following property translates the condition R.M e1 / \ Q2 ¤ f0g in terms of matrices. Property 4.18. Let M 2 SL.2; R/  fIdg. The following conditions are equivalent. (i) R.M e1 / \ Q2 ¤ f0g. (ii) There exists u 2 U  fIdg such that M uM 1 2 SL.2; Z/.

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Proof. Set M D

a b  c d

and u.t/ D

M u.t/M

1

215

1 t  01



. We have

1  act D c 2 t

 a2 t : act C 1

This expression implies that the conditions (i) and (ii) are equivalent. We deduce from Theorem 4.17 and Property 4.18 the following corollary. Corollary 4.19. Let M 2 SL.2; R/. Then either for some u 2 U  fIdg, SL.2;Z/ ŒM 

D SL.2;Z/ ŒM u;

or SL.2;Z/ ŒM U

D SL.2; Z/n SL.2; R/:

It is obvious that the action of U by right translation on the quotient space SL.2; Z/n SL.2; R/ defines a flow. Corollary 4.19 says that the non-wandering set of this flow is the full space and, more precisely, that the trajectories are periodic or dense.

4.3 Dynamics of the horocyclic flow In this section, we give a geometrical interpretation of Corollary 4.19. Let .z; u Å/ be in the unit tangent bundle T 1 H of H. Consider the arc length parametrization r W R ! H (i.e. d.r.t /; r.t 0 // D t 0  t for t 0 > t ) of the oriented geodesic associated to .z; u Å/. We have r.0/ D z

and

Å: r 0 .0/ D u

Such a geodesic defines two points in H.1/, u.C1/ D lim r.t / t!C1

and

u.1/ D lim r.t /: t !1

We associate to .z; u Å/ the horocycle H passing through z centered at the point u.C1/, oriented as follows: H D ˇ.R/,  where ˇ W R ! H is the arc length parametrization Å is a direct basis of Tz H (Figure 7, u.0/ D z). of H such that ˇ.0/ D z and ˇ 0 .0/; u For example, if z D a C ib and u Å D .0; 1/, then ˇ.s/ D a C sb C i b for all s 2 R (Figure 8). For each t , consider the map hzt W T 1 H ! T 1 H defined by   Å// D ˇ.t /; vÅ.t / ; hzt ..z; u 1 where vÅ.t/ belongs to Tˇ.t/ H and satisfies: .ˇ 0 .t /; vÅ.t // is a direct basis (Figure 9, u D .z; u Å/). By construction, for any g 2 G and real number t we have

g B hzt D hzt B g:

./

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Françoise Dal’Bo ˇ(s), s < 0 u(0)

H Æ u

ˇ(s) s> 0

H u(+ 1)

Figure 7 1 Æ u

a + ib

ˇ(s) = a + sb + ib s> 0 a

Figure 8

Recall (Corollary 1.6) that the map F W G ! T 1 H defined by F .g/ D g.i; u Å/, where u Å D .0; 1/, is a homeomorphism satisfying F .g 0 g/ D g 0 F .g/:   Consider the matrix u.t/ D 10 1t : Observe that, for any t 2 R, F .u.t// D hzt .F .Id//: Using the relation ./ we obtain for any g 2 G, F .gu.t// D g hzt .F .Id//: Hence, using ./ we have F .gu.t// D hzt .F .g//:

u u



ht (u) t> 0



ht (u), t > 0

H

Figure 9

./

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217

Since G acts transitively on T 1 H, it follows that F 1 .hzt ..z; vÅ/// D F 1 ..z; vÅ//u.t / for any t 2 R and any .z; vÅ/ 2 T 1 H. We deduce from this relation the following properties: Property 4.20. (i) The map hzt W T 1 H ! T 1 H is a homeomorphism. (ii) The map from .R; C/ to the group of homeomorphisms of T 1 H which sends t z to h t is a morphism of groups. (iii) The map hz W R  T 1 H ! T 1 H, defined by z u/ D hzt .u/; h.t; is continuous. Definition 4.21. The map hz is called the horocyclic flow on T 1 H. Consider now the subgroup  D PSL.2; Z/ and denote by  1 the projection of T H onto T 1 S D nT 1 H (see [5] for a topological description of T 1 S ). Using relation ./, we obtain that the flow hz induces a flow h on T 1 S defined for any .z; vÅ/ 2 T 1 H by h t . 1 ..z; vÅ/// D  1 .hzt ..z; vÅ///: 1

(In Figure 10, u D .z; u Å/ and u0 D .z 0 ; uÅ0 /.) u π1

T 1Δ u

π 1 (u)

π 1 (u )

H

Figure 10.  D PSL.2; Z/.

Since the map F W G ! T 1 H satisfies the relation ./, it induces a homeomorphism ˆ W nG ! T 1 S; defined by ˆ.g/ D F .g/: Moreover, for any real number t we have ˆ.gu.t// D h t . 1 .F .g///:

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It follows that a trajectory hR . 1 ..z; vÅ/// for .z; vÅ/ 2 T 1 H, is dense in T 1 S if and only if gU is dense in nG, where F .g/ D .z; vÅ/. Since nG is homeomorphic to SL.2; Z/n SL.2; R/, we deduce from Corollary 4.19 and from Property 4.18 the following theorem, called Hedlund’s Theorem [13]: Theorem 4.22. Let .z; vÅ/ 2 T 1 H. If v.C1/ belongs to Q [ f1g, then for some real number t ¤ 0, h t .. 1 ..z; vÅ/// D  1 ..z; vÅ//; (i.e.,  1 ..z; vÅ// is hR -periodic). Otherwise, hR . 1 ..z; vÅ/// D T 1 S: It follows from Theorem 4.22 that a bounded hR -trajectory in T 1 S is necessarily periodic. Moreover, consider the family . 1 ..i t; vÅt /// t >0 of elements in T 1 S with v t .C1/ D 1. According to Theorem 4.22, each  1 ..i t; vÅt // is hR -periodic. Notice that if t ¤ t 0 , then hR  1 ..it; vÅt // \ hR  1 ..i t 0 ; vÅt0 // D ¿ since  2  satisfies .1/ D 1 if and only if  is a translation. It follows that the set of hR -periodic trajectories in T 1 S is not countable. Following the same proof, this theorem can be extended to a discrete subgroup of G of finite type H . In this general context T 1 H is replaced by the set T 1 H.H / of .z; vÅ/ 2 T 1 H such that v.C1/ belongs to the limit set L.H / of H . This set is clearly H -invariant and invariant by the horocyclic flow. One proves that its projection on H nT 1 H is the non-wandering set of the flow hR on H nT 1 H. If we replace the condition v.C1/ belongs to Q [ f1g by v.C1/ is fixed by a parabolic transformation of H , and T 1 S by H nT 1 H.H /, we obtain (see [7]) for any .z; vÅ/ 2 T 1 H.H /: either  1 ..z; vÅ// is hR -periodic, or hR . 1 ..z; vÅ/// D H nT 1 H.H /: If H nT 1 H is compact, it is well known that L.H / D H.1/ and that H does not contain parabolic transformations ([7]). In this case one obtains that all hR -trajectories are dense in H nT 1 H.

5 Bounded geodesic trajectories We focus now on some trajectories of the geodesic flow on T 1 S . Unlike the horocyclic flow, we will see that the topological behavior of this flow is more complicated. More precisely, we show in this section the following differences: – The set of periodic geodesic trajectories in T 1 S is countable. – There exist bounded geodesic trajectories in T 1 S which are not periodic.

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5.1 Introduction to the geodesic flow Let .z; u Å/ be in T 1 H. Consider the arc length parametrization r W R ! H of the oriented geodesic associated to .z; u Å/. We have r.0/ D z

Å: r 0 .0/ D u

and

Let gz W R  T 1 H ! T 1 H be the map defined by

gz.t; .z; u Å// D .r.t /; r 0 .t //:

Å//. By construction, for any h 2 G and t in R we have Set gz.t; .z; u Å// D gzt ..z; u hgzt ..z; u Å// D gzt .h..z; u Å///:

./

As for the horocyclic flow, the map gzt W T H ! T H can be interpreted as a map on G. More precisely, consider the homeomorphism 1

1

F W G ! T 1H defined by F .g/ D g.i; u Å/, where u Å D .0; 1/. Let us introduce the matrix   t =2  t=2  0 e : d e D 0 e t =2 For any t 2 R we have

   F d e t=2 D gzt .F .Id//:

Since F .h0 h/ D h0 F .h/ (Corollary 1.6) for any h and h0 in G, we obtain    g t .F .Id//: F hd e t=2 D hz Hence

   F hd e t=2 D gzt .F .h//:

Since G acts transitively on T 1 H,  t g t ..z; vÅ/// D F 1 ..z; vÅ//d e 2 F 1 .z for any t 2 R and any .z; vÅ/ 2 T 1 H. From this relation we deduce the following properties. Property 5.1. (i) The map gzt W T 1 H ! T 1 H is a homeomorphism. (ii) The map from .R; C/ to the group of homeomorphisms of T 1 H which sends t to gzt is a morphism of groups. (iii) The map gz W R  T 1 H ! T 1 H defined by gz.t; .z; vÅ// D gzt ..z; vÅ// is continuous.

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Definition 5.2. The map gz is called the geodesic flow on T 1 H. Consider now the subgroup  D PSL.2; Z/. Recall that  1 denotes the projection of T 1 H onto T 1 S D nT 1 H. Using the relation ./, we obtain that the flow gz induces a flow g on T 1 S defined for any .z; vÅ/ 2 T 1 H by (Figure 11, v D .z; vÅ/) g t . 1 ..z; vÅ/// D  1 .z g t ..z; vÅ///:

gt (v) 1 gt (1 (v))

T 1Δ

v

1

 (v) H

0

1

2

Figure 11.  D PSL.2; Z/.

The map g t can also be interpreted in nG. More precisely, using the homeomorphism ˆ W nG ! T 1 S defined for any h 2 G by ˆ.h/ D  1 .F .h//; we obtain for any t 2 R,   t  ˆ hd e 2 D g t . 1 .F .h///:

5.2 Characterization of some trajectories In this section we use the terminology introduced for a general flow in Section 4.1. Consider the flow gR on T 1 S. Let .z; vÅ/ 2 T 1 H. Recall that the positive (resp. negative) trajectory gRC . 1 ..z; vÅ/// (resp. gR . 1 ..z; vÅ///) is divergent if and only if for some T 2 R the function from ŒT; C1/ (respectively .1; T ) into T 1 S which sends t to g t . 1 ..z; vÅ///) is a homeomorphism onto its image. The following proposition gives a characterization of such trajectories. Proposition 5.3. The positive geodesic trajectory gRC . 1 ..z; vÅ/// in T 1 S is divergent if and only if v.C1/ belongs to Q [ f1g: Proof. Recall that D fz 2 H j jzj > 1 and  1=2 6 Re z 6 1=2g is a fundamental domain for  (Section 3.2). Let C  H be the compact set of elements z 2 such that Im z 6 2. We know (Corollary 3.9) that the set of elements  2  such that .C / \ Œi; v.C1// ¤ ¿

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is infinite if and only if v.C1/ 2 R  Q. Let .z; vÅ/ 2 T 1 H and r W RC ! H be the arc length parametrization of the geodesic ray Œz; v.C1//. Since C is compact and the rays Œi; v.C1// and Œz; v.C1// have the same endpoints, we obtain that v.C1/ 2 R  Q if and only if there exist a non-bounded sequence .tn /n>0 of positive real numbers and a sequence .n /n>0 in  such that .n .r.tn ///n>0 converges. The last condition is equivalent to the fact that the sequence .g tn . 1 ..z; vÅ///n>0 converges, which is also equivalent to the fact that the positive trajectory gRC . 1 ..z; vÅ/// is not divergent. Corollary 5.4. The trajectory gR . 1 ..z; vÅ/// is divergent in T 1 S if and only if v.C1/ and v.1/ belong to Q [ f1g: Corollary 5.4 can be extended to a discrete subgroup H of G of finite type. In this general context, T 1 H is replaced by the set of .z; vÅ/ 2 T 1 H such that v.C1/ and v.1/ belong to the limit set L.H / of H . This set is clearly H -invariant and invariant by the geodesic flow. One proves that its projection on H nT 1 H is the non-wandering set of the flow gR on H nT 1 H. We obtain (see [7]) that an element  1 ..z; vÅ// of the non-wandering set of the flow gR is divergent if and only if v.C1/ and v.1/ are fixed by parabolic transformations of H . Let us now characterize the periodic trajectories of gR on T 1 S . By definition, if gRC . 1 ..z; vÅ/// is such a trajectory, there exists T > 0 such that gT . 1 ..z; vÅ/// D  1 ..z; vÅ//. In T 1 H, this condition is equivalent to the existence of a non-trivial  2  such that ..z; vÅ// D gzT ..z; vÅ//. The element  is hyperbolic since it fixes two different points v.C1/ and v.1/. Conversely, if v.C1/ and v.1/ are fixed by some non-trivial  2 , then  preserves the geodesic .v.1/v.C1// and hence ..z; vÅ// D gzT ..z; vÅ// for some T > 0. We thus obtain the following characterization. Proposition 5.5. The trajectory gR . 1 .z; vÅ// in T 1 S is periodic if and only if v.C1/ and v.1/ are fixed by some non-trivial element of . Notice that  contains infinitely many cyclic subgroups generated by a hyperbolic element, which are not conjugate in . This implies that the set of periodic geodesic trajectories in T 1 S is infinite. Moreover, unlike the case of the horocyclic flow, this set is countable. Clearly Proposition 5.5 holds for any discrete subgroup of G. Let .z; vÅ/ 2 T 1 H. If v.C1/ or v.1/ belongs to Q[f1g, then we deduce from Proposition 5.3 that the trajectory gR . 1 ..z; vÅ///  T 1 S is not bounded. We now give a characterization of unbounded semi-trajectories in terms of the approximation of their endpoints by points of .1/ D Q [ f1g. For  2 , write .z/ D .az C b/=.cz C d / with ad  bc D 1. Notice that the integer c is not zero if and only if  is not a translation. Set c D c. /.

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Proposition 5.6. Let .z; vÅ/ 2 T 1 H be such that v.C1/ is an irrational number. The positive trajectory gRC . 1 ..z; vÅ/// is not bounded if and only if there exists a sequence .n /n>0 in  with c.n / ¤ 0, and a sequence of positive real numbers .tn /n>0 with limn!C1 tn D C1 such that jv.C1/  n .1/j 6

1 2c 2 .n /tn

:

Proof. Recall that the fundamental domain of  can be decomposed as D C [ B; where B D fz 2 H I Im z > 2g \ and C D  B: From Proposition 2.3 we know that for any element g.z/ D .az C b/=.cz C d / of G with c ¤ 0, the image by g of the horocycle H t D fz 2 H I Im z D t g is the Euclidean circle tangent at g.1/ to the real axis and with Euclidean diameter 1=c 2 t . Hence, for any real number x, we have the following: Œi C x; x/ \ g.H t / ¤ ¿ if and only if (Figure 12) jx  g.1/j 6

1 : 2c 2 t

./ Ht

g(Ht ) 1/(tc2 (g))

H x

g (∞) Figure 12

Let z be on the geodesic .v.1/v.C1//. Set x D v.C1/. The fact that gRC . 1 ..z; vÅ/// is not bounded is equivalent to the fact that the projection .Œzx// of Œzx/ in S D nH is not bounded . Moreover, since the geodesic rays Œzx/ and Œx C i; x/ have the same endpoint, the set .Œzx// is not bounded if and only if .Œx C i; x// is not bounded. Using the fact that D C [ B, we obtain that .Œx C i; x// is not bounded if and only if there exists a sequence .n /n>0 in  with n .1/ ¤ 1, and a sequence of positive real numbers .tn /n>0 with limn!C1 tn D C1 such that Œx C i; x/ \ n .H tn \ B/ ¤ ¿:

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It follows from relation ./ that gRC . 1 ..z; vÅ/// is not bounded if and only if jx  n .1/j 6

1 : 2c 2 .n /tn

5.3 Bounded non-periodic trajectories In this section we give a geometrical construction of non-periodic bounded trajectories. Let us consider two hyperbolic elements  and  0 of  which have no common fixed points in H.1/. For example we can choose .z/ D .2z C 1/=.z C 1/ and  0 D t B  B t 1 , where t .z/ D z C 1. Consider the attractive fixed point x D  C of  and the repulsive fixed point y D  0  of  0 . Choose .z; vÅ/ 2 T 1 H such that v.1/ D x

and

v.C1/ D y:

Proposition 5.7. The trajectory gR . 1 ..z; vÅ///  T 1 S is bounded and not periodic. Proof. According to Proposition 5.5, if gR . 1 .0 z; vÅ/// is periodic, then there exists a hyperbolic element  00 2  such that  00 .x/ D x. Since  is discrete, it follows that  n = 00 m for some integers n and m. The same property is still true replacing  by  0 because  00 .y/ D y. This contradicts the fact that  and  0 have no common fixed points. Fix " > 0. Consider the closed "-neighborhood V of the geodesic .   C / preserved by  . Applying Property 1.13, one obtains z 2 .xy/ such that Œz; x/ is included in V . For the same reason, there exists z 0 2 .xy/ such that Œz 0 ; y/ is included in the "-neighborhood V 0 of the geodesic . 0   0 C / preserved by  0 . Since ..   C // and .. 0   0 C // are compact, .V / and .V 0 / are also compact. This implies that ..xy// is bounded and hence that the trajectory gR . 1 ..z; vÅ/// is included in a compact subset. Clearly this construction holds for any discrete subgroup of H admitting at least two hyperbolic transformations with no common fixed points.

6 Geodesic trajectories and Diophantine approximation Since E. Artin [1] we know that there are deep relations between number theory and geodesics in the modular surface S D nH. Following the way of Artin, we give some examples of these relations. More precisely, using the results of Section 5 we construct a bridge between the topology of the geodesic trajectories in T 1 S and two subjects in number theory: the theory of Diophantine approximations and the study of binary quadratic forms.

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6.1 Classical results in number theory In this section we give an overview of some classical results in number theory ([18], [14]). Let x be an irrational number. Since Q is dense in R, there exist sequences .pn =qn /n>0 of rational numbers converging to x. The theory of Diophantine approximations focuses on the speed of this convergence. More precisely, one of the problems consists in finding “the best decreasing” function ‰ W N ! R C such that limn!C1 ‰.n/ D 0 and ˇ ˇ ˇ p ˇ ˇx  n ˇ 6 ‰.jqn j/: ˇ qn ˇ

The continued fraction expansion of x plays a fundamental role in this problem. Recall that it is defined by ([18]) 1

Œn0 I n1 ; : : : ; nk  D n0 C n1 C

;

1 n2 C : : :

C

1 nk1 C

1 nk

where the sequence .ni /i>0 is given by the following algorithm: x0 D x, n0 D E.x0 /, where E.x/ denotes the integral part of x, and for any i > 1, xi D 1=.xi1  ni1 /

and

ni D E.xi /:

This sequence converges to x. Using a subsequence, Lejeune-Dirichlet proved that for any irrational number x, there exists a sequence .pn =qn /n>0 of rational numbers with qn > 0 and limn!C1 qn D C1 such that ˇ ˇ ˇ p ˇ 1 ˇx  n ˇ 6 : ˇ qn ˇ 2qn2

For some irrational numbers x the function ‰.n/ D 1=n2 is the “best one” in the sense that there exists a c < 1 such that for any p 2 Z and q in N we have jx  p=qj > c=q 2 :

p For example if x D 2, then one can take c D 1=4. This is not the case for arbitrary irrational numbers. For example, let ` be the Liouville number given by the series with general term 1=10nŠ . Set qn D 1=10nŠ and write pn =qn for the sum of the n first terms of this series. The sequence .pn =qn /n>0 satisfies the following inequalities: 0 0 j there exists .pn =qn /n>1 2 Q such that jx  pn =qn j 6 c=qn2 and

lim jqn j D C1g:

n!C1

Clearly, we have

1 : 2 Using this quantity, we decompose R into two disjoint subsets according to the fact that .x/ equals zero or not. 0 6 .x/ 6

Definition 6.1. An irrational number x is badly approximated if .x/ > 0: p Clearly, 2 is badly approximated, but ` is not. The fact that x is badly approximated can be characterized in terms of its continued fraction expansion. More precisely, we have (for a proof, see for example [5]): Proposition 6.2. Let x be an irrational number and Œn0 I n1 ; : : : ; nk ; : : :  its continued fraction expansion. We have .x/ > 0 if and only if the sequence .ni /i>0 is bounded. In particular, x is badly approximated if the sequence .ni /i>0 is almost periodic. Such real numbers are well understood [18]: Proposition 6.3. Let x be an irrational number and Œn0 I n1 ; : : : ; nk ; : : :  its continued fraction expansion. The sequence .ni /i>0 is almost periodic if and only if x is quadratic (i.e. the real number x is a solution of a polynomial equation AX 2 C BX C C D 0, where A, B, C are integers and A ¤ 0). If x is fixed by a nontrivial hyperbolic element  2 , then clearly x is quadratic. The converse is also true ([7], [18]): Proposition 6.4. Let x ¤ y be two distinct irrational numbers. The real numbers x and y are solutions of the same polynomial equation AX 2 C BX C C D 0, where A, B, C are integers, if and only if there exists a nontrivial element  2  such that .x/ D x and .y/ D y.

6.2 Dynamical characterization of badly approximated numbers In this section, we explain the dynamical approach to badly approximated numbers introduced by E. Artin [1]. We return to the modular surface S D nH. Recall that  denotes the projection of H onto S , and that  1 denotes the projection of T 1 H onto T 1 S . Let x be an irrational number. Our goal is to characterize the fact that x is badly approximated in terms of the geodesic ray .Œi; x//  S .

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Let Œn0 I n1 ; : : : ; nk     be the continued fraction expansion of x. Suppose that the sequence .ni /i>0 is almost periodic. According to Propositions 6.3 and 6.4, there exists a hyperbolic transformation  2  fixing x. Let y ¤ x be the other fixed point of . For any .z; u Å/ 2 T 1 H we know that the geodesic trajectory gR . 1 .z; u Å/// is periodic (Proposition 5.5). In particular the geodesic ray .Œz; x//  S is compact. Using Property 1.13, one obtains that .Œi; x// is a bounded subset of S. The following theorem, also proved in [8], says that this topological property of the ray .Œi; x// characterizes more generally the fact that x is a badly approximated real number. Theorem 6.5. An irrational number x is badly approximated if and only if .Œi; x// is a bounded subset of S. Proof. Let x be an irrational number and let .i; u Å/ 2 T 1 H such that u.C1/ D x. 1 Recall that the positive trajectory gRC . ..z; u Å/// is bounded in T 1 H if and only if .Œi; x// is bounded in S. Applying Proposition 5.6 we thus obtain that .Œi; x// is not bounded if and only if there exist a sequence .n /n>0 in  with c.n / ¤ 0 and a sequence of positive real numbers .tn /n>0 with limn!C1 tn D C1 such that ju.C1/  n .1/j 6

1 2c 2 .n /tn

:

We finish the proof of this theorem by adding that the point n .1/ is a rational number. Moreover, let y 2 Q and write y D p=q with p and q relatively prime. Consider p 0 ; q 0 in Z such that pq 0 qp 0 D 1 and set .z/ D .pz C p 0 /=.qz C q 0 /. This Möbius transformation belongs to , and satisfies .1/ D p=q and c 2 . / D q 2 . As an application of this theorem, we obtain a construction of bounded, non-compact, positive geodesic trajectories in T 1 S which is different from the geometrical construction explained in Section 5. More precisely, let .ni /i>0 be an infinite bounded sequence of strictly positive integers which is not almost periodic, and consider the positive irrational number x given by the limit as k goes to C1 of 1

Œn0 I n1 ; : : : ; nk  D n0 C n1 C

:

1 n2 C : : :

C

1 nk1 C

1 nk

Let .i; u Å/ 2 T 1 H such that u.C1/ D x. Applying Theorem 6.5, we obtain that the positive geodesic trajectory gRC . 1 ..z; u Å/// is bounded. Moreover, since .ni /i>0 is not almost periodic, x is not fixed by any non-trivial element of  (Propositions 6.3 and 6.4) and hence, .Œz; x// is not compact (Proposition 5.5). We obtain:

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Corollary 6.6. Let .z; u Å/ 2 T 1 H. The geodesic trajectory gR . 1 ..z; u Å/// is bounded and not compact if and only if the real numbers x D u.1/ and y D u.C1/ are badly approximated irrational numbers and x and y are not solutions of the same polynomial equation AX 2 C BX C C D 0, where A, B, C are integers. This characterization implies in particular that the set of bounded and non-compact geodesic trajectories in T 1 S is not countable.

6.3 Application to small values of binary quadratic forms The theory of Diophantine approximations is closely related to the study of quadratic forms [15]. For example, if we associate to an irrational number x the binary quadratic form Qx defined by Qx .X; Y / D X Y  xY 2 ; it is straightforward that x is badly approximated if and only if 0 does not belong to the set Qx .Z2 /  f0g: Choose .z; u Å/ 2 T 1 H such that u.C1/ D x and u.1/ D 1. Applying Theorem 6.5, we obtain that the positive trajectory gRC . 1 ..z; u Å/// is bounded if and 2 only if 0 does not belong to the set Qx .Z /  f0g: The family of quadratic forms Qx is included in the set Q of indefinite and nondegenerate binary quadratic forms. Notice that a quadratic form Q is in Q if and only if it can be written as Q.X; Y / D .aX C bY /.cX C d Y / with ad  bc ¤ 0. We say that .z; u Å/ 2 T 1 H is associated to Q if d b and u.C1/ D ; a c with the convention u.1/ D 1 if a D 0, and u.C1/ D 1 if c D 0. Å//  T 1 S the Our goal is to characterize in terms of the trajectory of gR . 1 .z; u fact that Q.Z2 / contains small values. Clearly if Q.X; Y / D .aX C bY /.cX C d Y / 2 Q and a, b, c, d are rational numbers, then 0 does not belong to the set Q.Z2 /  f0g: u.1/ D

Definition 6.7. A binary quadratic form is rational if there exist a binary quadratic form Q0 with rational coefficients and a real number ¤ 0 such that Q D Q0 : Notice that Qx is rational if and only if x 2 Q. Let Q.X; Y / D .aX C bY /.cX C d Y / 2 Q and let b=a 2 Q [ f1g. Consider  2  such that .b=a/ D 1. If .z; u Å/ 2 T 1 H is associated to Q, then Å/// D gR . 1 ..z 0 ; uÅ0 /// with u0 .1/ D 1 and u0 .C1/ D .d=c/. It gR . 1 ..z; u follows that Q.Z2 /  f0g does not contain 0 if and only if gRC . 1 ..z; u Å/// is bounded.

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Moreover Q is rational if and only if d=c 2 Q [ f1g: In conclusion, the study of such a family of quadratic forms is the same as the study of the family .Qx /x2R . We say that Q.X; Y / D .aX C bY /.cX C d Y / 2 Q is semirational if b=a 2 Q [ f1g or d=c 2 Q [ f1g. Clearly Q.X; Y / D .aX C bY /.cX C d Y / 2 Q is not semirational if abcd ¤ 0, and b=a and d=c are not rational numbers. Proposition 6.8. Let Q.X; Y / D .aX C bY /.cX C d Y / be a non-semirational quadratic form in Q. The set Q.Z2 /  f0g contains 0 if and only if for any (or for one) .z; u Å/ 2 T 1 H which is associated to Q, the trajectory gR . 1 ..z; u Å/// is not bounded. Proof. We have Q.Z2 /  f0g contains 0 if and only if there exists a sequence of integers ..pn ; qn //n>0 with qn ¤ 0 such that lim Q.pn ; qn / D 0:

n!C1

This condition is hence equivalent to lim qn2



n!0

b pn C qn a



d pn C qn c



D 0:

Up to taking a subsequence we can suppose that lim qn2

n!0



b pn C qn a



D0

or

lim qn2

n!0



d pn C qn c



D 0:

It follows that the set Q.Z2 /  f0g contains 0 if and only if the irrational numbers b=a or d=c are not badly approximated. Using Theorem 6.5, it follows that Q.Z2 /  f0g contains 0 if and only if the ray .Œi; b=a// or .Œi; d=c// is not bounded in S. The last condition also holds if we replace the point i by any point z on the geodesic with endpoints b=a and d=c. Let us now give a geometric characterization of the fact that a quadratic form Q 2 Q is rational. If Q D Qx , and if .z; u Å/ 2 T 1 H is such that u.C1/ D x and u.1/ D 1, then, using Corollary 5.4, we obtain that Qx is rational if and only if the positive trajectory gRC . 1 ..z; u Å/// is divergent. More generally, we clearly have: Proposition 6.9. Let Q.X; Y / D .aX C bY /.cX C d Y / be a quadratic form in Q which is semirational. This quadratic form is rational if and only if for any (or for Å/// one) .z; u Å/ 2 T 1 H, which is associated to Q, the geodesic trajectory gR . 1 ..z; u is positively and negatively divergent. This characterization does not extend to all quadratic forms of Q. For example, Y  Y 2 is a rationalpform of Q which the quadratic form Q0 .X; Y / D X 2  Xp can be written as Q0 .X; p Y / D .X C 1 C p5=2Y /.X  2=1 C 5Y /. Notice that the real numbers 1 C 5=2 and 2=1 C 5 are fixed by the hyperbolic Möbius

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transformation .z/ D .z C 1/=.z C 2/ 2 . It follows from Proposition 5.5 that if .z; u Å/ 2 T 1 H is associated to Q0 , then gR . 1 .z; u Å// is periodic. Theorem 6.10. Let Q.X; Y / D .aX C bY /.cX C d Y / be a quadratic form in Q which is not semirational. This quadratic form is rational if and only if for any (or for Å/// one) .z; u Å/ 2 T 1 H, which is associated to Q, the geodesic trajectory gR . 1 ..z; u is periodic. Proof. Suppose that the geodesic trajectory gR . 1 ..z; u Å/// is periodic. According to Proposition 5.5 the real numbers u.C1/ and u.1/ are fixed by a hyperbolic element  2  (hence they are irrational numbers). Hence b=a and d=c are solutions of a polynomial equation AX 2 C BX C C D 0, where A, B, C are integers and A ¤ 0 (Proposition 6.4). It follows that Q.X; Y / D ac.X 2  B=AX Y C C =AY 2 /, which implies that the quadratic form Q is rational. Now, suppose that Q is rational. Since Q is not semirational, both b=a and d=c are irrational numbers. More precisely, they are solutions of a polynomial equation AX 2 C BX C C D 0, where A, B, C are integers and A ¤ 0. It follows from Proposition 6.4 that there exists a non-trivial element  2  fixing these two points. Applying Proposition 5.5 we obtain that the trajectory gR . 1 ..zÅ u/// is periodic. We deduce from Proposition 6.8 and Theorem 6.10 that a non-semirational quadratic form Q 2 Q is an irrational quadratic form such that 0 does not belong to Q.Z2 /f0g if and only if for any (or for one) .z; u Å/ 2 T 1 H, which is associated to Q, the trajectory gR . 1 ..z; u Å/// is bounded and not periodic. Applying Proposition 5.7 or Corollary 6.6, we obtain: Corollary 6.11. There exist irrational (and not semirational) quadratic forms Q 2 Q such that Q.Z2 / is not dense in R. Corollary 6.11 is false in dimension n > 3. More precisely, a famous theorem proved by G. Margulis and conjectured by A. Oppenheim ([20], [22], [23], [5]) says: Theorem 6.12. If Q is an irrational, non-degenerate and indefinite quadratic form on Rn with n > 2, then Q.Zn / is dense in R. We finish this exposition by giving an overview of the proof of Margulis. All proofs of this theorem use a link between the study of Q.Zn / and a dynamical system. Actually, consider the set Qn1 of indefinite and non-degenerate quadratic forms Q on Rn such that the determinant of the symmetric matrix associated to Q is 1. The group SL.n; R/ acts on this set as follows: MQ.v/ D Q.M v/; where M 2 SL.n; R/ and v 2 Rn . Fix a form Q0 2 Qn1 .

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The orbit of Q0 by SL.n; R/ is equal to Qn1 and its stabilizer is the group H corresponding to the intersection of the orthogonal group associated to Q0 with SL.n; R/ [5]. This implies that the map ‰ W H n SL.n; R/ ! Qn1 defined by ‰.HM / D MQ0 is a bijection. Moreover, for any Q 2 Qn1 and M 2 SL.n; Z/, we have MQ.Zn / D Q.Zn /: This remark implies that for Q D Q0 .M /, the set Q.Zn / is related to the SL.n; Z/orbit of HM 2 H n SL.n; R/, for SL.n; Z/ acting on the right on this quotient space. By duality, we obtain that the set Q.Zn / is related to the H -orbit of M SL.n; Z/ 2 SL.n; R/= SL.n; Z/, for the left action of H on this quotient space. For n D 2 and Q0 .X; Y / D XY , the group H is precisely the diagonal group D of SL.2; R/. The map M 7! M 1 conjugates the left action and the right action of D on the quotient space SL.2; Z/n SL.2; R/. Moreover, recall (Section 5.1) that this action is conjugate by ˆ to the action of the geodesic flow on T 1 S . We thus obtain a link between Q.Z2 / and the geodesic trajectory ˆ.SL.2; Z/M 1 D/, as we already showed in this section. For n D 3, the group H is not simultaneously diagonalizable. More precisely, the connected component of H containing the identity is isomorphic to the group PSL.2; R/, and thus contains a one-parameter subgroup N isomorphic to the group U introduced in Section 1. Any matrix of N is unipotent in the sense that it is conjugate to a triangular matrix with 1’s in the diagonal. In his proof, Margulis shows that if the Oppenheim conjecture is true for n D 3, then it is true for n > 3 [5]. Like for the action of U on SL.2; R/= SL.2; Z/ (Theorem 4.22), the topological dynamics of the left action of the group N on SL.3; R/= SL.3; Z/ is regular. The presence of the subgroup N in H plays a key role in the proof of Margulis to show that, unlike the case n D 2, a bounded H -orbit is periodic. As a consequence, he obtains that for all irrational quadratic forms Q 2 Q31 , the set Q.Z3 / contains small values and hence that Q.Z3 / D R [5].

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S. G. Dani and G. A. Margulis, Values of quadratic forms at integral points: an elementary approach. Enseign. Math. (2) 36 (1990), no. 1–2, 143–174.

[10] S. G. Dani and A. Nogueira, On orbits of SL.2; Z/ and values of binary quadratic forms on positive integral pairs, J. Number Theory 95 (2002), no. 2, 313–328. [11] W. H. Gottschalk and G. A. Hedlund, Topological dynamics. Amer. Math. Soc. Colloq. Publ. 36, Amer. Math. Soc., Providence, RI, 1955. [12] S. Gallot, D. Hulin, and J. Lafontaine, Riemannian geometry. Universitext, SpringerVerlag, Berlin 1993. [13] G. A. Hedlund, Fuchsian groups and transitive horocycles. Duke Math. J. 2 (1936), no. 3, 530–542. [14] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers. 5th ed., The Clarendon Press/Oxford University Press, New York 1979; French transl. by François Sauvageot, Vuibert, Paris 2007. [15] J. Itard, Arithmétique et théorie des nombres. Que sais-je 1093, Presses Universitaires de France, Paris 1963. [16] S. Katok, Fuchsian groups. Chicago Lect. Math., University of Chicago Press, Chicago, IL, 1992. [17] A. Katok and V. Climenhaga, Lectures on surfaces. Stud. Math. Libr. 46, Amer. Math. Soc., Providence, RI, 2008. [18] A. Y. Khintchine, Continued fractions. Translated by Peter Wynn, P. Noordhoff Ltd., Groningen 1963. [19] A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems. Cambridge University Press, New York 1995. [20] G. A. Margulis, Formes quadratiques indéfinies et flots unipotents sur les espaces homogènes. C.R. Acad. Sci. Paris Ser. I Math. 304 (1987), no. 10, 249–253. [21] G. A. Margulis, Dynamical and ergodic properties of subgroup actions on homogeneous spaces with applications to number theory. In Proceedings of the International Congress of Mathematicians (Kyoto 1990), Math. Soc. Japan, Tokyo 1991, 193–215. [22] A. Oppenheim, The minima of indefinite quaternary quadratic forms. Ann. of Math. (2) 32 (1931), no. 2, 271–298.

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[23] A. Oppenheim, Values of quadratic forms. I. Quart. J. Math., Oxford Ser (2) 4 (1953), 54–59. [24] S. J. Patterson, Diophantine approximations in Fuchsian groups. Philos. Trans. Roy. Soc. London Ser. A 282 (1976), no. 1309, 527–563. [25] A. N. Starkov, Dynamical systems on homogeneous spaces. Transl. Math. Monogr. 190, Amer. Math. Soc., Providence, RI, 2000. [26] G. Troessaert andA. Valette, On values at integer points of some irrational, binary quadratic forms. In Essays on geometry and related topics, Monogr. Enseign. Math. 38, Enseignement Math., Genève 2001, 597–610.

Introduction to origamis in Teichmüller space Frank Herrlich Institut für Algebra und Geometrie, Karlsruhe Institue of Technology (KIT) 76128 Karlsruhe, Germany email: [email protected]

Contents 1 2

Motivation . . . . . . . . . . . . . . . . . Definition and characterization of origamis 2.1 Combinatorial definition . . . . . . . 2.2 Coverings of the punctured torus . . . 2.3 Monodromy . . . . . . . . . . . . . 2.4 Subgroups of F2 . . . . . . . . . . . 3 Teichmüller disks . . . . . . . . . . . . . . 3.1 Translation structures . . . . . . . . . 3.2 Variation of the translation structure . 3.3 Teichmüller disks . . . . . . . . . . . 4 Veech groups . . . . . . . . . . . . . . . . 4.1 Teichmüller and moduli space . . . . 4.2 The affine group . . . . . . . . . . . 4.3 Veech groups of origamis . . . . . . 4.4 Characteristic origamis . . . . . . . . 4.5 Congruence groups . . . . . . . . . . 5 An example: the origami W . . . . . . . . References . . . . . . . . . . . . . . . . . . . .

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1 Motivation An origami is in the first place a combinatorial object. We shall give in Section 2 four different characterizations, not all of them combinatorial. Step by step we shall discover more and more structure on these objects: • An origami determines a Riemann surface and even a surface with a translation structure. • The translation structure can be varied in a natural way; this yields a complex one-parameter family of Riemann surfaces.

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• The parameter space can be identified with the upper half plane, and the Riemann surfaces come along with a natural marking. • Thus from an origami, we obtain a map from the upper half plane into a Teichmüller space; this map turns out to be an isometric and holomorphic embedding. • The stabilizer in the mapping class group of this embedded upper half plane is a Fuchsian group, more precisely a lattice in PSL2 .R/. It is called the Veech group of the origami and can also be characterized by affine diffeomorphisms. • The image of the embedded upper half plane in the moduli space of Riemann surfaces is an affine algebraic curve, possibly with singularities; it is called the origami curve to the given origami. • Every origami is defined over a number field. Thus the (absolute) Galois group x Gal.Q=Q/ acts on origami curves, and hence ultimately on origamis. Because of the last property, P. Lochak considered these objects as one-dimensional analogs of dessins d’enfants; in [13], he proposed the name “origami” for them. A dessin d’enfants is also a combinatorial object, namely a graph with certain properties on a surface. It determines a Riemann surface which is defined over a number field, and thus a zero-dimensional arithmetic subvariety of the moduli space. For dessins d’enfants as well as for origamis, the dream would be to understand the Galois action on the combinatorial objects well enough to obtain new structural insights of the Galois group. In particular, one would like to understand how close the Galois group is to the so called Grothendieck–Teichmüller group. So far, this is but a dream, but at least there are several interesting relations between origamis and dessins d’enfants. Some of them are described in [11]. Although not under this name, origamis have been known since the pioneering work of Thurston and the seminal paper of Veech [19] in the late 1980s. They were – and still are – often studied in the context of Teichmüller disks and Teichmüller curves. In Veech’s work, they provide examples of polygonal billiard tables where the dynamical behaviour is “optimal”, i.e. the same as on the classical rectangle.

2 Definition and characterization of origamis 2.1 Combinatorial definition Take finitely many euclidean unit squares and glue them in such a way that • each left edge is glued to a right edge; • each upper edge is glued to lower edge; • the resulting closed surface X is connected.

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235

Example 1. Torus E with marked point. b

! a

Example 2. Quaternion origami. n

–

nn

/

//

!

nn

= =

– //

= =

=–

=

n

=–

= /

Note that the genus g of X can be calculated with Euler’s formula: for this we need the number of squares, edges and vertices on the surface obtained by the gluing. If d denotes the number of squares, we have 2d edges on X since each square has four edges, and each edge belongs to exactly two squares. The number of vertices is 4d before gluing, but their number on X depends on the way the squares are glued: we call it n and obtain d n C 1: 2  2g D d  2d C n D n  d or g D 2 In Example 2 we find g D 3.

2.2 Coverings of the punctured torus An origami as above comes with a map p W X ! E from the surface X to a torus E: p maps each square to the torus as in Example 1. The identifications that were made to construct the origami X are compatible with the identifications made to construct the torus E. Indeed a right edge is glued to a left edge and an upper edge is glued to a lower edge. The map p is unramified outside the vertices of the squares, hence ramified over at most one point on E which we call 1. Conversely, given a covering p W X ! E unramified outside 1, we get an origami as follows: every connected component of p 1 .E  .a [ b// is an open square, and the gluing of the squares is given by taking the closure of the components in X. We take this characterization as our official definition: Definition 2.1. a) An origami O of genus g and degree d is a covering p W X ! E from a connected closed oriented surface X of genus g to the torus E which is ramified at most over one point 1 in E.

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b) Origamis O1 D p1 W X1 ! E and O2 D p2 W X2 ! E are called equivalent if there is a homeomorphism ' W X1 ! X2 such that p1 D p2 B '.

2.3 Monodromy Recall that the fundamental group of the punctured torus is a free group on two generators. To be more precise, let E ? D E  f1g. Then with notation as in the following picture, we have 1 .E ? ; M / D F2 .x; y/. Note that the commutator xyx 1 y 1 is a loop around 1.

! x

"

P

x

y



!

y

M

M

For an origami O D .p W X ! E/ of degree d , let p 1 .M / D fM1 ; : : : ; Md g. We get a monodromy map m W 1 .E ? ; M / ! Sd D SymfM1 ; : : : ; Md g as follows: for  2 1 .E ? ; M / we set m./.i/ D j if the lift of  , that starts in Mi , ends in Mj . In this way, the origami O induces a homomorphism m D mO W F2 ! Sd . Let a D m.x/

and

b D m.y/:

The fact that X is connected implies that the subgroup of Sd generated by a and b acts transitively on the set f1; : : : ; d g. If we number the elements of p 1 .M / in a different way, a and b are replaced by their conjugates by the permutation that describes the renumbering. If O is given by squares, each square contains exactly one of the Mi . The lift of x that starts in Mi joins this point to the corresponding point in the next square to the right. Thus a (resp. b ) is the permutation of the labels given by passing to the right (resp. upper) neighbouring square. Conversely, given a and b in Sd , generating a transitive subgroup, we obtain an origami of degree d by labeling d squares with the numbers 1; : : : ; d and gluing the right edge of square i to the left edge of square a .i /, and its upper edge to the lower edge of square b .i /. Summing up we have shown Proposition 2.2. There is a bijection between equivalence classes of origamis of degree d and conjugacy classes of pairs .a ; b / in Sd that generate a transitive subgroup. Example. Taking up the second example in Section 2.1 and labelling the squares as shown below, we find a D .1 2 3 4/ .5 6 7 8/ and b D .1 7 3 5/ .2 6 4 8/.

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/

n

nn

–6=

=– 8 = =

1 = = 5 – //

2

nn

//

3

237

4 n

= 7 =– /

Exercise. Determine the subgroup of S8 generated by a and b . We shall clarify the meaning of this group in general at the end of the next subsection.

2.4 Subgroups of F2 An origami O D .p W X ! E/ restricts to an unramified covering p W X ? ! E ? where E ? D E  f1g as above and X ? D X  p 1 .1/. By the universal property of the universal covering u W Ez ! E ? , there is a unique unramified covering q W Ez ! X ? such that u D p B q. The group of deck transformations of the universal covering is in a natural way idenz ? / Š 1 .E ? / Š F2 . Untified with the fundamental group, thus we have Deck.E=E ? ? z z ?/ der this identification, U D Deck.E=X / Š 1 .X / is a subgroup of Deck.E=E of index d D degree.p/. Conversely, any subgroup U of finite index d of F2 determines an unramified covering p W X ? ! E ? of degree d . This map can be extended in a unique way to a ramified covering p W X ! E, where X is a closed surface. We have Proposition 2.3. There is a bijection between equivalence classes of origamis of degree d and conjugacy classes of subgroups of F2 of index d . By the rank-index formula for subgroups of a free group, any subgroup of F2 of index d is free of rank d C 1. On the other hand, the fundamental group of X ? is free of rank 2g C n  1, where g is the genus of X and n the number of punctures, i.e. n D jp 1 .1/j. Thus we have d C 1 D 2g C n  1, which confirms the Euler characteristic count of Section 2.1: g D d n C 1. 2 An explicit set of free generators for 1 .X ? / as a subgroup of 1 .E ? / D F2 .x; y/ can be found as follows. Represent the origami by a simply connected rectangular polygon P (which need not be planar); this can be achieved by beginning with an arbitrary square and then inductively gluing new squares to one of the free edges of the polygon obtained so far (according to the gluing of the origami). The final polygon P has 2.d C 1/ free edges e1 ; : : : ; ed C1 ; e10 ; : : : ; ed0 C1 , and the origami is obtained by gluing each ei to ei0 . Now fix a base point M inside P , e.g. the midpoint of a square. For each i , there is a unique path i in P from M to the midpoint of ei which is composed of horizontal

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and vertical connections of midpoints of squares, and a unique path i0 of the same type from the midpoint of ei0 to M . Together they represent a closed path on X ? , i.e. an element ui in 1 .X ? ; M /. Clearly u1 ; : : : ; ud C1 generate 1 .X ? ; M /. As a word in x and y, ui is obtained by replacing each horizontal piece of i and i0 by x or x 1 , depending on the orientation of the piece, and vertical pieces by y or y 1 . In the following example, the element ui is given by the word xyxyx 2 :

i >

ei



M

i0 ei0

>

The relation to the description in Section 2.3 is as follows: Let m D mO be the monodromy homomorphism to O and let  be an element of 1 .E ? /. Then m. /.1/ D 1 if and only if  lifts to a closed path in X ? , i.e. if and only if  2 1 .X ? / D U . It follows that the squares of the origamis correspond bijectively to the cosets of U in F2 . Under this correspondence, a is the right multiplication by x on the cosets, and b the right multiplication by y. Definition 2.4. An origami O is called normal (or Galois) if U is a normal subgroup of F2 . Equivalently, an origami O D .p W X ! E/ is normal if and only if p is a normal covering. Remark 2.5. For a normal origami, Gal.X=E/ D Deck .X ? =E ? / Š F2 =U Š ha ; b i  Sd : Example. The origami W is normal. The Galois group is the quaternion group of order 8. This can be checked e.g. by solving the exercise at the end of Section 2.3. A careful analysis of this example can be found in [8].

3 Teichmüller disks 3.1 Translation structures Let O D .p W X ! E/ be an origami. We can use the squares that constitute O as chart maps in the following way:

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• for every x 2 X which is in the interior of a square, we take this (open) square as a chart around x; • for every x 2 X that lies on an edge e of a square, but is not a vertex, we take as chart an open disk in the union of the two squares adjacent to e that contains x but no vertex. For the charts defined so far, the transition maps between different charts are translations in the plane. Considering the plane as the complex plane C, they are (very special!) holomorphic functions. Thus we have defined a structure of Riemann surface on X ? . It is known from the general theory of Riemann surfaces that it can be extended in a unique way to a structure of Riemann surface on the whole of X. In our situation, this construction is very explicit: Let v 2 X be a vertex of a square, i.e. v 2 p 1 .1/. Consider a small loop around v on X starting in horizontal direction, say. Since it also ends in horizontal direction, the number of squares that the loop crosses must be a multiple of 4, hence of the form 4k for some k  1. This number k is also the ramification index ep .v/ of the covering p in the point v. Thus in suitable local coordinates, p is given by z 7! z k in a neighbourhood of v. Summing up we have: Proposition 3.1. An origami O D .p W X ! E/ determines a structure of Riemann surface on X, and a translation structure on X ? . Definition 3.2. We call a surface S together with an atlas a translation surface if outside a finite subset †  S, the transition maps are translations, and for each v 2 †, the chart maps are of the form z 7! z kv for some positive integer kv . Note that a translation structure on a surface X  can be lifted in a unique way to any unramified covering Y  ! X  . Thus in particular, the universal covering Ez of E ? is endowed with a translation structure. More generally, let p W S ! S 0 be a ramified covering, where S is a translation surface. Then S 0 inherits a structure of translation surface: Let †S 0 D p 1 .†S / [ framification points of pg. Then the restriction p W S 0 †S 0 ! S †S is unramified, and for every v 2 †S 0 the exponent kv in Definition 3.2 can be taken to be kv D ep .v/  kp.v/ .

3.2 Variation of the translation structure We have seen in the last section how to define, for an origami O D .p W X ! E/, a structure of translation surface on X, and that this structure is completely determined by (p and) the translation structure on E. For the latter, we considered E as a square with opposite sides glued. We get a similar, but in general different, translation structure on E if we replace the square by an arbitrary parallelogram (say of area 1, to keep the volume of E fixed).

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Such parallelograms correspond bijectively to the points in the upper half plane H. Thus any  2 H induces a structure of Riemann surface on X. Moreover there is a natural marking coming from the identity map on X: Recall that a marked Riemann surface is an equivalence class of pairs .X; f / where X is a Riemann surface and f W X0 ! X is a diffeomorphism from a fixed reference surface X0 of the same topological type as X; two pairs .X; f / and .Y; g/ are considered equivalent, if g B f 1 W X ! Y is homotopic to a biholomorphic map. The set of all marked Riemann surfaces of a fixed genus g and a fixed number n  0 of distinguished points on it is the Teichmüller space Tg;n . From the above considerations we obtain Remark 3.3. Every origami O D .p W X ! E/ induces a map O W H ! Tg;n , where g is the genus of X and n D jp 1 .1/j. A proof of the following theorem can be found e.g. in [11], Section 3.2. The arguments given there were essentially known already to Teichmüller himself: Theorem 3.4. For an origami O, the map O W H ! Tg;n is a) injective, b) isometric (w.r.t. the hyperbolic metric on H and the Teichmüller metric on Tg;n ), c) holomorphic (w.r.t. the natural complex structure on Tg;n ). Definition 3.5. A map with the properties of Theorem 3.4 is called a Teichmüller embedding. The image O D O .H/ is called a Teichmüller disk (or complex geodesic) in Tg;n . Proof. a) follows from b). For b), we interpret O as follows: Any parallelogram of area 1 in the upper half plane with 0 as bottom left corner is the image of the unit square under an R-linear map of determinant 1, i.e. given by a matrix A 2 SL2 .R/. Two matrices A and B give the same complex structure if and only if they differ by a conformal map, i.e. a rotation, in other words if AB 1 2 SO2 .R/. Thus we can consider O equivalently as a map SO2 .R/n SL2 .R/ ! Tg;n . The identification of SO2 .R/n SL2 .R/ with H is  given by A 7! AN1 .i/, where a matrix ac db 2 SL2 .R/ acts on H in the usual way azCb as fractional linear transformation: z 7! czCd . Now recall that the Teichmüller distance between two marked Riemann surfaces .X; f / and .Y; g/ in Tg;n is defined as log K, where K is the minimal dilatation of a quasiconformal map h W X ! Y which is isotopic to g B f 1 . It is a well-known exercise in quasiconformal maps that an affine map has minimal dilatation within its isotopy class. Therefore the Teichmüller distance between O .A/ and O .B/ is equal to log K for the dilatation K of h D AB 1 . Explicitly, K D 1C , where  D ffzzN . 1

Introduction to origamis in Teichmüller space

p If e.g. AB 1 is the matrix M D shows  D

K1 , KC1

and the dilatation is

K 0

0

p1 K K1 1C KC1 K1 1 KC1

241

 for some K > 1, a short calculation D K. On the other hand, M 1 .i / D

Ki, and the hyperbolic distance between i and Ki also equals log K. For c), we use the property of Tg;n of being a moduli space: The construction at the beginning of this section provides us, for every  2 H, with a marked Riemann surface X . The union of all the X is in a natural way a complex manifold X O such that the projection X O ! H, which sends all points of X to  , is a proper holomorphic map. The universal property of a moduli space is that in such a situation, where we are given a holomorphic family of marked Riemann surfaces, the map from the base H of the family to the moduli space Tg;n , which maps the point  2 H to the point in Tg;n which represents the isomorphy class of the fibre X , is holomorphic.

3.3 Teichmüller disks The definition of the Teichmüller embedding O for an origami O is a special case of a more general construction: Let X be a compact Riemann surface of genus g and ! a holomorphic quadratic differential on X. This means that ! is a global section of the square of the canonical ˝2 bundle on X, i.e. ! 2 H 0 .X; X /. Equivalently, we can consider ! as an element of the Riemann–Roch vector space L.2KX /, where KX is a canonical divisor on X. In a local coordinate z, ! can be described as ! D f dz 2 with a holomorphic function f ; the transition functions between local coordinates z and zQ are given by f dz 2 D fQ d zQ 2 with f D fQ  . ddzzQ /2 . Note that for g D 0, i.e. for X D P 1 .C/, there are no nonzero holomorphic quadratic differentials. If g D 1, X is isomorphic to the structure sheaf OX , hence ˝2 also X Š OX , and there is, up to multiplication by a scalar, only one holomorphic quadratic differential. Explicitly, if X is the elliptic curve with equation y 2 D is holomorphic, and therefore ! x.x  1/.x  / for some 6D 0; 1, the 1-form dx y can be taken to be

dx 2 . y2

˝2 / is a For g  2, it follows from the Riemann–Roch theorem that H 0 .X; X 3g  3 dimensional complex vector space. Now let † be the (finite!) set of zeroes of ! in X . For P 2 X  D X  † choose a simply connected neighbourhood UP contained in X  . The map ' W UP ! C, RQp RQp Q 7! P ! D P f dz is a chart; here the integration is along an arbitrary path from P to Q in U . Since UP is simply connected and does not contain any zero of P p !, ! is well defined in UP up to sign. The transition map between UP and UP 0 is R P0 p obtained by adding c D P ! for a fixed path from P to P 0 , and possibly a change of sign. Thus the transition maps are of the form ' 0 D ˙' C c. Such an atlas is called a flat structure on X  , and X endowed with this structure is called a flat surface.

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Remark 3.6. If ! D 2 is the square of a holomorphic 1-form, the associated flat structure on X is a translation structure. Example 3. 1. If X D E1 is the elliptic curve with equation y 2 D x.x  1/.x C 1/, we find back the translation structure induced on the torus by the Euclidean unit square. This can be seen using the universal covering E1 D C=ƒ with the standard lattice ƒ D ZCZi: the holomorphic differential dz is invariant under ƒ and hence descends to dx on E1 . Therefore the translation structure is induced by the fundamental y domain for ƒ, which is the unit square. 2. On an origami O D .p W X ! E/, the translation structure is pulled back from E via p. Hence it corresponds to the differential .p  .E //2 , where E D dx . y The flat structure induced by a holomorphic quadratic differential on a Riemann surface X can be varied in essentially the same way as for an origami: For a matrix A 2 SL2 .R/, we obtain a new flat structure on the surface underlying X . The same argument as above gives Theorem 3.7. Let ! be a holomorphic quadratic differential on a compact Riemann surface of genus g  1. The construction just described gives a map ! W H D SO2 .R/nSL2 .R/ ! Tg;n ; where n is the number of zeroes of !. The map ! is a Teichmüller embedding.

4 Veech groups 4.1 Teichmüller and moduli space Recall that an origami O D .p W X ! E/ defines a Teichmüller embedding O W H ! Tg;n and a Teichmüller disk O D O .H/  Tg;n . We want to study the image C.O/ of O in Mg;n . Recall from Section 3.2 that a point in Tg;n represents a pair .X; f /, where X is a Riemann surface of genus g with n distinguished points on it and f is a diffeomorphism from the reference surface X0 to X. The mapping class group g;n D DiffeoC .X0 /=DiffeoC 0 .X0 / acts on Tg;n by ˛.X; f / D .X; f B ˛/. Note that this is well defined: if ˛ and ˇ represent the same element in g;n , ˛ B ˇ 1 is homotopic to the identity, and thus .X; f B ˛/ and .X; f B ˇ/ are equivalent pairs, hence define the same point in Tg;n . It is a crucial fact that the action of g;n on Tg;n is properly discontinuous. The orbit space is the moduli space Mg;n of Riemann surfaces of genus g with n distinguished points. The points of Mg;n correspond bijectively to the classes under biholomorphic maps of such Riemann surfaces.

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Fact. Mg;n is a quasiprojective variety. This means that there is a projective variety of which Mg;n is an open subvariety x g;n of stable curves (for the Zariski topology). The Deligne–Mumford moduli space M x g;n is is such a compactification of Mg;n , and Knudsen and Mumford proved that M in fact projective; see [6] for an introduction to these results. A well-known example of this fact is M1;1 which is isomorphic to the (complex) affine line A1 Š C. The isomorphism is given by the famous j -invariant: every (complex) elliptic curve can be represented by a Weierstrass equation y 2 D x 3 C Ax C B with complex numbers A and B satisfying  D 4A3 C 27B 2 6D 0; the parameters A and B resp. A0 and B 0 define isomorphic elliptic curves if and only if 3 03 their j -invariants agree, i.e. if j.A; B/ D 1728 4A D 1728 4A D j.A0 ; B 0 /.  0 The map from O to its image C.O/ in Mg;n clearly factors through the quotient by its stabilizer Stab.O/ D Stabg;n .O / in the mapping class group. The following proposition tells us that C.O/ is almost determined by Stab.O/: Proposition 4.1. For an origami O, the map qO W O =Stab.O/ ! C.O/ is birational. In fact, this result holds for arbitrary Teichmüller disks. It is due to the fact that g;n acts properly discontinuously on Tg;n . For origamis, this result has recently been strengthened by G. Schmithüsen [17]. To explain her result, note that the covering p W X ! E, which defines the origami O, can be considered as a family of coverings of elliptic curves (through the variation of the translation structure). It can be shown that this family induces a holomorphic map pO W C.O/ ! M1;1 ; pO maps the Riemann surface defined by a particular translation structure to the elliptic curve with the translation structure p . /. On M1;1 there are two special points E0 and E1728 corresponding to the elliptic curves y 2 D x 3  1 (for j D 0) and y 2 D x 3  x (for j D 1728). They are the only elliptic curves with nontrivial automorphisms. Now the result is Theorem 4.2 (Schmithüsen [17]). The map qO is an isomorphism outside p01 .E0 / and p01 .E1728 /.

4.2 The affine group Recall that an origami O D p W X ! E defines a translation structure on X  . Definition 4.3. Let X and Y be translation surfaces. A diffeomorphism f W X ! Y is called affine if there are coverings of X by charts .Ui ; zi / and of Y by charts .Vi ; wi /

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such that f .Ui /  Vi and on Ui , f is given by an affine map zi 7! Ai zi C bi , where Ai 2 GL2 .R/ and bi 2 R2 . Remark 4.4. If f W X ! Y is an affine diffeomorphism of translation surfaces, the matrix Ai is independent of i. Definition 4.5. Let X be a translation surface. a) Aff C .X/ is defined to be the set of orientation preserving affine diffeomorphisms f W X ! X. b) D W Aff C .X/ ! GLC 2 .R/; f 7! A, is a well-defined group homomorphism. c) Trans.X/ D ker.D/ is called the group of translations of X. d) .X/ D D.Aff C .X//  GLC 2 .R/ is called the Veech group of X. For an origami O, we define the Veech group .O/ to be the Veech group .X  / of the translation surface X  D X  p 1 .1/. Note that every affine diffeomorphism of X  has a unique extension to a diffeomorphism of X that maps the set p 1 .1/ of marked points to itself. The compact surface X and therefore also X  has finite area. Since the area has to be preserved by the affine diffeomorphisms, the Veech group .O/ D .X  / is contained in SL2 .R/. Example 4. For the “baby” (or trivial) origami E D id W E ! E, the Veech group is .E/ D SL2 .Z/ D Aff C .E  /. To see this, consider the universal covering R2 of E: The translation structure on 2 R coming from the “square” one on E is the usual euclidean structure. The affine maps for this translation structure are the familiar euclidean affine maps. Such an affine map descends to E if and only if it preserves the lattice Z C iZ, i.e. if and only if the matrix part is in SL2 .Z/. Proposition 4.6 (Earle and Gardiner [3]). For an origami O D p W X ! E we have Aff C .X  / Š Stab.O/: For the proof recall that every f 2 Aff C .X  / uniquely extends to a diffeomorphism of X. Thus we obtain a homomorphism  W Aff C .X  / ! g;n . It is clear that its image is contained in Stab.O/ and not very difficult to show that  is injective. The hardest part of the proof is to show that  is surjective onto the stabilizer Stab.O/. Under the isomorphism of the proposition, the translations correspond to the pointwise stabilizer Stabpw .O/ of O . Remark 4.7. For an origami O, Trans.O/ D Trans.X  / Š Stabpw .O/ is a finite group. For every point .Y; h/ 2 O , it is contained in the automorphism group of the Riemann surface Y .

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Corollary 4.8. .O/ Š Stab.O/=Stabpw .O/: In view of Proposition 4.1 this corollary states that the Riemann surface H= .O/ is birationally equivalent to the algebraic curve C.O/ in Mg;n . Again, this statement holds in greater generality, namely for all Teichmüller disks  whose image C./ in Mg;n is closed. By a theorem of Smillie (cf. [18], §5, for a proof), this happens if and only if the Veech group of  is a lattice.

4.3 Veech groups of origamis Using the definition of an origami by gluing squares, and also Example 4, one finds that the Veech group of an origami is a subgroup of SL2 .Z/. Less obvious is the fact that this subgroup is ”large”: Proposition 4.9 (Veech; Gutkin and Judge [5]). For an origami O, the Veech group .O/ is a subgroup of SL2 .Z/ of finite index. The following is a more precise result, which is the basis for an explicit calculation of Veech groups of origamis, since it allows for an algorithmic approach. Theorem 4.10 (Schmithüsen [14]). For an origami O, let U Š 1 .X  / be the subgroup of F2 Š 1 .E  / induced by the covering p W X ! E. Then .O/ D ˇ .StabAutC .F2 / .U //; where ˇ W Aut C .F2 / ! OutC .F2 / Š SL2 .Z/ is the canonical homomorphism. Sketch of proof. Consider H as the universal covering of E  (and hence of X  , too), and endow it with the translation structure induced by the square(s). Clearly .O/  .H/. A crucial step now is to show that .H/ D SL2 .Z/ (and not larger!). The idea for this is as follows: Let C ! E be the universal covering. Its restriction to E  is an unramified covering w W C  ƒ ! E  , where ƒ D Z C i Z. Hence the universal covering H ! E  factors through an unramified covering h W H ! C  ƒ. Schmithüsen shows that h is the “developing map” for the translation structure on H. As a consequence, .O/ consists of all matrices A 2 SL2 .Z/ which are of the form A D D.fO/ for some fO 2 Aff C .H/ that descends to X  . To characterize those A, identify F2 D 1 .E  / with the group Gal.H=E  / of deck transformations of the universal covering. Then Aff C .H/ can be identified with AutC .F2 / by sending fO to the automorphism  7! fO . / D fO B  B fO1 . The other crucial step in the proof is to show that fO descends to X  if and only if fO .U / D U . Note that Proposition 4.10 is a corollary of this theorem.

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4.4 Characteristic origamis Recall that a subgroup U of a group G is called characteristic if it is fixed by all automorphisms of G, i.e. '.U / D U for all ' 2 Aut.G/. Note that free groups have many characteristic subgroups in the following sense: Remark 4.11. Let U be a subgroup of Fn of finite index (for some n  1). Then U contains a characteristic subgroup of finite index. Proof. Clearly the intersection of all '.U /, where ' runs through all automorphisms of Fn , is characteristic. Since Aut Fn is finitely generated, this intersection is finite, and a finite intersection of finite index subgroups of a finitely generated group has finite index itself. We call an origami O characteristic if the corresponding subgroup U.O/ D 1 .X  / of F2 is characteristic. As a consequence of Theorem 4.11, the Veech group of a characteristic origami is SL2 .Z/. Together with Remark 4.12 we obtain the following surprising fact: Corollary 4.12. There are infinitely many origamis of genus g  2 whose Veech group is equal to SL2 .Z/. Perhaps even more surprising, there are also examples of non-normal origamis with Veech group SL2 .Z/. One of the first characteristic origamis discovered, and the smallest nontrivial one, is the quaternion origami W . That W is characteristic can be seen using the following remark. Remark 4.13. Let U  F2 be a normal subgroup of finite index, and let G D F2 =U . Then U is characteristic if for any two pairs .a; b/ and .a0 ; b 0 / of generators of G there is an automorphism  2 Aut.G/ such that  .a/ D a0 and  .b/ D b 0 . Proof. Giving a pair .a; b/ of generators is equivalent to giving a surjective homomorphism h W F2 ! G, namely h.x/ D a, h.y/ D b. Then for any ' 2 Aut.F2 /, h' D hB' W F2 ! G is also surjective. Hence by assumption, there exists  2 Aut.G/ such that h' D  B h. It follows that U D ker.h/ D ker. B h/ D ker.h' / D ker.h B '/ D ' 1 .U /: In the case of the quaternion origami W , the quotient group G is the classical quaternion group Q8 D f˙1; ˙i; ˙j; ˙kg. Except for 1 and 1, all elements of Q8 have order 4, and any two of them that are not inverse to each other generate Q8 . Clearly every such pair can be mapped to .i; j/ by an automorphism of Q8 . In [7] we give an explicit construction of a characteristic subgroup contained in a given (normal) finite index subgroup. Applied to the smallest origami of genus > 1,

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the L2;2 , this construction first yields the stairlike origami of degree 6 as normal origami; the corresponding characteristic origami has degree 108 and was studied in detail in [1].

4.5 Congruence groups It is a very natural question to ask which subgroups of SL2 .Z/ of finite index are Veech groups of origamis. The complete answer to this question is still open, but there are substantial partial results. Most of them have to do with congruence groups: Definition 4.14. A subgroup  SL2 .Z/ is called a congruence group if contains .n/ for some n  1, where     ˚ .n/ D ac db  10 01 mod n  SL2 .Z/ is the kernel of the projection pn W SL2 .Z/ ! SL2 .Z=nZ/. The smallest n such that .n/  is called the level of the congruence group. In her thesis [15], G. Schmithüsen proved that most congruence subgroups of SL2 .Z/ arise as Veech groups of origamis. The precise result is 



Theorem 4.15. Let n  1 and B D B1 [    [ Bk any partition of .Z=nZ/2 satxB  SL2 .Z=nZ/ the stabilizer of B, and let isfying B1 D f.0; 0/g. Denote by xB /. B D pn1 . Then there exists an origami OB with .OB / D B . The proof uses cleverly chosen coverings of the “trivial” n n origami. From this theorem, Schmithüsen deduces Corollary 4.16. For any prime p > 11, any congruence group of level p is the Veech group of an origami. For each of the primes 2, 3, 5, 7, 11, the same holds with one possible exception. For subgroups of .2/, Ellenberg and McReynolds showed in [4]: Theorem 4.17. Every subgroup of .2/ of finite index that contains Veech group of an origami.

 1

0 0 1



is the

There are also results for noncongruence groups: Proposition 4.18 (Schmithüsen; Hubert and Lelièvre [12]). The group .L2;n / is a noncongruence group for n  3.

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For the first example n D 3 in this series, the Veech group .L2;3 / has index 9 in SL2 .Z/. There are also other origamis whose Veech group is known to be noncongruence, see [16]. On the other hand, there is not a single subgroup of SL2 .Z/ of finite index for which it would be known that it cannot be the Veech group of an origami.

5 An example: the origami W In this section, we take up the discussion of the quaternion origami W that was first introduced in Section 2.1 and later mentioned at several places. For more details we refer to [8]. nnn

////

– k j /

1 ////

n

///

i

1 i

nnn

///

nn

//

k

j –

n

/

nn

=– k = =

/

=

//

1 = = j –

nn

//

n

– k =

1 i

i nn

n

= j =–

//

/

Recall that the corresponding (Riemann) surface XW has genus 3 and that the Galois group of the covering p W XW ! E is the quaternion group Q8 . Since f˙1g is a (normal) subgroup of Q8 , p factors through q W XW ! XW =f˙1g. Remark 5.1. a) XW =f˙1g Š E. b) p D Œ2 B q, where Œ2 is the multiplication by 2 on the elliptic curve E. This remark follows from the fact that the left half of the above figure is a fundamental domain for the action of f˙1g on XW : /

–

=

//

B

B /

=

–

/

which is equivalent to

//

=

=

–

– /

//

//

The next observation is Remark 5.2. The elliptic involution Œ1 on E lifts to an automorphism  on XW of order 2.   0 This remark is equivalent to saying that 1 0 1 is contained in the Veech group .W /.

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Note that  is obtained by rotating each square by , and then regluing the squares. It has four fixed points on XW , namely the centers of the squares labelled 1, 1, k and k. Thus  can also be described as rotation by an angle of  around one (in fact, any) of these points. Proposition 5.3. The automorphism group Aut.W / is a group of order 16, generated by Q8 and . The center of Aut.W / is cyclic of order 4, generated by c D k   . Geometrically, c is the rotation by an angle of  around the vertices. In particular, c has four fixed points. The quaternion origami W is one of the rare examples of a Teichmüller curve for which the equation is explicitly known: Proposition 5.4. The origami curve C.W / is the image in M3 of the family W W y 4 D x.x  1/.x  /;

2 P 1 .C/  f0; 1; 1g:

The map pW W C.W / ! M1;1 is induced from the fibrewise morphism p W W ! E , .x; y/ 7! Œ2  .x; y 2 /, where E is the elliptic curve given by the equation y 2 D x.x  1/.x  /. Idea of proof. Since C has four fixed points, the genus g of the quotient surface XW =hci can be calculated by the Riemann–Hurwitz formula: We have 2  3  2 D 4  .2g  2/ C 4  .4  1/;

hence 4 D 8  .g  1/ C 12:

Thus g D 0, and XW is a cyclic cover of P of degree 4 with four ramification points, each of order 4. Applying an automorphism of P 1 , we may assume that the branch points of this covering are 0, 1, 1 and some 2 P 1 .C/  f0; 1; 1g. Thus W has an equation of the form 1

W W y 4 D x "0 .x  1/"1 .x  /" ; where each of the "i is either 1 or 3. Looking at the combinatorial description of W in terms of squares one sees that the monodromy is the same around each of the four branch points. In other words, the "i are all equal (and thus can be chosen to be 1). Besides , Aut.W / also contains the involutions  D i and  D j. Proposition 5.5. For each of the six involutions ˛ 2 f˙; ˙; ˙g, the quotient surface W =h˛i is isomorphic to the elliptic curve E1 W y 2 D x.x  1/.x C 1/ D x 3  x: In particular, the quotient surface is independent of . Proof. Since c commutes with ˛ for each choice of ˛, it descends to an automorphism of order 4 with two fixed points on W =h˛i. The only elliptic curve having such an automorphism is E1 .

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From this proposition, one can deduce the following Corollary 5.6. The Jacobian of W is isogenous to E E1 E1 , i.e. there is a homomorphism with finite kernel and finite cokernel between the two abelian varieties. This is a prominent case of a “fixed part” in a family of Jacobians. On the one hand, there is only a very limited list of 1-parameter families of Jacobians with a fixed part of codimension one. On the other hand, there are not so many origamis known which have a nontrivial fixed part; see [2] for more examples, and also for counter-examples. Proposition 5.7. a) C.W / Š H= .W / D H= SL2 .Z/ Š A1 . x 3 is isomorphic to P 1 . b) The closure C.W / in M c) The unique point in C.W /  C.W / corresponds to the stable curve with two irreducible components, both isomorphic to E1 , that intersect (transversally) in two points. Proof of b) and c). As explained in detail in [10], going to the boundary in Mg can be achieved by replacing the square by rectangles that become thinner and thinner. This is equivalent to contracting the center lines of the squares in the original origami: nnn

////

nn

//

--- --- --- --/

n

///

///

n

/

--- --- --- --////

nnn

//

nn

The contraction results in two irreducible components, both nonsingular of genus 1. The two dotted lines yield two points of intersection of these components. The automorphism c has two fixed points on each component, and acts as an automorphism on both of them. Once again, we now use the fact that the only elliptic curve admitting such an automorphism is E1 . Because of its extraordinary properties we called W in German “eierlegende Wollmilchsau1 ”. Perhaps the most important property is the following: Theorem 5.8. C.W / intersects infinitely many other origami curves. W was the first origami for which this result was discovered. To my knowledge, it is still the only one for which it has been proved. Sketch of proof. The projection  W W ! W =h˛i (cf. Proposition 5.5) is ramified over four points, namely the fixed points of ˛. If we can find an isogeny ' W E1 ! E1 that maps these four points all to the same point, ' B  is an origami. 1An

idiom for allrounder.

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Any isogeny on E1 is multiplication by a suitable n. Thus it suffices to show that for infinitely many different values of , all four critical points are torsion points. In fact, the four critical points of  on E1 form an orbit under the automorphism cN induced by c. Choose one of the two fixed points of cN as the origin of the group structure on E1 . Then cN preserves the n-torsion points for each n, and it suffices to find one critical point that is a torsion point. Explicit calculation shows that for each n  3 and each n-torsion point P on E1 , there is 2 P 1  f0; 1; 1g such that P is a critical point of  . Corollary 5.9. For each torsion point P 2 E1 (of order n  3) there is an origami DP of degree 2n and genus 3 such that C.DP / intersects C.W /. It is a nice challenge to describe the origami DP in terms of squares. By its construction, DP is a double covering of a “trivial” n n-origami. More precisely, DP consists of two copies of the trivial n n-origami, glued in such a way that we have four ramification points that form an orbit under rotation by 90°. Example 5. For n D 3, there are two different possibilities for the ramification points (here the leaf is changed at the highlighted edges): c v v

w

b

a

v v

h g

k l w

i jf w wc e d a b

d d v

v

w

a

b

v v

l k w

g j wi e w d c h f b a

c

In [8] we also determined explicitly the Veech group of DP : it is a congruence group of level 2n if n is odd, and of level n if n is even. Finally, in [9] we studied the configuration of all the origami curves C.DP / in M3 : Theorem 5.10. The closure in M3 of the set of all origami curves C.DP /, where as above, P is an n-torsion point on E1 and DP is the associated origami, is a three-dimensional algebraic subvariety H .

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z be the Hurwitz space of coverings X ! E of degree 2, More precisely, let H where X has genus 3 and E is an elliptic curve, which are ramified over four points that are, for a suitable choice of the origin on E, of the form P , P , Q, Q. Then z has four irreducible components, three of which consist of hyperelliptic Riemann H surfaces X. z. Then H is the image in M3 of the non-hyperelliptic component of H

References [1] O. Bauer, Das 108er Origami. Diploma thesis, Karlsruhe 2005. [2] O. Bauer, Familien von Jacobischen von Origamikurven. PhD thesis, Universität Karlsruhe, 2009. [3] C. Earle and F. Gardiner, Teichmüller disks and Veech’s F -structures, Contemp. Math. 201 (1997), 165–189. [4] J. Ellenberg and D. McReynolds, Arithmetic Veech sublattices of SL.2; Z/. Duke Math. J., to appear. arXiv:0909.1851 [5] E. Gutkin and C. Judge, Affine mappings of translation surfaces. Duke Math. J. 103 (2000), 191–212. [6] J. Harris and I. Morrison, Moduli of curves. Grad. Texts in Math. 187, Springer-Verlag, New York 1998. [7] F. Herrlich, Teichmüller curves defined by characteristic origamis. Contemp. Math. 397 (2006), 133–144. [8] F. Herrlich and G. Schmithüsen, An extraordinary origami curve. Math. Nachr. 281 (2008), no. 2, 219–237. [9] F. Herrlich and G. Schmithüsen, A comb of origami curves in the moduli space M3 with three dimensional closure. Geom. Dedicata 124 (2007), 69–94. [10] F. Herrlich and G. Schmithüsen, The boundary of Teichmüller disks in Teichmüller and in Schottky space. In Handbook of Teichmüller theory, Vol. I, ed. by A. Papadopoulos, IRMA Lect. Math. Theor. Phys. 11, Eur. Math. Soc., Zürich 2007, 293–349. [11] F. Herrlich and G. Schmithüsen, Dessins d’enfants and origami curves. In Handbook of Teichmüller theory, Vol. II, ed. by A. Papadopoulos, IRMA Lect. Math. Theor. Phys. 13, Eur. Math. Soc., Zürich 2009, 767–809. [12] P. Hubert and S. Lelièvre, Prime arithmetic Teichmüller disks in H 2 . Israel J. Math. 151 (2006), 281–321. [13] P. Lochak, On arithmetic curves in the moduli spaces of curves. J. Inst. Math. Jussieu 4 (2005), no. 3, 443–508. [14] G. Schmithüsen, An algorithm for finding the Veech group of an origami. Experiment. Math. 13 (2004), 459–472. [15] G. Schmithüsen, Veech groups of origamis. PhD thesis, Universität Karlsruhe, 2005.

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[16] G. Schmithüsen, Origamis with noncongruence Veech groups. In Proc. 34th Symp. on Transformation Groups, Wing Co., Wakayama 2007, 31–55. [17] G. Schmithüsen, Generalized imprimitive Veech groups in Outer space. Manuscript 2010. [18] J. Smillie and B. Weiss Finiteness results for flat surfaces: A survey and problem list. In Partially hyperbolic dynamics, laminations, and Teichmüller flow, ed. by G. Forni et al., Fields Inst. Commun. 51, Amer. Math. Soc., Providence, RI, 2007, 125–137. [19] W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards. Invent. Math. 97 (1989), no. 3, 553–583.

Five lectures on 3-manifold topology Philipp Korablev and Sergey Matveev Department of Mathematics, Chelyabinsk State University Kashirin Brothers Street, 129, Chelyabinsk , 454001, Russia e-mail: [email protected] Department of Mathematics, Chelyabinsk State University Kashirin Brothers Street, 129, Chelyabinsk, 454001, Russia e-mail: [email protected]

Contents 1

Presentation of 3-manifolds . . . . . . . . . . . 1.1 Triangulations . . . . . . . . . . . . . . . 1.2 Heegaard splittings . . . . . . . . . . . . . 1.3 Surgery presentation . . . . . . . . . . . . 1.4 Spine presentation . . . . . . . . . . . . . 2 Turaev–Viro invariants . . . . . . . . . . . . . . 2.1 Construction . . . . . . . . . . . . . . . . 2.2 One example of the Turaev–Viro invariant . 3 JSJ-decomposition . . . . . . . . . . . . . . . . 4 Normal surfaces . . . . . . . . . . . . . . . . . 4.1 Normalization procedure . . . . . . . . . . 4.2 Matching system . . . . . . . . . . . . . . 4.3 Fundamental surfaces . . . . . . . . . . . 5 Algorithmic classification of Haken 3-manifolds 5.1 Introduction . . . . . . . . . . . . . . . . 5.2 Hierarchies and extension moves . . . . . 5.3 Proof of Theorem 15 . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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This article is based on the notes taken by the first author of lectures given by second author during the Geometry Master Class at the University of Strasbourg in April–May 2009.  Partially supported by RFBR grant No. 11-01-605.  Partially supported by RFBR & CNRS grant No. 10-01-91056 and the Program of Basic Research of Mathematical Branch of RAS.

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1 Presentation of 3-manifolds 1.1 Triangulations Usually triangulation means decomposition into triangles or, more generally, decomposition into simplices. Example 1. The 2-dimensional torus T 2 D S 1  S 1 can be presented as a square with pairwise identification of opposite sides and thus can be decomposed into two triangles by inserting a diagonal (Figure 1).

Figure 1. Presentation of a torus T 2 as a rectangle with a diagonal.

The above decomposition is not a triangulation in the usual sense since the condition “the intersection of any two simplices is a common face of each of them” does not hold. However, for our purposes it is convenient to neglect that condition and use the term triangulation for arbitrary decompositions into simplices as well (see definition below). Definition 1. Let D D f1 ; 2 ; : : : ; n g be a finite set of disjoint tetrahedra and let ˆ be a set f'1 ; '2 ; : : : ; '2n g of affine homeomorphisms between triangular faces of the tetrahedra such that every face has a unique counterpart. The pair .D; ˆ/ is called an identification scheme. Let us identify now all the faces of the tetrahedra via f'j ; 1  j  2ng. The resulting polyhedron is called the quotient space and denoted by M.D; ˆ/. We will refer to the above presentation of M.D; ˆ/ as a triangulation of M.D; ˆ/. Theorem 1. The quotient space M D M.D; ˆ/ of an identification scheme .D; ˆ/ is a 3-manifold if and only if .M / D 0. Proof. If a point x 2 M corresponds to a point in the interior of a tetrahedron or to a pair of points on the faces, the existence of a ball neighborhood of x is obvious. Suppose that x comes from identifying points x1 ; x2 ; : : : ; xk which either are vertices of tetrahedra or lie inside edges. The link of each point xi in the corresponding tetrahedron is a polygon: a biangle if xi lies inside an edge, and a triangle if it is a vertex. The link of the corresponding point of M is obtained from those polygons by pairwise identifications of their edges. It follows that it is a closed connected

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surface. Note that if xi lies inside an edge, then the corresponding surface is glued from biangles. It follows that the surface is either S 2 or RP 2 . We may conclude that M is a singular manifold in the followingSsense: M can be constructed from a genuine 3-manifold N with boundary @N D i Fi (which is obtained from M by cutting off all cone neighborhoods of singular points) by adding cones over all connected components Fi of @N . Let us calculate the Euler characteristic of M . Using the well-known formula .A [ B/ D .A/ C .B/  .A \ B/ and the fact that the Euler characteristic of the cone over any polyhedron is 1, we get .M / D .N / C m  .@N /, where m is the number of connected components Pm of @N . Since 2.N / D .@N /, we have 2.M / D 2m  .@N / and 2.M / D iD1 .2  .Fi //. Taking into account that .Fi /  2 and .Fi / D 2 if and only if Fi is a sphere, we get that M is a 3-manifold if and only if .M / D 0.

1.2 Heegaard splittings Definition 2. A Heegaard splitting of a closed orientable 3-manifold M is a presentation of M as a union M D H1 [ H2 of two handlebodies H1 and H2 such that H1 \ H2 D @H1 D @H2 . Lemma 1. Any closed orientable 3-manifold M admits a Heegaard splitting. Proof. Let T be a triangulation of M . Consider a regular neighborhood H1 D N.T .1/ / of the 1-dimensional skeleton T .1/ of T . Then H1 is a handlebody since it consists of ball neighborhoods of all vertices connected by index 1 handles (one handle for each edge). Let us show that H2 D M n Int H1 is a handlebody too. Indeed, it consists of balls (one ball inside each tetrahedron) joined by plates (one plate for each triangle). See Figure 2.

T1

Figure 2. The complement M n Int H1 consists of balls joined by plates.

Let h W @H ! @H be a self-homeomorphism of the boundary of a handlebody H . Then h determines a closed 3-manifold M obtained by gluing two copies of H via h.

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Moreover, M has a natural Heegaard splitting M D H1 [ H2 , where H1 ; H2 are the images of the copies of H under gluing. This construction is universal in the following sense: any Heegaard splitting H1 [ H2 of a given 3-manifold M can be obtained in that way. Indeed, the desired homeomorphism h W @H ! @H can be obtained by choosing identifications '1 W H ! H1 , '2 W H ! H2 and setting h D '21 '1 j@H . We may conclude that there is a close relation between Heegaard splittings and homeomorphisms @H ! @H . Note that isotopic homeomorphisms @H ! @H determine actually the same splitting of the same 3-manifold. So it is not surprising that the following notion plays an important role in 3-manifold topology. Definition 3. Let F be a closed orientable surface. Then the group of all orientationpreserving homeomorphisms F ! F modulo homeomorphisms isotopic to the identity is called the mapping class group of F and denoted by HC .F /. Definition 4. Let c  F be a simple closed curve on a surface F and N.c/  S 1  I be a small regular neighborhood of c. A homeomorphism c W F ! F is called a twist along c if (1) c is the identity outside N.c/; (2) c inside N.c/ is given by the formula re i' 7! re i.'C2.r1// , where .r; '/; 0  '  2, 1  r  2, is a parametrization of N.c/. In non-formal terms c can be described as the full rotation of one boundary component of N.c/ keeping the other fixed (Figure 3).

c c Figure 3. Homeomorphism c .

Theorem 2. Let F be a closed orientable surface. Then the mapping class group HC .F / is generated by twists. Our next goal is to sketch the proof of this theorem (see [16]). Definition 5. Two simple closed curves a, b in F are called c-equivalent (notation: a c b) if one curve can be transformed to the other by a sequence of twists. Lemma 2. Any two non-separating curves a, b on a connected orientable surface F are c-equivalent. Proof. We will prove the lemma in several steps.

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1. Suppose that a, b intersect each other transversally at exactly one point. It is easy to verify that a .b .a// D b. So a and b are c-equivalent. 2. Suppose that a, b are disjoint. Then there exists a simple closed curve d which intersects each of them at exactly one point (see Figure 4 for the cases when the complement to a [ b is connected and not connected). So, a and d are c-equivalent and b and d are c-equivalent, then a and b are c-equivalent too.

a

a

b

d

d b Figure 4. d intersects each of a and b at one point.

3. Now we proceed by induction on the number k of points in a \ b. The base of induction is given by k D 1; 0, see items 1, 2 above. Assuming that #.a \ b/ < k implies a c b, consider the case #.a \ b/ D k > 1. Let l be a subarc of b such that l \ a consists of the endpoints M , N of l. Let m1 , m2 be two complementary subarcs of a having the same endpoints (Figure 5). Then the simple closed curves l [ m1 and l [ m2 have a common arc l and no other common points. At least one of those curves does not split F since otherwise a would split F . Denote this non-splitting curve by d . Then d either does not intersect a or intersects it at one point. Clearly, the number of crossing points of b and d is strictly less than k (see Figure 5 for two cases of approaching l to a: from one side and from both sides). By the induction assumption, d c a and d c b. Therefore, a c b.

M

M d

m2

l

m1

N

d N

m1

b

m2

a Figure 5. Constructing of a new curve d .

l

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Proof of Theorem 2. Let h 2 HC .F / be a given homeomorphism. Choose disjoint simple closed curves m1 ; : : : ; mg  F which cut F to a disc with 2g  1 holes (Figure 6). Our goal is to return each curve h.mi / to mi using twists.

m1

m2

mg

Figure 6. Meridian system m1 ; : : : ; mg .

Since m1 and h.m1 / do not split F , these curves are c-equivalent. It means that there exists a product 1 of twists such that 1 h.m1 / D m1 . A little care is needed to get that 1 h induces the identity map m1 ! m1 . In exactly the same way we can construct a product of twists 2 such that 2 1 returns h.m2 / to m2 . Note that 2 can be constructed so as to be fixed on m1 . Doing so further, we construct a product of twists  such that  returns all h.mi / to mi . Let us now cut F along all mi . We get a disc D with 2g  1 holes. Then  determines a homeomorphism N W D ! D fixed on the boundary. It is known that any such homeomorphism can be represented by an element of the pure braid group B2g1 with 2g  1 strings. Moreover, the standard generators of B2g1 correspond to twists. It follows that h and hence h can be represented as product of twists.

1.3 Surgery presentation Definition 6. Let N.K/ be a regular neighborhood of a knot K  S 3 . Then a simple closed curve l  @N.K/ is called a longitude of K if l crosses a meridian of N.K/ at exactly one point. Definition 7. Let K be a knot in S 3 . By an integer framing of K we mean a choice of a longitude on @N.K/. By an integer framing of a link L  S 3 we mean integer framings of all components of L. Definition 8. Let L D L1 [ [ Ln be a framed link in S 3 . We say that a 3-manifold M is obtained from S 3 by surgery along L if M can be obtained as follows: for each i we cut off the solid torus N.Li / and glue it back such that the meridian of N.Li / is taken to the chosen longitude. Theorem 3. Any closed orientable 3-manifold admits a surgery presentation. Proof. Let M1 D H1 [h1 H2 ; M2 D H1 [h2 H2 be Heegaard splittings of the same genus such that h1 D h2 l , where l is a twist along a simple closed curve l  @H1 .

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Claim. M2 can be obtained from M1 by surgery along a framed knot in M1 . Indeed, pushing l to the interior of H1 , we get a knot K  H1 . Let N.K/ be a regular neighborhood of K and let A D S 1  I be an annulus joining l and @N.K/ (Figure 7). Consider an annulus twist ' W H1 n Int N.K/ ! H1 n Int N.K/, which consists in cutting H1 n Int N.K/ along A, twisting one of the boundaries of the cut by 360° and gluing the boundaries back. Then 'j@H1 D l is a twist along l, and '[email protected]/ is a twist along the longitude A \ N.K/. @H1

l K

A N.K/

Figure 7. Pushing of l into H1 .

Let Mi0 D H2 [hi .H1 n Int N.K//, i D 1; 2. Then the formula ´ '.x/; x 2 H1 n Int N.K/; ˆ.x/ D x; x 2 H2 ; determines a homeomorphism of M10 ! M20 (Figure 8). Since ' takes the meridian of N.K/ to a longitude of N.K/, we may conclude that M2 is obtained from M1 by a surgery along K. The claim is proved. id

H2 h1 D h2 l H1

H2 h2

'

H1

Figure 8. Pull-back of the homeomorphisms.

In order to prove the theorem, consider a Heegaard splitting M D H1 [h2 H2 and a Heegaard splitting S 3 D H1 [h1 H2 of S 3 of the same genus. By Theorem 2, h1 can be represented as h2 , where  is a superposition of twists along simple closed curves in @H1 . The effect of any additional twist factor in  results in a surgery along a knot in the corresponding 3-manifold. We may suppose that all those knots are disjoint and lie in H1  S 3 . Therefore, a sequence of twists gives a sequence of surgeries along a collection of disjoint knots in S 3 , i.e. a surgery along a link in S 3 .

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1.4 Spine presentation Definition 9. A compact polyhedron P is called simple if the link of any point x 2 P is homeomorphic to one of the following 2-dimensional polyhedra (Figure 9): (1) a circle (then x is called non-singular); (2) a circle with a diameter (then x is a triple point); (3) a circle with tree radii (then x is a true vertex).

Nonsingular point

Triple point

True vertex

Figure 9. Three types of neighborhoods.

We will call regular neighborhoods of true vertices butterflies. The body of each butterfly consists of four segments having a common endpoint, and it has six wings. Each wing spans two segments, and each pair of segments is spanned by exactly one wing. Any simple polyhedron P is naturally stratified. Each stratum of dimension 2 (a 2-component) is a connected component of the set of non-singular points. Strata of dimension 1 consist of open or closed triple lines, and dimension 0 strata are true vertices. Sometimes it is convenient to think of true vertices as transverse intersection points of triple lines. The set of all singular points of P is called the singular graph of P and denoted by S.P /. Definition 10. A simple polyhedron P is called special if it satisfies the following conditions: (1) Each 1-stratum of P is an open 1-cell. (2) Each 2-component of P is an open 2-cell. Definition 11. Let M be a compact 3-manifold with boundary. A polyhedron P  M is called a spine of M if M n P is homeomorphic to @M  .0; 1. By a spine of a closed 3-manifold M we mean a spine of M n Int.B 3 /, where B 3 is a 3-ball in M . A spine P is called special if P is a special polyhedron. Theorem 4. Any 3-manifold M has a special spine. Proof. Let T be a triangulation of a 3-manifold M . Consider a handle decomposition of M generated by T . This means the following: we replace each vertex with a handle

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Bi of index 0, each edge with a handle Cj of index 1, and each triangle with a handle Pk of index 2 S (Figure 2). The rest of M consists of index 3 handles. Let P D i;j;k @Bi [ @Cj [ @Pk be the union of the boundaries of all those handles. Then P is a special polyhedron and is indeed a special spine of M with several punctures, one puncture (removed open ball) for each handle. It remains to show that if M with m > 1 punctures has a special spine, then M with m  1 punctures also has a special spine. Since there are at least two punctures, there exist two distinct punctures B1 , B2 separated by a 2-component C1 of P (Figure 10). Now we fuse B1 and B2 into one ball so as to get a special spine of M with one puncture less. We do it by arc construction illustrated in Figure 10. C0

C2

B1

B1 C0

C1

C3 B2

E

C1 B1

Figure 10. Arc construction: we cut out two discs and attach an annulus and 2-cell E.

Finally, we show why each 2-component of the new spine is a 2-cell. There are two new 2-components: E (which is a 2-cell) and C 0 . The latter is obtained by joining C2 and C3 with a strip (the annulus used in the arch construction cut along an arc in @E). Therefore C 0 is a 2-cell assuming that C2 ; C3 are different 2-cells. To see that the proviso always holds, one may use the fact that we have started with two distinct balls separated by the 2-component C1 . Indeed, C2 differs from C3 since they separate different pairs of balls. After a few such steps we get a special spine P 0 of once punctured M . If M is closed, then we are done. If not, we slightly push P 0 into the interior of M and use the arch construction again to unite the ball and a component of M n P 0 homeomorphic to F  Œ0I 1/, where F is a connected component of @M . Example 2. A special spine P of the lens space L7;2 is shown in Figure 11. We draw only a regular neighborhood N of the singular graph S.P / in P . For restoring the whole special polyhedron we need to attach to N disks along all boundary curves of @N . Let us describe now two special polyhedra E and E 0 with boundary. E is a typical neighborhood of an edge in a simple polyhedron. It consists of a “cap” and a “cup” joined by a segment, with three attached wings (Figure 12). E 0 is the union of the lateral surface of a cylinder, a middle disc, and three wings (Figure 12). Note that there is a natural identification of @E with @E 0 .

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Figure 11. Special spine of L7;2 .

T

T 1

Figure 12. T -move.

Definition 12. The elementary move T on a simple polyhedron P consists in removing a proper subpolyhedron E  P and replacing it by E 0 . Notice that T increases the number of true vertices in a polyhedron by one, while the inverse move T 1 decreases it. Also note that we start and finish with simple polyhedra that have more than one true vertex each. This restriction is not burdensome since there are only four special spines with one vertex. The corresponding 3-manifolds are: S 3 , S 2  I and lens spaces L4;1 , L5;2 (Figure 13). Theorem 5 ([17]). Let P and Q be special polyhedra with at least two true vertices each. Then the following holds: (1) If P and Q are special spines of the same 3-manifold, then one can transform P into Q by a finite sequence of moves T ˙1 . (2) If one can transform P into Q by a finite sequence of moves T ˙1 and one of them is a special spine of a 3-manifold, then the other is a special spine of a homeomorphic manifold. In particular, homeomorphic special polyhedra determine homeomorphic 3-manifolds.

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a)

b)

c)

d)

Figure 13. Special spines of : a) L4;1 , b) L5;2 , c) S 3 , d) S 2  I .

2 Turaev–Viro invariants 2.1 Construction We divide the construction of the Turaev–Viro invariants into seven steps. (1) Fix an integer N 1 and consider the set C D f0; 1; : : : ; N  1g of integers. We will think of them as representing colors. (2) To each integer i D 0; 1; : : : ; N  1 assign a complex number wi called the weight of i . (3) Let E be a butterfly (i.e. a neighbourhood of a true vertex). Recall that it has six wings. We will color the wings by colors from the palette C in order to get different colored butterflies. The butterfly admits exactly N 6 different colorings. We consider colored butterflies up to color preserving homeomorphisms. ˇ ˇ ˇ j kˇ (4) To each colored butterfly we assign a complex number ˇ li m ˇ called the 6j n symbol (or the weight of the colored butterfly). Here i; j; k; l; m; n 2 C . Colors i, j and k correspond to the wings, attached to one edge, and l, m and n are the opposite wings (Figure 14).

j n

l

k i m

Figure 14. A colored butterfly.

(5) Let P be a special polyhedron, V .P / the set of its vertices, and C.P / the set of its 2-cells. Let Col.P / D f W C.P / ! C g be the set of all possible colorings

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of P . To each coloring  2 Col.P / assign a weight w./ by the rule ˇ ˇ Y Y ˇ i j k ˇ ˇ ˇ w.c/ : w./ D ˇ l m n ˇ v

v2V .P /

c2C.P /

(6) Finally, the weight of P is given by the formula X w./: w.P / D 2Col.P /

Definition 13. Let M be a 3-manifold. Consider a special spine P of M with 2 vertices, and define an invariant T V .M / by the formula T V .M / D w.P /, where w.P / is the weight of P . Theorem 6 ([20]). Let M be a 3-manifold and P be a special spine of M . If the 6j -symbols and weights wi are solutions to the system ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ i j k ˇ ˇ i j k ˇ Xˇ i m n ˇ ˇ j l n ˇ ˇ k l m ˇ ˇ ˇ ˇ 0 ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ l m n ˇ ˇ l m0 n0 ˇ D ˇ z n0 m0 ˇ ˇ z n0 l 0 ˇ ˇ z m0 l 0 ˇ wz ; z2C

then T V .M / does not depend on the choice of P . Therefore, T V .M / is a well-defined 3-manifold invariant. Note that the system in the previous theorem does not depend on M . The theorem actually says that any solution to the system induces an invariant of 3-manifolds. Proof. It is sufficient to show that w.P / is invariant with respect to T -moves. Let a special polyhedron P2 be obtained from a special polyhedron P1 by exactly one T -move, i.e. by removing a fragment E and inserting a fragment E 0 (Figure 12). For any coloring  of P1 , let Col .P2 / be the set of colorings of P2 that coincide with  on P1 n E D P2 n E 0 . Since only one 2-cell of the fragment E 0 (the middle disc) has no common points with @E 0 , the set Col .P2 / can be parameterized by the color z of this 2-cell. It follows that the set Col .P2 / consists of N colorings z , 0  z  N  1. Because of distributivity, the equation of the system in the theorem that P corresponds to the 9-tuple .i; j; k; l; m; n; l 0 ; m0 ; n0 / implies the equality w./ D z w.z / (Figure 15). To see this, multiply both sides of the equation by the constant factor that corresponds to the contribution made to the P weights by the exteriors of the fragments. Summing up the equalities w./ D z w.z / over all colorings of P1 , we get w.P1 / D w.P2 /.

2.2 One example of the Turaev–Viro invariant Set N D 2, so C consists of two colors f0; 1g. We will call them white and black. Let w0 D 1 and w1 D ", where " is one of the solutions to the equation "2 D " C 1. In

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m i

m

l

n

k

j

D

P z2C

m0 n0

i

n z

l j

m0 n0

l0

k

l0

Figure 15. Geometric presentation of equations.

order to assign weights to colored butterflies, we divide them into two groups. The first group consists of butterflies having at least one edge with two white and one black adjacent wings. All butterflies of that group have weights (i.e. symbols) 0. It means that if at some edge of a colored special polyhedron two white and one black wings meet together, then the weight of the coloring is 0. So there is no need to consider such colorings at all. All other butterflies belong to the second group. Their weights are given in Figure 16:

1

1

" 2

"1

"1

"2

Figure 16. Weights of colored butterflies.

Theorem 7. The previous selection of weights and 6j -symbols defines an invariant. Proof. We need to verify that the weight of a special polyhedron is preserved under T -moves. Let P be a spine of M and E  P be a regular neighborhood of an edge e. Let us perform the T -move along e. Consider a coloring  of E before this move and the induced colorings of E 0 after it (Figure 17) It is evident that the contributions to the total weights of complements to E and E 0 are the same. So we only need to verify that X !./jE D !.; z/jE 0 ; z2C

where z is a color (black or white) of an additional disc. It is easy to see that !./jE D P 1 1 " 2 " 2 , and z2C !./jE 0 D "3 1 C "3 ". After some manipulations we get !./jP D "2 D " C 1 D !./jE 0 . In a similar way we can verify invariance of !./ for all other colorings of E and E 0 . See [17] for details.

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black or white

e

T

Figure 17. Example of colorization.

3 JSJ-decomposition JSJ-decompositions of 3-manifolds had been discovered by W. Jaco–P. Shalen and K. Johannsen [10], [12]. Recall that a 3-manifold M is irreducible if any embedded 2-sphere in M bounds a 3-ball. Also, M is called prime if it is not a nontrivial connected sum of two other manifolds. Any prime manifold is irreducible, and the only closed orientable 3-manifold which is reducible and prime is S 2  S 1 . Theorem 8 (Kneser–Milnor [18]). Any closed orientable 3-manifold M can be presented in the form M D M1 # M2 # # Mn # k.S 2  S 1 /; where M1 ; : : : ; Mn are irreducible and k.S 2  S 1 / denotes a connected sum S 2  S 1 # # S 2  S 1 of k exemplars of S 2  S 1 . Moreover, this presentation is unique. Uniqueness in the previous theorem means the following: if M has two presenta0 0 0 tions M D M1 #M2 # #Mn #k1 .S 2 S 1 / and M D M1 #M2 # #Mm #k2 .S 2 S 1 /, 0 0 then n D m, k1 D k2 and fM1 ; : : : ; Mn g D fM1 ; : : : ; Mm g as sets. The meaning of the Kneser–Milnor theorem is that by working with 3-manifolds we can quite often reduce ourselves to irreducible ones. Let M be a closed orientable irreducible 3-manifold. There is no sense to cut it along spheres since all of them are trivial. Hence it would be natural to try more complicated surfaces. We begin with tori. Definition 14. A surface F in a 3-manifold M is called compressible if there is a disc D  M such that D \ T D @D and @D is a nontrivial curve in F . If there are no such discs, F is called incompressible.

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Definition 15. Let T  M be an incompressible torus in a closed 3-manifold M . Then T is called rough if any other incompressible torus T 0  M is virtually disjoint from T , i.e. there is an isotopy h t W M ! M such that h0 is the identity and T \h1 .T 0 / D ;. Definition 16. A collection T1 ; : : : ; Tn  M of disjoint rough tori is called a JSJsystem if (1) Ti and Tj are non-parallel for any i ¤ j ; (2) The collection is maximal, i.e. any other collection of disjoint rough tori satisfying (1) consist of not more than n tori. Theorem 9. Any manifold M has a JSJ-system. This system is unique up to isotopy. Proof. Existence of a JSJ-system is easy. Indeed, if M contains no rough tori, then the JSJ-system is empty. If M contains one, we will insert into M new and new rough tori such that each next torus is disjoint and non-parallel to all previous tori. The process stops since any 3-manifold can contain only finitely many disjoint nonparallel incompressible surfaces. Let us show that the resulting system T D fT1 ; T2 ; : : : Tn g is maximal and unique up to isotopy. Let T 0 D fT10 ; T20 ; : : : ; Tm0 g be any system of disjoint non-parallel rough tori in M . Since each torus Ti0 2 T 0 is rough, it can be shifted away from all tori of T . Furthermore, since the system T is maximal, each Ti must be parallel to one of the tori of T . It follows that the systems T 0 is isotopic to T or to a part of T . The next theorem is the result of efforts of many mathematicians. The last step was made by G. Perelman, see http://en.wikipedia.org/wiki/Geometrization_conjecture. Theorem 10. Let M be a closed orientable 3-manifold containing no rough tori. Then the following holds: (1) If M contains an essential torus, then M is a Seifert manifold; (2) If M does not contain essential tori, then M is either hyperbolic or a small Seifert manifold, which is fibered over S 2 with  3 exceptional fibers.

4 Normal surfaces 4.1 Normalization procedure Let T be a triangulation of a 3-manifold M . Definition 17 ([15], [5]). A proper surface F  M is called normal if F is in general position with respect to T and the following hold: (1) The intersection of F with every tetrahedron consists of discs. Those discs are called elementary.

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(2) The boundary of every elementary disc crosses at least one edge and crosses each edge at most once It is easy to show that any tetrahedron can contain elementary discs of seven different types: four triangles and three quadrilaterals (Figure 18).

Figure 18. Seven types of elementary discs.

Normal surfaces are very important for investigation of 3-manifolds. Let M be a 3-manifold. Then the set N of all normal surfaces in M possesses the following two informal properties: (1) N is informative since it contains representatives of all interesting classes of surfaces in M . In other words, we do not lose important information if we restrict ourselves to considering only normal surfaces. Certainly, the notions of interesting class and important information depend heavily on the problem we are trying to solve; (2) N admits a more or less explicit description. In order to confirm item 1, let us show how to transform any closed surface F in M into a normal one. To that end we describe normalization moves N1 –N4 on F . N1 : Suppose that the intersection of F with a tetrahedron  of T admits a nontrivial compressing disc D  , which meets F along @D. Then we compress F along D, i.e. cut F along @D and fill in two new boundary circles by two parallel copies of D. N2 : Suppose that F \  admits a nontrivial boundary compressing disc D, which meets F [ @ along @D such that (a) @D \ F is an arc in F ; (b) D \ @ is a segment of an edge e of  not contained in @M . Then we use D to eliminate two points of F \ e by an isotopy of F (Figure 19). N3 : Suppose that a component of F \  is a 2-sphere. Then we remove it. N4 : Suppose that F has a spherical component intersecting a triangle of T along a circle and consisting of two discs in the tetrahedra adjacent to the triangle. Then we remove this component.

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F D e

e Figure 19. Reducing e \ F by an isotopy.

Let us analyze the behavior of F under moves N1 –N4 from a purely topological point of view, without taking into account the triangulation. N1 compresses a tube, N2 is an isotopy, N3 and N4 consist in removing an inessential 2-sphere. Theorem 11. Any general position closed surface F in a triangulated 3-manifold M can be transformed into a normal surface F 0 by a sequence of moves N1 –N4 . Proof. Let us apply to F moves N1 –N4 as long as possible. Each move simplifies either F or the intersection of F with the tetrahedra of the triangulation. So we stop. Let us show that the resulting surface is normal. Indeed, the intersection of F 0 with any tetrahedron  consists of incompressible surfaces (otherwise we could apply N1 ). Any proper incompressible surface in  is a union of discs and spheres, but we cannot have spheres because of N3 . It remains to show that any component of @.F 0 \ T / crosses any edge no more than once. On the contrary, suppose that a component of @.F 0 \ T / crosses an edge e at least twice. Then there exists a boundary compressing disc D for F 0 \ T such that D \ @T is a subinterval of e. It follows that one can apply N2 , a contradiction. It is important to notice that if M is irreducible and F is incompressible and not a 2-sphere, then the normalization procedure determines an isotopy from F to F 0 .

4.2 Matching system Let T be a triangulation of a 3-manifold M . Denote by E1 ; E2 ; : : : ; En all types of elementary discs in all the tetrahedra (we consider the discs up to normal isotopy). Since each tetrahedron contains 7 types of elementary discs (4 triangles and 3 quadrilaterals), we have n D 7t , where t is the number of tetrahedra in T . We assign to those types integer variables x1 ; x2 ; : : : ; xn . In order to get equations, consider an angle of a common triangular face of two tetrahedra T1 and T2 . Each of the tetrahedra contains two types of elementary discs that intersect both sides of the angle: one triangle and one quadrilateral. Let Ei ; Ej  T1 and Ek ; El  T2 be those types. Then we write the equation xi C xj D xk C xl (Figure 20). Doing so for all angles of all triangles

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inside M , we get a system of 3m equations, where m is the number of those triangles (if M is closed, then m D 2t). xi

T1

xj

xk

xl

T2

Figure 20. The number of edges on glued faces must be equal.

Definition 18. The matching system is the system of equations of type xi C xj D xk C xl (one equation for each angle of each triangle) and inequalities xi 0 for 1  i  n. To each normal surface F we assign an n-tuple x.F / D .x1 ; x2 ; : : : ; xn / of nonnegative integer numbers in the following natural way: xi is the number of elementary discs (triangles or quadrilaterals) of type Ei in the intersection of F with the tetrahedra. It is obvious that the obtained tuple is a solution to the matching system. Definition 19. A nonnegative solution to the matching system is called admissible if for all three types of quadrilateral discs in the same tetrahedron, no more than one of the corresponding variables is positive. Since any two quadrilaterals in the same tetrahedron have nonempty intersection, only admissible solutions can correspond to embedded surfaces. Theorem 12. There exists a natural bijection between the set of admissible solutions to the matching system for a triangulated 3-manifold M and the set of normal surfaces in M . In this theorem we consider normal surfaces up to normal isotopy, which is invariant on all simplices of the triangulation. In order to realize a given admissible solution x D .x1 ; : : : ; xn /, we place into each tetrahedron the corresponding number of elementary discs so that the discs match along all triangle faces and thus form a normal surface. Already this theorem gives a good numerical description of the set N of all normal surfaces. Nevertheless, we want more.

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4.3 Fundamental surfaces Let E be a system of linear homogeneous equations and inequalities with integer coefficients. We do not require that the number of equations (inequalities) be equal to the number of variables. Quite contrary, the number of variables can be much greater than the number of equations and inequalities. Definition 20. A nonnegative integer solution x to E is called fundamental if it cannot be presented as x D y C z, where y, z are two other nonnegative nontrivial solutions. Theorem 13. The set of fundamental solutions to any system E of linear homogeneous equations and inequalities with integer coefficients is finite. Proof. Let Rn be a Euclidean space with coordinates .x1 ; : : : ; xn / which correspond to the variables of E. Denote by n1 the simplex in Rn with vertices .1; 0; : : : ; 0/; : : : ; .0; 0; : : : ; 1/. Let S be the set of all nonnegative solutions to E over the real numbers, L the support plane for n1 , and P D S \ L. Then we have: (1) P is the intersection of L with hyperplanes given by the above equations and half-spaces xi 0; (2) P is contained in n1 and hence is bounded. It follows that P is a convex polyhedron of dimension m  n  1. S can be considered as the union of straight rays that start at the origin and pass through points of P (Figure 21). The vertices of P have rational coordinates. Multiplying each vertex ki vi ı k1 v1

k3 v3

k2 v2 vi

v1

P v2

v3

O Figure 21. The solution space is the cone over P .

vi by the smallest number ki > 0 such that the coordinates of ki vi are integer, we get the set V D fki vi g of so-called vertex solutions. The vertex solutions are necessarily fundamental.

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Since P , as any convex polyhedron of dimension m, can be decomposed into msimplices without introducing new vertices, S can be presented as the union of infinite cones over m-simplices with vertices in V. It is sufficient to prove that each such cone contains only finitely many fundamental solutions. Let ı be an m-simplex with vertices w0 ; : : : ; wm 2 V and Sı  S the infinite cone over ı. Since any point of ı is a nonnegative linear combination of its vertices, P any integer point x 2 Sı can be presented in the form x D m ˛ iD0 i wi , where all ˛i are nonnegative. If one of the coefficients (say, ˛i ) is greater than 1, then x is not fundamental, since it can be presented as the nontrivial sum x D .x  .˛i  1/wi / C .˛i  1/wi of nonnegative integer solutions. We can conclude that all fundamental P solutions in Sı are contained in the compact set U D f m iD0 ˛i wi W 0  ˛i  1g. Since they are integers, there are only finitely many of them. Definition 21. A normal surface F is called fundamental if it corresponds to an admissible fundamental solution to the matching system. Let us describe a geometric interpretation of algebraic summation of solutions. Let x, y be two admissible solutions such that their sum is admissible, and let F1 , F2 be the corresponding normal surfaces. Shift F1 and F2 by a normal isotopy so that for each tetrahedron T every connected component of Fi \ T is either a flat triangle or a quadrilateral composed of two flat triangles. Then the intersection of any two elementary discs of F1 and F2 in T consists of no more than two arcs with endpoints in the interiors of triangular faces of T . Consider a double line c of F1 and F2 and decompose it into arcs which are connected components of the intersection of c with tetrahedra of the triangulation. Let l  c \ T be one of these arcs. It belongs to the intersection of two elementary discs D1 , D2 (pieces of F1 and F2 ) in T . There are two cut-and-paste operations along l: we cut the discs along l and glue them again in one of the two possible ways. The operations are called switches. The regular switch along l produces two elementary discs of the same types as D1 , D2 . In the case of irregular switch we get at least one disc that crosses an edge twice, i.e. does not produce a normal surface. See Figure 22, where we illustrate the case of a triangle and a quadrilateral. irregular

regular

Figure 22. Regular and irregular switches.

Note that regular switches along all arcs in c agree in the triangle faces and thus give a global regular switch along c. Now let us perform regular switches along all

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double lines of F1 \ F2 . By construction, the resulting surface F is normal and x.F N / D x.F N 1 / C x.F N 2 /. We will write F D F1 C F2 and call F the geometric sum of F1 and F2 . Note that x.F N / depends only on x.F N 1 /, x.F N 2 /. Thus the normal isotopy class of F is defined correctly although the relative position of F1 , F2 (in particular, the number of curves in F1 \ F2 ) can vary when we shift the surfaces by normal isotopies. If they are disjoint, then the sum is their union. The following theorem is a direct corollary of the definition of fundamental surface. Theorem 14. Any normal surface can be presented as a geometric sum of fundamental surfaces. The theory of normal surfaces is used extensively in algorithmic topology. Algorithms based on it most often follow the General Scheme described below. Suppose that we wish to solve a problem about a given 3-manifold M . General Scheme. 1. Reduce the problem at hand to one of the existence in M of a surface with some specific characteristic property; 2. Choose a triangulation of M and show that if M contains at least one characteristic surface F , then there exists a normal characteristic surface; 3. Show that if there is a normal characteristic surface, then there is a fundamental characteristic surface. One possible way to do that is to prove that if a non-fundamental characteristic surface F D F1 C F2 is not fundamental, then at least one of F1 ; F2 is also characteristic; 4. Construct an algorithm to decide whether or not a given surface is characteristic. Assume that all four steps of the General Scheme are carried out. Then the algorithm that solves the problem works as follows: (1) Choose a triangulation T of M . (2) Write down the corresponding matching system of linear equations. (3) Find the finite set of fundamental solutions. (4) Realize the fundamental solutions by normal surfaces. (5) Test each of the obtained fundamental surfaces for being characteristic. It follows that M contains a characteristic surface (i.e. that the problem in question has a positive answer) if and only if at least one of the fundamental surfaces is characteristic. The following questions can be answered by the above approach. (1) Is a given link in S 3 splittable, i.e. it can be divided into two nonempty sublinks by a 2-sphere? (2) Is a given knot trivial? (3) Is a given 3-manifold M irreducible? (4) Does M contain an essential incompressible surface?

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5 Algorithmic classification of Haken 3-manifolds 5.1 Introduction The following is known as the recognition problem for 3-manifolds: Does there exist an algorithm to decide whether or not two given 3-manifolds are homeomorphic? Why is this problem important? The reason is that a positive answer would imply the existence of an algorithmic classification of 3-manifolds. Indeed, one can easily construct an algorithm which enumerates step by step all compact 3-manifolds. Using it, we could create a list M1 ; M2 ; : : : of all 3-manifolds without duplicates by inquiring if each next manifold has been listed before. It is this list that is considered as a classifying list of 3-manifolds. Certainly, this is a classification in a very weak sense. The knowledge that the classifying list exists would not help to answer many possible questions. It is the proof of existence that is important since it gives a deeper understanding of the intrinsic structure of 3-manifolds. In this section we describe a positive solution of the above problem for Haken manifolds, see the definition below. This case is especially important since it implies a positive solution of the algorithmic classification problem for knots, one of the most intriguing problems of low-dimensional topology. Definition 22. A compact irreducible 3-manifold M is called a Haken manifold if it contains an essential surface (by an essential surface we mean a connected incompressible, boundary incompressible surface in M which is not parallel to @M and is not a 2-sphere). For example, for any nontrivial knot K  S 3 its complement space S 3 n Int N.K/, where N.K/ is a tubular neighborhood of K, is a Haken manifold. Theorem 15 (The Recognition Theorem). There is an algorithm to decide whether or not two given Haken 3-manifolds are homeomorphic. Corollary 1. There exists an algorithmic classification of Haken 3-manifolds. Corollary 2. There exists an algorithmic classification of knots and links in S 3 . Formally speaking, the second corollary does not follow from the Recognition Theorem since links (in contrast to knots) are not determined by their complements. Nevertheless, there is a stronger version of the Recognition Theorem. It has actually the same proof and takes into account boundary patterns, where a boundary pattern is a fixed graph on the boundary of a given 3-manifold. Since any link is determined by its complement together with a boundary pattern consisting of meridians of the link components, we get an algorithmic classification of links.

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The history of the positive solution of the recognition problem for Haken 3manifolds is very interesting. In 1962 W. Haken suggested an approach for solving the problem [6]. However, this approach contained a conceptual gap. Thanks to efforts of several mathematicians, by the early seventies a crucial obstacle was singled out, and, when in 1978 G. Hemion overcame it [7], it was broadly announced that the problem has been solved [13], [22]. Later on many topologists used extensively this result, but there appeared no paper contained a complete proof of that result. All papers and even books ([13], [22], [14], [8]) devoted to this subject were written according to the same scheme: they contained informal descriptions of Haken’s approach, of the obstacle, of Hemion’s result, and the claim that these three ingredients were sufficient. However, it turned out that there is another obstacle of similar nature that cannot be overcome by the same tools as the first one. Luckily the second obstacle was overcome in [17] by using an algorithmic version of W. Thurston’s theory of surface homeomorphisms [1].

5.2 Hierarchies and extension moves Let us describe the main idea of the proof of the Recognition Theorem. Suppose that we invented an algorithmic procedure which, given a 3-manifold M , constructs a canonical skeleton of M , which by definition is a special spine PM of M or of several times punctured M . Saying “canonical”, we mean that PM depends only on the topological type of M . In other words, we wish that if two manifolds are homeomorphic, then so are their canonical skeleta. This would prove the Recognition Theorem since by Theorem 5 homeomorphic skeleta determine homeomorphic 3manifolds and the recognition problem for special polyhedra is easily solvable. A key role in the construction of skeleta is played by the notion of hierarchy. The original definition treats hierarchy as a process of cutting a given 3-manifold M along essential surfaces such that the resulting 3-manifold is a disjoint union of balls [21]. We will use the modern approach: instead of cutting M along surfaces we insert them into the manifold. Such insertions are called extension moves. One can always achieve that the result of extension moves, i.e. the union P of all surfaces used by the construction, is a skeleton of M . The advantage of this approach is that there is no need to keep in memory all intermediate steps of the construction since P carries all the information about M . Example 3. Let us construct a canonical skeleton for M D T 2  I , where T 2 D S 1  S 1 is a 2-dimensional torus. The first inserted surface consists of two tori @M D T 2  f0; 1g. The second surface is an annulus A of type A D C  I , where C is a nontrivial simple closed curve in T 2 . Any other essential annulus A0 in M is equivalent to A in the sense that there is a homeomorphism M ! M taking A0 to A, so there is essentially only one extension move of that type. The complementary space V of @M [ A is a solid torus. We equip it with the boundary pattern D @A, which consist of longitudes of V . The third inserted surface is a meridional disc D

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of V which intersects A along two disjoint radial segments. D is also determined up to homeomorphisms M ! M taking A to A, so the move of that type is also unique. The resulting polyhedron P D @M [ A [ D can be taken as a canonical skeleton of M (Figure 23).

Figure 23. Canonical skeleton of T 2  I .

The above example shows how one can solve the recognition problem for that particular manifold M D T 2  I . Given an arbitrary Haken 3-manifold M 0 , we use normal surface theory to construct an essential annulus in A0  M 0 and an essential disc D 0  M 0 such that @M 0 [A0 D @D 0 and @D 0 crosses triple lines of @M 0 [A0 at two points. If there are no such annuli or discs, or if the polyhedron @M 0 [ A0 [ D 0 differs from the polyhedron P constructed in the example, then M 0 and M are different. Otherwise they are homeomorphic. It turns out that the idea of algorithmic construction of canonical skeleta illustrated in Example 3 works in many other cases. Let us say that a simple spine P  M is admissible if it contains @M and each 2-component of P is incompressible. We will assume that the closure Q of any connected component of M n P is a 3-manifold such that Q \ P D @Q. We call Q a chamber of P and equip it with the boundary pattern  D @Q \ S.P /. Let .Q; / be a chamber of an admissible polyhedron P  M and let F be a proper orientable incompressible surface in Q. We may assume that F separates Q since otherwise we may replace F by two parallel copies of F . Definition 23. The transition from P to P [ F is called extension move on P . We plan to do the following. Let P0 D @M if @M ¤ ; and P0 is an essential surface in M if M is closed (here is the only place where we use the assumption that M is Haken). Then we apply to P0 extension moves (that is, erect inside chambers new and new partition walls) as long as possible until getting a skeleton. The first difficulty is that at each step there may be infinitely many possibilities to insert an essential surface, i.e. to perform an extension move. If we use all of them, we get infinitely many different hierarchies and thus infinitely many different skeleta, which is inappropriate for algorithmic procedures. Fortunately, we do not have to take terminal skeleta for all hierarchies. It turns out that one can subject the process of inserting new surfaces (i.e. of erecting new partition

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walls) to so strong restrictions that for any Haken manifold we get only a finite set of terminal skeleta. This would be sufficient for proving the Recognition Theorem since comparing finitely many special polyhedra is algorithmic. Let us describe three basic types E1 –E3 of allowable extension moves. Consider a simple subpolyhedron P of a Haken 3-manifold M and a chamber Q of P . E1 : Inserting an essential torus. Suppose that there is an essential torus T  Int Q. Then we replace P by P [ T . Let us show that Q contains only finitely many essential tori up to homeomorphisms of Q fixed at @Q. Indeed, consider a JSJ-system for Q. It decomposes Q into Seifert manifolds and manifolds without essential tori. All essential tori in Seifert manifolds are vertical (i.e. consist of fibers). Therefore they can be represented by simple closed curves in the base surface, i.e. by their projections. It remains to note that for any surface F there are only finitely many types of curves in F (up to homeomorphisms F ! F ). For example, any surface of genus four contains three types of nontrivial curves: (i) non-separating, (ii) separating F into punctured surfaces of genus 1 and 3, (iii) separating F into two punctured surfaces of genus 2. The next move is similar to E1 , but instead of tori we insert proper annuli. We call a proper annulus A in a chamber Q of P clean if @A does not intersect the boundary pattern  of Q. There are two types of annuli. A clean essential annulus A  Q is called longitudinal if for any other clean essential annulus A0  Q there is an isotopy h t W .Q; / ! .Q; /, 0  t  1, such that h0 D id and h1 .A0 / \ A is either empty or consists of middle circles of the annuli. If A is not longitudinal, then it is called transverse. It means that there is a clean essential annulus A0  Q such that A \ A0 consists of radial segments and cannot be destroyed by isotopy .Q; / ! .Q; /. E2 : Inserting a longitudinal annulus. Suppose that there is a longitudinal annulus A  Int Q. Then we replace P by P [ A. A relative version of the JSJ-decomposition theorem (involving boundary patterns, rough clean annuli, and a finiteness theorem for proper arcs in surfaces) tells us that Q contains only finitely many longitudinal annuli up to homeomorphisms Q ! Q fixed on @Q. There is a drastic difference between longitudinal and transverse annuli since the number of non-equivalent transverse annuli in Q, considered up to homeomorphisms .M; P / ! .M; P /, may be infinite. For example, let Q be an I -chamber, i.e. a direct or twisted product of a surface and an interval. Then any two non-isotopic vertical essential annuli in Q may be different, even if there exists a homeomorphism h W Q ! Q taking one annulus to the other. The reason is that in general you cannot extend h to a homeomorphism M ! M . Definition 24. Let F be a proper surface in a chamber .Q; /. Then .F / D .F / C #.@F \ / is called the pattern complexity (or, in abbreviated form, pcomplexity) of F .

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E3 : Inserting a minimal essential surface. Suppose that all essential surfaces in Q have positive p-complexity. Then we choose an essential surface of minimal p-complexity and insert it into Q. Let us show that Q contains only finitely many essential surfaces of minimal pcomplexity and all of them can be constructed algorithmically. Indeed, it follows from the assumption that all fundamental surfaces have positive p-complexities. On the other hand, p-complexity is additive with respect to summation of surfaces. It means that for any integer N there are only finitely many combinations of fundamental surfaces having total p-complexity N and hence only finitely many essential surfaces of p-complexity N . Therefore we may try N D 1; 2; : : : until getting the level containing the first essential surface (which exists since @Q ¤ ;). Then we can enumerate all essential surfaces of that p-complexity. The following proper connected surfaces have nonpositive p-complexity: (1) Spheres, projective planes, and clean discs (they have negative p-complexity). (2) Clean annuli, clean Möbius bands, tori, Klein bottles and proper discs whose boundary circles cross the pattern at exactly one point (they have p-complexity 0). How to get rid from all of them? Recall that M and hence each chamber .Q; / are irreducible and boundary irreducible. It means that Q cannot contain essential surfaces of negative p-complexity. We may also assume that there are no discs of p-complexity 0 and that all possible E1 and E2 moves have been already performed. So Q contains no essential tori (hence no Klein bottles) and no longitudinal annuli. The difficulty is that we cannot get rid of transverse annuli. The good news is that the only chambers which contain transverse annuli but no other surfaces of nonnegative p-complexity are I -chambers, that is, direct or twisted products of a surface and an interval.

5.3 Proof of Theorem 15 Let .M; / be a given Haken manifold. Denote by P0 the boundary of M (if M is closed, P0 is empty). Let us apply to P0 step by step extension moves E1 –E3 . Doing so, at each step we multiply the pair .M; Pi / in several number of exemplars to be able to realize separately all possible extensions. When this branched process stops, we get a finite set of admissible polyhedra in M . Let us analyze their chambers. If a chamber Q contains no transverse annuli, then @Q is a sphere (otherwise we could apply extension moves further). By irreducibility of M , Q is ball. If Q contains a transverse annulus, then Q is an I -chamber of type z I , where F is an orientable surface and G  z I is the orientable twisted F  I or G  product of a non-orientable surface G and an interval. The union U of all I -bundle chambers can be organized into chains of four types. Direct and Stallings chains consist of direct I -bundle chambers. The difference is that direct chains are not closed and each intersects @M along two exemplars of the base surface. The Stallings chains are closed and are contained inside M . Here is

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twisted

direct

Figure 24. I -bundle chambers form chains of four types: direct, twisted, Stallings, and quasiStallings.

a right moment to recall that any Stallings manifold Mh with fiber F has a form .F  I /=h , where h W F  f0g ! F  f1g is determined by an orientation-preserving homeomorphism F ! F , which we denote also by h. Twisted and quasi-Stallings chains consist both of direct and twisted I -bundle chambers. Again, the difference is that twisted chains intersect @M while the quasiStallings do not. By definition, a quasi-Stallings manifold has a form .F  I /=.˛;ˇ / , where F is an orientable surface and ˛ W F  f0g ! F  f0g, ˇ W F  f1g ! F  f1g are orientation-reversing involutions without fixed points. In order to terminate the process of construction of simple skeleta of M we have to subdivide those chains of I -chambers into balls. For non-closed chains this can be done easily since they are I -bundles, and the structure of I -bundles is known well. For Stallings and quasi-Stallings chains this is a difficult problem, which cannot be solved within the above idea of dividing chambers by controlled insertion of new surfaces. The conclusion is that one should solve the recognition problem for Stallings and quasi-Stallings manifolds independently, by other methods. Moreover, it suffices to solve the problem in the fiber-preserving sense, i.e. suggest an algorithm to determine if two given Stallings or quasi-Stallings manifolds are fiber-preserving homeomorphic. Let us consider these cases separately. It is easy to see that two Stallings manifolds Mh , Mg are fiber-preserving homeomorphic if and only if their monodromy homeomorphisms f; g W F ! F are conjugate (the latter means that there exists a homeomorphism h W F ! F such that hf h1 D g). The conjugacy problem and hence the recognition problem for Stallings manifolds was solved by G. Hemion. Theorem 16 (G. Hemion [7], [8]). Suppose that homeomorphisms f; g of a surface F onto itself admit no essential periodic curves. Then one can construct finitely many homeomorphisms hi W F ! F , 1  i  m such that any homeomorphism h W F ! F conjugating f to g has the form h D hi f n , where 1  i  m and n is an integer.

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Note that hi f n conjugates f to g if and only if so does hi . Therefore, Theorem 16 reduces the conjugacy problem of f , g to testing finitely many homeomorphisms hi , which can be easily done algorithmically. Let us describe a solution of the recognition problem for quasi-Stalling manifolds. It is easy to see that any quasi-Stallings manifold M˛;ˇ admits a two-sheeted covering p W M˛ˇ ! M˛;ˇ by a Stallings manifold whose monodromy is the product of involutions ˛, ˇ. So manifolds M˛;ˇ , M˛0 ;ˇ 0 can be fiber-preserving homeomorphic only if so are the corresponding Stallings manifolds M˛ˇ , M˛0 ˇ 0 . Suppose that they are. It follows from the Hemion’s Theorem that there is a conjugating homeomorphism of the form h D gf n , where f D ˛ˇ and g is one hi . It remains to answer the following question: given hi and f , does there exist n such that hi f n conjugates ˛ to ˛ 0 , ˇ to ˇ 0 and hence determines a homeomorphism M˛;ˇ ! M˛0 ;ˇ 0 ? This question can be easily reduced to the following problem. Problem. Given homeomorphisms f; g W F ! F , can we decide algorithmically whether or not g is isotopic to an integer power of f ? If the answer is affirmative, even only for homeomorphisms admitting no periodic curves, we get a recognition algorithm for quasi-Stallings manifolds. It turns out that the answer is affirmative indeed. The proof is based on the Thurston theory of surface homeomorphisms [19] and references therein. We derive from it a few facts needed for the proof. The main fact is that if a homeomorphism f W F ! F has no periodic curves and .F / < 0, then f is isotopic to a pseudo-Anosov homeomorphism. Therefore, one can assign to f a number .f / called the stretching factor. It possesses the following properties (see [19], [2]): (1) .f / is an algebraic number greater than 1; (2) .f n / D .f /jnj for any integer n ¤ 0; (3) isotopic homeomorphisms have the same stretching factor; Moreover, it is proved in [1] that the stretching factor can be calculated algorithmically. One can write a computer program that assigns to any f a matrix with nonnegative integer elements such that .f / is the maximal eigenvalue of that matrix. Let us show how stretching factors help us to solve the above problem and thus to finish the proof of Theorem 15. Let f , g be given, say, as compositions of Dehn twists. We calculate .f / and .g/. Since .f / > 1 by property (1), one can find an integer number N such that N .f / > .g/. It follows from properties (2), (3) that if an integer power f n of f is isotopic to g, then jnj < N . So to answer the question whether or not g is isotopic to an integer power of f it suffices to test all integer numbers n between N and N for possessing the required property. Note that if such an n does exist, then it is unique. Indeed, if g D f n and g D f m , then n D m since f is not periodic.

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References [1]

M. Bestvina and M. Handel, Train-tracks for surface homeomorphisms. Topology 34 (1995), no. 1, 109–140.

[2]

A. J. Casson and S. A. Bleiler, Automorphisms of surfaces after Nielsen and Thurston. London Math. Soc. Stud. Texts 9, Cambridge University Press, Cambridge 1988.

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A. T. Fomenko and S. V. Matveev, Algorithmic and computer methods for three-manifolds. Math. Appl. 425, Kluwer Academic Publishers, Dordrecht 1997.

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C. McA. Gordon and J. Luecke, Knots are determined by their complements. J. Amer. Math. Soc. 2 (1989), no. 2, 371–415.

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W. Haken, Theorie der Normalflächen. Acta Math. 105 (1961), 245–375.

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W. Haken, Über das Homöomorphieproblem der 3-Mannigfaltigkeiten. I. Math. Z. 80 (1962), 89–120.

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G. Hemion, On the classification of homeomorphisms of 2-manifolds and the classification of 3-manifolds. Acta Math. 142 (1979), no. 1–2, 123–155.

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G. Hemion, The classification of knots and 3-dimensional spaces. Oxford Sci. Publ., The Clarendon Press, Oxford University Press, New York 1992.

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W. Jaco, Lectures on three-manifold topology. CBMS Regional Conf. Ser. in Math. 43, Amer. Math. Soc., Providence, R.I., 1980.

[10] W. Jaco and P. B. Shalen, A new decomposition theorem for irreducible sufficiently large 3-manifolds. In Algebraic and geometric topology, Part 2, Proc. Sympos. Pure Math. XXXII, Amer. Math. Soc., Providence, R.I., 1978, 71–84. [11] W. Jaco and P. B. Shalen, Seifert fibered spaces in 3-manifolds. Mem. Amer. Math. Soc. 21 (1979), no. 220,. [12] K. Johannson, Homotopy equivalences of 3-manifolds with boundaries. Lecture Notes in Math. 761, Springer-Verlag, Berlin 1979. [13] K. Johannson, Topologie und Geometrie von 3-Mannigfaltigkeiten. Jahresber. Deutsch. Math.-Verein. 86 (1984), no. 2, 37–68. [14] K. Johannson, Classification problems in low-dimensional topology. In Geometric and algebraic topology, Banach Center Publ. 18, PWN, Warsaw 1986, 37–59 [15] H. Kneser, Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten. Jahresber. Deutsch. Math.-Verein. 38, 248–260 (1929) [16] W. B. R. Lickorish, A finite set if of generators for the homotopy group of a 2-manifold. Proc. Cambridge Philos. Soc. 60 (1964), 769–778. [17] S. V. Matveev, Algorithmic topology and classification of 3-manifolds. Second edition, Algorithms Comput. Math. 9, Springer-Verlag, Berlin 2007. [18] J. Milnor, A unique factorisation theorem for 3-manifolds, Amer. J. Math. 84 (1962), 1–7. [19] W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. [20] V. G. Turaev and O.Ya. Viro, State sum invariants of 3-manifolds and quantum 6j -symbols. Topology 31 (1992), no. 4, 865–902.

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[21] F. Waldhausen, On irreducible 3-manifolds which are sufficiently large. Ann. of Math. (2) 87 (1968), 56–88. [22] F. Waldhausen, Recent results on sufficiently large 3-manifolds. In Algebraic and geometric topology, Part 2, Proc. Sympos. Pure Math. XXXII, Amer. Math. Soc., Providence, R.I., 1978, 21–38.

An introduction to globally symmetric spaces Gabriele Link Institut für Algebra und Geometrie, Karlsruhe Institue of Technology (KIT) 76128 Karlsruhe, Germany email: [email protected]

Contents 1

Generalities on symmetric spaces . . . . . . . 1.1 Geometric definition . . . . . . . . . . . 1.2 The group of isometries . . . . . . . . . 1.3 Algebraic point of view . . . . . . . . . 1.4 Geodesics and curvature . . . . . . . . . 1.5 Examples . . . . . . . . . . . . . . . . . 1.6 The Killing form . . . . . . . . . . . . . 1.7 Decomposition of symmetric spaces . . . 2 Symmetric spaces of non-compact type . . . . 2.1 Flats and rank . . . . . . . . . . . . . . 2.2 Roots and root spaces . . . . . . . . . . 2.3 Iwasawa decomposition . . . . . . . . . 2.4 The space of maximal flats . . . . . . . . 2.5 Weyl group and opposition involution . . 2.6 Cartan decomposition and Cartan vector . 3 The geometry at infinity . . . . . . . . . . . . 3.1 The geometric boundary of S . . . . . . 3.2 The Furstenberg boundary . . . . . . . . 3.3 The Bruhat decomposition . . . . . . . . 3.4 Visibility at infinity . . . . . . . . . . . . 3.5 Busemann functions and distances . . . . References . . . . . . . . . . . . . . . . . . . . . .

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Introduction The present text provides lecture notes for a course on symmetric spaces given in the framework of the “Semaine spéciale M2 : Géométrie et théorie des groupes” held at the Institut de Recherche Mathématique Avancée in Strasbourg from April 28 to  Supported

by the FNS grant PP002-102765 and the DFG grant LI 1701/1-1.

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May 3, 2008. It intends to give an accessible introduction to the theory of Riemannian symmetric spaces with an emphasis on those of non-compact type. Since the excellent textbook [H] by S. Helgason on the subject contains complete proofs of all relevant results way beyond the scope of this introduction, we content ourselves here with merely stating certain results, giving precise references for the more ambitious reader. We want to give the reader a guideline through a part of the landscape, trying to motivate the steps we take and illustrating the basic results by means of a detailed treatment of particular important examples. For a deeper understanding, the reader is strongly encouraged to study the books by Helgason [H], Eberlein [E] and also Borel [B1] and Wolf [Wo]. The plan of the text is as follows: Section 1 gives an overview on the geometry and algebraic coding of arbitrary globally symmetric spaces. In Section 2 we investigate more precisely the case of symmetric spaces of non-compact type which, in particular, are manifolds of non-positive sectional curvature. Their theory is intimately related to the theory of semi-simple Lie groups, so we describe the important Iwasawa and Cartan decompositions of such spaces. Finally in Section 3 we study the geometry at infinity of globally symmetric spaces of non-compact type: like any Hadamard manifold these spaces can be compactified by adding a sphere at infinity. Due to the rich structure of symmetric spaces, this geometric boundary can be described more precisely: we give a parametrization of boundary points in terms of the Cartan decomposition, relate it to the Furstenberg boundary and show how the Bruhat decomposition helps to describe pairs of boundary points which can be joined by a geodesic. In the last section we study Busemann functions and see how they can be used to obtain invariant Finsler metrics on the differentiable manifold underlying the symmetric space. Acknowledgments. The author is grateful to the organizers of the “Semaine spéciale” in Strasbourg for the opportunity to give this lecture series. She also warmly thanks her thesis advisor Enrico Leuzinger for introducing her to the beautiful theory of symmetric spaces.

1 Generalities on symmetric spaces In this section we begin with a definition of Riemannian symmetric spaces and deduce many important properties from it. We will see that such manifolds have a huge group of isometries which acts transitively. Moreover, any simply connected symmetric space S is diffeomorphic to a homogeneous space G=K, where G is a connected Lie group with an involutive automorphism whose fixed point set is essentially the compact subgroup K  G. This algebraic coding allows to describe the geometry in Lie algebraic terms: we will see that geodesics are projections to G=K of certain one-parameter subgroups of G, the curvature tensor is described by Lie brackets, and totally geodesic submanifolds correspond to Lie triple systems. In particular, the Levi-Civita connection remains

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the same when endowing the differentiable manifold S with a different Riemannian structure with respect to which S is also a symmetric space. Towards the end of this introductory section we will see that after dividing out a direct factor isomorphic to a Euclidean motion group, the isometry group G becomes semi-simple; in this way the problem is reduced to the study of certain involutive automorphisms of semi-simple Lie algebras.

1.1 Geometric definition Let S be a connected Riemannian manifold and x 2 S. The  geodesic symmetry  sx at x is the local diffeomorphism defined by sx .y/ ´ expx  idTx S .exp1 .y// . x Definition 1.1. S is called locally symmetric, if sx is a local isometry for all x 2 S. If sx is a global isometry for all x 2 S , then S is called (globally) symmetric. Examples. S D En , Sn , Hn are globally symmetric, and any quotient nS, where   Is.S / is a discrete, torsion free group of isometries of S , is locally symmetric. Theorem 1.2 ([H], Theorem IV.5.6). A simply connected locally symmetric space is globally symmetric. Notice that this theorem in particular implies that the Riemannian universal cover of a locally symmetric space is globally symmetric. Conversely, every locally symmetric space is a quotient of a globally symmetric space by a discrete, torsion free group of isometries isomorphic to the fundamental group. In these notes we will only be concerned with globally symmetric spaces. Let d denote the distance function on S induced from the Riemannian metric. Proposition 1.3. If S is globally symmetric, then S is complete and homogeneous. Proof. For completeness we show that all geodesics are defined on R. The claim then follows from the theorem of Hopf–Rinow. Let c be a geodesic in S and suppose there exists b 2 R such that c is defined on .a; b/ for some a < b, but not at b. Take " D ba and consider the geodesic 4 symmetry sx at x ´ c.b  "/. Then c.b/ D sx .c.b  2"// exists, hence c is defined at parameter b which contradicts our assumption. By the Theorem of Hopf–Rinow and completeness we know that for any pair of points x; y 2 S and t ´ d.x; y/ there exists a geodesic c W R ! S such that c.0/ D x and c.t/ D y. Then y D sc.t=2/ B sx .x/, i.e. S is homogeneous. Remark 1.4. Notice that the isometry sc.t =2/ B sx in the above proof belongs to the connected component Iso .S/ of the identity in Is.S /. Hence we have shown that the (possibly smaller) group Iso .S/ acts transitively on S.

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For x, y 2 S we denote by cx;y the unique unit speed geodesic emanating from x which contains y. With this notation we have   (1.1) sx .y/ D sx cx;y .d.x; y// D cx;y .d.x; y//:

1.2 The group of isometries We first state an important rigidity property of isometries of a Riemannian manifold which we will need in the sequel. For a diffeomorphism ˆ of a Riemannian manifold M we denote by Dˆ W TM ! TM its differential. Lemma 1.5 (Rigidity of isometries; [dC], Lemma 4.2). Let ˆ and ‰ be two local isometries of a connected Riemannian manifold S. Assume that at some point x we have ˆ.x/ D ‰.x/ and Dx ˆ D Dx ‰. Then ˆ D ‰. Moreover, the group of isometries of a Riemannian manifold satisfies the following properties: Theorem 1.6 ([H], Theorem IV.2.5). Endowed with the compact-open topology the isometry group Is.S/ of a Riemannian manifold S is a locally compact topological transformation group of S. Moreover, for all x 2 S the isotropy subgroup Is.S /x ´ fg 2 Is.S / W g.x/ D xg at x is compact. In the sequel we assume that S is globally symmetric, and denote by G ´ Iso .S / the identity component of Is.S/. We fix o 2 S and let K ´ fg 2 Iso .S / W g.o/ D og be the compact isotropy subgroup of G at o. Then by Remark 1.4 we have S D G.o/ ´ fg.o/ W g 2 Gg. Theorem 1.7 ([H], Lemma IV.3.2 and Theorem IV.3.3 (i), (ii)). The topological group G has an analytic structure compatible with the compact-open topology in which it is a connected Lie transformation group of S . Moreover, G=K is analytically diffeomorphic to S, and K contains no non-trivial normal subgroup of G. Notice that by Theorem II.2.6 of [H] a topological group has at most one analytic structure compatible with its topology with respect to which it is a Lie group. In the remainder of this section we will have a look at the geodesic symmetry in S . Lemma 1.8. If x 2 S and k 2 Is.S/x then the geodesic symmetry at x satisfies sx B k D k B sx : Proof. Let z 2 S arbitrary. If t ´ d.x; z/ D d.k.x/; k.z// D d.x; k.z//, then by (1.1) and the fact that g.cx;z / D cg.x/;g.z/ for any isometry g 2 Is.S / we get sx .k.z// D cx;k.z/ .t / D k.cx;z .t // D k.sx .z//.

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Lemma 1.9. Let x 2 S and g 2 Is.S/ be such that x D g.o/. Then sx D g B so B g 1 . Proof. Let y 2 S arbitrary, t ´ d.x; y/ and z ´ g 1 .y/. Then sx .y/ D cx;y .t / D cg.o/;g.z/ .t / D g.co;z .t // D g.so .co;z .t ///; hence the claim follows from co;z .t/ D z D g 1 .y/.

1.3 Algebraic point of view We have seen that globally symmetric spaces are diffeomorphic to G=K, where G is a connected Lie group and K  G the isotropy subgroup at some point. One natural question concerns the reverse statement: which homogeneous spaces are symmetric spaces? Before we address this question we need some more facts relating a globally symmetric space S to the connected Lie group G ´ Iso .S / and its Lie algebra g. Denote e W g ! G the exponential mapping of g into G, and e 2 G the identity element in G. We fix a base point o 2 S , let K  G be the isotropy subgroup of G at o, and consider the geodesic symmetry so at o. The automorphism  2 Aut.G/ of G defined by .g/ ´ so Bg Bso1 is an involution, i.e.  2 is the identity idG 2 Aut.G/. We set G  ´ fg 2 G W  .g/ D gg; and we denote by .G  /o the identity component of G  . Notation. For simplification, we will in the sequel omit the “B” when referring to composition of group elements in G. Moreover, the action of G on S will be denoted by a dot “”. Proposition 1.10. .G  /o  K  G  . Proof. Let k 2 K. Then so k so1  o D o D k  o and Do .so k so1 / D  idTo S B Do k B . idTo S / D Do k; hence by rigidity of isometries so k so1 D k and therefore K  G  . Next let g 2 .G  /o . Then there exists a path p W Œ0; 1 ! .G  /o such that p.0/ D e and p.1/ D g. Now o D so  o gives so p.t /  o D so p.t /so1  o D  .p.t //  o D p.t /  o for all t 2 Œ0; 1, i.e. p.t /  o is a fixed point of so for all t 2 Œ0; 1. But o is an isolated fixed point of so , hence necessarily p.t /  o D o for all t 2 Œ0; 1. In particular we have g  o D p.1/  o D o which implies g 2 K. In order to give a condition under which a homogeneous space is symmetric, we recall some facts from the theory of Lie groups and Lie algebras.

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Let G be a connected Lie group with Lie algebra g. Then for h 2 G the conjugation map I.h/ W G ! G, g 7! hgh1 is an isomorphism of Lie groups. We denote by Ad.h/ ´ De .I.h// W g ! g its differential at the identity e 2 G. Ad.h/ is a Lie algebra automorphism, hence in particular ŒAd.h/X; Ad.h/Y  D Ad.h/ŒX; Y 

for all X; Y 2 g:

Moreover, we have the following useful formula: e Ad.h/X D he X h1

for any h 2 G; X 2 g:

(1.2)

The map AdG W G ! GL.g/, h 7! Ad.h/ is an analytic group morphism which is called the adjoint representation of G. Definition 1.11. .G; K/ is called a Riemannian symmetric pair if G is a connected Lie group, K  G a closed subgroup such that AdG .K/ is a compact subgroup of GL.g/ and if there exists an analytic involutive automorphism  of G such that .G  /o  K  G  : Notice that if S is a globally symmetric space, G D Iso .S / and K  G the isotropy subgroup of an arbitrary point x 2 S , then .G; K/ is a Riemannian symmetric pair with respect to the analytic involutive automorphism of G induced by the geodesic symmetry at x. In this case we call .G; K/ the Riemannian symmetric pair associated to .S; x/. The following theorem in particular answers the question raised in the introduction. Proposition 1.12 ([H], Proposition IV.3.4). If .G; K/ is a Riemannian symmetric pair and  any analytic involutive automorphism of G such that .G  /o  K  G  , then G=K is a globally symmetric space with respect to any G-invariant Riemannian metric. If  W G ! G=K denotes the natural projection and so the geodesic symmetry at o D .K/ D eK 2 G=K, then so B  D  B : In particular, so is independent of the choice of the G-invariant Riemannian metric. The following proposition shows that under very general conditions the automorphism  is completely determined by its set of fixed points G  . Proposition 1.13 ([H], Proposition IV.3.). Let .G; K/ be a Riemannian symmetric pair, k the Lie algebra of K and z the Lie algebra of the center of G. If k \ z D f0g, then there exists exactly one analytic involutive automorphism  of G such that .G  /o  K  G  . We remark that for semi-simple Lie groups G we have z D f0g, hence clearly k \ z D f0g. Moreover, if .G; K/ is the Riemannian symmetric pair associated to a

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globally symmetric space S with base point o 2 S , then K contains no non-trivial normal subgroup of G by Theorem 1.7; hence in this case the analytic involutive automorphism  of G induced by the geodesic symmetry at o is the only one satisfying .G  /o  K  G  . We next look at the Cartan involution ‚ W g ! g defined as the differential ‚ ´ De  of  at the identity e 2 G. Since ‚2 D Idg one can look at the eigenspace decomposition g D k ˚ p of g with respect to the eigenvalues C1 and 1 respectively. This decomposition is called the Cartan decomposition of g with respect to ‚. Moreover, ‚ is a Lie algebra automorphism and we have the Cartan relations Lemma 1.14. Œk; k  p, Œk; p  p, Œp; p  k. Proof. We prove Œk; p  p, the other inclusions are similar. Let X 2 k, Y 2 p arbitrary. Then ‚ŒX; Y  D Œ‚X; ‚Y  D ŒX; Y  D ŒX; Y ; i.e. ŒX; Y  belongs to the 1-eigenspace of ‚.

1.4 Geodesics and curvature Now let S be a globally symmetric space with base point o 2 S and .G; K/ the associated Riemannian symmetric pair. By our remark following Proposition 1.13 there exists exactly one analytic involutive automorphism  of G with .G  /o  K  G  , so the Cartan decomposition g D k ˚ p is uniquely determined. Let  W G ! S, g 7! g  o, denote the natural map, e W g ! G the Lie group exponential mapping, De  W g ! To S the differential of  at the identity e 2 G, and expo W To S ! S the Riemannian exponential mapping. The importance of the Cartan decomposition of g is reflected in the following Theorem 1.15 ([H], Theorem IV.3.3 (iii)). De  jp W p ! To S is an isomorphism (of vector spaces with Lie bracket), and ker.De  / D k. Moreover, we have   .e X / D expo De .X / for any X 2 p: (1.3) This shows in particular that the Riemannian exponential map exp W T S ! S of a globally symmetric space does not depend on its Riemannian metric; for any Ginvariant Riemannian metric on S Š G=K the exponential map is the same! Moreover, this immediately shows how geodesics in S look like: Corollary 1.16. The geodesic c  S emanating from o with tangent vector De  .X / 2 To S , X 2 p, is given by c.t / D e tX  o; t 2 R:

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Notice that if c  S is an arbitrary geodesic, then by transitivity of the action of G there exists g 2 G such that c.0/ D g  o. So g 1  c is a geodesic emanating from o 2 S and therefore of the form e tX  o for some X 2 p. This shows that for every geodesic c  S there exist g 2 G and X 2 p such that c.t / D ge tX  o;

t 2 R:

The following theorem describes the curvature tensor and totally geodesic submanifolds of S . Notice that for the curvature tensor we use the definition from [H]; in the book [dC] by do Carmo the curvature tensor is defined with the opposite sign. Theorem 1.17 ([H], Theorem IV.4.2, Theorem IV.7.2). (1) The curvature tensor Ro evaluated in To S is given by   Ro .De .X/; De .Y //De .Z/ D De   ŒŒX; Y ; Z ; X; Y; Z 2 p: (2) Totally geodesic submanifolds through o are of the form e q  o, where q  p is a Lie triple system, i.e. ŒŒq; q; q  q. In particular – as we have already seen for the Riemannian exponential mapping exp W T S ! S – the curvature tensor and the totally geodesic submanifolds of S do not depend on the given Riemannian metric on S . These facts also follow from the following Theorem 1.18 ([H], Corollary IV.4.3). The Levi-Civita connection on G=K is the same for all G-invariant Riemannian structures on G=K.

1.5 Examples (1) SL.n; R/= SO.n/. Consider the connected Lie group G D SL.n; R/ which is the group of all .nn/matrices with determinant 1 and entries in R. On G we consider the involutive automorphism  W G ! G, g 7! .g t /1 . Then G  D fg 2 G W .g t /1 D gg D fg 2 G W g t g D eg D SO.n/ DW K: The Lie algebra g D sl.n; R/ consists of all .n  n/-matrices with trace 0 and entries in R, and the Cartan involution ‚ W sl.n; R/ ! sl.n; R/ is given by ‚.X / ´ X t for X 2 sl.n; R/. So the Cartan decomposition of an element X 2 sl.n; R/ is the well-known unique decomposition X D 12 .X  X t / C 12 .X C X t / of a matrix into its anti-symmetric and symmetric part. If so.n/ denotes the Lie algebra of K D SO.n/, and sym0 .n/ the set of symmetric .n  n/-matrices of trace zero with entries in R, we therefore have sl.n; R/ D so.n/ ˚ sym0 .n/:

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Denote o D eK 2 G=K the base point and consider the positive definite symmetric bilinear form hX; Y i ´ Tr.X  Y /;

X; Y 2 To .G=K/ Š p  g:

(1.4)

As we will see more precisely in Section 1.7, this scalar product on To .G=K/ can be naturally extended by G-left-translations to a G-invariant Riemannian metric on G=K. The set Pos1 .n/ of positive definite symmetric .n  n/-matrices with determinant 1 and entries in R can be identified with G=K as follows: it is a standard fact from elementary linear algebra that any matrix p 2 Pos1 .n/ can be written as a matrix product p D b t b for some b 2 SL.n; R/. With the action of g 2 SL.n; R/ on Pos1 .n/ given by g  p ´ g t pg, p 2 Pos1 .n/, G acts transitively on Pos1 .n/. If we choose the .n  n/-identity matrix In as a base point o in Pos1 .n/, then SO.n/ D fg 2 G W g  In D In g. If n D 2, we can identify G=K endowed with the G-invariant Riemannian metric induced by hX; Y i ´ 2  Tr.X  Y /; X; Y 2 To .G=K/, and the real hyperbolic plane H2 ´ fx C iy W x 2 R y > 0g endowed with the metric ds 2 D .dx 2 C dy 2 /=y 2 . Indeed, SL.2; R/ acts transitively by isometries via linear fractional transformations on H2 , and SO.2/ is the isotropy subgroup of the point i 2 H2 . So Pos1 .2/ with a metric rescaled by the factor 2 can be identified with the hyperbolic plane .H2 ; ds 2 /. Exercise. Show that we need to have the factor 2 in equation (1.4) in order that .SL.2; R/= SO.2/; h; i/ is isometric to .H2 ; ds 2 /. (2) G=K , G  SL.n; R/ closed subgroup with G t D G . As involutive automorphism we take again  W G ! G, g 7! .g t /1 , so K D G \ SO.n/. If o D eK 2 G=K denotes the base point, then the positive definite bilinear form given by (1.4) on To .G=K/ extends to a G-invariant Riemannian metric on G=K. (a) The group G D SO.p; q/ of linear transformations leaving invariant the bilinear form Q.x; y/ D x1 y1      xp yp C xpC1 ypC1 C    C xpCq ypCq ; x; y 2 RpCq , on RpCq is invariant under transposition. Therefore if K D G \ SO.p C q/ D SO.p/  SO.q/ we get a symmetric space G=K. Let M.p; q/ denote the set of .pq/-matrices with entries in R. The Cartan decomposition of the Lie algebra of SO.p; q/ is given by so.p; q/ D k ˚p, where k D so.p/  so.q/  so.p C q/ is the Lie algebra of K and   ˚ p D B0t B0 W B 2 M.p; q/  sym0 .p C q/:

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In the particular case p D 1, this symmetric space with an appropriately rescaled metric is isometric to the hyperbolic space of dimension q. (b) The group G D Sp.2q; R/ of linear transformations leaving invariant the standard symplectic form !.x; y/ D x1 yqC1 C x2 yqC2 C    C xq y2q  xqC1 y1      x2q yq ; x; y 2 R2q , on R2q is invariant under transposition. If K D Sp.2q; R/ \ SO.2q/, then G=K is a symmetric space. The Cartan decomposition of the Lie algebra of Sp.2q; R/ is given by sp.2q; R/ D k ˚ p, where  ˚ A B  k D B W A; B 2 M.q; q/; At D A  so.2q/ t A is the Lie algebra of K and ˚   B p D BAt A W A; B 2 M.q; q/; At D A  sym0 .2q/: Recall that a complex structure on a real vector space V is an endomorphism J of V with the property J 2 D  idV . Moreover, if g 2 GL.V /, then g B J B g 1 is also a complex structure. Consider the set S2q of complex structures J on the symplectic vector space .R2q ; !/ such that the symmetric bilinear form defined by qJ .x; y/ ´ !.x; Jy/;

x; y 2 R2q ;

(1.5)

is positive definite. A complex structure with this property is called !compatible. G D Sp.2q; R/ acts naturally on S2q by conjugation, i.e. g  J ´ gJg 1 for g 2 G, J 2 S2q . Indeed, if g 2 Sp.2q; R/ then qgJ .x; y/ D !.x; gJg 1 y/ D !.g 1 x; Jg 1 y/ D qJ .g 1 x; g 1 y/; so qgJ is positive definite if qJ is. Moreover, this action is transitive. We choose as a base point o 2 S2q the !-compatible complex structure given by the matrix   0 Iq J0 ´ I (1.6) Iq 0 its associated symmetric bilinear form qJ0 is the standard scalar product in R2q . Then the isotropy subgroup of G at o is precisely the group K D Sp.2q; R/ \ SO.2q/, so S2q D Sp.2q; R/  o can be identified with G=K. Notice that in the particular case q D 1 we have Sp.2; R/ D SL.2; R/, so the subspace S2 of R2 with the appropriately rescaled metric can be identified with the hyperbolic plane .H2 ; ds 2 /. (c) The group G D SL.2; R/  SL.2; R/ acts by isometries on H2  H2 endowed with the product metric, and K D SO.2/  SO.2/ fixes the point o ´ .i; i/ 2 H2  H2 . So in this case the symmetric space G=K endowed

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with the G-invariant metric induced by hX; Y i ´ 2  Tr.X  Y /; X; Y 2 To .G=K/, is isometric to a product of hyperbolic planes H2  H2 . (3) SO.p C q/=.SO.p/  SO.q//. As before we denote  by Iq the  .q  q/-identity matrix and let s 2 SL.p C q; R/ Ip 0 be the matrix s D 0 Iq . For G D SO.p C q/ we consider the involutive automorphism  W G ! G, g 7! sgs. Then K D SO.p/  SO.q/ is a compact subgroup fixed by  . The Cartan involution ‚ W so.p C q/ ! so.p C q/ is given as follows: if A 2 M.p; p/, At D A, B 2 M.p; q/, D 2 M.q; q/, D t D D, then for   A B 2 so.p C q/ XD B t D we have   A B : ‚.X/ D Bt D So so.p C q/ D .so.p/  so.q// ˚ p, where ˚ 0 B   p ´ B W B 2 M.p; q/ : t 0 In this case, the symmetric bilinear form given by hX; Y i ´  Tr.X  Y /;

X; Y 2 To .G=K/ Š p  g;

(1.7)

is positive definite, and hence can be extended to a G-invariant Riemannian metric on G=K. Here the symmetric space G=K is the Grassmannian manifold of p-dimensional oriented subspaces of RpCq . In the particular case p D 1 this is the q-dimensional sphere, and the standard metric induced from the embedding into RqC1 is a scalar multiple of the above metric. (4) Compact Lie groups as symmetric spaces. Let G be a compact connected Lie group. Then the mapping  W G  G ! G  G;

.g1 ; g2 / 7! .g2 ; g1 /;

is an involutive automorphism of the product group G  G. The fixed point set of  is the diagonal  ´ f.g; g/ W g 2 Gg in G  G which is isomorphic to G and hence compact. The pair .G  G; / is a Riemannian symmetric pair and the coset space .G  G/= is diffeomorphic to the original group G via the mapping .G  G/= ! G, .g1 ; g2 / 7! g1 g21 . A Riemannian metric on .G  G/= is .G  G/-invariant if and only if the corresponding Riemannian metric on the group G is bi-invariant. Hence by Proposition 1.12, G is a globally symmetric space with respect to any bi-invariant Riemannian metric on G. The natural mapping of G  G onto G Š .G  G/=

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corresponds to  W G  G ! G, .g1 ; g2 / 7! g1 g21 . Recalling that the geodesic symmetry so at o ´ ./ D e is given by so B  D  B  we obtain so .g/ D g 1 for g 2 G. Exercise. Using Lemma 1.9 prove that for any h; g 2 G we have sh .g/ D hg 1 h. Next let g denote the Lie algebra of G, and e W g ! G the Lie group exponential mapping. Then the product algebra g  g is the Lie algebra of G  G, and the identity     .X; Y / D 12 .X C Y /; 12 .X C Y / C 12 .X  Y /;  12 .X  Y / gives the Cartan decomposition of gg into the two eigenspaces of ‚ D D.e;e/  . In particular, we have p ´ f.X; X/ W X 2 gg  gg. So if eO W gg ! G G denotes the Lie group exponential mapping, and exp W T G ! G the Riemannian exponential mapping of the symmetric space G, then (1.3) implies that for all X 2 g,   expo D.e;e/ .X; X/ D .eO .X;X/ / D  .e X ; e X / D e 2X : We conclude that the geodesics in the symmetric space G through the base point o D e are the one-parameter subgroups of the group G.

1.6 The Killing form For Lie groups the Killing form is an important and natural bilinear form on the Lie algebra. We will see that it also plays a very important role in the theory of globally symmetric spaces. In this section we will describe the Killing form and some of its properties. For that we need some more facts from the theory of Lie groups. Recall the adjoint representation Ad ´ AdG W G ! GL.g/ described in Section 1.3. The Lie algebra of GL.g/ is the vector space gl.g/ of all linear endomorphisms of g endowed with the bracket Œˆ; ‰ ´ ˆ B ‰  ‰ B ˆ for ˆ; ‰ 2 gl.g/. It is naturally identified with the tangent space of GL.g/ at the identity map idg . The adjoint representation ad W g ! gl.g/ of g is defined as the differential ad ´ De Ad of the map Ad W G ! GL.g/ at the identity of G. It can be shown that for X 2 g the endomorphism ad.X/ is given by ad.X/Y D ŒX; Y  for Y 2 g. Moreover we have the relation Ad.e X / D e ad.X / ; X 2 g; (1.8) where on the left-hand side e W g ! G denotes the Lie group exponential mapping and on the right-hand side e W gl.g/ ! GL.g/ denotes the exponential mapping given by the usual power series 1

X 1 1 ˆk ; e ˆ ´ Id Cˆ C ˆ B ˆ C 2 kŠ kD3

ˆ 2 gl.g/:

(1.9)

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The Killing form B of a Lie algebra g is the symmetric bilinear form on g defined by B W g  g ! R; .X; Y / 7! Tr.ad.X / B ad.Y //; where Tr W gl.g/ ! R is the canonical trace map. The following properties of the Killing form will turn out to be very useful: Proposition 1.19 ([H], II.6 (2)). (1) B.X; ŒY; Z/ D B.Y; ŒZ; X / D B.Z; ŒX; Y / for any X; Y; Z 2 g. (2) For any Lie algebra automorphism ˆ W g ! g we have B.ˆ.X /; ˆ.Y // D B.X; Y /, X; Y 2 g. Definition 1.20. A Lie algebra g over R is called semi-simple if its Killing form is non-degenerate. A Lie group is called semi-simple if its Lie algebra is. So if G is a connected semi-simple Lie group, we can construct from the Killing form a natural G-invariant semi-Riemannian metric on the analytic manifold G as follows: requiring that the Riemannian exponential mapping expe W Te G ! G at the identity e coincides with the Lie group exponential mapping e W g ! G we get a natural identification of the tangent space Te G at the identity e 2 G with the Lie algebra g of G. Let Qe be the non-degenerate symmetric bilinear form on Te G corresponding to the Killing form of g. If for g 2 G the map Lg 2 Aut.G/ denotes left multiplication by g on the analytic manifold G, then its differential at a point h 2 G is a linear map Dh Lg W Th G ! Tgh G. We define a non-degenerate symmetric bilinear form Qg on Tg G via   Qg .v; w/ ´ Qe .De Lg /1 .v/; .De Lg /1 .w/ ; v; w 2 Tg G: (1.10) Doing this for all g 2 G, we get a semi-Riemannian structure Q on G. Moreover, if g; h 2 G and v; w 2 Tg G are arbitrary, then using De Lhg D Dg Lh B De Lg we compute Qhg .Dg Lh .v/; Dg Lh .w//   .1.10/ D Qe .De Lhg /1 .Dg Lh .v//; .De Lhg /1 .Dg Lh .w//   D Qe .De Lg /1 .v/; .De Lg /1 .w/ .1.10/

D

Qg .v; w/:

So Q is indeed G-left-invariant. For an arbitrary (not necessarily semi-simple) Lie algebra g the following proposition will be very convenient in the sequel. Proposition 1.21 ([H], Proposition II.6.8). If u  g is a compactly embedded subalgebra with u \ z D f0g, then the Killing form Bju restricted to u is negative definite.

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Notice that g being semi-simple necessarily implies that the center z of g is trivial. So in this case the Killing form restricted to any compactly embedded subalgebra is negative definite. For the remainder of this section we let S be a globally symmetric space with base point o 2 S , .G; K/ the associated Riemannian symmetric pair, and g D k ˚ p the Cartan decomposition with e k D K. Then k is a compactly embedded subalgebra of g, and k \ z D f0g by the last assertion of Theorem 1.7. So from the previous proposition we know that the Killing form restricted to k is negative definite. We will now prove several useful lemmata. Lemma 1.22. k and p are orthogonal with respect to the Killing form. Proof. Let Z 2 k, X 2 p arbitrary, and ‚ W g ! g the Cartan involution. By definition of k and p we have ‚.Z/ D Z and ‚.X/ D X. Moreover, since ‚ is a Lie algebra automorphism, we have from Proposition 1.19 (2) B.Z; X / D B.‚.Z/; ‚.X// D B.Z; X / D B.Z; X/; which implies B.Z; X / D 0. Lemma 1.23. For all k 2 K we have Ad.k/p D p. Proof. Let Z 2 k be such that k D e Z , and X 2 p be arbitrary. Then 1

X1 .1.8/ .1.9/ Ad.k/X D Ad.e Z /X D e ad.Z/ X D X C ad.Z/X C ad.Z/i X: iŠ iD2

Since ad.Z/X D ŒZ; X 2 p, and inductively ad.Z/ X D ad.Z/i1 ŒZ; X  2 p for i  2 by Lemma 1.14, we get Ad.k/p  p. The reverse inclusion follows from g D k ˚ p and the fact that Ad.k/ W k ! k and Ad.k/ W g ! g are isomorphisms. i

For any x 2 S we denote by x W G ! S , g 7! g  x, the orbit map, and De x W g ! Tx S its differential at the identity e 2 G. We know from Theorem 1.15 that De o maps p isomorphically into To S. For x 2 S arbitrary, we have the following: Lemma 1.24. If x 2 S and g 2 G such that g  o D x, then De x maps Ad.g/p isomorphically into Tx S. Proof. For g 2 G D Iso .S/ we denote by Do g W To S ! Tgo S its differential at the base point o 2 S, and Lg , Rg 2 Aut.G/ left- and right-multiplication by g on G. Now let x 2 S and g 2 G be such that g o D x. From the definitions we immediately get o B Lg D g B o ; x D o B Rg :

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If X 2 g then by the above relations we have x .e tX / D o B Rg .e tX / D o B Lg B.Lg /1 B Rg .e tX /; „ ƒ‚ … DgBo

and from .Lg /1 B Rg .e tX / D g 1 e tX g D e t Ad.g d ˇˇ De x .X / D ˇ

1 /X

we conclude

d ˇˇ x .e tX / D ˇ

    1 go e t Ad.g /X D Do gBDe o Ad.g 1 /X : dt tD0 dt tD0   we therefore have X 2 ker De x if and only if Since Do g is an isomorphism  Ad.g 1 /X 2 ker De o D k. This is equivalent to ker De x D Ad.g/k. We know from Lemma 1.22 that g is the orthogonal direct sum of k and p. Since Ad.g/ W g ! g is a Lie algebra automorphism and hence by Proposition 1.19 (2) preserves the Killing form, we know that g D Ad.g/g ˇ can be decomposed into the orthogonal direct sum Ad.g/k ˚ Ad.g/p. Hence De x ˇAd.g/p is an isomorphism. Notice that if g; h 2 G satisfy g  o D h  o D x, then h1 g fixes o and therefore belongs to K, so by Lemma 1.23   Ad.g/p D Ad.h/ Ad.h1 g/p D Ad.h/p: ƒ‚ … „ Dp

This shows that the map Do x does not depend on the choice of g 2 G such that g  o D x. Moreover, the decomposition g D Ad.g/k ˚ Ad.g/p can be interpreted as the Cartan decomposition of g with respect to the involution induced by the geodesic symmetry sx at x D go 2 S; the isotropy subgroup of G at x is the compact subgroup e Ad.g/k D gKg 1 .

1.7 Decomposition of symmetric spaces We have seen in Section 1.3 that a globally symmetric space together with the choice of a base point o 2 S gives rise to a pair .g; ‚/, where g is the Lie algebra of the group of isometries Iso .S/, and ‚ the differential at the identity of the involutive automorphism  of G induced by the geodesic symmetry at o. Moreover, the set of fixed points of ‚ in g is a compactly embedded subalgebra. In this section we will have a look at such pairs. Definition 1.25. An orthogonal symmetric Lie algebra is a pair .l; &/, where l is a Lie algebra over R and & is an involutive automorphism of l such that u D fX 2 l W &X D Xg is a compactly embedded subalgebra of l. .l; & / is called effective if in addition u \ z D f0g, where z  l denotes the center of l. Notice that any pair .g; ‚/ coming from a globally symmetric space is effective by the last assertion of Theorem 1.7.

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Definition 1.26. Let .l; & / be an effective orthogonal symmetric Lie algebra with Killing form B, and l D u ˚ e the decomposition of l into the eigenspaces of & for the eigenvalue C1 and 1 respectively. Then .l; &/ is said to be of (1) compact type if l is compact and semi-simple; (2) non-compact type if l is non-compact and semi-simple, and if Bju is negative definite and Bje is positive definite; (3) Euclidean type if e is an abelian ideal in l. Notice that the proof of Lemma 1.22 shows that the subspaces u and e are orthogonal with respect to the Killing form. Moreover, Proposition 1.21 implies that the Killing form restricted to u is negative definite. We say that a pair .L; U / is associated with an orthogonal symmetric Lie algebra .l; & / if L is a connected Lie group with Lie algebra l, and U is a Lie subgroup of L with Lie algebra u. So we can define the type of a pair .L; U / according to the type of the effective orthogonal Lie algebra it is associated to. Similarly, the type of a globally symmetric space S is defined as the type of an associated Riemannian symmetric pair .G; K/ (which is naturally associated to an effective orthogonal symmetric Lie algebra .g; ‚/ as above). Notice that even though every choice of base point a priori gives rise to a different Riemannian symmetric pair, the types of all such pairs are the same: if instead of a base point o 2 S we take the base point x D g  o, g 2 G, then the Lie algebra g remains the same and only the involution ‚ on g is changed to Ad.g/‚. Example 1. SL.n; R/= SO.n/ is a symmetric space of non-compact type: g D sl.n; R/ is non-compact and semi-simple. Moreover, Bjk is negative definite by Proposition 1.21, and Bjp is a positive multiple of the positive definite symmetric bilinear form (1.4). Example 2. If G  SL.n; R/ is a closed subgroup invariant under transposition and K D SO.n/ \ G, then G=K is also a symmetric space of non-compact type, because g is non-compact and semi-simple, Bjk is negative definite by Proposition 1.21, and Bjp is a positive multiple of the positive definite symmetric bilinear form (1.4). Example 3. SO.p C q/=.SO.p/  SO.q// is a symmetric space of compact type: g D so.p C q/ is compact and semi-simple. Notice that in this case Bjp is a negative multiple of the positive definite symmetric bilinear form (1.7). Example 4. A compact connected semi-simple Lie group G Š .G  G/= is a symmetric space of compact type with respect to any metric induced by a G-bi-invariant metric on G. The corresponding orthogonal symmetric Lie algebra is .g  g; ‚/, where ‚.X; Y / ´ .Y; X/ for X; Y 2 g, and gg is compact and semi-simple. The next theorem gives a decomposition for effective orthogonal symmetric Lie algebras.

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Theorem 1.27 ([H], Theorem V.1.1). Let .l; &/ be an effective orthogonal symmetric Lie algebra. Then there exist ideals l0 , l and lC such that (1) l can be decomposed as a direct sum l D l0 ˚ l ˚ lC ; (2) l0 , l and lC are invariant under & and orthogonal with respect to the Killing form of l ; (3) the pairs .l0 ; &jl0 /, .l ; &jl / and .lC ; & jlC / are effective orthogonal symmetric Lie algebras of Euclidean type, compact type and non-compact type respectively. Let .L; U / be a pair associated with an effective orthogonal symmetric Lie algebra .l; & /, and l D u ˚ e the decomposition of l into the eigenspaces of & for the eigenvalues C1 and 1 respectively. In the proof of the above theorem, S. Helgason shows that for any Ad.U /-invariant positive definite symmetric bilinear form Q on e there exists an endomorphism ˆ of e such that Q.ˆ.X/; Y / D B.X; Y /

for any X; Y 2 e:

Moreover, if l is of Euclidean, compact or non-compact type, then all eigenvalues of ˆ are identically zero, strictly negative or strictly positive, respectively. This immediately implies the following Proposition 1.28. If .l0 ; &0 /, .l ; & /, .lC ; &C / are effective orthogonal symmetric Lie algebras of Euclidean, compact and non-compact type respectively, then the Killing form restricted to e0 , e , eC is identically zero, negative definite and positive definite, respectively. For the remainder of this section we let S be a globally symmetric space, o 2 S a base point, and .G; K/ the associated Riemannian symmetric pair. Let  W G ! S, g 7! g  o, denote the natural map, and g D k ˚ p the Cartan decomposition of the Lie algebra g of G with e k D K. From the previous proposition we know that if S is of compact type, then Bjp induces a scalar product Qo on To S , and if S is of non-compact type, then Bjp does. As performed in Section 1.6 for a semi-simple Lie group, we can extend this scalar product to a G-invariant Riemannian structure on S : for g 2 G we denote by Dg W T S ! T S the differential of the isometry g. For x 2 S we choose g 2 G such that x D g  o and define a scalar product Qx on Tx S via   (1.11) Qx .v; w/ ´ Qo .Do g/1 .v/; .Do g/1 .w/ ; v; w 2 Tx S: Notice that if h 2 G also satisfies h  o D x, then h1 g 2 K. Moreover, for any k 2 K the Killing form Bp restricted to p is Ad.k/-invariant. Since Qo .De  .X /; De  .Y // D B.X; Y /, and De  B Ad.k/ D Do k B De  we conclude that Qo is invariant under Do k for any k 2 K. Hence the assignment x 7! Qx is consistent and defines a Riemannian structure Q on S . This structure is G-invariant: indeed, if x; y 2 S , v; w 2 Tx S are arbitrary, and g 2 G such that y D g  x , h 2 G such that x D h  o,

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then y D gh  o and Do .gh/ D Dx g B Do h gives   .1.11/ Qy .Dx g.v/; Dx g.w// D Qe .Do .gh//1 B Dx g.v/; .Do .gh//1 B Dx g.w/   .1.11/ D Qe .Do h/1 .v/; .Do h/1 .w/ D Qx .v; w/: Hence both for symmetric spaces S of compact type and of non-compact type the Killing form canonically induces a G-invariant Riemannian structure on S . Conversely, any G-invariant Riemannian metric Q on a symmetric space S of compact or non-compact type is essentially determined by the restriction of the Killing form to p: since Qo BDe jp is an Ad.K/-invariant positive definite symmetric bilinear form on p, by the remark following Theorem 1.27 there exists an automorphism ˆ of p such that in To S D De .p/ we have Qo .De .ˆ.X//; De .Y // D B.X; Y / for all X; Y 2 p: Moreover, according to whether S is of compact type or of non-compact type, all eigenvalues of ˆ are strictly negative or strictly positive, and all eigenspaces of ˆ are invariant by Ad.K/. More details can be found e.g. in Section 2.3 of [E], Section 8.2 of [Wo] or Chapter V, §1 and §3 in [H]. We will now look at the sectional curvature of the globally symmetric space S . Let h; i denote the scalar product in p Š To S induced from the Riemannian metric. Recall from  Theorem 1.17 that for X; Y; Z 2 p the curvature tensor R.X; Y /Z ´ .De  /1 Ro .De .X/; De .Y //De .Z/ is given by ŒŒX; Y ; Z. Given two linearly independent vectors X; Y 2 p, the sectional curvature .hX; Y i/ of the two-plane hX; Y i in To S spanned by De .X/ and De  .Y / is defined by .hX; Y i/ ´ p

hR.Y; X/X; Y i hX; X ihY; Y i  hX; Y i2

:

We have the following Theorem 1.29 ([H], Theorem V.3.1). Let S be a globally symmetric space with associated Riemannian symmetric pair .G; K/ such that K  G is connected and closed, and Q an arbitrary G-invariant Riemannian metric. (1) If S is of compact type, then S has non-negative sectional curvature. (2) If S is of non-compact type, then S has non-positive sectional curvature. (3) If S is of Euclidean type, then the sectional curvature of S is identically zero. Proof. Notice that by G-invariance of the metric it suffices to prove the claim for arbitrary two-planes in To S Š p. Recall that h; i denotes the scalar product on p induced by the Riemannian structure Q. We first prove (1) and (2). By the remark following Proposition 1.28 there exists an automorphism ˆ of p with all eigenvalues strictly positive such that hˆ.X/; Y i D B.X; Y /, X; Y 2 p, according to whether S is of compact type or of non-compact type.

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Now choose an arbitrary two-plane E in To S and a basis X; Y 2 p of .De  /1 .E/ satisfying hX; X i D hY; Y i D 1 and hX; Y i D 0. Then the sectional curvature of E is given by .E/ D hR.Y; X/X; Y i D hŒŒY; X; X ; Y i D hY; ŒŒY; X ; X i:

(1.12)

Let k 2 K be such that Ad.k/Y is an eigenvector of ˆ with eigenvalue say ˇ > 0. Since the scalar product is invariant under Ad.k/ we get .E/ D hAd.k/Y; Ad.k/ŒŒY; X; Xi D ˇ 1 hˆ.Ad.k/Y /; Ad.k/ŒŒY; X ; X i: If S is of compact type, we therefore have by Ad.k/-invariance of the Killing form and Proposition 1.19 (1) ˇ .E/ D B.Ad.k/Y; Ad.k/ŒŒY; X ; X / D B.Y; ŒŒY; X ; X / D B.ŒY; X; ŒX; Y / D B.ŒX; Y ; ŒX; Y /  0; because B is negative definite on k and ŒX; Y  2 k. Similarly, for S of non-compact type we get ˇ .E/ D B.ŒX; Y ; ŒX; Y / 0. The claim then follows from the fact that all eigenvalues of ˆ are positive. If S is of Euclidean type, then p is an abelian ideal in g. So for all X; Y 2 p we have R.Y; X/X D ŒŒY; X; X D 0, hence by (1.12) .E/ D 0 for any two-plane E  To S . Notice that for the non-compact type, the hypothesis that K is connected and closed is always satisfied; for the compact type, K is always closed but not necessarily connected. We finally state the de Rham decomposition Theorem 1.30 ([H], Proposition V.4.2). Let S be a globally symmetric space. Then S D S0  S  SC ; where S0 is a symmetric space of Euclidean type, S a symmetric space of compact type, and SC a symmetric space of non-compact type. This theorem implies that in order to understand arbitrary globally symmetric spaces it suffices to study symmetric spaces of Euclidean, compact and non-compact type separately. Since the symmetric spaces of Euclidean type are isometric to Euclidean spaces by Theorem 1.29 (3), the interesting classes of symmetric spaces are those of compact or of non-compact type.

2 Symmetric spaces of non-compact type In this section we will study the structure of globally symmetric spaces of non-compact type which are known to be non-positively curved by Theorem 1.29. Moreover, it

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follows from Definition 1.26 that the connected component of the identity of the isometry group is a semi-simple Lie group. More precisely, we have the following Proposition 2.1 ([E], Proposition 2.1.1). The connected component of the identity of the isometry group of a globally symmetric space of non-compact type is a semi-simple Lie group with trivial center and without compact factor. We will explain the classical decompositions of semi-simple Lie groups and relate them to the geometry of the symmetric space. In this way we can understand how totally geodesically embedded Euclidean spaces and the so-called horocycles sit inside our manifold. For the remainder of this text we will assume that S is a globally symmetric space of non-compact type, o 2 S a base point, G D Iso .S / and K  G the compact isotropy subgroup at o. Let g D k ˚ p be the Cartan decomposition, and De  W p ! To S the isomorphism given in Theorem 1.15. We will assume that the Riemannian structure of S is induced by the Killing form Bjp restricted to p; from the remark following Proposition 1.28 we know that this is not a severe restriction.

2.1 Flats and rank Definition 2.2. A k-flat in S is a totally geodesic k-dimensional submanifold isometric to Rk . The rank r of S is defined as the maximal natural number r for which an r-flat exists in S . An r-flat is called a (maximal) flat. Notice that a 1-flat is simply a geodesic. Moreover, if a symmetric space of noncompact type has an upper negative bound on its sectional curvature, then it is of rank one. The rank one symmetric spaces of non-compact type are completely classified: they are precisely the hyperbolic spaces over the reals, complex numbers and quaternions, and the hyperbolic plane over the Cayley numbers. Every other symmetric space of non-compact type is of rank bigger than one and therefore possesses totally geodesically embedded Euclidean planes. We next address the following question: how does an r-flat F through the base point o 2 S look like in terms of Lie algebras? We first remark that F is totally geodesic, hence by Theorem 1.17 (2) it necessarily has the form F D e q  o for a Lie triple system q  p. Moreover, the sectional curvature restricted to F equals zero. Hence for all X; Y 2 q Š To F such that B.X; Y / D 0, B.X; X/ D B.Y; Y / D 1 we have the condition 0 D .hX; Y i/ D B.ŒX; Y ; ŒX; Y /: Again, the fact that B is negative definite on k implies that ŒX; Y  D 0. Hence q  p has to be an abelian subspace. Now let a  p be a maximal abelian subspace of dimension r D rank.S /. Then F D e a  o is a maximal flat in S. Since G acts by isometries on S, every set of the

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form g  F , g 2 G, is also a maximal flat. We will see later on that every flat in S is necessarily a G-translate of F . Example 1. For SL.n; R/= SO.n/ we know from Section 1.5 that p D sym0 .n/  sl.n; R/, the set of symmetric .n  n/-matrices of trace zero with entries in R. A maximal abelian subspace a of p is the set of diagonal matrices of trace zero, i.e. ˚  P a D Diag.t1 ; : : : ; tn / W t1 ; : : : ; tn 2 R; niD1 ti D 0 : We have seen that the set of positive definite symmetric .n  n/-matrices Pos1 .n/ with determinant one is diffeomorphic to SL.n; R/= SO.n/, where the SL.n; R/-action on Pos1 .n/ is given by g  p ´ g t pg, p 2 Pos1 .n/, g 2 SL.n; R/. The base point o 2 Pos1 .n/ is the fixed point of SO.n/, hence the identity matrix In 2 Pos1 .n/. So we get a maximal flat ˚  P F D e a  o D Diag.e 2t1 ; : : : ; e 2tn / W t1 ; : : : ; tn 2 R; niD1 ti D 0 (2.1)  ˚ Q D Diag. 1 ; : : : ; n / W 1 ; : : : ; n > 0; niD1 i D 1 in S, and the rank equals n  1. Exercise. Every flat F in SL.n; R/= SO.n/ is isomorphic to the Euclidean vector space Rn1 , hence for x; y; z 2 F the angle †x .y; z/ between the vectors pointing from x to y and from x to z is well-defined. Using formula (1.4), show that for x; y; z in a common flat F with †x .y; z/ D =2 we have d.y; z/2 D d.x; y/2 C d.x; z/2 . Example 2a. For SO.p; q/=.SO.p/  SO.q//, p q, a maximal abelian subspace a of p is given by ˚ 0 D  aD D 0 W D D .dij / 2 M.p; q/; dij D 0 for i ¤ j : In particular, the rank equals p D minfp; qg. Example 2b. For Sp.2q; R/=.SO.2q/ \ Sp.2q; R// a maximal abelian subspace a of p is given by  ˚  0 aD D 0 D W D D Diag.t1 ; : : : ; tq /; t1 ; : : : ; tq 2 R : In particular, the rank equals q. As in Section 1.5 we consider the set S2q of !-compatible complex structures on the symplectic vector space .R2q ; !/ with the Sp.2q; R/-action by conjugation. Choosing as a base point o 2 Sp.2q; R/ the !-compatible complex structure defined by the matrix J0 given in (1.6), S2q is diffeomorphic to Sp.2q; R/=.SO.2q/ \ Sp.2q; R//. The following set F is a maximal flat in S2q : ˚    0 I   1  A 0 W A D Diag.e t1 ; : : : ; e tq /; t ; : : : ; t 2 R F D e a  o D A0 A01 Iq 0 q 1 q 0 A ˚  0 A2   t1 tq D ; : : : ; e /; t ; : : : ; t 2 R W A D Diag.e 1 q 2 A 0  ˚  0 Diag.1 ;:::;q / D W 1 ; : : : ; q > 0 : Diag. 1 ;:::; 1 / 0 1

q

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Example 2c. A maximal flat in H2  H2 is simply a set f.c1 .t1 /; c2 .t2 // W t1 ; t2 2 Rg; where ci is a geodesics in the i-th H2 -factor for i D 1; 2. Lemma 2.3. Every geodesic is contained in at least one flat. Proof. If c  S is a geodesic, then by the remark following Corollary 1.16 there exists g 2 G and X 2 p such that c.t / D ge tX  o, t 2 R. Take a maximal abelian subspace a  p which contains X. Then c is contained in the flat ge a  o. Definition 2.4. Let X 2 p, and Zg .X/ ´ fY 2 g W ŒY; X  D 0g the centralizer of X in g. The vector X is called regular if Zg .X / \ p is maximal abelian, and singular otherwise. Notice that if X 2 p is singular, then dim.Zg .X / \ p/ > r. The following lemma in particular shows that regular vectors exist. Lemma 2.5 ([H], Lemma V.6.3 (i)). Let a  p be a maximal abelian subspace. Then there exists an element H 2 a such that Zg .H / \ p D a. Theorem 2.6. If a, a0 are maximal abelian subspaces of p, then there exists k 2 K such that a0 D Ad.k/a. Proof. Choose H 2 a, H 0 2 a0 regular. Recall that B denotes the Killing form on g, and consider  the bounded differentiable map f W K ! R defined by f .k/ D B Ad.k/H; H 0 . Let k0 2 K be one of its critical points. Then for any Z 2 k we have d ˇˇ d ˇˇ 0D ˇ f .k0 e tZ / D ˇ B.Ad.k0 e t Z /H; H 0 / dt tD0 dt tD0 d ˇˇ d ˇˇ D ˇ B.Ad.k0 / Ad.e tZ /H; H 0 / D B.Ad.k0 / ˇ .Ad.e t Z /H /; H 0 / dt tD0 dt t D0 D B.Ad.k0 /.ad Z/H; H 0 / D B.Ad.k0 /ŒZ; H ; H 0 / D B.Ad.k0 /Z; ŒAd.k0 /H; H 0 /: From Lemma 1.23 we know that Ad.k0 /H 2 p, so by Lemma 1.14, ŒAd.k0 /H;H 0  2 k. Since both Ad.k0 /Z and ŒAd.k0 /H; H 0  belong to k, Z 2 k is arbitrary, and the restriction of the Killing form B to k is negative definite by Proposition 1.21, we conclude that ŒAd.k0 /H; H 0  D 0. Since H 0 is regular, every element in g which commutes with H 0 is contained in a0 , so Ad.k0 /H 2 a0 . Since a0 is abelian, every element in a0 commutes with Ad.k0 /H , and therefore every element in Ad.k01 /a0 commutes with H . Since a D Zg .H / \ p we conclude that Ad.k01 /a0  a. Exchanging the role of a and a0 in the argument above we conclude that there exists k 2 K such that Ad.k/a  a0 . Hence Ad.k/ Ad.k01 /a0  Ad.k/a  a0 ;

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which shows that Ad.k/ Ad.k01 /a0 D a0 D Ad.k/a. Notice that this theorem in particular implies that all maximal abelian subspaces of p have the same dimension r D rank.S/. Lemma 2.7. The vector X 2 p is regular if and only if the geodesic c  S defined by c.t/ ´ e tX  o, t 2 R, is contained in precisely one flat. Proof. Suppose X 2 p is regular, i.e. a ´ Zg .X / \ p is maximal abelian, and c is contained in more than one flat. Since every flat through the base point o is of the 0 form e a  o with a0  p a maximal abelian subspace, we may assume that c  e a  o 0 and c  e a  o, a0  p maximal abelian, a0 ¤ a. Since X 2 a0 and a0 is abelian we conclude that every element in a0  p commutes with X , hence is contained in Zg .X / \ p D a. But a0  a and dim a0 D dim a D rank.S / then imply a0 D a, a contradiction. Conversely assume that c is contained in precisely one flat, say e a  o for a  p maximal abelian. Suppose Zg .X/ \ p is not maximal abelian. Then a  Zg .X / \ p and there exists X 0 2 Zg .X/ \ p such that X 0 … a. Choose a0  p maximal abelian such that X 0 2 a0 . Then by the choice of X 0 we have a0 ¤ a, and X 0 2 Zg .X / implies ŒX 0 ; X  D 0, hence X 2 a0 . We conclude that c is contained in the two different flats 0 e a  o and e a  o, a contradiction. Example 1. X 2 p is regular if and only if all its eigenvalues are distinct. Let us look in the case n D 3 at the singular vector X D Diag.1; 1; 2/ 2 p D sym0 .3/. An easy computation shows that for any 2 R the element k. / ´  cos  sin  0  sin  cos  0 0 0 1

2 K D SO.3/ satisfies Ad.k. //X D X .

In particular, there exists a one-parameter family of flats containing the geodesic c defined by c.t / D e tX  o, t 2 R. Example 2b. X 2 p is regular if and only if all its eigenvalues are distinct and different from zero. In the case q D 2 the vector X D Diag.1; 1; 1; 1/ 2 p is singular: clearly every element in K of the form 0 1 cos sin

0 B  sin cos

C C ; 2 R; kDB @ cos sin A 0  sin cos

satisfies Ad.k/X D X . For 1 ; 2 > 0 we set  1 cos2 C 2 sin2

A .1 ; 2 / ´ .2  1 / sin cos

 .2  1 / sin cos

: 1 sin2 C 2 cos2

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  t Then the geodesic c.t / ´ e tX  o D et0I e0 I2  S4 belongs to each of the 2 following flats parametrized by 2 R: n  o 0 A .1 ;2 / W ; > 0  S4 : 1 1 1 2 A . ; / 0 1 2

Similarly, the vector Y D Diag.1; 0; 1; 0/ is invariant by Ad.k/ for any k 2 K of the form 0 1 1 0 0 0 B0 cos 0 sin C C ; 2 R: kDB @0 0 1 0 A 0  sin 0 cos

! e 2t 0 So for all 2 R the geodesic c.t / ´ e tY  o D contained in the flat 80 0 0 ˆ < 0 . 1 2 / sin  cos  2 B @ 1 0 ˆ 1 : 1 2 2

0 2 sin C  cos  2

0

1 0 0  1 sin2  2 cos2  2

0

0

0

.2  1 / sin  cos 

1

0

e 2t 0

0 1

1

0

 S4 is

9 > =

C A W 1 ; 2 > 0 : > ;

2

Example 2c. A geodesic c in H2  H2 is of the form c.t / D .c1 .t cos /; c2 .t sin //; where ci are geodesics in the i-th H2 -factor, i D 1; 2, and 2 Œ0; =2. The geodesic c is regular if the parameter is contained in the open interval .0; =2/, c is singular if 2 f0; =2g. In other words, c is singular if and only if its projection to one of the factors is a point.

2.2 Roots and root spaces Recall that ‚ W g ! g is the Cartan involution, and B W g  g ! R the Killing form on g. For this section we will use the following positive definite bilinear form on g hhX; Y ii ´ B.X; ‚.Y //;

X; Y 2 g;

which is obtained from the Killing form by “changing sign on k". Notice that on p the scalar product hh; ii coincides with the one induced by Bjp (which by our assumption made at the beginning of Section 2 determines the Riemannian structure on S ). Lemma 2.8. The operator ad X, X 2 p, is self-adjoint on g with respect to hh; ii. Proof. Let X 2 p. We show that for all Y; Z 2 g hh.ad X/Y; Zii D hhY; .ad X /Zii:

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Indeed, we have hhŒX; Y ; Zii D B.ŒX; Y ; ‚Z/ D B.‚Z; ŒX; Y /; using Proposition 1.19 (1), we conclude hhŒX; Y ; Zii D B.‚Z; ŒY; X/ D‚X

‚…„ƒ D B.Y; Œ X ; ‚Z/ D B.Y; Œ‚X; ‚Z/ D B.Y; ‚ŒX; Z/ D hhY; ŒX; Zii:  Corollary 2.9. If a  p is maximal abelian, then fad H W H 2 ag is a commutative family of self-adjoint operators on g with respect to hh; ii. This corollary in particular implies that g decomposes into an orthogonal direct sum of common eigenspaces with respect to hh; ii. This motivates the following Definition 2.10. A linear map ˛ W a ! R is called a root of the pair .g; a/ if g˛ ´ fX 2 g W ŒH; X D ˛.H /X for all H 2 ag ¤ f0g. The subspace g˛ of g is then called a root space. It is easy to see that a  g0 , where the subscript 0 denotes the trivial linear map. We will write † for the set of non-trivial roots. We have #† < 1, and M g˛ : (2.2) g D g0 ˛2†

Recall that ‚ W g ! g denotes the Cartan involution. It is easy to see that for all X 2 g we have X C ‚X 2 k, and X  ‚X 2 p. Lemma 2.11. For all ˛ 2 † we have ‚g˛ D g˛ : Proof. Let X 2 g˛ . Then for all H 2 a we have ŒH; X  D ˛.H /X . Moreover, H 2 p, i.e. ‚H D H . We conclude, using the fact automorphism,   that‚ is a Lie algebra  ŒH; ‚X  D Œ‚.„ƒ‚… ‚H /; ‚X D ‚ ŒH; X  D ‚  ˛.H /X D ˛.H /‚X . DH

Lemma 2.12. We have Œg˛ ; gˇ  ´ fŒX; Y  W X 2 g˛ ; Y 2 gˇ g  g˛Cˇ

for all ˛; ˇ 2 †:

Proof. Let X 2 g˛ , Y 2 gˇ . Then for H 2 a we have by the definition of ad W g ! gl.g/ and the Jacobi identity .ad H /ŒX; Y  D ŒH; ŒX; Y  D ŒX; ŒY; H   ŒY; ŒH; X  D ŒX; ˇ.H /Y   ŒY; ˛.H /X  D ˇ.H /ŒX; Y   ˛.H /ŒY; X  D .˛ C ˇ/.H /ŒX; Y :



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Lemma 2.13. H 2 a n f0g is regular if and only if ˛.H / ¤ 0 for any ˛ 2 †. Proof. Let H 2 a n f0g. First assume that H is regular, i.e. Zg .H / \ p is maximal abelian. Since a  Zg .H / this implies a D Zg .H / \ p. Suppose there exists ˛ 2 † such that ˛.H / D 0. Then for all X 2 g˛ we have ŒH; X  D ˛.H /X D 0, hence g˛  Zg .H /. Similarly we have g˛ 2 Zg .H /, and therefore X  ‚X 2 Zg .H / \ p D a for all X 2 g˛ , a contradiction. Conversely suppose ˛.H / ¤ 0 for all ˛ 2 †, and Zg .H / \ p is not maximal abelian. Then there exists Y 2 p  g, Y … a, Y ¤ 0 such that ŒH; Y  D 0. For ˛ 2 † denote by Y˛ the projection of Y to g˛ . Then X X ŒH; Y˛  D ˛.H /Y˛ : 0 D ŒH; Y  D ˛2†

˛2†

Since ˛.H / ¤ 0 for all ˛ 2 † and g is a direct sum of the g˛ , this implies Y˛ D 0 for all ˛ 2 †, a contradiction to Y ¤ 0. Corollary 2.14. If areg denotes the set of regular vectors in a, then [ ker.˛/: areg D a n ˛2†

Definition 2.15. A Weyl chamber in a is a connected component of areg . Notice that a Weyl chamber is isomorphic to an open Euclidean cone in a. In the sequel Eij will denote a quadratic matrix which has a 1 at the position i -th row, j -th column, and zeros everywhere else. The size of the matrix will be taken so that it fits into the frame of the example considered. Example 1. For SL.n; R/= SO.n/ we consider H D Diag.t1 ; t2 ; : : : ; tn / 2 a. An easy calculation shows that .ad H /Eij D ŒH; Eij  D .ti  tj /Eij : Hence we have n.n  1/ non-zero roots, and g0 D a. In particular X sl.n; R/ D a C R  Eij : i¤j

A Weyl chamber in a is e.g. given by  ˚ P aC ´ Diag.t1 ; t2 : : : ; tn / W niD1 ti D 0; t1 > t2 >    > tn : Example 2a. Recall the set of .p  q/-matrices with values  that M.p; q/ denotes  in R. For SO.2; 3/= SO.2/  SO.3/ we have 9 8 0 1 t1 0 0 0 = < 0 t2 0 A W t1 ; t2 2 R Š R2 : a D H.t1 ; t2 / ´ @ t1 0 ; : 0 t2 0 0

0

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Here we have 8 roots: if H D H.t1 ; t2 / then a calculation shows that with ˛1 , ˛2 defined by ˛1 .H / D t1 , ˛2 .H / D t2 , the set of roots is given by f˛1 ; ˛2 ; ˛1 ; ˛2 ; ˛1 C ˛2 ; ˛1  ˛2 ; ˛1 C ˛2 ; ˛1  ˛2 g: The corresponding root spaces are   g˛1 D R  E15 C E51 C E35  E53 ;   g˛2 D R  E25 C E52 C E45  E54 ;

  g˛1 D R  E15 C E51  E35 C E53 ;   g˛2 D R  E25 C E52  E45 C E54 ;

  g˛1 C˛2 D R  E12  E21 C E23 C E32  E14  E41  E34 C E43 ;   g˛1 ˛2 D R  E12  E21 C E23 C E32 C E14 C E41 C E34  E43 ;   g˛1 C˛2 D R  E12  E21  E23  E32  E14  E41 C E34  E43 ;   g˛1 ˛2 D R  E12  E21  E23  E32 C E14 C E41  E34 C E43 : A Weyl chamber in a is for example aC D fH.t1 ; t2 / 2 a W t1 > t2 > 0g.   Example 2b. For Sp.4; R/= SO.4/ \ Sp.4; R/ we have a D fH.t1 ; t2 / ´ Diag.t1 ; t2 ; t1 ; t2 / W t1 ; t2 2 Rg Š R2 : If H D H.t1 ; t2 / then with ˛1 , ˛2 defined by ˛1 .H / D t1 , ˛2 .H / D t2 , the set of roots is given by f2˛1 ; 2˛2 ; 2˛1 ; 2˛2 ; ˛1 C ˛2 ; ˛1  ˛2 ; ˛1 C ˛2 ; ˛1  ˛2 g: Here the corresponding root spaces are g2˛1 D R  E13 ; g˛1 C˛2 g˛1 C˛2

g2˛2 D R  E24 ; g2˛1 D R  E31 ; g2˛2 D R  E42 ;     D R  E14 C E23 ; g˛1 ˛2 D R  E12  E43 ;     D R  E21  E34 ; g˛1 ˛2 D R  E32 C E41 ;

and a possible Weyl chamber in a is aC D fH.t1 ; t2 / 2 a W t1 > t2 > 0g. Notice that even though at first sight this root system looks different from the one in Example 2a, the root systems are isomorphic: taking instead of ˛1 , ˛2 the roots 1 , 2 defined by 1 .H / D t1 C t2 and 2 .H / D t1  t2 , the set of roots equals f1 C 2 ; 1  2 ; 1  2 ; 1 C 2 ; 1 ; 2 ; 2 ; 1 g; and the corresponding root spaces obviously are g˙1 D g˙.˛1 C˛2 / ; g˙.1 C2 / D g˙2˛1 ;

g˙2 D g˙.˛1 ˛2 / ; g˙.1 2 / D g˙2˛2 :

The fact that the two root systems are the same has a deeper reason: the Lie algebras so.2; 3/ and sp.4; R/ are isomorphic (see e.g. Chapter X, §6.4 (ii) in [H]). Remark 2.16. The roots of the pair .g; a/ form a root system in the finite dimensional vector space a over R according to the following definition.

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Definition 2.17 ([H], X.3.1). Let V be a finite dimensional vector space over R and R  V a finite set of non-zero vectors. R is called a root system in V , and its members are called roots if (1) R generates V ; (2) for each ˛ 2 R there exists a reflection s˛ along ˛ leaving R invariant; (3) for all ˛, ˇ 2 R the number m˛ˇ determined by s˛ ˇ D ˇ  m˛ˇ ˛ is an integer, i.e. m˛ˇ 2 Z.

2.3 Iwasawa decomposition For this section we fix a Weyl chamber aC  a, and denote by r ´ dim a the rank of S . We will need the following subset of the set of roots † of the pair .g; a/ †C ´ f˛ 2 † W ˛.H / > 0 for all H 2 aC g: Definition 2.18. A root is called simple if it cannot be written as a sum ˛ D ˇ C , where ˇ, 2 †C . By Theorem III.V.7 in [H] there exist a set of r simple roots ‡ ´ f˛1 ; ˛2 ; : : : ; ˛r g and c1 ; : : : ; cr 2 N [ f0g such that ˛D

r X

ci ˛i

for all ˛ 2 †C :

iD1

Such a set is called a fundamental set of roots. Moreover, we have † D †C t .†C /: We next put nC ´

X

g˛ ;

˛2†C

which is a nilpotent Lie algebra by Lemma 2.12 and the fact that † is a finite set. Set C N C ´ e n which is a unipotent subgroup of G. The following theorem is called the Iwasawa decomposition: Theorem 2.19 ([H],Theorem IX.1.3). We have g D k ˚ a ˚ nC . If A D e a , then the mapping K  A  N C ! G, .k; a; n/ 7! kan is a diffeomorphism. As a consequence we have S D G  o D N C A  o which is sometimes called the “foliation by flats”. N C -orbits in S are called horocycles.

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Example 1. We consider the set †C of positive roots with respect to the Weyl chamber ˚  P aC D Diag.t1 ; t2 ; : : : ; tn / W niD1 ti D 0; t1 > t2 >    > tn : Let ˛i denote the simple root determined by ˛i .Diag.t1 ; t2 ; : : : tn // D ti  tiC1 for i 2 f1; 2; : : : ; n  1g. Then a fundamental set of roots is precisely the set ‡ D f˛1 ; ˛2 ; : : : ; ˛n1 g. The nilpotent Lie algebra nC is the set of upper triangular .n  n/-matrices with zeros in the diagonal, and N C is the group of upper triangular .n  n/-matrices with 1’s in the diagonal. If DiagC .n/ denotes the set of diagonal .n  n/-matrices with positive entries and determinant one, we have SL.n; R/ D SO.n/  DiagC .n/  N C :

(2.3)

This decomposition simply comes from the Gram–Schmidt orthonormalization procedure in linear algebra. In H2 the foliation by flats is simply the foliation by geodesics given by       t=2 0 1 x 1 x e  i D  e t i D e t i C x; x; t 2 R:  0 1 0 1 0 e t=2 Any fixed t 2 R determines a horocycle fe t i C x W x 2 Rg.   Example 2b. In Sp.4; R/= SO.4/ \ Sp.4; R/ we take the Weyl chamber ˚  aC D Diag.t1 ; t2 ; t1 ; t2 / W t1 > t2 > 0 : Then the set of positive roots in the notation from the previous section is given by †C D f2˛1 ; 2˛2 ; ˛1 C ˛2 ; ˛1  ˛2 g. The roots 2˛1 and ˛1 C ˛2 are not simple; a fundamental set of roots is ‡ D f˛1  ˛2 ; 2˛2 g. Here we have ´ ! μ 0 a x z nC D

0

0 0

z 0 a

y 0 0

W a; x; y; z 2 R :

2.4 The space of maximal flats Let a  p be a maximal abelian subspace. In Section 2.1 we have seen that every set of the form ge a  o, g 2 G, is a maximal flat in S . The following theorem shows that every maximal flat can be written in this way. Theorem 2.20. All maximal flats are conjugate in S, i.e. the space of maximal flats is homogeneous. Proof. We have to show that for arbitrary maximal flats F , F 0 in S there exists g 2 G such that F 0 D g  F . Since G acts transitively on S we may assume that o 2 F . Pick x 2 F 0 and let g 2 G be such that g  x D o. Replacing F 0 by g  F 0 we may further assume that o 2 F 0 . Now let a, a0  p be maximal abelian subspaces

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such that F D e a  o and F 0 D e a  o. We show that there exists k 2 K such that 0 F 0 D k  F , i.e. e a  o D ke a  o. This is equivalent to the existence of k 2 K such that a0 D Ad.k/a, hence the claim follows from Theorem 2.6. Example 1. Recall from (2.1) that in Pos1 .n/ the set ˚  Q F D Diag. 1 ; : : : ; n / W 1 ; : : : ; n > 0; niD1 i D 1 is a maximal flat containing the base point o D In . Given an arbitrary point p 2 Pos1 .n/, how does a flat through o containing p look like? Since p 2 Pos1 .n/ is diagonalizable, there exists k 2 SO.n/ such that the matrix kpk 1 is diagonal with positive entries and determinant 1. With the action of SL.n; R/ on Pos1 .n/ given by g  p ´ g t pg, p 2 Pos1 .n/, g 2 SL.n; R/, we conclude kpk 1 D .k 1 /t pk 1 D k 1  p 2 F; hence F 0 ´ k  F is a flat in Pos1 .n/ through o containing p. Notice that the matrix k 2 SO.n/ above is not unique. Conjugating with an appropriate element w 2 SO.n/ we can arrange that wkpk 1 w 1 is a diagonal matrix Diag. 1 ; : : : ; n / such that 1  2      n . This motivates the definition of the Weyl group in the following section. Example 2c. If F is a flat in H2  H2 , there exist unit speed geodesics c1 , c2 in the two factors such that F D f.c1 .t1 /; c2 .t2 // W t1 ; t2 2 Rg. Since SL.2; R/ acts simply transitively on the set of unit speed geodesics of H2 , there exists .g1 ; g2 / 2 SL.2; R/SL.2; R/ such that g1 c1 .t1 / D e t1 i and g2 c2 .t2 / D e t2 i for all t1 ; t2 2 R.

2.5 Weyl group and opposition involution We have seen that a maximal flat F  S is an isometric copy of Rr , where r denotes the rank of S. Hence abstractly its full isometry group would be O.n/ Ë Rn . However, the induced isometries of F (i.e. the isometries of F in G D Iso .S /) are generated by all translations, but only finitely many rotations. These rotations are encoded in the so-called Weyl group of S . We denote by M the centralizer, and by M  the normalizer of a in K, i.e. M D fk 2 K W Ad.k/H D H for all H 2 ag; M  D fk 2 K W Ad.k/H 2 a for all H 2 ag: Definition 2.21. The Weyl group of S is the factor group M  =M . The Weyl group is finite and satisfies the following properties which are proved in Chapter VII of [H]. Proposition 2.22 ([H], VII.2). (1) W leaves invariant the configuration of singular hyperplanes.

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(2) W is finitely generated by reflections s˛ (in the walls of a fixed Weyl chamber). (3) W acts simply transitively on the set of Weyl chambers of a flat with apex o. Remark 2.23. If S is a rank one symmetric space, then the Weyl group is isomorphic to Z=2Z. Given a Weyl chamber aC  a, there exists a unique element w 2 W such that every representative mw 2 M  of w satisfies Ad.mw /aC D aC ´ fH 2 a W H 2 aC g: If aC denotes the closure of the Weyl chamber aC , then this element defines a map

W aC ! aC H 7!  Ad.mw /H;

(2.4)

which is called the opposition involution. Notice that is an isometry which is the identity if and only if Ad.mw / D  ida . Example 1. For Pos1 .n/ the Weyl group is isomorphic to the group of permutations of n-tuples. The element w corresponds to the permutation which maps the n-tuple .t1 ; t2 ; : : : ; tn / to .tn ; tn1 ; : : : ; t1 /. For the closed Weyl chamber  ˚ P aC D Diag.t1 ; t2 ; : : : ; tn / W niD1 ti D 0; t1  t2      tn  sym0 .n/ the opposition involution is given by  

Diag.t1 ; t2 ; : : : ; tn / D Diag.tn ; tn1 ; : : : ; t1 /: Example 2b. For S2q the Weyl group is isomorphic to the subgroup of permutations of 2q-tuples generated by transpositions among the first q elements and the transpositions .k; k C q/, 1 k q. The element w here acts as  ida , so D idaC . Example 2c. For H2  H2 the Weyl group is isomorphic to Z=2Z  Z=2Z. The opposition involution is again the identity. Definition 2.24. By abuse of notation a Weyl chamber in S is defined to be a set of C the form ge a  o, where aC is a Weyl chamber in a and g 2 G.

2.6 Cartan decomposition and Cartan vector This section provides a refinement of the Cartan decomposition of the Lie algebra g of G studied in Section 1.3. We fix a Weyl chamber aC  a in a maximal abelian subalgebra a of p and denote by aC its closure. The following theorem implies that symmetric spaces of non-compact type are Weyl chamber isotropic.

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Theorem 2.25. pD

[

Ad.k/aC :

k2K

Proof. Let X 2 p be arbitrary. By Theorem 2.6 there exists k 2 K such that Ad.k/X 2 a. The claim now follows from the fact that the Weyl group acts transitively on the set of Weyl chambers of a. Since S is complete, the Theorem of Hopf–Rinow implies that the base point o 2 S can be joined to any point  x 2 S by a geodesic. Moreover, since De  W p ! To S is an isomorphism and expo De .X/ D e X o, we have S D e p o. So S is diffeomorphic to Rdim p and we obtain from the previous theorem polar coordinates for S: C

Corollary 2.26. S D e p  o D Ke a  o. If x D ke H  o 2 S, we will call k 2 K an angular projection and H 2 aC the Cartan projection of x. It can be shown (see e.g. [H], Theorem IX.1.1) that the Cartan projection of a point x is unique, whereas its angular projection in general is not. Using the fact that S Š G=K, we get the Cartan decomposition of G: C

Corollary 2.27. G D Ke a K. The following definition plays an important role in the theory of higher rank symmetric spaces: Definition 2.28. Given x; y 2 S , we choose g 2 G such that g  x D o. The Cartan vector H.x; y/ 2 aC of the ordered pair of points .x; y/ 2 S  S is defined as the Cartan projection of g  y. Notice that the definition of the Cartan vector H.x; y/ does not depend on the choice of g 2 G such that g  x D o. Indeed, if h 2 G also satisfies h  x D o, then hg 1 2 K. So if g  y D ke H  o we get h  y D hg 1 g  y D hg 1 k e H  o: „ƒ‚… 2K

Furthermore, the length of the Cartan vector of the ordered pair of points .x; y/ 2 S S is simply the distance d.x; y/. In particular, if S is a rank one symmetric space, then the Cartan vector reduces to the Riemannian distance. Hence in higher rank symmetric spaces the Cartan vector is a natural generalization of the Riemannian distance function. We remark that H.y; x/ D .H.x; y//, where is the opposition involution (2.4). This can be seen as follows: let h 2 G be such that h  x D o and h  y D e H.x;y/  o, and mw 2 M  be a representative of w 2 W . Then g ´ e H.x;y/ h 2 G satisfies

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g  y D o, and we have g  x D e H.x;y/ h  x D e H.x;y/  o D .mw /1 e .H.x;y// mw  o D .mw /1 e .H.x;y//  o; hence the Cartan projection H.y; x/ of g  x equals .H.x; y//.

3 The geometry at infinity In this section we will describe the geometry at infinity of a globally symmetric space S of non-compact type. From Section 1.1 and Theorem 1.29 we know that S is in particular a Hadamard manifold, i.e. a complete simply connected Riemannian manifold of non-positive sectional curvature. Therefore S is homeomorphic to Rdim S and can be compactified by attaching its so-called geometric boundary. Due to the rich algebraic structure of globally symmetric spaces, this boundary can be described much more precisely than it is possible for general Hadamard manifolds. In particular, there exists a natural quotient of a dense subset of the geometric boundary which is called the Furstenberg boundary; for rank one symmetric spaces these two boundaries coincide. On the other hand, we are led to study the pairs of points in the geometric boundary which can be joined by a geodesic. For rank one symmetric spaces all pairs of distinct boundary points can be joined by a geodesic; in the higher rank setting the flats destroy this property. However, the Bruhat decomposition will allow us to describe the pairs of boundary points which can be joined by a geodesic. Finally we will introduce Busemann functions which serve as a tool in the construction of G-invariant Finsler pseudo-distances on S. When studying the action of discrete groups on S , these pseudo-distances play a key role for the construction of generalized Patterson–Sullivan measures (see e.g. [A], [L2]). However, we will not touch on this subject here. Recall the notation from Section 2: o 2 S denotes the base point, G D Iso .S / and K  G the compact isotropy subgroup at o. Let g D k ˚ p be the Cartan decomposition and assume that the Riemannian structure of S is induced by the Killing form restricted to p. Moreover, we will fix once and for all a Weyl chamber aC in a maximal abelian subalgebra a of p. Throughout this section all geodesics and geodesic rays are supposed to have unit speed.

3.1 The geometric boundary of S Definition 3.1. We say that two geodesic rays c1 , c2 are equivalent if d.c1 .t/; c2 .t//

is bounded as t ! 1:

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The geometric boundary @S of S is defined as the set of geodesic rays in S modulo this equivalence relation. In order to topologize the space Sx ´ S [ @S, we introduce the following sets: x for " > 0; R  1, x 2 S and  2 @S let CR;" x;  S be the truncated cone x CR;" x; ´ fy 2 S W d.x; y/ > R; d.cx; .R/; cx;y .R// < "g in Sx, where cx; denotes the unique unit speed geodesic emanating from x 2 S in the class of  2 @S (compare (1.1) for the definition of cx;y , y 2 S ). Definition 3.2 ([Ba], Chapter II). The cone topology on Sx is the topology generated by the open sets in S and these truncated cones. If not stated otherwise, convergence in S [ @S will always mean convergence with respect to the cone topology. The relative topology on @S turns the geometric boundary into a topological space. The isometry group of S has a natural action by homeomorphisms on the geometric boundary. If g 2 G, and  2 @S is represented by a geodesic ray c in S, then g   is the class of the geodesic ray g  c in S . Notice that this assignment does not depend on the choice of the geodesic ray c in the class of : indeed, if c 0 is a ray different from c representing , then d.c.t /; c 0 .t// is bounded as t tends to infinity. Since g is an isometry, d.g  c.t /; g  c 0 .t// is bounded as t tends to 1, hence g  c 0 is equivalent to g  c and therefore represents the same point in the geometric boundary. It is well-known that the geometric boundary endowed with the cone topology is homeomorphic to the unit tangent space of an arbitrary point x 2 S . If p1 and aC 1 denote the set of vectors of length 1 with respect to the Killing form in p and aC respectively, then @S Š To1 S Š p1 D Ad.K/aC 1: In particular, a tuple .k; H / 2 K  aC 1 defines a unique point in @S by taking the class of the geodesic ray c.t / ´ ke H t  o, t > 0. Conversely, given a point  2 @S there exists k 2 K and H 2 aC 1 such that  is the class of the geodesic ray c.t / ´ ke H t  o, t > 0. In this case we write  D c.1/. By the Cartan decomposition, H is uniquely determined by , whereas k is only determined up to right multiplication by an element in the centralizer of H in K. We call k an angular projection, and H the Cartan projection of , and we will write  D .k; H /. If r D rank.S/ > 1, we define the regular boundary @S reg as the set of classes reg D @S. with Cartan projection in aC 1 . If rank.S/ D 1, we use the convention @S Notice that G   D K   for any  2 @S . Furthermore, G acts transitively on @S if and only if rank.S/ D 1.

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Example 1. For n D 2, the symmetric space SL.2; R/= SO.2/ can be identified with H2 . The geometric boundary of the hyperbolic plane H2 is the set R [ f1g which is homeomorphic to the sphere S1 . In general, for n  2, a point  in the geometric boundary of S D Pos1 .n/ determines an eigenvalue-flag pair as follows: let X D X./ 2 p Š To S be the unit vector such that the geodesic ray c.t / ´ e Xt  o, t > 0, satisfies c.1/ D . Let f 1 ; 2 ; : : : ; l g be the l n distinct eigenvalues of X, arranged so that 1 > 2 >    > l . For 1 i l let Ei be the eigenspace of X in Rn for the eigenvalue i , mi ´ dim Ei , and Vi the direct sum of the eigenspaces fEj W 1 j i g. We thus obtain a flag of subspaces V1  V2      Vl D Rn . Notice that X 2 p D sym0 .n/ P implies Tr.X/ D 0, hence liD1 mi i D 0. Moreover, B.X; X / D 1 translates into P the condition liD1 mi 2i D 1. Hence to each point  2 @S we have associated a vector ./ D . 1 ; 2 ; : : : ; l / 2 Rl and a flag F ./ D .V1 ; V2 ; : : : ; Vl / in Rn subject to the above conditions. Such a pair will be called an eigenvalue-flag pair. The group G D SL.n; R/ acts naturally on the flags in Rn : if g 2 G, then a subspace V  Rn is mapped by g to the subspace g  V  Rn of the same dimension. So a flag F D .V1 ; V2 ; : : : ; Vl / is mapped to the flag g F ´ .g V1 ; g V2 ; : : : ; g Vl /. We can therefore consider the action of G D SL.n; R/ on the set of eigenvalue-flag pairs given by g  . ; F / ´ . ; g  F /;

g 2 G:

For 1 j n we denote by e.j / the j -th standard basis vector in Rn . Suppose ˚  P X./ 2 aC D Diag.t1 ; t2 ; : : : ; tn / W t1      tn ; niD1 ti D 0 : Let f 1 ; 2 ; : : : ; l g, 2 l n, be the distinct eigenvalues of X./ in decreasing order, Pi and mi , 1 i l, the multiplicity of the eigenvalue i . Then with di ´ j D1 mj the flag F ./ D .U1 ; U2 ; : : : ; Ul / is given by Ui D spanR .e.1/; : : : ; e.di //;

1 i l:

(3.1)

This flag will be called the standard flag in Rn determined by .m1 ; m2 ; : : : ; ml /. Conversely, given an integer l with 2 l n, a flag of subspaces F D .V1 ; V2 ; : : : ; Vl / in Rn , mi ´ dim Vi  dim Vi1 , 1 i l, and a vector D . 1 ; 2 ; : : : ; l / 2 Rl satisfying 1 > 2 >    > l ;

l X iD1

mi i D 0

and

l X

mi 2i D 1;

iD1

then there exists a unique point  2 @S such that ./ D and F ./ D F as follows. Let H 2 sym0 .n/ be the diagonal matrix with entries 1 ; 2 ; : : : ; l occurring according to their multiplicities. We then choose an element k 2 K D SO.n/ such that k  F is the standard flag in Rn determined by .m1 ; m2 ; : : : ; ml /; this is possible because we can choose an orthonormal basis for V1 , and if i  2 we can extend the orthonormal basis of Vi1 to an orthonormal basis of Vi . Moreover, different choices

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of such k 2 K are equal up to left multiplication by an element in K which preserves the standard flag in Rn determined by .m1 ; m2 ; : : : ; ml /; hence all possible choices of k 2 K define the same ray c.t / ´ k 1 e H t  o, t > 0, and we can set  D c.1/. So we have seen that the geometric boundary @S of S D Pos1 .n/ is identified with the set of eigenvalue-flag pairs. Notice that  2 @S reg if and only if l D n. Moreover, this identification is G-equivariant: it can be shown that for g 2 G and  2 @S with corresponding eigenvalue-flag pair . ./; F .// we have     .g  /; F .g  / D g  ./; F ./ : Example 2b. Consider the space S D S2q of !-compatible complex structures on .R2q ; !/ with base point o 2 S2q given by the matrix (1.6). A point  in the geometric boundary @S2q is uniquely determined by an element X D X./ 2 p1 such that co; .t / D e Xt  o. Now X 2 sym0 .2q/ \ sp.2q; R/ implies that X possesses q pairs of eigenvalues . ;  / with  0. Denote by f 1 ; 2 ; : : : ; l g the l q distinct positive eigenvalues of X, arranged so that 1 > 2 >    > l > 0. For 1 i l let Ei  R2q be the eigenspace of X for the eigenvalue i , mi ´ dim Ei , and Wi the direct sum of the eigenspaces fEj W 1 j i g. Notice that the subspaces Wi , 1 i l, are isotropic, i.e. Wi  Wi! ´ fx 2 R2q W !.x; y/ D 0 for all y 2 Wi g: We so obtain a flag of isotropic subspaces W1  W2  : : :  Wl of the symplectic vector space .R2q ; !/. Since B.X; X / D 1 we further have the condition P 2 kiD1 mi 2i D 1. Hence to each point  2 @S2q we have associated a vector ./ D . 1 ; 2 ; : : : ; l / 2 Rl and a flag F ./ D .W1 ; W2 ; : : : ; Wl / of isotropic subspaces in R2q . The symplectic group Sp.2q; R/ maps an isotropic subspace of .R2q ; !/ to an isotropic subspace of .R2q ; !/. So G D Sp.2q; R/ acts naturally on the set of isotropic flags as above: if g 2 G and F D .W1 ; W2 ; : : : ; Wl / is an isotropic flag, then g  F is the isotropic flag g  F ´ .g  W1 ; g  W2 ; : : : ; g  Wl /. As in the previous example we will consider the action of G D Sp.2q; R/ on the set of pairs of positive eigenvalues and isotropic flags given by g  . ; F / ´ . ; g  F /;

g 2 G:

Let e.j / denote the j -th standard basis vector in R2q , 1 j 2q, and assume that ˚   0 X./ 2 aC D D 0 D W D D Diag.t1 ; t2 ; : : : ; tq /; t1  t2      tq : If f 1 ; 2 ; : : : ; l g, 1 l q, are the distinct positive eigenvalues of X./ P in decreasing order, mi denotes the multiplicity of the eigenvalue i , and di ´ ji D1 mj , 1 i l, then the flag F ./ D .U1 ; U2 ; : : : ; Ul / is given by Ui D spanR .e.1/; : : : ; e.di //;

1 i l:

(3.2)

This flag will be called the isotropic standard flag in .R ; !/ determined by .m1 ; m2 ; : : : ; ml /. 2q

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Notice that for 1 i l the eigenspace Ei for the eigenvalue i is given by Ei D spanR .e.di 1 C 1/; : : : ; e.di //, where we used the convention d0 D 0; moreover, the eigenspace Ei for the eigenvalue  i is Ei D spanR .e.di1 C q C 1/; : : : ; e.di C q//. If dl < q, then the eigenspace E0 for the eigenvalue 0 is the symplectic subspace E0 D spanR .e.dl C 1/; : : : ; e.q/; e.dl C q C 1/; : : : ; e.2q//. Conversely, given an integer l with 1 l q, a flag of isotropic subspaces F D .W1 ; W2 ; : : : ; Wl / in R2q , mi ´ dim Wi  dim Wi1 , 1 i l, and a vector D . 1 ; 2 ; : : : ; l / 2 Rl such that l X 1 > 2 >    > l > 0 and 2  mi 2i D 1; iD1

then there exists a unique point  2 @S such that ./ D and F ./ D F as follows: let D 2 M.q; q/ be the diagonal matrix with entries 1 ; 2 ; : : : ; l occurring P according to their multiplicities; if dl ´ liD1mi < q, the remaining q dl diagonal 0 entries are filled with zeros. Set H ´ D 0 D . Consider the standard scalar product 2q q0 in R which – as we have seen in Section 1.5 – is given by q0 .x; y/ ´ !.x; J0 y/;

x; y 2 R2q ;

where J0 denotes the complex structure given by the matrix (1.6). Notice that if dl D q, then the subspace L ´ Wl is necessarily Lagrangian, i.e. L D L! . If dl < q we choose a Lagrangian subspace L of R2q such that Wl  L. We take an orthonormal basis fb1 ; : : : ; bm1 g of W1 with respect to q0 and extend it inductively to an orthonormal basis of W2 ; : : : ; Wl ; L. By a standard procedure in symplectic linear algebra (see e.g. Section 1.1 in [dS]) we can inductively extend the orthonormal basis fb1 ; b2 ; : : : ; bq g of L to an orthonormal basis fb1 ; : : : ; b2q g of R2q such that !.bi ; bj / D ıi;j q for all i; j 2 f1; 2; : : : ; 2qg. Hence one can find k 2 K D SO.2q/ \ Sp.2q; R/ such that k  bi D e.i / for all 1 i 2q, which implies that k  F is the isotropic standard flag determined by .m1 ; m2 ; : : : ; ml /. Notice that if k 0 2 K, k 0 ¤ k, maps F to the isotropic standard flag determined by .m1 ; m2 ; : : : ; ml /, then k 1 k 0 2 K has to leave invariant the eigenspaces of H . This precisely translates to the fact that Ad.k 1 k 0 /H D H , hence any such k determines the same geodesic ray c.t / ´ k 1 e H t  o, t > 0, and we set  D c.1/. So we have seen that @S can be identified with the set of pairs of positive eigenvalues and isotropic flags. Notice that  2 @S reg if and only if l D q. If X./ possesses only one positive eigenvalue > 0, then the flag of isotropic subspaces reduces to a qdimensional Lagrangian subspace L D L! of .R2q ; !/. The condition B.X; X / D 1 moreover implies D p12q . As before one can show that the identification described above  is G-equivariant:  if g2 G and  2 @S with corresponding pair . ./; F .// then .g  /; F .g  / D g  ./; F ./ . Example 2c. If S D H2  H2 , for i D 1; 2 we denote by @Si Š S1 the geometric boundary of the i -th H2 -factor. Then the regular geometric boundary @S reg can be

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identified with @S1  @S2  .0; =2/. There are two singular boundary strata, one isomorphic to the boundary of the first factor @S1 and one isomorphic to the boundary of the second factor @S2 . Hence   @S Š @S1 t @S2 t @S1  @S2  .0; =2/ :

3.2 The Furstenberg boundary Now let us see what happens if we forget about the aC 1 -factor. Recall that M is the centralizer of a in K and consider the projection  B W @S reg ! K=M; .k; H / 7! kM: Definition 3.3. We define the Furstenberg boundary @F S as  B .@S reg /. The Furstenberg boundary has a natural differentiable structure arising from the Lie group structure of K. Geometrically it can be described as the set of equivalence classes of Weyl chambers in S (see [M]), where two Weyl chambers in S are equivalent if and only if they have bounded Hausdorff distance. The following lemma relates the cone topology to the topology of K=M . It is a corollary of Lemma 2.9 in [L1]. Lemma 3.4. A sequence .n /  @S reg converges to  D .k; H / 2 @S reg in the cone topology if and only if  B .n / converges to kM in K=M and the Cartan projections of n converge to H in aC 1. Hence  B is continuous, and rank.S/ D 1 if and only if  B is a homeomorphism. Moreover, the projection  B induces an action of G by homeomorphisms on the Furstenberg boundary K=M D  B .@S reg /. More precisely, if G D KAN C is the Iwasawa decomposition from Section 2.3 with A D e a , and  I the natural projection  I W G ! K=M; g D kan 7! kM; then we have the following 0 Lemma 3.5. Let g 2 G and  D .k; H / 2 @S with k 2 K and H 2 aC 1 . If k 2 K I 0 0 is such that  .gk/ D k M , then g   D .k ; H /. In particular, if  2 @S reg , then g   B ./ D  B .g  / D k 0 M .

Proof. Consider the unit speed geodesic c ´ co; , i.e. c.t / D ke H t  o for t 2 R. We write gk D k 0 an with k 0 2 K, a 2 A and n 2 N C . In order to prove that g  c.t/ converges to .k 0 ; H / 2 @S as t ! 1, we let R  1 and " > 0 arbitrary. For t > R we denote by c t the geodesic emanating from o passing through g  c.t /. If

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s t ´ d.o; g  c.t //, then by the triangle inequality js t  t j d.o; g  o/. Using the convexity of the distance function we estimate for t > R C d.o; g  o/  R d.k 0 e H s t  o; g  c.s t // C d.g  c.s t /; c t .s t // d.k 0 e HR  o; c t .R//

st  R D d.k 0 e H s t  o; gke H s t  o/ C d.g  c.s t /; g  c.t // st  R D d.k 0 e H s t  o; k 0 ane H s t  o/ C d.c.s t /; c.t // „ ƒ‚ … st

R st



Djs t t jd.o;go/

d.o; an  o/ C d.o; g  o/

since d.e H s  o; ane H s  o/ d.o; an  o/ for all s > 0. From s t ! 1 as t ! 1 we get d.k 0 e HR  o; c t .R// < " for t sufficiently large. Hence g   D .k 0 ; H /. Example 1. For S D Pos1 .n/, n  2, the Furstenberg boundary @F S is identified with the space of regular flags in Rn , i.e. the set of flags F D .V1 ; V2 ; : : : ; Vn / such that dim Vi  dim Vi1 D 1 for all 1 i n. Example 2b. For S D S2q , the Furstenberg boundary @F S is identified with the space of complete isotropic flags in R2q , i.e. the set of flags F D .W1 ; W2 ; : : : ; Wq / such that Wi  R2q is an isotropic subspace and dim Wi  dim Wi1 D 1 for all 1 i q. Example 2c. For S D H2  H2 , the Furstenberg boundary @F S is isomorphic to @S1 @S2 , where @Si denotes the geometric boundary of the i -th H2 -factor, i 2 f1; 2g.

3.3 The Bruhat decomposition The main reference for this section is [Wa], Chapter 1.2. Given the Iwasawa decomposition G D KAN C from Section 2.3, we consider the closed subgroup P D MAN C  G. Any subgroup in G conjugate to P is called a minimal parabolic subgroup. The homogeneous space G=P can be identified with the Furstenberg boundary K=M via the bijection N W G=P ! K=M; gP 7!  I .g/: The Bruhat decomposition gives a cell decomposition of G=P , hence induces a cell decomposition of the Furstenberg boundary which we will describe geometrically in the next section. Recall that the factor group W D M  =M is the Weyl group of the pair .g; a/. We denote by w 2 W the unique element such that Ad.mw /.aC / D aC for any

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representative mw of w in M  , and put

X

n ´ Ad.mw /nC D

g˛ :

˛2†C

For w 2 W represented by mw 2 M  we set uw ´ nC \ Ad.mw /n  nC

and

Uw ´ e u w :

The Bruhat decomposition of G with respect to the minimal parabolic subgroup P is the disjoint union G G N C mw P D Uw mw P: (3.3) GD w2W

w2W

Notice that the orbit corresponding to w 2 W is parametrized by N C D Uw , and the restriction of the above bijection N to N C mw P defines a map  W N C ! K=M; n 7! .nm N w P /: Geometrically, this map can be interpreted in the following way: if n 2 N C , then C .n/ 2 K=M is the unique element such that the Weyl chamber .n/e a  o is C equivalent to the Weyl chamber ne a  o. The following property of the map  is well-known: Proposition 3.6 ([H], Corollary IX.1.9). The map  is a diffeomorphism onto an open submanifold of K=M whose complement consists of finitely many disjoint manifolds of strictly lower dimension. It follows that the orbit N C mw P is a dense and open submanifold of G=P . We N C mw P /  K=M a big cell of the will call a G-translate of the set .N C / D .N Furstenberg boundary. Example 1. For simplicity we treat the case n D 3. Recall the Iwasawa decomposition G D KAN C of G D SL.3; R/ from (2.3), where K D SO.3/, A denotes the set of diagonal matrices with positive entries in SL.3; R/, and N C the set of upper diagonal .n  n/-matrices with 1’s in the diagonal. Here the centralizer of a in K is the finite set M D fDiag.1; 1; 1/; Diag.1; 1; 1/; Diag.1; 1; 1/; Diag.1; 1; 1/g; and P D MAN C is a minimal parabolic subgroup. The Weyl group W is represented by the set of matrices n  0 1 0  1 0 0  e D I3 ; w1 D 1 0 0 ; w2 D 0 0 1 ;  0 0 1 0 0 1  0 1 0 0 1 0  0 0 1 o w3 D 1 0 0 ; w4 D 0 0 1 ; w D 0 1 0 010

100

1 0 0

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One easily computes the sets Uw as follows: Ue D feg, Uw D N C , n 1 x 0  n 1 x z  o o Uw1 D 0 1 0 W x 2 R ; Uw3 D 0 1 0 W x; z 2 R ; n 01 00 01  n 01 00 z1  o o Uw2 D 0 1 y W y 2 R ; Uw4 D 0 1 y W y; z 2 R : 00 1

00 1

Hence for SL.3; R/ the Bruhat decomposition yields 6 cells, one isomorphic to a point, two isomorphic to R, two isomorphic to R2 , and the maximal one isomorphic to R3 . Example 2b. We consider q D 2, and use the Iwasawa decomposition G D KAN C described in Section 2.3. Here the centralizer of a D fDiag.t1 ; t2 ; t1 ; t2 / W t1 ; t2 2 Rg in K is the finite set M D fI4 ; Diag.1; 1; 1; 1/; Diag.1; 1; 1; 1/; I4 g; and P D MAN C is a minimal parabolic subgroup. The following table gives a set of representatives w of the Weyl group W and describes the corresponding sets uw  nC : w

uw

e D Diag.1; 1; 1; 1/ 0 B B B @ 0 B B B @ 0 B B B @ 0 B B B @ 0 B B B @ 0 B B B @

0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0

1 1 0 0 0 0 0 C C C 0 0 1 A 0 1 0 1 0 0 0 0 0 1 C C C 0 1 0 A 1 0 0 1 0 0 1 0 0 0 C C C 1 0 0 A 0 1 0 1 1 0 0 0 1 0 C C C 0 0 1 A 0 0 0 1 0 0 1 0 1 0 C C C 1 0 0 A 0 0 0 1 0 1 0 1 0 0 C C C 0 0 0 A 0 0 1

w D



0 I2 I2 0



0 80 ˆ < B @ ˆ :

0 0

80 ˆ < B @ ˆ :

0 b 0 0

0 z 0

0 0

b 0

0 y

80 < @ 0 : 0

x y

y z 0 y z 0

0 0

b 0 0

C AWb2R

> ;

0 0

9 =

1 AWz2R

;

x 0 0 b

0

80 < @ 0 : 0

80 ˆ < B @ ˆ :

0

0

80 < @ 0 : 0

9 > =

1

b 0

0 0 0 0

9 > =

1 C A W b; x 2 R

> ;

9 =

1 A W y; z 2 R

;

9 = A W x; y; z 2 R ; 1

x y 0 b

y 0 0 0

9 > =

1 C A W b; x; y 2 R

> ;

nC

Hence for Sp.4; R/ the Bruhat decomposition yields 8 cells, one isomorphic to a point, two isomorphic to R, two isomorphic to R2 , two isomorphic to R3 and the maximal one isomorphic to R4 .

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Example 2c. We have seen that for S D H2  H2 the Furstenberg boundary is given by @F S D @S1  @S2 . Hence @F S Š .R [ f1g/  .R [ f1g/. Here the Bruhat decomposition is isomorphic to the decomposition @F S D f.1; 1/g t .f1g  R/ t .R  f1g/ t .R  R/:

3.4 Visibility at infinity If the rank of S equals one, or, more generally, if S is a Hadamard manifold with a negative upper bound on the sectional curvature, then any pair of distinct points in the geometric boundary can be joined by a geodesic. In a symmetric space of higher rank this fails to be true. In this section we will describe the set of points in the geometric boundary @S of a globally symmetric space S which can be joined to a given point  2 @S by a geodesic. Definition 3.7. The visibility set at infinity viewed from  2 @S is the set Vis1 ./ ´ f 2 @S j there exists a geodesic c such that c.1/ D ; c.1/ D g: It is clear that for g 2 Is.S/ we have Vis1 .g  / D g  Vis1 ./. Moreover, we have the following description of Vis1 ./, which is Proposition 2.21.13 (2) in [E]. We include a proof here for the convenience of the reader. Proposition 3.8. If StabG ./  G denotes the stabilizer of a point  2 @S, then Vis./ D StabG ./  co; .1/: Proof. By transitivity of the G-action and the fact that g  Vis1 ./ D Vis1 .g  / for g 2 G and  2 @S we can assume that  2 @S is stabilized by the minimal parabolic subgroup P D MAN C , so P  StabG ./. If  2 Vis1 ./, then there exists a geodesic c  S such that c.1/ D  and c.1/ D . Let g 2 G be such that c.0/ D g  o. Using the Iwasawa decomposition we may write g D nak with n 2 N C , a 2 e a and k 2 K. Then c.0/ D nak  o D na  o. The geodesic c0 defined by c0 .t / ´ .na/1  c.t / D a1 n1  c.t / satisfies c0 .1/ D c.1/ D  because N C and A stabilize . So c0 .0/ D .na/1  c.0/ D o implies c0 .t / D co; .t / for all t 2 R. We conclude  D c.1/ D na  c0 .1/ D na  co; .1/; hence  2 StabG ./  co; .1/. Conversely let p 2 StabG ./ and set  ´ p  co; .1/. If c is the geodesic defined by c.t / ´ p  co; .t / for t 2 R, then we have c.1/ D  and c.1/ D p  co; .1/ D p   D  because p fixes . Hence  2 Vis1 ./. The following lemma relates the visibility set of regular points to our coordinates introduced in Section 3.1 and the map  from Section 3.3. Even though it is a direct

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consequence of [L1], Corollary 2.15, we include the proof here for the convenience of the reader. Recall the definition of the opposition involution (2.4). Lemma 3.9. If  2 @S reg is stabilized by the minimal parabolic subgroup P  G and possesses the Cartan projection H 2 aC 1 , then Vis./ D f.k; .H // W kM 2 .N C /g: Proof. Let k 2 K be such that kM D .n/ with n 2 N C . Consider the geodesic c.t/ ´ n  co; .t / which satisfies c.1/ D  because N C stabilizes . If mw 2 M  is a representative of w 2 W , we have c.t / D ne H t  o D nmw e .H /t  o; hence by the property of the map  the geodesic rays c.t /, t > 0, and ke .H /t o, t > 0, are equivalent. This shows c.1/ D .k; .H //, so we conclude .k; .H // 2 Vis1 ./. Conversely, let  2 Vis1 ./ and choose a geodesic c in S with c.1/ D  and c.1/ D . From the proof of Proposition 3.8 we know that there exist n 2 N C and a 2 e a such that the geodesic c0 defined for t 2 R by c0 .t / ´ .na/1  c.t / D a1 n1  c.t/ satisfies c0 .t/ D co; .t / D e H t  o. Moreover, since A also stabilizes co; .1/ we conclude  D c.1/ D na  c0 .1/ D na  co; .1/ D n  co; .1/ D n  c0 .1/: If k 2 K is an angular projection of , then  D .k; .H //, and the geodesic ray ke .H /t  o, t > 0, is equivalent to the geodesic ray ne H t  o, t > 0. Hence by definition of the map  we have kM D .n/. Since the opposition involution preserves the set aC of regular elements in aC , this lemma in particular implies that the visibility set at infinity viewed from a regular boundary point is contained in the regular boundary. This allows the following Definition 3.10. The Bruhat visibility set viewed from  2 @S reg is the image of Vis1 ./ under the projection  B W @S reg ! K=M , i.e. VisB ./ D  B .Vis1 .//: We remark that if rank.S/ D 1, then VisB ./ Š Vis1 ./ D @S n fg for all  2 @S . In general, an immediate consequence of Lemma 3.9 is the fact that VisB ./ can be identified with the nilpotent Lie group N C or an arbitrary orbit N C  x, x 2 S. Moreover, all Bruhat visibility sets are open and dense submanifolds of K=M by Proposition 3.6. Example 1. Let S D Pos1 .n/, n  3. Assume first that  is stabilized by P D MAN C  G. Then there exist l 2 f2; 3; : : : ; ng, .m1 ; m2 ; : : : ; ml / 2 Nl with Pl l i D1 mi D n such that ./ D . 1 ; 2 ; : : : l / 2 R and F ./ D .U1 ; U1 ; : : : ; Ul / n is the standard flag in R determined by .m1 ; m2 ; : : : ; ml / via (3.1).

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For 1 i l we denote by Ui? the orthogonal complement of Ui and remark  that  ´ co; .1/ corresponds to the eigenvalue-flag pair ./; F ./ with ./ D ? . l ;  l1 ; : : : ;  1 / and F ./ D .Ul? ; Ul1 ; : : : ; U1? /. We will say that two flags F D .V1 ; V2 ; : : : ; Vl /, F 0 D .W1 ; W2 ; : : : ; Wk / are in opposition if k D l and Rn D Vi ˚ WliC1 , 1 i l  1. For  and  as above clearly F ./ and F ./ are in opposition. Moreover, if g 2 G D SL.n; R/, then g  F and g  F 0 are in opposition if and only if F and F 0 are. By Proposition 3.8 we have Vis1 ./ D StabG ./  , and g 2 G stabilizes  if and only if g leaves invariant each of the eigenspaces for the eigenvalues 1 ; 2 ; : : : ; l of X./ 2 p. This is equivalent to g  Ui D Ui for all 1 i l. So we conclude that  2 Vis1 ./ if and only if there exists g 2 StabG ./ such that ./ D .g  / D ./ D . l ;  l1 ; : : : ;  1 /; ? ; : : : ; g  U1? /. This second condition is and F ./ D F .g  / D .g  Ul? ; g  Ul1 satisfied if and only if F ./ and F ./ are in opposition. If  2 @S is arbitrary, there exists g 2 G such that g   is stabilized by P . So P there exist l 2 f2; 3; : : : ; ng, .m1 ; m2 ; : : : ; ml / 2 Nl with liD1 mi D n such that .g / D . 1 ; 2 ; : : : l / 2 Rl and F .g / D .U1 ; U1 ; : : : ; Ul / is the standard flag in Rn determined by .m1 ; m2 ; : : : ; ml / via (3.1). Since  2 Vis1 ./ if and only if g   2 Vis1 .g/, this shows that  2 Vis1 ./ if and only if ./ D . l ;  l1 ; : : : ;  1 /, and F ./ is in opposition to F ./. Summarizing we have  2 Vis1 ./ if and only if the following two conditions are satisfied:

(a) If ./ D . 1 ; 2 ; : : : ; l /, ./ D .1 ; 2 ; : : : ; k /, then k D l and i D  li C1 for all i 2 f1; 2; : : : lg. (b) F ./ and F ./ are in opposition. This immediately implies that  B ./ 2 VisB ./,  2 @S reg , if and only if the regular flags F ./ and F ./ are in opposition. Example 2b. Consider S D S2q for q  2. As before we first assume that  is stabilized by P D MAN C  G D Sp.2q; R/. Then there exist 1 l q, P .m1 ; m2 ; : : : ; ml / 2 Nl with liD1 mi q such that ./ D . 1 ; 2 ; : : : l / 2 Rl and F ./ D .U1 ; U2 ; : : : ; Ul / is the isotropic standard flag in .R2q ; !/ determined by .m1 ; m2 ; : : : ; ml / via (3.2).   0 I Let m 2 G be the element defined by the matrix Iq 0q . Here  ´ co; .1/   corresponds to the pair ./; F ./ with ./ D . 1 ; 2 ; : : : ; l / and F ./ D m  F ./. Moreover, each of the linear subspaces Ui ˚ m  Ui , 1 i l, is a symplectic subspace of .R2q ; !/, i.e. ! restricted to Ui ˚ m  Ui is non-degenerate. Motivated by this property we will say that two flags F D .V1 ; V2 ; : : : ; Vl /, F 0 D .W1 ; W2 ; : : : ; Wk / of isotropic subspaces are complementary if k D l and Vi ˚ Wi , 1 i l, is a symplectic subspace of .R2q ; !/. Notice that this necessarily implies dim Wi D dim Vi for all i 2 f1; 2; : : : ; lg. Clearly the isotropic flags F ./ and

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F ./ from above are complementary; moreover, if g 2 G, then g  F and g  F 0 are complementary if and only if F and F 0 are. Now Proposition 3.8 implies Vis1 ./ D StabG ./  , and g 2 G stabilizes  if and only if g leaves invariant each of the isotropic eigenspaces for the positive eigenvalues 1 ; 2 ; : : : ; l of X./. Hence we conclude that  2 Vis1 ./ if and only if there exists g 2 StabG ./ such that ./ D .g  / D ./ D ./, and F ./ D F .g  / D g  F ./. The latter condition is satisfied if and only if F ./ is complementary to F ./. If  2 @S is arbitrary, there exists g 2 G such that g   is stabilized by P . So P there exist l 2 f1; 2; : : : ; qg, .m1 ; m2 ; : : : ; ml / 2 Nl with liD1 mi q such that .g  / D . 1 ; 2 ; : : : l / 2 Rl and F .g  / D .U1 ; U1 ; : : : ; Ul / is the isotropic standard flag determined by .m1 ; m2 ; : : : ; ml / via (3.2). Since  2 Vis1 ./ if and only if g   2 Vis1 .g  / this shows that  2 Vis1 ./ if and only if .g  / D ./ D . 1 ; 2 ; : : : ; l / and F ./, F ./ are complementary. We conclude that  2 Vis1 ./ if and only if the following conditions are satisfied: (a) If ./ D . 1 ; 2 ; : : : ; l /, ./ D .1 ; 2 ; : : : ; k /, then k D l and i D i for all i 2 f1; 2; : : : lg. (b) The isotropic flags associated to  and  are complementary. This implies in particular that  B ./ 2 VisB ./,  2 @S reg , if and only if the complete isotropic flags of  and  are complementary. Example 2c. If S D H2  H2 we have seen that @S D @S1 t @S2 t @S reg ; where @S reg D @S1  @S2  .0; =2/, and @S1 , @S2 are the two singular boundary strata. For i 2 f1; 2g,  2 @Si , we have  2 Vis1 ./ if and only if  2 @Si and  ¤ . If  D .1 ; 2 ; / 2 @S reg , then  2 Vis1 ./ if and only if  D .1 ; 2 ; '/ 2 @S reg with 1 ¤ 1 , 2 ¤ 2 and ' D .

3.5 Busemann functions and distances In this final section we discuss Busemann functions and how they can be used to construct a family of G-invariant Finsler pseudo-distances on S for which the flats are isomorphic to a pseudo-normed vector space. For more details about G-invariant Finsler structures on symmetric spaces we refer the reader to P. Planche’s thesis ([P]). Let x; y 2 S ,  2 @S , and c  S a geodesic ray in the class of . We put   B .x; y/ ´ lim d.x; c.s//  d.y; c.s// : s!1

This number is independent of the chosen ray c, and the function B .; y/ W S ! R; x 7! B .x; y/;

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is called the Busemann function centered at  based at y (see also Chapter II of [Ba]). It satisfies the following properties: Proposition 3.11. For all  2 @S, x; y; z 2 S, g 2 G ´ Iso .S / we have (1) Bg .g  x; g  y/ D B .x; y/, (2) B .x; z/ D B .x; y/ C B .y; z/, (3) jB .x; y/j d.x; y/, (4) B .x; y/ D d.x; y/ if and only if  D cx;y .1/. Using Busemann functions we introduce an important family of (possibly nonsymmetric) pseudo-distances. Definition 3.12. Let  2 @S . We define the directional distance of the ordered pair .x; y/ 2 S  S with respect to the subset G    @S by BG W S  S ! R; .x; y/ 7! BG .x; y/ ´ sup Bg .x; y/: g2G

Notice that in rank one symmetric spaces G   D @S and for x; y 2 S we have d.x; y/ D BG .x; y/ D sup B .x; y/  Bcx;y .1/ .x; y/ D d.x; y/; 2@S

hence BG equals the Riemannian distance d for any  2 @S. In general, the corresponding estimate for the Busemann functions implies BG .x; y/ d.x; y/

for all  2 @S and all x; y 2 S:

Moreover, BG is a (possibly non-symmetric) G-invariant pseudo-distance on S (for a proof see [L1], Proposition 3.7), and we have BG .x; y/ D d.x; y/  sup cos †x .y; g/: g2G C

In particular, if G D Ke a K is a Cartan decomposition, H 2 aC 1 the Cartan projection of , and H.x; y/ 2 aC the Cartan vector of the ordered pair .x; y/ according to Definition 2.28, then   (3.4) BG .x; y/ D hhH ; H.x; y/ii D B H ; H.x; y/ for all x; y 2 S: This shows in particular that the flats of S are isomorphic to Rr endowed with a pseudo-norm. Moreover, from the remark following Definition 2.28 we know that BG .y; x/ D hhH ; .H.x; y//ii D hh .H /; H.x; y/ii; because is an involution and, by Ad.K/-invariance of the Killing form, preserves the scalar product. So BG is symmetric if and only if the Cartan projection H of

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 satisfies .H / D H . This clearly always holds when is the identity; so all the directional distances are symmetric e.g. in S2q , q  1, and H2  H2 . Example 2c. If S D H2  H2 we have seen that @S D @S1 t @S2 t @S reg ; where @S reg D @S1  @S2  .0; =2/, and @S1 , @S2 are the two singular boundary strata. If  D .1 ; 2 ; / 2 @S reg one can easily deduce from the definition of the Busemann functions that for x D .x1 ; x2 /, y D .y1 ; y2 / B .x; y/ D cos  B 1 .x1 ; y1 / C sin  B 2 .x2 ; y2 /: For the directional distance we therefore get by definition and the remark about rank one symmetric spaces BG .x; y/ D cos  d1 .x1 ; y1 / C sin  d2 .x2 ; y2 /; where for i 2 f1; 2g, di denotes the Riemannian distance in the i -th H2 -factor. Hence for  2 @S reg the directional distance is a proper distance function. If  2 @Si , i 2 f1; 2g, we similarly obtain B .x; y/ D B .xi ; yi /

and

BG .x; y/ D di .xi ; yi /:

In particular, B is symmetric, but only a pseudo-distance.

References [A]

P. Albuquerque, Patterson-Sullivan theory in higher rank symmetric spaces. Geom. Funct. Anal. 9 (1999), 1–28.

[Ba] W. Ballmann, Lectures on spaces of non-positive curvature. DMV Seminar, Band 25, Birkhäuser, Basel, 1995. [BGS] W. Ballmann, M. Gromov and V. Schroeder, Manifolds of non-positive curvature. Progr. Math. 61, Birkhäuser, Boston, Mass., 1985. [Be] Y. Benoist, Propriétés asymptotiques des groupes linéaires I. Geom. Funct. Anal. 7 (1997), 1–47. [B1] A. Borel, Semisimple groups and Riemannian symmetric spaces. Texts Read. Math. 16, Hindustan Book Agency, New Delhi 1998. [B2] A. Borel, Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem. J. Differential. Geom. 6 (1972), 543–560. [dS] A. C. da Silva, Symplectic geometry. In Handbook of differential geometry, Vol. II, Elsevier/North-Holland, Amsterdam 2006, 79–188. [dC] M. P. do Carmo, Riemannian geometry. Math. Theory Appl., Birkhäuser Boston, Mass., 1992.

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[E]

P. Eberlein, Geometry of nonpositively curved manifolds. Chicago Lectures in Math., University Chicago Press, Chicago, Ill., 1996.

[H]

S. Helgason, Differential geometry, Lie groups, and symmetric spaces. Corrected reprint of the 1978 original, Grad. Stud. Math. 34, Amer. Math. Soc., Providence, R.I., 2001.

[Kn] G. Knieper, On the asymptotic geometry of non-positively curved manifolds. Geom. Funct. Anal. 7 (1997), 755–782. [L1] G. Link, Limit sets of discrete groups acting on symmetric spaces. Dissertation, Karlsruhe, 2002. http://digbib.ubka.uni-karlsruhe.de/eva/2002/mathematik/9 [L2] G. Link, Hausdorff dimension of limit sets of discrete subgroups of higher rank Lie groups. Geom. Funct. Anal. 14 (2004), 400–432. [L3] G. Link, Geometry and dynamics of discrete isometry groups of higher rank symmetric spaces. Geom. Dedicata 122 (2007), 51–75. [M] G. D. Mostow, Strong rigidity of locally symmetric spaces. Ann. of Math. Stud. 78, Princeton University Press, Princeton, N.J., 1973. [P]

P. Planche, Géométrie de Finsler sur les espaces symétriques. Ph.D. Thesis, Geneva, 1995. http://www.unige.ch/math/biblio/these/thesepl.ps

[Wa] G. Warner, Harmonic analysis on semisimple Lie groups I. Grundlehren Math. Wiss. 188, Springer-Verlag, Berlin 1972. [Wo] J. A. Wolf, Spaces of constant curvature. McGraw-Hill Book Co., New York 1967.

Geometry of the representation spaces in SU.2/ Julien Marché Centre de Mathématiques Laurent Schwartz École Polytechnique Route de Saclay, 91128 Palaiseau Cedex, France email: [email protected]

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1 Examples of representation spaces . . . . . . . . . . . 1.1 Generalities on SU(2) . . . . . . . . . . . . . . 1.2 Generalities on representation spaces . . . . . . 1.3 Basic examples . . . . . . . . . . . . . . . . . . 1.4 Some representation spaces of knot complements 2 Differentiable structure and twisted cohomology . . . 2.1 Two examples . . . . . . . . . . . . . . . . . . 2.2 The general case . . . . . . . . . . . . . . . . . 2.3 Applications . . . . . . . . . . . . . . . . . . . 2.4 Reidemeister torsion . . . . . . . . . . . . . . . 3 Gauge theory . . . . . . . . . . . . . . . . . . . . . . 3.1 Principal bundles and flat connections . . . . . . 3.2 Sections and connection forms . . . . . . . . . . 3.3 de Rham cohomology and isomorphisms . . . . 4 Chern–Simons theory . . . . . . . . . . . . . . . . . 4.1 The Chern–Simons functional . . . . . . . . . . 4.2 Construction of the prequantum bundle . . . . . 4.3 Examples . . . . . . . . . . . . . . . . . . . . . 5 Surfaces of higher genus . . . . . . . . . . . . . . . . 5.1 Trace functions and flat connection along a curve 5.2 Global description of the moduli space . . . . . 5.3 Some applications . . . . . . . . . . . . . . . . 6 Introduction to geometric quantization . . . . . . . . . 6.1 Spin structures . . . . . . . . . . . . . . . . . . 6.2 Lagrangian foliations . . . . . . . . . . . . . . . 6.3 Bohr–Sommerfeld leaves . . . . . . . . . . . . . 6.4 Going further . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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Introduction Representations of fundamental groups of manifolds of dimension 2 and 3 in a compact Lie group have a long history. In the case of surfaces, they appeared after the development of Teichmüller theory, for instance to classify holomorphic vector bundles. In the case of 3-manifolds, they were used to help distinguishing 3-manifolds as knot complements, or for geometrization purposes. Then, in the eighties, representations of surfaces were much studied for their symplectic properties until Witten discovered a deep relationship between these representation spaces and the Jones polynomial of knots, via Chern–Simons quantum field theory. This field is still very lively and these notes were written as a preparation for understanding this relationship. We planned to give a quick review of the geometric aspects of the representation spaces. In a first part, we give some examples of representation spaces for surfaces and knots. They will help the reader to understand the second part where we introduce the basic tool in order to understand the differential geometry of representation spaces: twisted (co)homology. We give a brief account of the symplectic structure on surfaces and on Reidemeister torsion. We introduce gauge theory in the third part and review the symplectic structure of surface representations in this context. This third part is mostly an introduction to the fourth where we explain the basic constructions of Chern–Simons theory and its applications to the geometry of representation spaces. The fifth part studies in more detail the representation space of a closed surface of genus at least 2 by introducing trace functions and a related integrable system. In the last part, we give an introduction to geometric quantization, insisting on Lagrangian fibrations and spin structures. With the help of some examples, we treat the case of representation spaces of surfaces.

1 Examples of representation spaces 1.1 Generalities on SU(2) x> D We define the group SU.2/ as the group of matrices M 2 M.2; C/ satisfying M M 1 and det M D 1. We can write this set alternatively as  n o 2 2 ˛ ˇN W ˛; ˇ 2 C; j˛j C jˇj D 1 : SU.2/ D ˇ ˛N This last description shows that SU.2/ is topologically a sphere S 3 . We often look at SU.2/ as the unit sphere in the space of quaternions H where the standard generators are the following:       i 0 0 1 0 i iD ; jD ; kD : 0 i 1 0 i 0

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These generators also form a basis of the Lie algebra su.2/ of SU.2/. Observe the following elementary  fact: given  M 2 SU.2/, there is a unique ' 2 Œ0;  such that i' e 0 M is conjugate to 0 ei' . We call ' the angle of M . It can be easily computed via the formula Tr.M / D 2 cos.'/. We define ang.M / D arccos.Tr.M /=2/.

1.2 Generalities on representation spaces Let  be a finitely presented group. For our purposes, we let  be the fundamental group of a compact manifold or of a finite CW-complex. We denote by R.; SU.2// the set of homomorphisms from  to SU.2/. As  is finitely generated, say by t1 ; : : : ; tn , any element  2 R.; SU.2// is determined by the image of the generators. This gives an embedding of R.; SU.2// into SU.2/n sending  to the family .Ai D .ti //in . Moreover, let R1 ; : : : ; Rm be a generating family of relations for . A family .Ai / defines a representation if and only if Rj .Ai / D 1 for all j 2 f1 : : : mg. Here are two easy consequences. (1) R.; SU.2// is a topological space (as a subspace of SU.2/n ). (2) R.; SU.2// is a real algebraic variety as SU.2/ is algebraic and the relations are algebraic maps. More precisely, SU.2/ may be defined as f.x; y; z; t/ 2 R4 W x 2 C y 2 C z 2 C t 2 D 1g putting ˛ D x C iy and ˇ D z C it . It is easy to verify that these structures do not depend on the choices of generators and relations. We say that two representations , 0 are conjugate if there is M 2 SU.2/ such that 0 D MM 1 . We denote by M.; SU.2// the set of conjugacy classes of representations, or the quotient R.; SU.2//= SU.2/. This construction allows us to define M.X; SU.2// D M.1 .X/; SU.2// for any topological space X . The ambiguity in 1 .X / is precisely described by a conjugation and hence disappears in the quotient. We deduce the following consequences for M.X; SU.2// where 1 .M / is finitely generated: (1) M.; SU.2// is a topological space (quotient topology). (2) M.; SU.2// can be given a structure of a real algebraic variety, but we will not deal with this topic in these notes.

1.3 Basic examples Let us remove SU.2/ from our notation. Our first example is M.S 1 /, which is the set of conjugacy classes of SU.2/. It is homeomorphic to Œ0;  via the angle map. The two boundary points correspond to the conjugacy classes of the central elements ˙1. Let X D S 1 _ S 1 . Then M.X/ D f.A; B/ 2 SU.2/2 g= SU.2/ is the set of conjugacy classes of pairs of matrices. Let ' and be the  angles of A and B rei' 0 spectively. One can suppose up to conjugation that A D e0 ei' and that there

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 i exists P 2 SU.2/ such that B D P e 0 ei0 P 1 . Multiplying P on the right by     e iy 0 e ix 0 does not change B whereas multiplying P on the left by con0 e iy 0 e ix   N jugates B by a matrix which does not act on A by conjugation. Given P D ˇ˛ ˛Nˇ ,      i.xCy/ N i.xy/  ix iy e 0 0 we compute e0 eix P e0 eiy D ˛e i.yx/ ˇi.xCy/ . By setting x C y D ˇe

˛e N

 arg.˛/; x  y D arg.ˇ/ one can suppose that ˛ and ˇ are real and nonnegative. We   2 i e Cˇ 2 e i ˛ˇ.e i e i / . This formula describes all possible compute B D ˛˛ˇ.e i e i / ˇ 2 e i C˛ 2 e i values of B where ˛; ˇ  0 and ˛ 2 Cˇ 2 D 1. To show that all these pairs .A; B/ are not conjugate, we compute Tr.AB/ D e i' .˛ 2 e i Cˇ 2 e i /Ce i' .ˇ 2 e i C˛ 2 e i / D 2˛ 2 cos.' C / C 2ˇ 2 cos.'  /. Let  2 Œ0;  be the angle of AB. We see that cos./ is a convex combination of cos.' C / and cos.'  /. We deduce that  belongs to the interval Œj'  j; min.' C ; 2  '  /. We can sum up our computations in the following proposition: Proposition 1.1. The map from M.S 1 _ S 1 / to f.'; ; / 2 Œ0; 3 W ' C C   2; '  C ;  ' C ;   ' C g sending .A; B/ to the triple .ang.A/; ang.B/; ang.AB// is a homeomorphism. The symmetry between ', and  can be explained by replacing X with a pair of pants † that is a disk with two holes. The three angles ', ,  are the angles of the three boundary components, and the inequations they satisfy are symmetric with respect to these coordinates. In Figure 1 is represented the moduli space M.S 1 _ S 1 / where axes correspond to the angles of A, B and AB. Notice that the corners of this tetrahedron correspond to central representations whereas its boundary corresponds to abelian representations.

Figure 1. The moduli space of a free group with two generators.

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Our last easy but important example is the torus S 1  S 1 . One sees that M.S 1  S 1 / D f.A; B/ 2 SU.2/2 W AB D BAg= SU.2/. The .'; / 2 R=2Z    R=2Z to the representation defined by  map sending i' i 0 and D e 0 ei0 is surjective as two commuting matrices are A D e0 ei' simultaneously diagonalizable. Moreover, if two pairs .'; / and .' 0 ; 0 / are conjugate then either .' 0 ; 0 / D .'; / or .' 0 ; 0 / D .';  /. One deduces that M.S 1  S 1 / is homeomorphic to the torus .R=2Z/2 quotiented by the involution .'; / 7! .';  /. This is a sphere with four conical points of angle  corresponding to representations with values in f˙1g. This moduli space appears as the boundary of the tetrahedron in Figure 1. With the same proof, we can show that M.S 1  S 1  S 1 / is homeomorphic to .R=2Z/3 =I where I.'; ; / D .';  ; /.

1.4 Some representation spaces of knot complements Let us look at two families of knots whose fundamental groups have the property of being presented by two generators and one relation, which makes the study of their representation space much simpler than the general case. Before going further, let us introduce some terminology. Definition 1.2. Let X be a topological space and  2 R.X / a representation. (1) If  takes values in f˙1g, we say that  is central. (2) If the image of  is contained in an abelian subgroup of SU.2/, we say that  is abelian. (3) In the other cases  is said to be irreducible. These definitions are invariant by conjugation hence we will use the same terminology for elements of M.X/. A central representation is given by a homomorphism from 1 .X / to Z=2. These representations are in bijection with H 1 .X; Z=2/. An abelian representation is given by a homomorphism from 1 .X / to S 1 , hence these representations are in bijection with H 1 .X; S 1 /. However, as in the case of the torus, the inversion map I W S 1 ! S 1 induces a map I on H 1 .X; S 1 / and conjugacy classes of abelian representations are in bijective correspondence with H 1 .X; S 1 /=I . 1.4.1 Torus knots. Let F .'; / D ..2 C cos '/ cos ; .2 C cos '/ sin ; sin '/ be the standard embedding of .R=2Z/2 in R3 . Given two positive and relatively prime integers a and b, we define the torus knot T .a; b/ as the image of the embedding t 7! F .at; bt /. One can see in Figure 2 the example of T .5; 2/. Let us look at this knot in S 3 D R3 [ f1g. The complement of T .a; b/ cut along the torus supporting the knot has two components which are solid tori. Their

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Figure 2. A torus knot.

intersection C is the complement of the knot inside the torus which is an annulus, hence connected. Then the van Kampen theorem asserts that Ga;b D 1 .S 3 n T .a; b// D hu; vjua D v b i. One can show indeed that the generator of 1 .C / has order a in one side and b in the other, which gives the expression of Ga;b . Let  W Ga;b ! SU.2/ be an irreducible representation. Then .ua / D .v b / commutes with the image of  and hence has to belong to the center. We deduce that .u/2a D .v/2b D 1: Then, the angle of .u/ takes the values k=a for 0 < k < a and the angle of .v/ take the values l=b for 0 < l < b. As .ua / D .1/k D .v b / D .1/l one has k D l mod 2. The angle of .uv/ determines the representation and takes its values in the non trivial interval Œjk=a  l=bj; min.k=a C l=b; 2  k=a  l=b/. Hence the irreducible part of M.Ga;b / is a disjoint union of .a  1/.b  1/=2 arcs. The ends of these arcs are made of abelian representations which we describe now. To find abelian representations, one can suppose that .u/ and .v/ are diagonal with angle ' and respectively. Then the angles should satisfy a' D b mod 2. We obtain all solutions by taking ' D bt , D at and letting t in Œ0; . The ends of the irreducible segments are equally distributed on the reducible segment as in Figure 3 where we see on the left the abstract moduli space of the trefoil knot T .3; 2/, and on the right the way it is embedded in the tetrahedron. 1.4.2 Two-bridge knots. A two-bridge knot is a knot in R3 which is in Morse position relatively to some coordinate and has only 2 maxima and minima. All two-bridge knots can be put in a standard projection called Schubert normal form, see [BZ95], [Sch56]. Let a and b be integers such that a is positive, b is odd and the inequality a < b < a holds. Formally, we define the projection of the two-bridge

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Figure 3. Moduli space of the trefoil.

knot B.a; b/ as the unique diagram obtained by gluing two copies of the disc in the left hand side of Figure 4 by a diffeomorphism of the boundary circle sending pk to

p1 : : :

pb1

p0

p2b1 : : : Figure 4. A two-bridge knot.

pbk for k 2 Z=2aZ. We drew on the right of the figure the example of the knot B.5; 3/. Let Ha;b be the fundamental group of the complement of B.a; b/. Then it has the following presentation: Ha;b D hu; v j wu D vwi. In this formula, w D ue1 v e2 : : : v ea1 , where for all k, we set ek D .1/bkb=ac .

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The proof is based on the Wirtinger presentation of knot groups. Removing the two under-bridges, that is, the two copies of the segment joining p0 to pb , we obtain two disjoint arcs which correspond to the generators u and v. Using the Wirtinger relation at each crossing, one can label all the remaining arcs in the projection and the labeling is consistent providing that the relation wu D vw is satisfied. This explains the presentation of Ha;b . For a precise proof, see [BZ95], [Sch56]. As usual, a representation  is determined by the traces x D Tr .u/ D Tr .v/ (because u and v are conjugate) and y D Tr .uv/. The equality uw D wv is equivalent to Tr.uwv 1 w 1 / D 2 and this equality converts into a polynomial in x and y thanks to the following lemma. Lemma 1.3. For all A; B 2 SU.2/ one has Tr.AB/ C Tr.AB 1 / D Tr.A/ Tr.B/. For any word W in A, B there is a polynomial in three variables FW such that Tr.W / D FW .Tr.A/; Tr.B/; Tr.AB//. Proof. The Cayley–Hamilton identity gives B 2  Tr.B/B C 1 D 0. Multiplying by AB 1 and taking the trace, we get the first identity. We prove the second assertion by induction on the length of W by applying the first identity in a convenient way, see for instance [CS83]. We finally proved that there exists a family of polynomials Fa;b 2 ZŒx; y such that M.Ha;b / D fx; y 2 R2 W Fa;b .x; y/ D 0; x 2  2  y  2g. The inequality x 2  2  y  2 is the trace of the inequalities we viewed in the case of S 1 _ S 1 . The equality y D 2 holds if and only if uv D 1 and y D x 2  2 if and only if uv 1 D 1. This last equality occurs precisely for abelian representations of Ha;b . The next proposition proven in [Le91] simplifies the computation of Fa;b . Proposition 1.4. Let w be the word associated to Ha;b and set wn to be the word w P with the n first and n last letters removed. Then Fa;b D .a1/=2 .1/n Fwn where nD0 we set F1 D 1. For instance, the figure eight knot 4.1 is B.5; 3/ and then one computes w D uv 1 u1 v and F5;3 D x 2 y  y 2  2x 2 C 3. The torus knot T .5; 2/ D 5:1 D B.5; 1/ has w D uvuv and F5;1 D y 2  y  1. The corresponding representation spaces are shown in Figure 5.

2 Differentiable structure and twisted cohomology Let X be a topological space whose fundamental group is finitely presented. Let 1 .X / D ht1 ; : : : ; tn j R1 ; : : : ; Rm i be a presentation. We recall that M.X / is the quotient of R.X/ by an action of SU.2/ and that R.X / is identified to the preimage of 1 by the map R W SU.2/n ! SU.2/m defined by R.Ai / D .Rj .Ai //. The latter

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Figure 5. Moduli space of 4.1 and 5.1.

space is a submanifold of SU.2/ provided that R is a submersion on the preimage of 1. The purpose of this part is to use this argument in a systematic way. For what concerns the quotient, we see that the stabilizer of a representation  has the following form: (1) SU.2/ if  is central. (2) S 1 if  is abelian. (3) f˙1g if  is irreducible. For a good geometric quotient, we need the stabilizer to be constant and, moreover, the biggest part of the moduli space will correspond to the smallest stabilizer. All these conditions will be easily readable in the twisted cohomology we introduce now. Definition 2.1. Let W be a finite CW-complex with a 0-cell as basepoint. Denote by z the universal covering of W and by A the ring ZŒ1 .W /. Then, the cell complex W z ; Z/ is naturally a left A-module where 1 .W / acts by deck transformations. C .W Given any left A-module E, we define   z /; E/ and H  .W; E/ D H  C  .W; E/ ; C  .W; E/ D HomA .C .W z / ˝A E and H .W; E/ D H .C .W; E// : C .W; E/ D C .W

2.1 Two examples Let us look at the case W D S 1 . Its universal cover is R with Z acting by translations. We define eQ0 D f0g and eQ1 D Œ0; 1. These cells project respectively on the 0-cell and the 1-cell of S 1 . As A-modules we have C0 .S1 / D AeQ0 and C1 .S1 / D AeQ1 . Identifying A with ZŒt ˙1 , we compute that @eQ1 D .t  1/eQ0 .

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An A-module is nothing more than an abelian group E with an automorphism ' corresponding to the action of t . The twisted (co)homology of the circle is then computed from the following complexes: C 0 .S 1 ; E/ ' E

d

/ C 1 .S 1 ; E/ ' E;

C0 .S 1 ; E/ ' E o

@

C1 .S 1 ; E/ ' E;

where @v D '.v/  v and d is obtained from the formula d D .1/jjC1  B @. We deduce from it the isomorphisms H 0 .S 1 ; E/ D H1 .S 1 ; E/ D ker.Id '/ and H 1 .S 1 ; E/ D H0 .S 1 ; E/ D coker.Id '/. As a first application, consider M.S 1 /, that is, the set of conjugacy classes in SU.2/. Pick g 2 SU.2/ not central, and consider the map cg W SU.2/ ! SU.2/ defined by cg .h/ D hgh1 . We identify once and for all the tangent space of SU.2/ at g to su.2/ via the map which associates to a path g t such that g0 D g the derivative dtd j t D0 g t g01 . Using this identification, we compute D1 cg ./ D dtd j t D0 e t  ge t  g 1 D   Adg . From this computation, we see that the tangent space of the SU.2/-orbit through g is the image of the map   Adg . Hence, the tangent space of M.S 1 / at Œg is the cokernel of this map. We can interpret it as H 1 .S 1 ; Adg / where Adg is a notation for the vector space su.2/ with automorphism Adg . This point is indeed very general. Before attacking the general case, let us look at the case of W D S 1 _    _ S 1 , a pointed union of n circles. Then, its fundamental group is free, say 1 .W / D ht1 ; : : : ; tn i and R.W / D SU.2/n . We have M.W / D SU.2/n = SO.3/ and the action is free on irreducible representations. Hence for an irreducible representation , one has the following identification: TŒ M.W / D coker D1 c where c .h/ D .h1 h1 ; : : : ; hn h1 / and i D .ti /. We compute as before D1 c ./ D .  Ad1 ; : : : ;   Adn /. z is a regular 2n-valent tree, and one can choose eQ0 as a On the other hand, W lift of the basepoint and oriented edges eQ11 ; : : : ; eQ1n starting from eQ0 and representing t1 ; : : : ; tn . As before, we have @eQ1i D .ti  1/eQ0 . We define again the A-module Ad as su.2/ with ti acting as Adi C 0 .W; Ad / D su.2/; C 1 .W; Ad / D su.2/n and d D D1 c ./: At irreducible representations, we compute H 0 .W; Ad / D ker d0 D f 2 su.2/ W Adi  D ; 1  i  ng D f0g as Ad  is irreducible as a representation of 1 .W / in su.2/. Hence, as before we have an identification TŒ M.W / D H 1 .W; Ad /.

2.2 The general case Suppose W is a 2-dimensional CW-complex of the following form: (1) one 0-cell lifted to eQ0 ,

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(2) n 1-cells lifted to oriented edges eQ1i for i  n starting at eQ0 , (3) m 2-cells lifted to polygons eQ2j for j  m with a base point at eQ0 . For each 2-cell eQ2j , one can read starting at eQ0 a word in the generators ti represented by the 1-cells. Denoting by Rj these words, we get a presentation of 1 .W / given by 1 .W / D ht1 ; : : : ; tn jR1 ; : : : Rm i: Let R W SU.2/n ! SU.2/m be the map defined for an n-tuple  D .1 ; : : : ; n / by R.1 ; : : : ; n / D .R1 ./; : : : ; Rm .//. The space R.W / D R1 .1; : : : ; 1/ is smooth at  if R is a submersion at . Let us compute the differential of R by supposing " m D 1. We write R./ D i"11 : : : ikk for il 2 f1; : : : ; ng and "l D ˙1. D R.i / D D

d j tD0 .e t1 i1 /"1 : : : .e t k ik /"k dt ´ X "1 : : : "l1 i "l : : : "k il1 l il "lC1 " i"11 : : : ill il ilC1 i1

l

ik

"

: : : ikk

if "il D 1; if "il D 1: "

(2.1)

i

l1 l On the other hand, @eQ2 is the sum over l of either ti"11 : : : til1 eQ1 if "l D 1 or

"

i

ti"11 : : : till eQ1l if "l D 1. Taking the adjoint of this map to obtain a map from C 1 .W; Ad / to C 2 .W; Ad /, we get exactly the same expression as in (2.1). In conclusion, the following diagram commutes: T R.W / 

D R

/ su.2/m



C 1 .W; Ad /

d1





/ C 2 .W; Ad /.

The map R is a submersion at  if and only if d1 is surjective, or equivalently if H 2 .W; Ad / D 0. The argument on the pointed union of circles can be repeated exactly and shows that the action of SU.2/ by conjugation at  is locally free if d0 is injective which amounts to say that H 0 .W; Ad / D 0. If both conditions are satisfied, then the quotient M.W / is a manifold at Πand the tangent space is identified with ker d1 = im d0 D H 1 .W; Ad /.

2.3 Applications The interest of the language of (co)homology is to use its tools, namely exact sequences, Poincaré duality and universal coefficients. We refer to [Ha02] for general reference on twisted (co)homology. First, we can define relative (co)homology of pairs as in the untwisted case and it fits into a long exact sequence as usual. For what concerns Poincaré duality, we have the following generalization: given a compact and oriented n-manifold with

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boundary and an A-module E, the cap product with the fundamental class ŒM  gives an isomorphism H k .M; E/ ' Hnk .M; @M I E/: For what concerns universal coefficients, let W be a finite CW-complex, R be a principal ring and E an RŒ1 .W /-module which is free as an R-module. Then there is an exact sequence 0 ! Ext.Hk1 .W; E/; R/ ! H k .W; E  / ! Hk .W; E/ ! 0: The proof is the usual one applied to the free complex C .W; E/ whose dual can be identified with C  .W; E  / thanks to our hypotheses. We very often use the wellknown fact that the Euler characteristic of a complex and of its homology are the same. We deduce from it that the Euler characteristic of H  .W; Ad / is 3 .W / because the twisted complex is obtained from the standard one by tensoring by su.2/ which has dimension 3. 2.3.1 Surfaces. Let † be a closed surface and  2 R.†/ be an irreducible representation. The RŒ1 .†/-module Ad is free over R of dimension 3 and its dual can be identified to itself thanks to the invariant Killing form hA; Bi D Tr ABx > . We deduce that H 2 .†; Ad / ' H0 .†; Ad / ' H 0 .†; Ad / ' H 0 .†; Ad / D 0. Hence irreducible representations are smooth points of M.†; SU.2//. Written differently, Poincaré duality states that the following pairing is non-degenerate. H 1 .†; Ad /  H 1 .†; Ad /

[

/ H 2 .†; Ad ˝ Ad /

h;i

/ H 2 .†; R/

R

/R:

This gives a non-degenerate 2-form ! on the irreducible part of M.†/. We will show later that it is a closed form, and hence that M.†/ is symplectic. 2.3.2 The torus case. This case is not covered by the previous one because all representations  on a torus are abelian. Before computing the corresponding cohomology a pair group, recall that M.S 1  S 1 / is covered by the map sending   .'; / to the i' e 0 representation '; sending the first generator to ' D D e i' and the 0 e i'   i second one to  D e 0 ei0 D ei . Then H 0 .S 1  S 1 ; Ad / ' H 2 .S 1 S 1 ; Ad / ' ker.Id  ' / \ ker.Id  /.

In the quaternionic basis i, j, k, Ad' D

1 0 0 0 cos.2'/  sin.2'/ 0 sin.2'/ cos.2'/

. The subspace fixed by

this matrix is generated by i provided that 2' … 2Z. We deduce from this that the rank of H 2 .S 1  S 1 ; Ad / is constant equal to 1 if ' or is not in Z, or equivalently if '; is not central. One can apply the constant rank theorem to state that M.S 1 S 1 / is actually a manifold at all non-central representations. A computation shows that the pull back of ! in the coordinates ', is d' ^ d .

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2.3.3 3-manifolds with boundary. Let M be a 3-manifold with nonempty boundary, and  2 R.M /. The following exact diagram is a piece of the sequence of the pair .M; @M /, where vertical isomorphisms are given by Poincaré duality: H 1 .M; Ad / 

˛



H 2 .M; @M I Ad /

ˇ

/ H 1 .@M; Ad / 

ˇ

/ H 2 .M; @M I Ad /



/ H 1 .@M; Ad /

˛





/ H 1 .M; Ad / .

We read from this diagram that rk ˇ D rk ˛  . Standard linear algebra says that rk ˛  D rk ˛ and the exactness of the first line gives rk ˛ D dim ker ˇ. One deduces from it that dim H 1 .@M; Ad / D rk ˇ C dim ker ˇ D 2 rk ˛ whereas rk H 1 .M; Ad / D rk ˛ C dim ker ˛ D 12 dim H 1 .@M; Ad / C dim ker ˛. This implies that the three following properties are equivalent: (1) The map H 1 .M; Ad / ! H 1 .@M; Ad / is injective. (2) The map H 1 .M; @M I Ad / ! H 1 .M; Ad / vanishes. (3) dim H 1 .M; Ad / D 12 H 1 .@M I Ad /. We deduce from these computations the following result. Let  2 R.M / be a representation. We call it regular if it is irreducible and satisfies the equivalent properties above. Theorem 2.2. Regular representations are smooth points of R.M / and the restriction map M.M / ! M.@M / is a Lagrangian immersion when restricted to regular representations and corestricted to irreducible (non central in the torus case) representations. Proof. Let  be a regular representation in R.M /, and consider a 2-dimensional CWcomplex W on which M retracts. Then, the differential d1 in the complex C  .W; Ad / is the derivative at  of equations defining R.M /. The corank of this derivative is equal to the dimension of H 2 .W; Ad / D H 2 .M; Ad /: we will see that the conditions for a representation to be regular are equivalent to the condition that H 2 .M; Ad / is as small as possible, and hence ensures that R.M / is smooth at . As  is irreducible, H 0 .M; Ad / D 0 and an Euler characteristic computation gives .M / dim su.2/ D  dim H 1 .M; Ad / C dim H 2 .M; Ad /. We deduce from this formula that H 2 .M; Ad / is as small as possible if and only if the same is true for H 1 .M; Ad /. We have proved that dim H 1 .M; Ad / D 12 dim H 1 .@M / C dim ker ˛ where ˛ is the natural map from H 1 .M; Ad / to H 1 .@M; Ad /. By assumption, the restriction of  to the boundary is in the smooth part of R.M /, hence the dimension of H 1 .@M; Ad / does not change. The regularity condition is dim ker ˛ D 0 and hence is equivalent to the fact that the dimension of H 1 .M; Ad / is as small as possible. We proved the first part of our assumption. We also see that the equation dim ker ˛ D 0 implies that dim H 1 .M; Ad / D 12 dim H 1 .@M; Ad /.

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Moreover, ˛ is the derivative at Œ of the restriction map r W M.M / ! M.@M / which becomes an immersion. Finally, let us show that the image of ˛ is isotropic by looking at the diagram above. Let u; v 2 H 1 .M; Ad /. Then, one may show that !.˛.u/; ˛.v// D 0. But this can be written as hˇ .u# /; ˛.v/i where u# is Poincaré dual to u and h; i is the duality pairing. Recall that im ˇ D .ker ˇ/? D .im ˛/? ; this implies precisely that !.˛.u/; ˛.ˇ// vanishes. Hence, the image of ˛ is Lagrangian and the theorem is proved. Let us look at the following example. As seen in Section 1.4, the moduli space of the trefoil knot is a segment (abelian representations) with another segment attached to it. The restriction map sends this space to the representation space of the torus that we represent as a union of two stacked-up squares (known as the pillow case). The image of this map is represented on the left of Figure 6. The restriction map is an immersion for all regular points but is not injective. The case of the figure eight knot is presented on the right. In that case, the moduli space is the union of a segment (abelian representations) and a circle (irreducible representations). The restriction map is an immersion at all regular points and the map fails to be injective at one point.

Figure 6. Moduli spaces of 3.1 and 4.1 restricted to the boundary.

2.4 Reidemeister torsion Given a real vector space V of dimension n, we can form the line det V D ƒn V . We consider this line as even if n is even and odd if n is odd. This convention will play a role with the implicit use of the isomorphism V ˝ W ! W ˝ V sending v ˝ w to .1/jvjjwj w ˝ v where jvj and jwj are the degrees of v and w respectively. With that convention, one can write det.V ˚ W / D det.V / ˝ det.W / as this identification depends on the order of V and W up to a sign prescribed by the convention. The following considerations rely on the fact that for any short exact sequence 0 ! U ! V ! W ! 0 there is a canonical isomorphism det U ˝ det W D det V . This isomorphism is defined by sending u1 ^    ^ ui ˝ w1 ^    ^ wj to z1 ^    ^ w zj where w z is any lift of w in V . u 1 ^    ^ ui ^ w

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Given a finite complex of finite dimensional real vector spaces C  , we define k det C  D det C 0 ˝ .det C 1 /1 ˝    ˝ .det Ck /.1/ . We define the determinant of its cohomology by the same formula. Lemma 2.3. There is a canonical isomorphism (which involves the sign rule) det C  D det H  : Proof. Define for all i 2 Z, Zi D ker di and B i D im di . The exact sequence 0 ! Z i ! C i ! B i ! 0 gives det C i D det Z i ˝ det B i and the exact sequence 0 ! B i 1 ! Z i ! H i ! 0 gives det Z i D det B i1 ˝ det H i . Removing the ˝ sign, we compute det C 0 .det C 1 /1 det C 2 : : : D det Z 0 det B 0 .det Z 1 det B 1 /1 det Z 2 det B 2 : : : D det B 1 det H 0 det B 0 .det B 0 det H 1 det B 1 /1 det B 1 det H 2 det B 2 : : : : One can see that all factors det B i appear twice with opposite signs and then cancel, proving the proposition. Let W be a finite CW-complex and  be a representation in R.W /. For all cells, z . For each k, number we choose an orientation and a lift to the universal covering W nk 1 the lifted k-cells as eQk ; : : : ; eQk . Recall that the twisted cochain complex of dimension k is L i C k .W; Ad / D eQk su.2/: ink

3 su.2/ the SU.2/-invariant volume element  D i ^ j ^ k. We define Choose in ƒV  again k D ink eQki , and ƒ D 0 ˝ 1 1 ˝    2 det C .W; Ad /. With the isomorphism provided by the lemma, we obtain an element in det H  .W; Ad / that we denote by the same letter. One can check easily that this element does not depend on the choice of the lift of the cell because SU.2/ fixes , but changing the orientation of a cell or its numbering do change ƒ up to a sign. To remove this ambiguity, we do the same construction for the complex C  .W; R/ by taking the same cells with the same orientation and order, and replacing  with 1 2 det R. Doing so, one gets an element T 2 det C  .W; R/ D det H  .W; R/. We call the quotient ƒ=T 2 det H  .W; Ad / det H  .W; R/1 the Reidemeister torsion at . It remains to understand how this quotient changes when we change the cell decomposition but one can show that it does not change under cell subdivision and collapsing, see [Tu02]. This implies that the torsion only depends on W up to simple homotopy. Let us give some applications: in the case of a surface † and an irreducible representation  2 R.†/, one obtains the equalities H 0 .†; R/ D R, H 2 .†; R/ D R and H 0 .†; Ad / D H 2 .†; Ad / D 0. By considering the standard generators of the determinant of these spaces, one sees that the torsion reduces to an element of

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.det.H 1 .W; Ad ///1 det H 1 .W; R/. This element does not giveus any  new infor g 1 3g3 ! ! mation as one can show that it is equal to the Liouville volume form .3g3/Š gŠ where g is the genus of † (supposed at least equal to 2) and !; ! are the symplectic forms on H 1 .†; R/ and H 1 .†; Ad / respectively. In the case of a regular representation in R.M /, we have H 0 .M; Ad / D 0 and 2 H .M; Ad / D H 2 .@M; Ad / as one can deduce from the commutative diagram below with exact lines: / H 2 .M /

H 2 .M; @M / 



H 1 .M /

0



/ H 2 .@M /

/ H 3 .M; @M / D 0



/ H 1 .M; @M / :

Then, one has H 2 .@M / D H 0 .@M / . This is 0 if the genus of @M is greater than 2, and in the case of a torus, it is one-dimensional. One can choose a preferred generator of this space, showing that det H  .M; Ad / ' H 1 .M; Ad /1 . Choosing an element T in det H  .M; R/, one gets a well-defined element ƒ 2 H 1 .M; Ad /1 which may be interpreted as a volume form on the regular part of M.M /.

3 Gauge theory Let M be a compact manifold of dimension at most 3. In this section, we denote SU.2/ by G as it can be replaced by any Lie group here. The Lie algebra of G will be denoted by G .

3.1 Principal bundles and flat connections Definition 3.1. A principal G-bundle over M is a fiber bundle  W P ! M with a right action of G on P such that G acts freely and transitively on each fiber. Two such bundles are isomorphic if there is a G-equivariant bundle isomorphism lifting the identity of M . Definition 3.2. A principal bundle with flat structure .P; F / is a principal bundle  W P ! M and a foliation F of P which is G-equivariant and such that the restriction of  to each leaf is a local diffeomorphism. Again two such pairs are isomorphic if the bundles are isomorphic and the foliations correspond through this isomorphism. Such a flat G-bundle is often described by covering M with open sets Ui on which we can find sections si of P whose images lie in the same leaf (we will say that these sections are flat). On the intersection of two such open sets Ui and Uj , the two sections

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differ by the action of a locally constant map gi;j W Ui \ Uj ! G. These data are sufficient to reconstruct the flat G-bundle up to isomorphism. Take a point p in a flat G-bundle P and write x D .p/. Then, any path W Œ0; 1 ! M such that .0/ D p lifts uniquely to a path Q ! P if one asks that Q .0/ D p and that Q stays locally on the same leaf. If .1/ D x, then there exists a unique g 2 G such that Q .1/ D pg. The assignment 7! g does not depend on the homotopy class of and gives rise to a homomorphism Holp W 1 .M; x/ ! G. The conjugacy class of this representation does not depend on p and x; moreover, we have the following fundamental result: Theorem 3.3. The holonomy map gives a bijection between isomorphism classes of flat G-bundles and M.M; G/. Proof. We construct the inverse map in the following way. Let x be a basepoint in M and  a representation of 1 .M; x/. Then, denote by M  G the quotient of z  G by the equivalence relation .m; g/  . :m; . /g/ for all m 2 M; g 2 G, M

2 1 .M; x/. The map .m; g/ D m and the action .m; g/:h D .m; gh/ provide M  G with a G-bundle structure. The foliation F is the quotient of the foliation of z  G whose leaves are M z  fgg for g 2 G. One can check that these constructions M are inverse to each other, which proves the theorem.

3.2 Sections and connection forms On manifolds M of dimension at most 3 and for connected and simply connected groups G, all G-bundles on M are trivial, that is, isomorphic to M  G. To prove this, it is sufficient to find a section s of any G-bundle  W P ! M . The map M  G ! P sending .m; g/ to s.m/g will be the desired isomorphism. Let W be a CW-complex homotopic to M and let  W P ! W be a G-bundle. Then, one can choose arbitrarily a section over the 0-skeleton of W . For each 1-cell, we extend the section of P given at the ends, using the fact that the fiber (isomorphic to G) is connected. At the boundary of each 2-cell, there is some section that we can extend along the cell as G is simply connected. Finally, we deduce from the fact that 2 .G/ D 0 that the section also extends to the 3-cells and hence to W , which proves the assumption. The existence of sections gives us another practical viewpoint on flat G-bundles that we explain now. Let .P; F / be a flat G-bundle. Given a section s of P , one can encode the foliation F with a 1-form A on M with values in G . We define it in the following way: let x be a point in M and h be the map defined at a neighborhood of x with values in G such that the map m 7! s.m/h.m/ describes the leaf of F passing at s.x/ as suggested in Figure 7. Then, we set Ax D Dx h 2 G . Any other section s 0 is obtained from s by the right action of a map g W M ! G. We write for short s 0 D s g . If the leaf passing at s.x/ is the image of the map sh for

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F P

h

s

 M

x Figure 7. From foliations to connection forms.

some map h defined around x with values in G, then by G-invariance, the image of the leaf passing through s.x/g.x/ is the image of the map m 7! s.m/h.m/g.x/. This map can be rewritten as sgg 1 hg.x/. The 1-form Ag associated to the section s g is then the derivative at x of the map m 7! g 1 .m/h.m/g.x/, that is, g 1 Dx hg  g 1 Dx g. We then obtain Ag D g 1 Ag C g 1 dg. Nevertheless, any section s and 1-form A 2 1 .M; G / do not necessarily define a flat structure but only a G-invariant distribution of subspaces in TP transverse to the fibers of . We would like to give a condition on A for this to hold. In order to be tangent to a foliation, the distribution has to be integrable, that is, we have to verify the Frobenius condition that if two vector fields belong to this distribution, their bracket also belongs to it. To verify this condition, consider two vector fields X , Y on M . The 1-form A defines uniquely a 1-form AQ 2 1 .P; G / whose kernel is the invariant Q and distribution. This 1-form is characterized by the equations Rg AQ D g 1 Ag s  AQ D A, where Rg is the action of a fixed element g on P . The vector fields X and Y Q Xz / D A. Q Yz / D 0 extend uniquely to horizontal vector fields Xz and Yz on P such that A. z z and  X D X;  Y D Y . These vector fields belong to the distribution defined by Q Xz ; Yz / D 0. The A and their bracket belongs to the distribution if and only if A.Œ Q z z z Q z z Q z Q z z Q Xz ; Yz / D 0. equation dA.X; Y / D X:A.Y /  Y :A.X/  A.ŒX ; Y / gives dA. Let ,  be two elements of G and v ; v be the vector fields coming from the infinitesimal action of G on P , for instance v .p/ D dtd j t D0 pe t  . Then, by conQ  / D  and hence dA.v Q  ; v / D ŒA.v Q  /; A.v Q  /. We deduce that the struction A.v 1 Q D 0 is true when applied to pairs of horizontal (resp. vertical) identity dAQ C 2 ŒAQ ^ A vector fields. One checks that this is again true for horizontal and vertical vector Q D 0 holds. Pulling it back by s, we get fields and hence, the identity dAQ C 12 ŒAQ ^ A 1 dA C 2 ŒA ^ A D 0 which is the flatness equation for A. This is a necessary and sufficient condition for the distribution defined by A to be integrable. These considerations may be summarized in the following proposition:

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351

Proposition 3.4. Let M be a manifold and G a Lie group such that all G-bundles on M are trivial. Then the set of isomorphism classes of flat G-bundles is isomorphic to the set of connections A 2 1 .M; G / satisfying dA C 12 ŒA ^ A D 0 up to the action of the gauge group given by Ag D g 1 Ag C g 1 dg.

3.3 de Rham cohomology and isomorphisms Given a trivialized flat G-bundle .P; s; A/ where s is a section of P and A is a flat connection, we define the twisted de Rham complex as the complex  .M; G / with the differential dA .˛/ D d˛ C ŒA ^ ˛. The flatness equation implies that dA2 D 0. We denote by HA .M; G / the cohomology of this complex. It is related to the twisted cohomology via the following de Rham theorem. Let M be a compact manifold homeomorphic to a cell complex W and .P; s; A/ be a flat G-bundle. Fixing p 2 P over a basepoint x in M gives a holonomy representation  2 R.1 .M; x/; G/. z is contractible, the flat G-bundle induced on M z is trivial. As the universal cover M g z Q Q Hence, there is a map g W M ! G such that A D 0 where A is the connection z . Moreover, we can suppose that g.x/ D 1. Let ˛ 2 k .M; G / form induced on M be a cocycle. Then we denote by I.˛/ 2 RC k .W; Ad / the cocycle which associates associates to a lifted k-cell eQk the integral eQk g 1 ˛g. Theorem 3.5. The map I is a chain map which induces for all k an isomorphism from HAk .M; G / to H k .M; Ad /. When k D 1, we can interpret I as the derivative of the holonomy function. More precisely, let M be a manifold and .P; F t / be a 1-parameter family of foliations on the same G-bundle P . Assuming the existence of a section s W M ! P , the family of foliations gives a family of connection 1-forms A t satisfying dA t C 12 ŒA t ^ A t  D 0. t Let A D A0 and suppose that this family is smooth. Then ˛ D dA j satisfies dt t D0 1 d˛ C ŒA ^ ˛ D 0. Hence, it represents an element Œ˛ of HA .M; G /. Let be loop in 1 .M; x/; then Hol A t 2 G and dtd j t D0 Hol A t .Hol A/1 D z I.˛/. /. To see this, it is sufficient to do the computation in the universal cover M after having trivialized A. Let † be an oriented surface and  be an irreducible representation corresponding to a flat connection A. Then, the tangent space of M.†/ at Œ is isomorphic to H 1 .†; Ad / ' HA1 .†; G /. The cup-product in cohomology corresponds to the exte1 2 rior product of forms in De R Rham cohomology. Then, the form !A W HA .†; G / ! R is given by !A .˛; ˇ/ D † h˛ ^ ˇi. Remarking that A does not appear in this formula is the key point for showing that ! is a closed 2-form. The following proposition is a technical ingredient for relating gauge theoretical arguments to the study of representation spaces.

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Proposition 3.6. Let M be a compact manifold of dimension at most 3, and U be a contractible open set in M.M; G/ consisting of regular representations. Then, there is a smooth map A W U ! 1 .M; G / such that for all 2 U , A is a flat connection whose holonomy is in the class of . Proof. With our assumption, the quotient map R.M / ! M.M / is a fibration over U as U consists in regular representations. As U is contractible, there is a smooth section  W U ! R.M / of the quotient map. Fix a point x in M and consider the z  G  U=  where we set . :m;  . /g; /  .m; g; / for all product P D M

2 1 .M; x/. This construction gives a flat G-bundle over M  U such that for each 2 U , the holonomy of P over M  f g is given by  . Moreover, M  U is homotopic to M and P has to be trivial as a G-bundle. Taking a smooth section s W M  U ! P , we pull back the flat structure of P to a flat connection A on M  U . The restriction of A to each slice A  f g gives the connection A that we are looking for. As a first application of gauge theory, we finally prove that Mreg .†/ is a symplectic manifold. Proposition 3.7. Let † be a closed surface. The non-degenerate 2-form ! on the regular part of M.†/ is closed, and hence symplectic. Proof. Let us show it on any open set U as in Proposition 3.6. Let A W U ! 1 .M; G / be the map given by this proposition. The remark following Theorem 3.5 implies that the derivative of A at is a map T M.†/ ! 1 .M; G / taking values in ker dA . Considering its class in HA1 .†; G /, one gets the inverse de Rham isomorphism. We R conclude that the form ! on U is the pull-back of the form !.˛; ˇ/ D † h˛ ^ ˇi on 1 .†; G /. The latter expression is a constant 2-form on an infinite dimensional space. It is closed in the sense that for any smooth maps X, Y , Z from 1 .†; G / to itself, the following identity holds: 00

d!.X; Y; Z/00 D X:!.Y; Z/  Y:!.X; Z/ C Z:!.X; Y / C !.X; ŒY; Z/ C !.Y; ŒZ; X / C !.Z; ŒX; Y / D 0:

This identity, pulled back to U implies that ! is closed.

4 Chern–Simons theory The term Chern–Simons theory usually refers to the study of secondary characteristic classes on flat bundles. Indeed, by Chern–Weyl theory, we know that given a G-bundle  W P ! M , we can compute the characteristic classes (Chern, Euler and Pontryagin classes) of associated bundles by integrating invariant polynomials in the curvature of some connection of P . The existence of flat connections implies the vanishing of all

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characteristic classes. Chern and Simons introduced some primitives of the Chern– Weyl classes giving non trivial invariants of flat G-bundles. For our purposes, we will reduce Chern–Simons theory to the following constructions: (1) Given a closed 3-manifold M , we construct a locally constant map CS W M.M / ! R=4 2 Z. (2) Given a closed surface †, we obtain a hermitian line bundle with connection .L; j  j; r/ over the regular part of M.†/ such that the curvature of r is the symplectic form !. We call this bundle the prequantum bundle of † and denote it by L† . (3) Given a 3-manifold M with boundary, we obtain a flat lift CS of the restriction map M.M / ! M.@M / to the prequantum bundle of @M : L@M p8 p CS p p p  p / M reg .@M /: M reg .M /

4.1 The Chern–Simons functional Let M be a 3-manifold possibly with boundary,  W P ! M a trivializable G-bundle and F a flat structure on P . Given a section s W M ! P , one obtains R a flat connection 1 A by the procedure described in Section 3. One set CS.A/ D 12 M hA ^ ŒA ^ Ai. 1 Recall that A belongs to  .M; G / so that A ^ A ^ A belongs to 3 .M; G ˝3 /. Applying the antisymmetric map .X; Y; Z/ 7! hX; ŒY; Zi to the coefficients, one obtains the 3-form hA ^ ŒA ^ Ai which can then be integrated. The main point is to compute how this functional changes when changing the section. Given a map g W M ! G one has Z Z 1 1 g 1 1 hg Ag ^ g dgi  hg 1 dg ^ Œg 1 dg ^ g 1 dgi: CS.A / D CS.A/ C 2 @M 12 M The proof is a direct consequence of Stokes formula, with the use of some formulas forms on Lie groups, see [Fr95]. Denote by W .g/ the term R for1differential 1 1 hg dg ^ Œg dg ^ g 1 dgi. It is also called the Wess–Zumino–Witten func12 M tional. By definition, the formR g 1 dg is equal to g  where is the left Maurer–Cartan 1 form on G. Hence W .g/ D M g  where D 12 h ^ Œ ^ i is the Cartan 3-form on G. Assuming G D SU.2/, we deduce from this expression that W .g/ mod 4 2 depends only on the restriction of g to @M . Indeed, given another 3-manifold N with an oriented diffeomorphism R ' W @N ! @M and a map h W N ! G such that g B' D h, we consider the integral M [.N / f  where f stands for g on M and h onRN . This integral is equal to W .g/  W .h/. On the other hand, it is equal to .deg f / G D 4 2 deg f 2 4 2 Z. This proves that if M has no boundary, then CS.Ag / D CS.A/ mod 4 2 .

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Hence, the map CS W M.M; SU.2// ! R=4 2 Z is well defined when M has no boundary. In order to show that it is locally constant, recall that if A t is a smooth family of flat t connections with A0 D A then the derivative ˛ D dA j satisfies d˛ C Œ˛ ^ A D 0. dt Rt D0 R d CS.A t / 1 1 Moreover, dt j tD0 D 4 M h˛ ^ ŒA ^ Ai D  2 M h˛ ^ dAi. On the other hand, h˛^dAi D dh˛^AiChd˛^Ai. Moreover hd˛^Ai D hŒ˛^ A^Ai D hŒA^A^˛i D 2hdA^˛i. This implies the identity h˛ ^dAi D dh˛ ^Ai: R 1 t/ Hence, one has d CS.A j D hA ^ ˛i. tD0 dt 2 @M In the case where M has no boundary, this proves that the Chern–Simons function is locally constant on M.M /.

4.2 Construction of the prequantum bundle Let † be a closed surface. Recall that for G D SU.2/, all principal G-bundles are trivial, hence flat structures are encoded by flat connections A 2 1[ .†; G / (that is, A satisfies dA C 12 ŒA ^ A D 0). Moreover, two connections represent the same element of M.†/ if and only if they are related by the action of the group  of maps from † to G. Consider the finer equivalence relation where A and Ag are considered to be equivalent if CS.A/ D CS.Ag /. New equivalence classes form a bundle over the old ones with fiber R=4 2 Z. This is the construction of the prequantum bundle. Let us give another point of view of the same construction, technically more appropriate. Set L D 1[ .†; G /  R=2Z and define an action of the gauge group on L by the formula .A; /g D .Ag ; C c.A; g// where Z 1 1 hg 1 Ag ^ g 1 dgi  c.A; g/ D W .g/: 4 † 2 Then, c is a cocycle in the sense that for any flat connection A and gauge group elements g, h one has c.A; gh/ D c.A; g/ C c.Ag ; h/. Consider the quotient map L= ! 1[ .†; G /=  D M.†/. By using the local sections of the projection 1[ .†; G / ! M.†/ given by Proposition 3.6 and the fact that the gauge group acts freely on connections encoding irreducible representations, one finds that the above quotient is actually a principal fiber bundle over M reg .M / with fiber R=2Z. The prequantum line bundle L is the fiber bundle associated to L=  with the representation of R=2Z on C given by :z D e i z. It is naturally a hermitian line bundle. Let us show that this bundle has a connection with curvature !. On the trivial bundle L ! 1[ .†; G / there is a natural connection given by the R 1 expression d   where  is the 1-form given by A .˛/ D 4

† hA ^ ˛i. One can check directly that this form is equivariant and hence defines a connection on the 1 !. The quotient L. The curvature of this connection is the derivative of , that is, 2

1 same formula is true on the quotient as both symplectic forms ! on [ .†; G / and M.†/ correspond in the quotient.

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We can give the third and last application: given a 3-manifold with boundary M , and a flat connection A on it, we consider the element .A; CS.A/=2/ 2 L. Given g W M ! G, the connection Ag is sent to the equivalent element .Ag ; CS.Ag /=2/. Hence, this map also denoted by CS is well defined from M.M / to L@M R . Moreover, 1 t/ given a smooth family A t , we already computed d CS.A D .˛; hA ^ ˛i), dt 4 @M dA t where ˛ D dt . This derivative is in the kernel of the connection d  , which shows that CS.A t / is a parallel lift over L@M over the restriction of A t to @M as asserted.

4.3 Examples 4.3.1 Closed 3-manifolds. Let us look at some examples of Chern–Simons invariants for closed manifolds M . Recall that we constructed a locally constant map CS W M.M / ! R=4 2 Z. The trivial representation is obtained as the holonomy of the connection A D 0. In that case, one has CS.A/ D 0. For less trivial examples, consider some manifolds obtained as a quotient of S 3 D SU.2/ by a finite subgroup H , foro instance the lens spaces L.p; 1/ given by Hp D n  e 2ik=p 0 0 e 2ik=p

; k 2 Z=pZ , or the quaternionic manifold Q8 given by H D f˙1; ˙i; ˙j; ˙kg. In these cases M D SU.2/=H and there is a natural non-trivial flat bundle P given by the quotient of SU.2/  SU.2/ by the equivalence relation .g1 ; g2 /  .g1 h; h1 g2 / for h 2 H . The map  W P ! M is the first projection. A section is given by s.g/ D .g; g 1 /. OneR computes that the connection associated to that section is g 1 dg. Hence, CS.A/ D S 3 =H D 4 2 =jH j. Another easy example is M D S 1  S 1  S 1 . In that case, all representations are abelian, hence all flat connections are equivalent to connections with values in Ri. As hi; Œi; ii D 0, one has necessarily CS.A/ D 0 for all A. This is compatible with the fact that M.M / is connected and CS is locally constant.

4.3.2 The torus case. Let us give a finite dimensional construction for the prequantum bundle L over the torus † D S 1  S 1 . Consider the map F W R2 ! M.†/ sending .'; / to the representation '; W Z2 ! SU.2/ where '; .a; b/ D exp.i.a' C b //. The fibers of F are the orbits of the action of the group H D Z2 Ì Z=2Z where the first factor acts by translation and the second one by inversion. One can lift the map F to 1[ .†; G / by sending .'; / to the connection A'; D i.'ds C dt / where s; t are coordinates of the two S 1 factors identified to R=Z. The action of H is realized by a gauge group action as follows. The translation by .2k; 2l/ is given by the action of gk;l .t; s/ D exp.2i.ks C lt //, whereas the inversion is given by the action of the constant map g D j. These actions lift to the trivial bundle L DR R2  R=2Z in the following way: 1 1 .'; ; /gk;l D .' C 2k; C 2l; C 4

† hA'; ; gk;l dgk;l i/ as one can show gk;l D .' C 2k; C 2l; C that W .gk;l / D 0 (see [Ma07]). We obtain .'; ; / 'l  k/ and .'; ; /j D .';  ; /. We recognize an action of H on L. The

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quotient produces a bundle over the quotient of R2 by H , smooth over non central representations. This gives an elementary construction of the prequantum bundle in that case which is very useful for computing Chern–Simons invariants for knot exteriors. 4.3.3 Some knot complements. Let us give an application of these constructions to the case of a knot complement. In that case, the manifold M is the complement of a tubular neighborhood of a knot in S 3 . Its boundary is identified with S 1  S 1 . The Lagrangian immersion r W Mreg .M / ! M reg .S 1  S 1 / is shown in Figure 6. What kind of information can we extract from the existence of a lift CS W Mreg ! LS 1 S 1 ? If we have a closed loop in M reg , then we showed that it lifts to LS 1 S 1 . In other terms, the holonomy of L along r. / is trivial. This holonomy is easy to compute as e iS=2 where S is the symplectic area enclosed by r. /: hence S has to be an integral multiple of 4 2 . We can check this in the two examples of Figure 6; more generally, this shows for instance that r. / cannot be a small oval.

5 Surfaces of higher genus Till now, we did not say much about surfaces with positive genus although they are very important and interesting. Let † be such a surface of genus g: we showed that M.†/ splits into an irreducible and an abelian part and that the irreducible part is a smooth symplectic manifold of dimension 6g  6, with a prequantum bundle L ! M.†/. To answer simple questions about the topology of that space or its symplectic volume, we introduce a family of functions called trace functions which give to M.†/ the structure of an integrable system.

5.1 Trace functions and flat connection along a curve Let a 1-dimensional connected submanifold of † which does not bound a disc. We will call a curve. We associate to the map h W M.†/ ! Œ0;  by the formula h .Œ/ D ang . /. This gives a well-defined and continuous function on M.†/, smooth where it is different from 0 and . We will compute the Hamiltonian vector field X associated to this map and compute its flow, showing that it is 4-periodic. We will give later on an interpretation of this flow in terms of twisting of flat bundles on † along . Our way of understanding such constructions uses heavily a lemma on the normalization of a flat connection along a curve that we state here without proof. Lemma 5.1. Let † be a compact oriented surface and ˆ W S 1 Œ0; 1 be an orientable embedding. Set D ˆ.S 1  f1=2g/ and let U be a contractible open set in Mreg .†/. Suppose that for all in U , the representation indexed by takes non-central values

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on . Then there is a smooth map A W U ! 1[ .†; su.2// and a smooth map  W U ! G such that (1) the connection A represents ; (2) ˆ A D . /dt where t is the coordinate identifying S 1 to R=Z. We say that a flat connection A on † is normalized along † if there exists  2 G such that ˆ A D dt . In that case, the holonomy of A along is equal to exp./ and one has h .A/ D p1 kk mod  where kk2 D h; i. 2 The aim of this section is to identify the Hamiltonian vector field of h that is, the vector field X on Mreg .†/ such that iX ! D dh . We give a de Rham lift of this vector field assuming that all connections are normalized along . This description allows us to compute its flow and its lift to the prequantum bundle. Proposition 5.2. In the setting of Lemma 5.1, a lift of X at A normalized such that ˆ A D dt is given by the connection ˆ . p '.s/ds/ where s is the coordinate of 2kk Œ0; 1 and ' is a function with support in Œ0; 1 and integral 1. Proof. Let A be a normalized flat connection. One needs to prove the equality !.X ; Y / D dh .Y / for all Y in the tangent space of ŒA. Thanks to Lemma 5.1, we can normalize all connections in the neighborhood of A, hence one can suppose that all tangent connections ˛ are such that ˆ ˛ D dt for some  2 su.2/. R One has !.X ; ˛/ D S 1 Œ0;1 h'.s/ds p ^ dt i D ph;i . On the other hand, dh .˛/ D

h;i p . 2kk

2kk

2kk

This proves the formula.

The same proof gives that the Hamiltonian flow of h during a time T sends the normalized connection A to ˆT .A/ D A C TX . This shows in particular that this flow is periodic as A C 4X D Ag for p Z s 2 2 /: '.u/du g.t; s/ D exp. kk 0 In this formula, g is a smooth function which is equal to 1 outside the image of ˆ. In the case where is separating, one can replace 4 by 2 and the function g will still be well defined, being equal to 1 on one side and to 1 on the other side. This shows that the Hamiltonian flow of h for separating curves is 2-periodic although it is 4-periodic for non separating curves. Let us give a geometric interpretation of these flows. Let .P; F / be a flat SU.2/-bundle over † and be a curve on †. The holonomy of F along is a transformation of the fiber which we suppose to be non-central. y with two circles on the boundary. Cutting † on , we get a new closed surface † y ! † be the gluing map. The flow ˆt .P; F / is obtained as a quotient of Let  W † the form   .P; F /=  t .

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To give a precise formula for  t , we orient and take a point p on  1 . /. The holonomy along in the positive direction sends p to pg. Let C (resp.  ) be y which respects (resp. does not respect) the orientation of . the component of @† C  Let p , p be the preimages of p in the corresponding fibers. Then, byp definition p C exp. p t  /  t p  where  is the unique element of su.2/ with kk < 2 2 such 2kk that g D exp./. We claim that there is a unique isomorphism of flat SU.2/-bundles  t W  1 . C / !  1 .  / which extends the previous formula. We obtain this description easily from the previous one by integrating the connection A in directions transverse to .

5.2 Global description of the moduli space Given a closed surface of genus g > 1, the maximal number of disjoint nonhomotopic to a point and pairwise nonhomotopic curves is 3g  3. Let . i /i2I be such a family. It decomposes the surface into pairs of pants in the sense that the complement of the curves i is a disjoint union of 2g  2 discs with two holes. It is convenient to construct from this decomposition a trivalent graph . The set of vertices denoted by V ./ corresponds to pairs of pants and edges to curves. An edge is incident to a vertex if the corresponding curve bounds the corresponding pair of pants. An example is shown in Figure 8 for g D 2.

Figure 8. Pants decomposition of a genus 2 surface.

Consider the map h W M.†/ ! Œ0; I given by  7! .hi .//i2I . This is an integrable system in the sense that it is a maximal set of Poisson commuting functions. We sum up the properties of this map in the following proposition. Theorem 5.3. The image of h is the polyhedron  consisting of the .˛i /i2I such that for any trivalent vertex v of  the following relation holds: j˛i  ˛j j  ˛k  min.˛i C ˛j ; 2  ˛i  ˛j /

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if i, j , k are incident to v 2 V ./. Set M B .†/ D h1 ..0; /I /. Then, the flows ˆti commute on MB .†/ and cover the fibers of h. Proof. Let v be a vertex of  and  an element of M.†/. Then, restricting  to the pair of pants Pv encoded by v 2 V ./, we obtain an element of M.Pv /. Let i , j , k be the edges incident to v; then ˛i , ˛j , ˛k are the angles of  on the boundary components of Pv . Proposition 1.1 tells us that they need to satisfy the inequalities of the theorem. This explains why the image of h is in . Let us give a short explanation of the remaining part. Consider an element ˛ in . Then, we know that there exist corresponding representations in the pants Pv for all v, unique up to conjugacy. One can realize these representations as flat bundles. For each edge in , we glue the corresponding boundary curves. The holonomies of the flat bundles are conjugate, proving that one can glue them into a flat bundle on †. This proves the surjectivity of h. In the case when all these holonomies are non central, then all possible ways of gluing these bundles are described by the Hamiltonian flow of the corresponding angle function. By construction these flows commute and cover the fibers of h. Let us describe more precisely the fiber of h. Fix an element  of MB .†/ and set ˛ D h./. The joint Hamiltonian flow of the functions .hi / gives an action of RI on the fibre h1 .˛/ by the formula t: D ˆt11 : : : ˆtNN ./ for any numbering of the elements of I . The kernel of this action is precisely described by the lattice 4ƒ  RI where we set ƒ D VectZ fei ; ev W i 2 I; v 2 V ./g: In this formula ei is the basis element with coordinates .ıij /j 2I whereas we set ev D .ei C ej C ek /=2 where i , j , k are the edges incident to v. We already showed that the flows ˆ are 4-periodic, which explains why ei belongs to ƒ for all i . We prove in the same way that 4ev is in the kernel of the action. Let i , j , k be three curves bounding Pv and A be a flat connection representing , normalized in the neighborhood of the three curves. Then, after applying the three flows during a time 2, we obtain a gauge equivalent connection where the gauge element is equal to 1 in the interior of Pv , 1 in the exterior and is given in the standard neighborhoods of the three curves by the same formulas as in Proposition 5.2. This shows that 4ƒ belongs to the kernel of the action; we refer to [JW94] for the proof that these lattices are actually equal.

5.3 Some applications There are plenty of applications of this description as it is very precise for geometric and symplectic aspects. As an example, let us describe the symplectic structure in this setting and give a formula for the symplectic volume of M.†/.

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Proposition 5.4. Let s W Int./ ! MB be a Lagrangian section of h over the interior of the polyhedron . The map ˆ W Int./RI =ƒ ! MB defined by ˆ.˛; t / D t:s.˛/ P 1  is a diffeomorphism on h .Int.// and we have ˆ ! D i d˛i ^ dti . One deduce from this formula that the volume of M.†/ is equal to the volume of the dense open subset h1 .Int.// which is equal to Vol.RI =4ƒ/ Vol./. Moreover, we have Vol.RI =4ƒ/ D Vol.RI =4ZI /=Œƒ; ZI  which is finally equal to .4/3g3 =Œƒ; ZI . To compute the index of ZI in ƒ we notice that it is equal to the dimension of C 1 .; Z2 / divided by coboundaries. We find Œƒ; ZI  D dim H 1 .; Z2 / D 2g . Hence Vol.M.†// D .2/3g3 22g3 Vol./:

6 Introduction to geometric quantization In this section, we introduce some basic objects of geometric quantization. It is a procedure which associates to a symplectic manifold M with extra structure a vector space Q.M / called “quantization of M ”. By construction, some functions on M act on Q.M / with commutation relations prescribed by the Poisson bracket. Of course a good example to keep in mind is T  Rn whose quantization is L2 .Rn / and where position, momentum and Hamiltonian operators are quantization of the coordinates and the energy. In full generality, our construction is naive and not well motivated but produces at least vector spaces and operators. We compute them in the case of moduli spaces and describe the so-called Bohr–Sommerfeld leaves which coincide with the spectrum of curve operators in Chern–Simons topological quantum field theory as initiated by Witten in [Wi89]. A detailed introduction to geometric quantization can be found in [GS77], [BW97]. The computation of Bohr–Sommerfeld fibers was done in [JW92]. In these notes, we obtain it in a more direct way and take into account the metaplectic correction.

6.1 Spin structures Recall that for all n, there is a group Pin.n/ sitting in the following exact sequence 1 ! Z2 ! Pin.n/ ! O.n/ ! 1: This extension is characterized by the two following features: over SO.n/, it is the unique connected 2-fold covering (universal for n > 2) and any lift in Pin.n/ of a reflection in O.n/ has order 2. As GL.n/ retracts on O.n/ there is a unique group f GL.n/ sitting in the exact sequence f 1 ! Z2 ! GL.n/ ! GL.n/ ! 1 and which is isomorphic to Pin.n/ when restricted to O.n/.

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On a vector space V of dimension n, we denote by R.V / the set of bases of V . It z / with a is a homogeneous space over GL.n/. We call spin structure on V a set R.V z f free transitive action of GL.n/ and a map p W R.V / ! R.V / intertwining the actions f of GL.n/ and GL.n/. Spin structures on V form a category Sp.V / where all objects are isomorphic with precisely two isomorphisms. This category is equivalent to the category with one object and automorphism group Z2 . Given two vector spaces V; W , there is a functor F from Sp.V /  Sp.W / to z /; R.W z // to GL.n z /  R.W z /= . The f C m/  R.V Sp.V ˚ W / sending .R.V f equivalence relation is generated by .hg; hsv ; sw /  .g; sv ; sw / for h 2 GL.n/ and f .hg; sv ; hsw /  .g; sv ; sw / for h 2 GL.m/. This functor F is equivalent to the functor trivial on objects and sending Z2  Z2 to Z2 via addition. Let us give two generalizations of this construction. In the first one, we consider an exact sequence 0 ! U ! V ! W ! 0. Choosing a section s W W ! V gives an isomorphism from U ˚ W to V sending .u; w/ to u C s.w/, and hence as before a functor Fs W Sp.U /  Sp.W / ! Sp.V /. This functor depends on s but only up to a unique natural transformation, so this dependence is not relevant for categorical purposes. Our last generalization consists in a comparison between spin structures on a complex and on its cohomology, equivalent to Lemma 2.3. Lemma 6.1. Given a finite dimensional complex C  D C 0 !    ! C n , we define L L Sp.C  / D Sp. i C i /, and Sp.H  / D Sp. i H i .C  //. There is an equivalence of categories Sp.C  / ! Sp.H  /, well defined up to natural transformation. Proof. In the setting of the proof of Lemma 2.3, choose objects in Sp.H i / and Sp.B i /. The exact sequence 0 ! B i1 ! Z i ! H i ! 0 give a functor from Sp.B i1 /  Sp.H i / to Sp.Z i / whereas the exact sequence 0 ! Z i ! C i ! B i !Q0 gives a i i i functor from Sp.Z i /Sp.B L i/ to Sp.C /. One obtains finally an element of i Sp.C / which we send to Sp. i C /. Applying a non-trivial automorphism on the element of Sp.B i / does not change the result as this element appears twice in the result. This shows the lemma since all functors are well defined up to natural transformation. f Definition 6.2. A spin structure on a manifold M of dimension n is a left GL.n/z z principal bundle R.M / on M with a bundle map  W R.M / ! R.M /, the GL.n/f bundle of framings on M which intertwines the actions of GL.n/ and GL.n/. In short, it is a smooth collection of spin structures for all tangent spaces of M . Two f spin structures are isomorphic if there is an isomorphism of GL.n/-bundles commuting with the projections on the framing bundle. We denote by Sp.M / the category of spin structures on a manifold M . The simplest example is the circle, on which there are two isomorphism classes of spin structures that we obtain in the following way.

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(1) Identify all tangent spaces of R=Z with R and take the same spin structure on this “constant” tangent space. (2) Consider S 1  R2 and consider the induced spin structure, using the trivial spin structure in R2 as above and the equivalence Sp.T S 1 /  Sp.N / ' Sp.R2 jS 1 / where N is the (trivial) normal bundle of S 1 in R2 . z 1 / is a trivial covering of R.S 1 / whereas the covering In the first case, the bundle R.S is non-trivial in the second case. Much more generally, a manifold M admits a spin structure if and only if its second Stiefel–Whitney class w2 .M / vanishes and in that case, isomorphism classes of spin structures form an affine space directed by H 1 .M; Z2 /. In the case of moduli spaces of closed surfaces, Lemma 6.1 gives us the following proposition. Proposition 6.3. Let † be a closed surface. There is a well-defined spin structure on M reg .†; SU.2//. More precisely, we define a functor F W Sp.H  .†; R/ ˝ G / ! Sp.Mreg .†; SU.2//. Proof. This is a direct consequence of Lemma 6.1 as we have an equivalence of categories Sp.H  .†; Ad //  Sp.C  .†; Ad //. The second category does not depend on  since  appears only in the differentials. Hence, all these categories are identified together. In the case where  is irreducible, the first category reduces to Sp.H 1 .†; Ad //  Sp.TŒ M.†//. On the contrary, if  is the trivial representation, the first category reduces to Sp.H  .†; G // D Sp.H  .†; R/ ˝ G / hence proving the proposition.

6.2 Lagrangian foliations The main ingredient in the geometric quantization of symplectic manifolds .M; !/ (for real polarizations) is a Lagrangian foliation, that is, an integrable distribution of Lagrangian subspaces of TM . For our purposes,we will suppose that this foliation may have singularities and that its leaves are the regular fibers of a map  W M ! B which is a submersion over a dense open subset of B. Asking that the fibers are Lagrangian is equivalent to asking that all functions on M written as f B  for f W B ! R Poisson commute. Let us give a list of classical examples to keep in mind: (1) Let V be a symplectic vector (or affine) space, then any linear Lagrangian subspace L of V gives such a foliation and the fibration is given by the quotient V ! V =L. (2) Given a manifold M , its cotangent space T  M is a symplectic manifold foliated by the individual cotangent spaces. The fibration we are looking at is the natural projection  W T  M ! M . This model corresponds physically to the canonical quantization of M .

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(3) Let .V; !; q/ be a symplectic plane with a positive quadratic form q. The level sets of q give a foliation of V minus its origin. The map q W V ! RC is our desired fibration with 0 as a singular fiber. This model is referred to as the harmonic oscillator. (4) Let .E; q/ be an oriented 3-dimensional euclidian space. Let ˛ be a positive number and S D q 1 .˛ 2 / be the sphere of radius ˛. Then the sphere S inherits a symplectic structure where symplectic frames at x are by definition couples .v; w/ tangent to S such that the determinant of .v; w; x/ is ˛ 2 . Any linear form  on E restricts to a Lagrangian fibration S ! R with two singularities, its maximum and minimum. The example we are interested in is M.†; SU.2//. Given a pants decomposition of † with cutting curves . i /i2I , we get a map h W M.†/ ! Œ0; I . We showed in the last section that this map takes its values in a polyhedron   Œ0; I and that it is a Lagrangian fibration over the interior of .

6.3 Bohr–Sommerfeld leaves Let .M; !/ be a symplectic manifold. We require that there is a prequantum bundle L over M , a spin structure s 2 Sp.M / and a Lagrangian fibration  W M ! B possibly with singularities. The first condition is a symplectic one: a prequantum bundle exists if and only if !=2 is an integral class in H 2 .M; R/. The second is topological as a spin structure exists if and only if w2 .M / D 0. In both cases, there is no uniqueness unless H 1 .M; Z/ D 0. The geometric quantization process gives a way for constructing a Hilbert space from these data which is finite-dimensional if M is compact and on which functions factorizing with the projection  W M ! B have a natural quantization as operators. Moreover, quantizations coming from different Lagrangian foliations can be compared by using a pairing introduced by Blattner, Kostant and Sternberg. 6.3.1 Half-form bundle and quantization. Let L be a Lagrangian submanifold of a symplectic manifold .M; !/ with spin structure s. In this section we will define a bundle det 1=2 L, the square root of the line bundle of volume elements on L. We start with some preliminaries. P Consider on R2n the symplectic form i dxi ^ dyi where we used standard coordinates .x1 ; : : : ; xn ; y1 ; : : : ; yn /. Any basis .e1 ; : : : ; en / of Rn can be extended to a unique symplectic basis .e1 ; : : : ; en ; f1 ; : : : ; fn / of R2n adapted to the decomposition R2n D Rn ˚ Rn . This gives a map from GL.n/ to GL.2n/. Pulling back the f c bundle GL.2n/ ! GL.2n/, we get a new group GL.n/ with a projection to GL.n/. This group is called the metalinear group and has the following down to earth description: it is isomorphic as a group to the direct product GLC .n/  Z4 with the projection to GL.n/ given by the map .A; x/ ! .1/x A. There is an important mor-

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p c phism det1=2 W GL.n/ ! C defined by det 1=2 .A; x/ D det.A/i x . Its square is the pull-back of the usual map det W GL.n/ ! R. z / and suppose it Let V be any symplectic vector space V with a spin structure R.V  is symplectomorphic to L ˚ L (such a symplectomorphism is equally described by a pair of transverse Lagrangians L and L0 , where L0 is the image of L ). We can define a complex line det 1=2 .L/ in the following way. Pulling-back the spin structure of V y to L as above, we get a metalinear structure on L, i.e. a set R.L/ homogeneous under 1=2 y c GL.n/. We define det .L/ D R.L/  C=  where .hs; z/  .s; det1=2 .h/z/ for c any h in GL.n/. As expected we have a natural isomorphism det 1=2 .L/˝2 ' det.L/. We are ready to define the half-form bundle of a Lagrangian submanifold in a spin symplectic (called metaplectic) manifold .M; !; s/. Let L be such a submanifold, and L0 be a Lagrangian subbundle of TM jL transverse to T L. Topologically, there is neither obstruction nor choice to find such a subbundle. Applying the preceding construction to all tangent spaces T Ll for l 2 L gives the line bundle det 1=2 .L/ that we are looking for. We will often denote it by ı for short. In the case where L is a regular fibre of a Lagrangian fibration  W M ! B, there is an isomorphism between the cotangent space T  L at any point l 2 L and the tangent space TB at .l/. In other words, the tangent and cotangent spaces of Lagrangian fibres are naturally trivialized. This isomorphism comes from the identification of T  L with the normal bundle NL via ! and the derivative of . Thanks to this isomorphism, one can give a flat structure to T L and to all its associated bundles like det.L/ or det1=2 .L/. This explains why the half-form bundle ı is indeed a flat bundle on L. The underlying connection on it is often called the Bott connection. Definition 6.4. Let .M; !; L; s/ be an enriched symplectic manifold and let  W M ! B be a Lagrangian fibration. Then we say that a fibre L of  is a Bohr–Sommerfeld fibre if there are non-trivial covariant flat sections of L ˝ ı over L. The space of all these sections is denoted by H .L; L ˝ ı/. If L is connected, then H D H .L; L ˝ ı/ is 1-dimensional. If moreover L is compact, then there is a Hermitian structure on the dual H  D H .L; L ˝ ı  / 1=2  inducing one on H . Indeed, if  is a section .L/ then N is a R of ı2 D det 2 N non-negative density on L. We set js ˝ j D L jsj . Let BS be the subset of B parametrizing Bohr–Sommerfeld fibres. Then to each b in BS we have a complex line H . 1 .b/; L ˝ ı/ (provided that  1 .b/ is connected). We denote by H .M; L ˝ ı/ the space of sections of this line bundle over BS – roughly speaking, this is the space of sections of L ˝ ı over M covariantly constant along the fibres of . This rather ad hoc definition is motivated by the Rfact that H .M; L ˝ ı/ has a natural inner product defined by hs1 ˝ 1 ; s2 ˝ 2 i D BS hs1 ; s2 i1 S2 . In this formula, BS is the subset of B describing Bohr–Sommerfeld fibres supposed to be a codimension 0 submanifold of B whereas 1 S2 is a density on BS and then is ready to be integrated. There is a family of possibilities from compact fibres and discrete

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Bohr–Sommerfeld set to non-compact fibres and codimension 0 Bohr–Sommerfeld set. This is best understood with examples. 6.3.2 Standard examples (1) If .V; !/ is a symplectic vector space, we construct a prequantum bundle L by taking the trivial bundle V  C with connection d   where  is a 1-form on V 1 !. Let L be a linear Lagrangian in V and take a spin structure such that d D 2

z R.V / on V . For auxiliary purposes, we also choose a Lagrangian L0 transverse to L. We explained in the last section that these data produce a complex line det 1=2 .L/ which is isomorphic to det 1=2 .L0 /. Moreover, sections of L constant along L are determined by their restriction to L0 : the quantization procedure reduces then to sections of det 1=2 .L0 / as all fibres are Bohr–Sommerfeld. This space is called the intrinsic Hilbert space of L0 as for any section  of it, the quantity N can be integrated and the space of integrable sections is a Hilbert space. Certainly, different choices of L0 give isomorphic spaces although the isomorphism has to be computed using parallel transport along L. Comparing the Hilbert spaces associated to different Lagrangian subspaces, L amounts in constructing the so-called metaplectic representation. (2) If M is any manifold, we can construct a prequantum bundle L on T  M by 1 taking the product T  M  C with connection d  2

 where  is the Liouville  1-form. A spin structure on T M induces a metalinear structure on M and finally a square root ı of the line bundle det.T  M /. The quantization associated to it is the set of sections of ı over M , that is, the intrinsic Hilbert space of M (again, all fibres are Bohr–Sommerfeld). (3) In the case of the harmonic oscillator .V; !; q/, we consider any prequantum bundle L over V (constructed as before) and a fixed spin structure on V . The fibres are the circles q 1 .˛/ for ˛ > 0. The holonomy of L along this leaf is e iA where A is the symplectic area enclosed by q 1 .˛/. The half-form bundle ı restricts to each leaf to a flat non-trivial line bundle. Its holonomy is then 1. Finally, Bohr–Sommerfeld fibres correspond to the values ˛ such that e iA D 1 where A D c˛ and c is the volume of the disc q 1 .Œ0; 1/. This forces ˛ to be of the form k=c where k is an odd integer, corresponding to the spectrum of the harmonic oscillator. (4) Finally, the case of the sphere is the most interesting one for us as it is the only compact example. Recall that we saw the sphere as the level set q 1 .˛ 2 / and our normalization is such that its symplectic volume is 2 ˛. Then, we are able to find a prequantum bundle L if and only if ˛ is an integer. Let  W V ! R be a (unit) linear form and choose a spin structure s on S (unique up to homotopy). Regular fibres of  are circles on which ı is as before a non trivial flat bundle. The holonomy of L along a fiber 1 .l/ is e iA where A is the symplectic area enclosed by the circle. The quantity A=2 is an integer if and only if l is an

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integer satisfying jlj  ˛ and l D ˛ mod 2. Hence, Bohr–Sommerfeld orbits correspond to integers l satisfying jlj  ˛ and l ¤ ˛ mod 2. 6.3.3 The case of moduli spaces. Consider the setting of Section 5. As before † denotes a closed surface of genus g and the aim of this section is to investigate the quantization of Mreg .†; SU.2// with its symplectic form ! and Chern–Simons bundle L. For more generality, we will consider an integer K called the level and multiply the symplectic form by K. This new symplectic form admits L˝K as a prequantum bundle. The spin structure is the one defined in Section 6.1 and the Lagrangian fibration comes from the map h W M.†/ ! Œ0; I sending Œ to .hi .//i2I where I indexes a maximal system of cutting curves . i /i2I . We need now to determine which fibers are Bohr–Sommerfeld, and to do this we must compute the holonomy of L˝K and ı along the fibres which are tori of the form RI =ƒ. Hence, it is sufficient to compute the holonomy of both bundles along the generators of ƒ. We do this in the following next two propositions. Proposition 6.5. Fix i 2 I and .˛i / in Int./. Pick  2 M B .†/ such that h./ D .˛i /. Let ei and ev be the generators of ƒ described in Section 5.2. – The holonomy of L˝K along the path ei is e 2iK˛i . – The holonomy along the path ev for edges i , j , k incident to the same vertex v is e iK.˛i C˛j C˛k / . Proof. Orient all curves i and choose a cylinder ‰i W S 1  Œ0; 1 ! † around them. Let A be a flat connection in 1[ .†; su.2// representing  and suppose that it is normalized along each cylinder: in formulas there are vectors i in su.2/ such that ‰i A D i dt . We showed in Proposition 5.2 that the Hamiltonian flow of hi changes A to AT D ˆTi A such that ‰i AT D dt  pT  '.s/ds. Suppose that at time T 2jj

the element of L is represented by .AT ; T / 2 1[ .†; su.2//  R=2Z with 0 D 0. d Then as in Section 4.2, this path in L is parallel if and only if . dT .AT ; T // D T0  R jj 1 hAT ; A0T i D 0. This equation reduces to T0 D  p which we solve putting 4 † 4 2

p

T D  Tpjj . A computation shows that .A4 ; 4 / is equivalent to .A;  2jj/. 4 2

p This proves the first result as jj D 2hi . To compute the second term, we only need to replace 4 with 2 and take into account thepcontributions of the three curves i ; j and k . The pair .A; 0/ is transported to .A;  22 .je j C jf j C jg j// which gives the second result of the proposition. It remains to compute the holonomy of the half-form bundle ı along the fibre which we do in the following proposition. Proposition 6.6. Let L be the distribution of Lagrangian subspaces in T M B .†/ given by L D ker D h. Denote by ı the bundle det 1=2 .L/ as before. Then the holonomy of ı is 1 along ei and 1 along ev .

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Proof. We prove this proposition by considering half-periods: suppose that two representations 1 ; 2 W 1 .†/ ! SU.2/ are the same when composed with the projection SU.2/ !SO.3/. Then, because Ad1 D Ad2 , we have TŒ1  M.†/ D TŒ2  M.†/. Moreover, if we start the flow ˆ at a representation  and follow it during a time 2, we reach the representation #  where # is the Poincaré dual of in H 1 .†; Z2 / D Hom.1 .†/; f˙1g/. This fact is a direct consequence of the expression of the flow in Section 5.2. Finally, the map h satisfies h. # / D ˙h./. This shows that the Lagrangian subbundles at  and #  are isomorphic so that we are allowed to compute the holonomy of ı along that path. If we show that this holonomy is 1 we prove the proposition as ei is a composition of 2 such paths and ev is a composition of 3 of them. As Int./ is convex, all fibers of h are isotopic and the holonomy of ı does not depend on which fiber we consider. Let us do the computation in the following case: let Ai , Bi be the standard generators of 1 .†/ satisfying the following relation with n even: 1 1 1 A1 B1 A1 1 B1 : : : An Bn An Bn D 1:

Then we define an element  of M B .†/ by setting .Ai / D i, .Bi / D j. Suppose that is the curve represented by A1 and set Œ t  D ˆt Œ. Then we set  t .B1 / D je i t =2 and the other values are unchanged. The cell decomposition of † consists in one 0-cell, 2n 1-cells and one 2-cell. We identify C0 .†; Ad t / and C2 .†; Ad t / with su.2/ and C1 .†; Ad t / with su.2/2n . The differentials dhi belong to H1 .†; Ad / and generate L , the dual of the Lagrangian subspace we are interested in. These elements are represented as twisted cycles by

i ˝ fi were fi is a non-zero element of su.2/ fixed by Ad.i / . This shows that all of them can be represented in C1 by vectors which do not depend on t . Let us denote them by Fi for i 2 I . In order to understand the metalinear structure on L, we need to extend this basis to a symplectic basis of H 1 .†/. We obtain in that way vectors Gi for i in I. We can choose them arbitrarily for t D 0 and modify them in the vicinity of

for t > 0. By this procedure, we can suppose that only the first two projections of the vectors Gi depend on t . Considering a fixed basis of C2 , we push it to C1 with the injective map @2 and get three vectors U1 , U2 , U3 . At the same time, we consider three vectors V1 , V2 , V3 in C1 whose image by @1 is a fixed basis of C0 . By construction we get a basis .V1 ; V2 ; V3 ; U1 ; U2 ; U3 ; Fi ; Gi / of C1 . This basis depends on t and defines for t 2 Œ0; 2 a closed path in GL.C1 /. The holonomy we are looking for is the homotopy class of this path. To compute this path, we use the decomposition of C1 into blocks corresponding to A1 , B1 on the  hand, and the other generators onthe other hand. First, we have P one @1 .i ; i / D i Ad t .Ai / i  i C Ad t .Bi / i  i . In particular, @1 .1 ; 1 / D i1 i1 C je i t =2 1 .je i t=2 /1 so that @1 .jz=2; ix=2/ D ix C jz for x 2 R and z 2 C so that we can set Vz1 D .0; i=2/; Vz2 D .j=2; 0/; Vz3 D .k=2; 0/.

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On the other hand, we compute @2 ./ D .ii1  ke i t=2 e it=2 k1 ; ke i t =2 e it =2 k 1  je i t =2 e it =2 j1 ; : : :/ where the dots mean the remaining components which do not depend on t . We obtain U1 D @2 .i/ D .2i; 0; : : :/, U2 D @2 .j/ D .j.e i t C 1/; 2je i t ; : : :/, U3 D @2 .k/ D .k.1  e i t /; 2ke i t ; : : :/. Writing the matrix of this basis we find a matrix M.t / with the property that its derivative is block triangular with on the diagonal a 2  2 rotation matrix R t of angle t followed by a zero matrix. This proves that in GL.C1 /, this path is not topologically trivial as the inclusion of GL2  GLn at the level of fundamental groups sends R t to the generator. This ends the proof in our case. In the case when separates † or when the genus is an odd integer, the result follows by considering an explicit example in each case and performing similar computations. Putting these results together, we find that the Bohr–Sommerfeld orbits are parai

/i2I where i are integers belonging to Œ1; K 1 which satisfy metrized by tuples . K the following conditions for all triples i, j , k of adjacent edges: (1) i  j C k , (2) i C j C k is odd, (3) i C j C k  2K. These conditions are referred to as quantum Clebsch–Gordan conditions and describe a basis of the quantization of M.†/ as constructed for instance in [BHMV].

6.4 Going further At this point, we defined the geometric quantization of the moduli space M.†/ as a finite dimensional Hilbert space depending on a Lagrangian fibration, itself depending on a pants decomposition of †. The theory can be developed in the following directions: (1) Some easy developments: count the number of Bohr–Sommerfeld fibers and compare to the Verlinde formula (the dimension of conformal blocks, see [BK01] or [BHMV]). Explain how the functions hi are quantized and act naturally on the quantization. This also provides a quantization of the Dehn twists acting on M.†/ with an explicit spectrum. (2) Given another pants decomposition, we have a new construction of the quantization which can be compared to the previous one. Suppose that the fibrations are transverse to each other. Then the half-forms sections can be intersected and give a pairing between the two quantizations called Blattner–Kostant–Sternberg pairing (BKS pairing, see [GS77], [BW97]). It is not clear whether this pairing gives a unitary isomorphism between the two quantizations. If so, it would give a satisfactory description of the quantization of M.†/ where all trace functions

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would be quantizable, and the action of the mapping class group of † would extend to the quantization. (3) A 3-manifold M bounding † produces a Lagrangian immersion M.M / ! M.†/ with a flat section CS of the bundle L and a volume form T on M.M /. If we manage to find a well-defined square root of this form, we obtain a semiclassical state associated to M . Using the BKS-pairing, this state may be viewed as a vector belonging to any quantization, see [JW94]. (4) All these data should fit in a Topological Quantum Field Theory (TQFT); that is, they may have functorial properties with respect to the gluing of 3-manifolds along their boundaries. Moreover, we expect that this TQFT appears as the semiclassical approximation of a family of TQFTs indexed by the integer K called level. These TQFTs may be constructed either by geometric quantization (with complex polarization, see [Wi89], [At90]) or with link polynomials and quantum groups (see [RT91], [BHMV]).

References [AB83]

M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces. Philos. Trans. Roy. Soc. London Ser. A 308 (1999), 291–326.

[At90]

M. F. Atiyah, The geometry and physics of knots. Cambridge University Press, Cambridge 1990.

[BK01]

B. Bakalov and A. Kirillov, Jr., Lectures on tensor categories and modular functors. Univ. Lecture Ser. 21, Amer. Math. Soc., Providence, RI, 2001.

[BW97]

S. Bates and A. Weinstein, Lectures on the geometry of quantization. Berkeley Math. Lect. Notes 8, Amer. Math. Soc., Providence, RI, 1997.

[BHMV]

C. Blanchet, N. Habegger, G. Masbaum and P. Vogel, Topological quantum field theories derived from the Kauffman bracket. Topology 34 (1995), 883–927.

[BZ95]

G. Burde and H. Zieschang, Knots. De Gruyter Stud. Math. 5, Walter de Gruyter, Berlin 1985.

[CS83]

M. Culler and P. B. Shalen, Varieties of group representations and splitting of 3manifolds. Ann. of Math. 117 (1983), 109–146.

[Fr95]

D. S. Freed, Classical Chern–Simons theory, 1. Adv. Math. 113 (1995), 237–303.

[Gol84]

W. Goldman, The symplectic nature of the fundamental groups of surfaces. Adv. Math. 54 (1984), 200-225.

[Gol86]

W. Goldman, Invariant functions on Lie groups and Hamiltonian flows on surface group representations. Invent. Math. 85 (1986), 263–302.

[GS77]

V. Guillemin and S. Sternberg, Geometric asymptotics. Math. Surveys 14, Amer. Math. Soc., Providence, RI, 1977.

[Ha02]

A. Hatcher, Algebraic topology. Cambridge University Press, Cambridge 2002.

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[JW92]

L. C. Jeffrey and J. Weitsman, Bohr–Sommerfeld orbits in the moduli space of flat connections and the Verlinde dimension formula. Comm. Math. Phys. 150 (1992), no. 3, 593–630.

[JW93]

L. C. Jeffrey and J. Weitsman, Half density quantization of the moduli space of flat connections and Witten’s semiclassical manifold invariants. Topology 32 (1993), no. 3, 509–529.

[JW94]

L. C. Jeffrey and J. Weitsman, Toric structures on the moduli space of flat connections on a Riemann surface: volumes and the moment map. Adv. Math. 106 (1994), no. 2, 151–168.

[Kl91]

E. P. Klassen, Representations of knot groups in SU.2/. Trans. Amer. Math. Soc. 326 (1991), no. 2, 795–828.

[La07]

F. Labourie, Representations of surface groups. Course at ETH-Zürich, Fall 2007.

[Le91]

T. T. Q. Le, Varieties of representations and their subvarieties of homology jumps for certain knot groups. Russian Math. Surveys 46 (1991), no. 2, 250–251. p J. Marché, The Kauffman bracket at 1. Preprint.

[Ma07]

[RSW89] T. R. Ramadas, I. M. Singer and J. Weitsman, Some comments on Chern–Simons gauge theory. Comm. Math. Phys. 126 (1989), no. 2, 409–420. [RT91]

N. Reshetikhin and V. Turaev, Invariant of 3-manifolds via link polynomials and quantum groups. Invent. Math. 103 (1991), 547–597.

[Sch56]

H. Schubert, Knoten mit zwei Brücken. Math. Z. 65 (1956), 133–170.

[Tu02]

V. Turaev, Torsions of 3-dimensional manifolds. Progr. Math. 208, Birkhäuser, Basel 2002

[Wi89]

E. Witten, Quantum field theory and the Jones polynomial. Comm. Math. Phys. 121 (1989), no. 3, 351–399.

Algorithmic construction and recognition of hyperbolic 3-manifolds, links, and graphs Carlo Petronio Dipartimento di Matematica Applicata, Università di Pisa Via Filippo Buonarroti, 1C, 56127 Pisa, Italy email: [email protected]

Contents 1 3-dimensional “objects” . . . . . 2 Hyperbolic structures . . . . . . . 3 Cusped manifolds . . . . . . . . 4 Complexity and closed manifolds 5 Geodesic boundary and graphs . . References . . . . . . . . . . . . . . .

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This survey article describes the algorithmic approaches successfully used over the time to construct hyperbolic structures on 3-dimensional topological “objects” of various types, and to classify several classes of such objects using such structures. Essentially, it reproduces the contents of a course given by the author at the “Master Class on Geometry” held in Strasbourg from April 27 to May 2, 2009. The author warmly thanks the organizers Norbert A’Campo, Frank Herrlich and (particularly) Athanase Papadopoulos for having set up this excellent activity and for having invited him to contribute to it.

1 3-dimensional “objects” The main objects of interest in 3-dimensional topology are 3-manifolds, namely topological spaces obtained by patching together portions of Euclidean 3-space. Depending on whether the patching is performed along continuous, differentiable, or piecewise-linear maps, one gets the three different categories of manifolds named TOP, DIFF, and PL, respectively. In higher dimension these categories can differ from each other in an essential way (for instance, one TOP manifold can have nondiffeomorphic DIFF structures), but in dimension 3 it has been known for a long time

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(see for instance the foundational work of Kirby and Siebenmann [30]) that the three categories are equivalent to each other. For this reason in the sequel we will use the DIFF and the PL approaches interchangeably, the former being more suited to the discussion of geometric structures, the latter to a combinatorial treatment. In addition we will always view manifolds up to the natural equivalence relation in the category in use, namely we will view two diffeomorphic or PL-equivalent manifolds as being just one and the same object. We address the reader to the by now classical introductions to the topic of 3-manifolds due to Hempel and to Jaco [28], [29]. The most general setting of an algorithmic classification of manifolds (or of other topological objects, as discussed below) consists of the following ingredients: • a combinatorial presentation of the objects under consideration, namely a way to associate a topological object to some finite set of data, so that, given a bound on the “complexity,” all the relevant sets of data can be recursively enumerated by a computer; • a set of moves on the combinatorial data, by repeated applications of which one is sure to relate to each other any two sets of data representing the same topological object; • certain invariants of the topological objects, using which one can (sometimes) prove an object is different from another one, and perhaps also show that two objects are the same (when the invariant is a complete one). In the rest of this section we will describe some combinatorial presentations of 3-manifolds and of other related 3-dimensional topological objects introduced below, together with the corresponding moves. In the next section we will illustrate the powerful invariants coming from the machinery of hyperbolic geometry, and in the subsequent sections we will discuss how the combinatorial approach and the use of the hyperbolic invariants can be (and has been) used to produce extremely satisfactory classification results. (Loose) triangulations of manifolds, and spines In the sequel all our manifolds will be 3-dimensional, connected, orientable, and compact (with or without boundary). Starting from the case of a closed manifold M , namely one with empty boundary, we will call (loose) triangulation of M a realization of M as the quotient of a disjoint union of standard tetrahedra under the action of a simplicial orientation-reversing pairing of the (codimension-1) faces. Note that a triangulation is not strictly a PL structure on M according to the original definition [52], because in M the tetrahedra can be self-incident and multiply incident to each other. However a loose triangulation in our sense can be transformed into a PL structure by subdivision. The next result (due to Matveev and to Piergallini, see [17], [39], [50] and the references quoted therein) describes the combinatorial approach to closed 3-manifolds using triangulations: Theorem 1.1. Let M be a closed orientable 3-manifold. Then the following holds. • Given v  1 one can find triangulations of M with v vertices.

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• Given v  1 and two triangulations of M with v vertices, both consisting of at least two tetrahedra, one can transform them into each other by repeated applications of the 2-to-3 move shown in Figure 1 (top), and its inverse. • One can transform any two triangulations of M into each other by repeated applications of the 2-to-3 and the 1-to-4 moves shown in Figure 1, and their inverses.

2-to-3

1-to-4

Figure 1. The 2-to-3 and the 1-to-4 moves on triangulations.

Remark 1.2. Enumerating by computer the triangulations of closed orientable manifolds is in principle easy, even if computationally demanding. For increasing n  1 one lists all the possible orientation-reversing pairings between the faces of n tetrahedra yielding a connected result, and one checks that in the quotient space the link of every vertex is the 2-sphere S 2 (to do which one only has to show that it has Euler characteristic 2). Here is a useful alternative viewpoint on triangulations. Let M have a triangulation, and consider the 2-skeleton of the cell subdivision dual to the triangulation, as suggested in Figure 2 (left).

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Figure 2. Duality between triangulations and special spines.

This gives a spine of M minus the vertices of the triangulation, namely a complex onto which this space collapses. This complex is actually a special polyhedron, namely one satisfying the following conditions: • it consists of non-singular surface points as in Figure 3 (left), of singular points giving triple lines as in Figure 3 (center), and of at least one singular vertex as in Figure 3 (right); • the connected components of the set of non-singular points are open discs.

Figure 3. Special polyhedra.

(In the sequel we will always call special spine of M a spine that is also a special polyhedron.) The construction can be reversed: using a technical notion of orientability for a special polyhedron (see for instance [6]) one uses Figure 2 (right) to associate to an orientable special polyhedron a set of tetrahedra and a pairing between their faces. As illustrated below, this does not always give a triangulation of a closed manifold, but one can check whether it does along the lines of Remark 1.2. The spine versions of the moves on triangulations are shown in Figure 4.

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2-to-3

1-to-4 Figure 4. The 2-to-3 and the 1-to-4 moves on special spines.

Ideal triangulations Turning to the case of a compact manifold M with non-empty boundary @M , one can adapt to M the notion of (loose) triangulation by calling ideal triangulation any of the following pairwise equivalent notions: • A realization of M minus its boundary as the space obtained by first gluing a finite number of disjoint tetrahedra along simplicial maps, and then removing the vertices. • A realization of the space X obtained from M by collapsing each component of @M to a point as the quotient of a disjoint union of tetrahedra under a simplicial pairing of the faces, in such a way that the quotient vertices correspond to the collapsed components of @M . • A realization of M as a gluing of truncated tetrahedra as in Figure 5, with gluings between the lateral hexagons induced by simplicial gluings of the non-truncated tetrahedra. For the next result we refer again to [17]: Theorem 1.3. Any compact orientable 3-manifold M with non-empty boundary admits ideal triangulations, and any two of them consisting of at least two tetrahedra can be transformed into each other by repeated applications of the 2-to-3 move shown in Figure 1 (top) and its inverse. Remark 1.4. It is actually quite easy to deduce Theorem 1.1 from Theorem 1.3. One only needs to remark that removing some number v of open 3-balls from a connected and closed M is a well-defined operation, from the result of which M can be reconstructed unambiguously by capping off the boundary spheres. Moreover the

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Figure 5. A truncated tetrahedron.

1-to-4 move of Figure 1 (bottom) is one that allows to increase by 1 the number of vertices of an ideal triangulation, and hence to increase by 1 the number of punctures in a punctured closed manifold represented by the triangulation. The dual viewpoint of special spines carries over to the case of manifolds with boundary, and the corresponding statement is actually even more expressive: Theorem 1.5. Each orientable compact 3-manifold with non-empty boundary admits special spines. Each orientable special polyhedron is the spine of a unique 3-manifold with nonempty boundary. Two special spines of the same 3-manifold with non-empty boundary, both having at least two vertices, are related to each other by repeated applications of the 2-to-3 move of Figure 4 (top) and its inverse. Remark 1.6. We have repeatedly excluded from our statements the triangulations consisting of one tetrahedron only (and, dually, the spines having one vertex only). This is not a serious issue, because only a small number of uninteresting manifolds are described by these triangulations or spines. Knots, links and graphs Besides manifolds, knots are the next main objects of interest in 3-dimensional topology. According to the basic definition, a knot is a tamely embedded circle in 3-space, but one can easily extend the situation by considering links, defined as disjoint unions of knots, and let the ambient manifold in which a link is embedded be an arbitrary closed one. This leads to considering pairs .M; L/, with closed M and L  M a link, that we will always view up to equivalence of pairs (in the appropriate category) without further mention. We then define a triangulation of a link-pair .M; L/ as a (loose) triangulation of M that contains L as a subset of its 1-skeleton. The next result was implicit in the work of Turaev and Viro [56] and was formally established by Amendola [2] (see also Pervova and the author [47] for more on spines of link-pairs):

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Theorem 1.7. Every link-pair .M; L/ with non-empty L admits triangulations with precisely one vertex on each component of L, and no other vertex. Any two such triangulations of .M; L/ consisting of at least two tetrahedra can be transformed into each other by repeated applications of the 2-to-3 move shown in Figure 1 (top), and of the inverse of this move applied when the edge that disappears with the move does not belong to L. A further category of objects that one deals with is given by the pairs .M; G/ where M is a closed 3-manifold and G  M is a graph, that is a 1-subcomplex of M . A triangulation of .M; G/ is one of M that contains G as a subcomplex of its 1-skeleton. The previous result holds also for these objects, with the requirement that the triangulation should have one vertex at each vertex of G and one on each knot component of G. Orbifolds We finally introduce orbifolds, defined as spaces having a singular differentiable structure locally defined as the quotient of Euclidean space under the action of a finite group of orientation-preserving diffeomorphisms. Since a finite orientable differentiable action is conjugate to a special orthogonal one, one sees that the local group acting can be assumed to be either cyclic, or dihedral, or the automorphism group of one of the Platonic solids. This implies that the support of a (closed, orientable, locally orientable) 3-orbifold is a closed orientable 3-manifold, in which the singular locus is a trivalent graph with edges labelled by integers and local aspect as in Figure 6.

n

n

2

3 2

2

4 3

2

5 3

2

3

Figure 6. Local aspect of a closed locally orientable 3-orbifold.

2 Hyperbolic structures In this section we review the definition of hyperbolic n-space, we summarize its main features, and we define the hyperbolic structures we will be interested in constructing on each of the types of topological 3-dimensional objects illustrated in the previous section. Hyperbolic n-space The n-dimensional hyperbolic space Hn can be defined as the only complete and simply connected Riemannian n-manifold having all sectional

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curvatures equal to 1, see [15]. For our purposes it will however be helpful to have at hand the following concrete models of this space: • The disc model, defined as the open unit disc B n D fx 2 Rn W kxk < 1g endowed with the metric dsx2 D

dx 2 4 .1  kxk2 /2

:

• The half-space model, defined as the upper half-space n D fx 2 Rn W xn > 0g C

endowed with the metric dsx2 D

dx 2 : xn2

• The hyperboloid model, defined as the hyperboloid ˚  n HC D x 2 R1;n W hxjxi.1;n/ D 1; x0 > 0 ; where R1;n is the Minkowski space RnC1 endowed with the metric hxjyi.1;n/ D x0 y0 C x1 y1 C    C xn yn I n the Riemannian metric on HC is given by the restriction of the metric hji.1;n/ n to the hyperplanes tangent to HC , on which hji.1;n/ is positive-definite.

The different models allow to single out some of the features of Hn that we will need below (see [54], [5], [51]): • As one sees very well from the disc model, Hn has a natural compactification obtained by adding the points at infinity, that constitute an .n  1/-dimensional sphere @Hn . • The geodesics of Hn ending at the point 1 in the half-space model nC are the vertical half-lines. • A horosphere, defined as a connected complete hypersurface orthogonal to all the geodesics ending at a given point of @Hn , called its center, if centered at 1 in n the C model is given by a horizontal hyperplane, so it is endowed with a natural Euclidean structure; moreover the horosphere together with its center bound a topological disc in the compactified hyperbolic space, called a horoball. • An isometry  of Hn must have fixed points either in Hn or in @Hn , and hence it must be of one of the following types: – elliptic, namely with fixed points in Hn ; in this case, assuming 0 is fixed in the disc model,  can be identified to an orthogonal matrix;

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– parabolic, namely with no fixed points in Hn and exactly one on @Hn ; in this case, assuming 1 is fixed in the half-space model,  can be identified n to an affine isometry of Euclidean space Rn1 acting horizontally on C ; in particular, if n D 3 and  preserves the orientation, it is just a horizontal translation; – hyperbolic, namely with no fixed points in Hn and exactly two on @Hn ; in n this case, assuming 0 and 1 are fixed in C , it has the form   A 0 .x/ D   x 0 1 with A 2 O.n  1/ and  > 1. Closed and cusped hyperbolic manifolds Let us temporarily drop our assumption that all manifolds should be compact, and take a possibly open n-dimensional manifold N . A hyperbolic structure on N can be defined in one of the following equivalent ways: • A complete Riemannian metric on N with all sectional curvatures equal to 1. • A complete Riemannian metric on N making it locally isometric to Hn . • An identification between N and the quotient of Hn under the action of a discrete and torsion-free group of isometries. • A faithful representation of 1 .N / into the group of the isometries of Hn having discrete and torsion-free image. To state the first main general result we need to introduce further notation. Given a Riemannian manifold N and " > 0, we define the "-thick part NŒ";C1/ of N as the set of x 2 N such that every loop based at x and having length at most " is null in 1 .N; x/, and the "-thin part N.0;" of N as the closure of the complement of its "-thick part. The following holds: Theorem 2.1. If a hyperbolic n-dimensional manifold N is non-compact but has finite volume then there exists " > 0 such that its "-thick part NŒ";C1/ is compact, and its "-thin part N.0;" is a disjoint union of components of the form †  Œ0; 1/, with † a closed Euclidean .n  1/-manifold. (This is a consequence of the more general Margulis lemma, according to which there exists a universal constant "n depending only on n such that for every ndimensional hyperbolic N , if "  "n then the "-thin part N.0;" has finitely many components all having an “easy” fundamental group.) Since the only closed orientable surface carrying a Euclidean structure is the torus T , this result implies that an orientable 3-dimensional finite-volume hyperbolic N is the union of a compact manifold M bounded by tori and a finite number of cusps based on tori, as suggested in Figure 7. Moreover N can be identified to the interior

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Figure 7. An allusive picture of a cusped hyperbolic 3-manifold.

of M . For this reason, with a slight abuse of terminology, we will say that M itself is hyperbolic, always meaning that the hyperbolic structure is actually defined on the interior of M , and that the toric boundary components of M give rise to cusps. The next general result is the following one: Theorem 2.2 (Mostow rigidity). If n  3, two finite-volume hyperbolic n-manifolds having isomorphic fundamental groups are isometric to each other. In particular, every n-manifold carries at most one finite-volume hyperbolic metric up to isometry. This deep theorem has the important consequence that any geometric invariant of a hyperbolic manifold, such as the volume or the length of the shortest geodesic for a closed manifold, is automatically a topological invariant. To state the next result, we need to recall that performing a Dehn filling of a torus boundary component T of a compact 3-manifold M consists in gluing to M the solid torus D 2  S 1 along a homeomorphism f W @.D 2  S 1 / ! T . The result of this operation depends only on the slope on T that becomes contractible in the attached D 2 S 1 , namely on the isotopy class on T of the simple non-trivial curve f .S 1  fg/. If M has several boundary components we will call Dehn filling of M any manifold obtained by performing this operation on some (possibly all) of the toric components of @M . The next general result shows that in dimension three, given a cusped hyperbolic manifold, one can produce a wealth of new ones: Theorem 2.3 (Thurston’s hyperbolic Dehn filling). Let M be a finite-volume hyperbolic 3-manifold with cusps based on tori T1 ; : : : ; Tk . Then for j D 1; : : : ; k there exists a finite set Ej of slopes on Tj such that every Dehn filling of M performed along slopes ˛1 ; : : : ; ˛k with ˛j 62 Ej is hyperbolic. Note that the theorem includes the case of the “empty” filling of some cusp (or several ones), that leaves the cusp as it is. We also remark in passing that one can define a natural topology on the space of hyperbolic manifolds and that taking a sequence of fillings of M in which on each cusp the length of the slope (defined for instance as the norm of its coordinates with respect to some fixed homological basis) tends to

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infinity, one gets a sequence of hyperbolic manifolds converging to M , with volumes converging from below to that of M . Hyperbolic manifolds with geodesic boundary When a compact 3-manifold M has boundary components which are not tori, one has no hope to construct on it or on its interior a finite-volume hyperbolic structure (an infinite-volume non-rigid one often exists on the interior, but this is a completely different story). In this case one allows the boundary of M to be part of the hyperbolic structure, in the form of a totally geodesic surface. To explain the matter in detail, we again temporarily remove the restriction that manifolds should be compact, and consider an arbitrary one N , possibly non-compact and with boundary, with the boundary itself possibly non-compact. We then say that N is hyperbolic with geodesic boundary if it has a complete finite-volume Riemannian structure locally modeled on open subsets of a half-space in hyperbolic space H3 . Mirroring N in its boundary we get the double D.N / of N , which is hyperbolic without boundary, so its universal cover can be identified to H3 . Moreover @N is a totally geodesic surface in D.N /, and the universal cover of N can be identified to the closure of any connected component in H3 of the complement of the family of disjoint planes in H3 that project in D.N / onto @N . This allows the following alternative description of a hyperbolic structure with geodesic boundary: • A hyperbolic structure with geodesic boundary on N corresponds to a realization of N as the quotient of the intersection H of a family of half-spaces in H3 under the action of a discrete and torsion-free group of isometries of H3 that leave H invariant. Let us now describe the thin part of a finite-volume hyperbolic 3-manifold N with geodesic boundary. Since D.N / is finite-volume hyperbolic without boundary, for a sufficiently small " > 0 the "-thin part of D.N / consists of cusps based on tori. Each such cusp is either disjoint from @N , in which case it gives rise to a toric cusp in N , or it is cut into two symmetric pieces by @N . It is then not too difficult to see that the corresponding portion of the thin part of N is an annular cusp, namely of type A  Œ0; C1/, with A a Euclidean annulus obtained by gluing together two opposite sides of a rectangle. This discussion implies that a finite-volume hyperbolic 3-manifold N with geodesic boundary compactifies to a certain M with a specified family of closed annuli A on @M , so that N is given by M minus A and the toric components of @M . Note that @M cannot contain spheres and no annulus in A can lie on a toric component of @M . In the sequel we will sometimes speak with a slight abuse of a hyperbolic compact .M; A/ to mean that a (complete and finite-volume, as always) hyperbolic metric is defined on M minus the union of A and all the toric boundary components of @M . Hyperbolic structures with geodesic boundary still enjoy Mostow rigidity, but only in the sense that each manifold can carry at most one such structure up to isometry: it is not true in this context that the fundamental group determines the structure, as shown by Frigerio [18].

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Links, orbifolds, and graphs For a link-pair .M; L/ with closed M a hyperbolic structure is simply one on the exterior of L in M , with one cusp for each component of L. Turning to a 3-orbifold, recall that the finite local action on R3 defining it can be assumed to be orthogonal, up to conjugation, and that the stabilizer of a point in the group of isometries of hyperbolic space is the orthogonal group. The notion of a hyperbolic structure on a closed 3-orbifold is then an obvious extension of those already defined: it is a complete finite-volume singular Riemannian metric locally given by the quotient of an open ball in H3 under a finite action of isometries fixing the center of the ball. Versions of the definition for orbifolds with cusps and/or with boundary exist but will not be referred to below. For a graph-pair .M; G/ we will consider three different types of hyperbolic structure. • With totally geodesic boundary: an ordinary hyperbolic structure on the exterior X of G in M ; note that the knot components of G give rise to toric cusps, whereas components with vertices give compact components of the boundary. • Of orbifold type: an orbifold hyperbolic structure on M with some admissible labelling of the edges of G by integers. • With parabolic meridians: a hyperbolic structure on .X; A/, where X is the exterior of G in M and A is a system of meridinal annuli of the edges of G; note that for such a structure there is one toric cusp for each component of G, one annular cusp for each edge joining two vertices (or a vertex to itself), and one thrice punctured sphere of geodesic boundary for each vertex of G. Hyperbolisation So far we have not explained for what reason one should hope a 3-dimensional manifold (or graph, or orbifold) to have a hyperbolic structure. We now discuss the obstructions to the existence of such a structure and state the extremely deep results according to which the absence of these obstructions is actually sufficient to guarantee hyperbolicity. To begin, we recall that an essential surface in a 3-manifold M is a properly embedded one whose fundamental group, under the inclusion, injects into that of M , and which is not parallel to the boundary. It is not too difficult to show that a hyperbolic manifold cannot contain essential surfaces with non-negative Euler characteristic (that is, spheres, tori, discs, or annuli). The following result has first been proved by Thurston [55] for Haken manifolds (those containing some essential surface), remained as a conjecture for a long time, and was eventually established by Perelman [44], [45], [46] (see also [7]): Theorem 2.4. If a compact 3-manifold M with (possibly empty) boundary consisting of tori does not contain any essential surface with non-negative Euler characteristic then M is either hyperbolic or a Dehn filling of P  S 1 , where P is the 2-sphere minus three open discs.

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(The reason why Dehn fillings of P  S 1 make an exception is that they are the only manifolds containing a 1 -injective immersed torus but no embedded essential one, thanks to a result of Casson and Jungreis [13].) The philosophy underlying the previous theorem is that cutting a manifold along a surface with non-negative Euler characteristic one gets a (possibly disconnected) simpler one, from which the original manifold can be reconstructed. Therefore hyperbolic manifolds and Dehn fillings of P  S 1 can be viewed as building blocks for general 3-manifolds. Hyperbolization holds, with the necessary adjustments, for manifolds with more general boundary (and annuli on this boundary), see [21], and for orbifolds (which requires in particular the introduction of the notion of essential 2-suborbifold), see [8], [14]. An important consequence of the hyperbolization theorem is that if a graph-pair .M; G/ admits a hyperbolic structure with totally geodesic boundary on its exterior then for any admissible labelling of the edges, which turns .M; G/ into an orbifold, .M; G/ admits a corresponding orbifold hyperbolic structure, and that if for some labelling of the edges .M; G/ admits an orbifold hyperbolic structure then it admits one with parabolic meridians.

3 Cusped manifolds We will now describe the algorithmic approach to the construction and recognition of cusped hyperbolic manifolds, carried out with extreme success by Callahan, Hildebrandt and Weeks [11]. Hyperbolic ideal tetrahedra Let us start from a compact 3-manifold M with nonempty boundary consisting of tori, and from an ideal triangulation T of M . The idea to hyperbolize M , which dates back to Thurston [54], is to choose a hyperbolic shape separately for each tetrahedron in T and then to ensure consistency and completeness of the structure induced on M . To spell out this idea we begin by defining a hyperbolic ideal tetrahedron as the convex envelope  in H3 of four non-aligned points in @H3 , endowed with the orientation induced by H3 . (Recall that three points on @H3 D P 1 .C/ are always aligned, namely there exists a geodesic plane having all three of them as points at infinity.) Intersecting  with a small enough horosphere centered at any of its vertices, one gets a Euclidean triangle, which gets rescaled if the horosphere is shrunk. Moreover one can see that two triangles lying on horospheres centered at distinct vertices have the same angle at the edge of  joining these vertices, which implies that the four triangles at the vertices of  are actually similar to each other, so  determines a similarity class of an oriented triangle in the plane, and the converse is also true. To be more specific, let us note that the oriented isometries of H3 act in a triply transitive way on @H3 , so without loss of generality we can assume in the half-space

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3 model C viewed as C  .0; C1/, that a positively oriented triple of vertices of  is .0; 1; 1/. This implies that the fourth vertex is some z 2 C with =.z/ > 0, 2 namely z 2 C . Then the hyperbolic structure of  is determined by z, that we will call module of  along the edge .0; 1/, see Figure 8. Moreover the modules of 

1

z

z’’ z’

z’ z

z’’ 0

z

1

Figure 8. Modules of a hyperbolic ideal tetrehedron. 1 along the other edges are as shown in the figure, with z 0 D 1z and z 00 D 1  z1 . In particular,  has the same module along any two edges opposite to each other. And, conversely, once an orientation and a pair of opposite edges have been fixed on 2 turns the tetrahedron into an ideal an abstract tetrahedron, the choice of any z 2 C hyperbolic one as in Figure 8.

Consistency and completeness Let us return to our M ideally triangulated by T , and assume that there are n tetrahedra 1 ; : : : ; n and k toric boundary components T1 ; : : : ; Tk . Choosing a hyperbolic structure on 1 ; : : : ; n corresponds to choosing 2 z1 ; : : : ; zn 2 C , that we can view as variables. Using again the fact that the isometries 3 of H act in a triply transitive way on @H3 , it is now easy to see that for any choice of z1 ; : : : ; zn the hyperbolic structure on the tetrahedra extends to the interior of the glued faces in M . We then have the following: Proposition 3.1 (Consistency equations). The hyperbolic structure given by z1 ; : : : ; zn extends along an edge e of T in M if and only if the product of all the modules along e of 1 ; : : : ; n (with j not contributing to the product if it is not incident to e, and possibly contributing in a multiple fashion if it is multiply incident to it) equals 1, and the sum of the arguments of these modules equals 2. Remark 3.2. If the product of the modules along an edge equals 1, then the sum of the arguments of these modules is a positive multiple of 2. Using this fact and the observation that .M / D 0, because @M consists of tori, one then sees that if the products of the modules along all edges of T equals 1, then the sum of the arguments

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always equals 2. This implies that consistency of the hyperbolic structure defined by z1 ; : : : ; zn translates into n algebraic equations. (See also below for the number of these equations.) Suppose now that z1 ; : : : ; zn satisfy the consistency equations along all the edges of T . Then each boundary torus Tj is obtained by gluing Euclidean triangles along similarities, and consistency ensures that the similarity structure on the triangles extends to the edges and the vertices. Summing up, z1 ; : : : ; zn induce a similarity structure on each Tj , and we have: Proposition 3.3. The hyperbolic structure on M defined by z1 ; : : : ; zn is complete if and only if the induced similarity structure on each Tj is actually Euclidean. To turn the completeness condition into equations, we note that a similarity structure on a torus T induces a representation (the holonomy) of 1 .T / into the group of complex-affine automorphisms of C. This representation is well-defined up to conjugation, so its dilation component  W 1 .T / ! C is well-defined, and of course T is Euclidean if and only if  is identically 1. If the similarity structure on T is obtained by gluing triangles with specified modules, and ˛ is a simplicial loop in the resulting triangulation, one can easily show that .˛/ is the product of the modules of the triangles that ˛ leaves to its left, as suggested in Figure 9. w12

˛ w10 w11 w9 w8 ˛ w7 w6 w5

w4 w3 w1 w2

H) .˛/ D w1    w12 ˛

˛

Figure 9. Computation of the dilation component of the holonomy of a simplicial loop.

Therefore: Proposition 3.4 (Completeness equations). For j D 1; : : : ; k let j and j be generators of 1 .Tj /. The hyperbolic structure on M defined by z1 ; : : : ; zn is complete if and only if for all j the product of the modules of the triangles on Tj that j leaves to its left equals 1, and the same happens for j .

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Remark 3.5. The images of j and j under the holonomy representation of 1 .T / associated to a similarity structure are commuting complex-affine automorphisms of C. The condition .j / D 1 means that the holonomy of j is a translation; if this translation is non-trivial then also j maps to a translation, therefore .j / D 1. This shows that the two conditions to impose on each Tj are “almost” equivalent to each other, so in practice one adds to the n consistency equations only k, and not 2k, completeness equations. Moreover it was shown by Neumann and Zagier [43] that if a solution exists then k of the consistency equations can be dismissed; in addition, the complete structure corresponds to a smooth point in the space of deformations of the structure, which is a k-dimensional algebraic variety. This fact can be exploited for instance to establish Theorem 2.3. To conclude the discussion on the construction of the hyperbolic structure on a would-be cusped manifold M , we note that using an arbitrary ideal triangulation T of M it is not true that a solution of the corresponding consistency and completeness equations always exists, even if M is actually hyperbolic. And, as a matter of fact, it is not even known that one T such that the corresponding equations have a genuine solution exists (despite the wrong statement in [5] that this follows from [16], see also below). However when one starts from a minimal triangulation of a hyperbolic M , namely one with a minimal number of tetrahedra, the solution always exists in practice. Weeks’ wonderful software SnapPea [61] is capable (among other things) to find a minimal triangulation of a given (a priori possibly non-hyperbolic) M , to seek for a solution of the corresponding equations, and also to deduce from patterns it sees in the triangulation the existence of topological obstructions to hyperbolicity. It is using these features (and the recognition machinery described in the rest of this section) that the census [11] of cusped manifolds triangulated by at most seven tetrahedra has been obtained. Canonical decomposition Once the hyperbolic structure on a cusped M has been constructed, the need naturally arises to recognize such an M , namely to be able to effectively determine whether M is the same as any other given cusped manifold. Several hyperbolic invariants, and chiefly the volume (which is easily computed from a hyperbolic ideal triangulation by means of the Lobachevski function, see [42]), can often distinguish manifolds, but different manifolds actually can have the same volume, as proved by Adams [1], and other invariants, so the need of a complete one remains. This complete invariant is provided by a result of Epstein and Penner [16], and it allows to perform the recognition very efficiently. We will first state this result informally and then provide the necessary details. The basic underlying idea is best described starting from an arbitrary compact Riemannian manifold X (of any dimension) with non-empty boundary. In this case one can define the cut-locus CutX .@X/ of @X in X as the set of points joined by more than one distance-minimizing path to @X. To visualize CutX .@X /, imagine that we start pushing all the components of @X towards the interior of X, all at the same pace.

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At some point some collision (or self-collision) will start occurring; we then fuse together the collided points, leave them still henceforth, and keep pushing the rest. Eventually we exhaust all the space available in M and we are left with CutX .@X / in the form of the membrane on which the collisions have taken place. (See Figure 10 for an allusive picture in dimension 2.) This description should make it obvious that

Figure 10. The cut locus of the boundary in a Riemannian surface as the result of pushing the boundary towards the interior as far as possible.

CutX .@X / is a compact subset of X onto which X retracts, and that it has dimension at least one less than that of X . Supposing X has dimension 3 one can in addition imagine that in a generic situation CutX .@X/ will be a special spine of X, and therefore that dual to it there will be a topological ideal triangulation of X . In more general contexts dual to CutX .@X/ there will be a decomposition of X into ideal polyhedra more complicated than tetrahedra. Turning to a cusped hyperbolic M , we first note that we cannot take CutM .@M /, because @M is at infinite distance from any point in the interior of M , since @M is not really part of the hyperbolic structure, but rather of its compactification. Recall however that each cusp of M has the form T  Œ0; 1/, where T is a flat torus, and

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more precisely the image in M of a horoball of H3 acted on by the Z ˚ Z lattice of the parabolic elements of 1 .M / fixing the center of the horoball. If we replace the cusp T  Œ0; C1/ with T  Œh; C1/ for some h > 0 we get a smaller cusp, with volume that tends to 0 as h ! C1. Therefore for sufficiently small v > 0 we can take disjoint cusps at each end of M all having volume v, and call M .v/ the complement in M of their interior. The following fact has a peculiarly hyperbolic nature, as we will explain before providing a detailed proof: Proposition 3.6. CutM .v/ .@M .v/ / is independent of v. To appreciate this result, consider the case of a Riemannian manifold X D T  2 Œ0; 1 , with metric ds.p;t/ D f .t /  d p2 C dt 2 , where d 2 is a flat metric on T giving it area 1, and f is a smooth increasing function such that f .t / D 1 for 0  t  13 and     f .t / D 2 for 56  t  1. Viewing T  0; 13 and T  56 ; 1 as the ends of X , we see that   ˚7  1 1  . they both have volume 13 , so X . 3 / D T  13 ; 56 , and Cut . 1 / @X . 3 / D T  12 3 X ˚     1 1  1 11 13 However X . 6 / D T  6 ; 12 and Cut . 1 / @X . 6 / D T  24 . X

6

Proof of Proposition 3.6. Assume that two liftings to H3 of volume-v cusps in M 3 (possibly of the same one) are the horoballs centered at 1 and at 0 in the C model of 3 2 H , namely one of them is a half-space O1 D Œh1 ; C1/  R and the other one is a Euclidean ball O2 of radius h22 centered at .0; 0; h22 /, so that its top point has height h2 . Since distinct liftings to H3 of the cusps in M are disjoint or coincide, one has h2 < h1 . Now suppose that the action of the Z˚Z lattice of parabolic elements of 1 .M / fixing 1 gives as a quotient of f0g  R2 a flat torus of area a1 . Note that a1 is independent of v, namely, if we change v then the height h1 changes but the area a1 does not. Moreover v is equal to the integral of the volume form x13 dx1 dx2 dx3 of H3 over 3

Œh1 ; C1/  A1 , where A1 is a parallelogram of area a1 , therefore v D

a1 . 2h2 1

Applying

the inversion with respect to the radius-1 at 0, which is a hyperbolic  1 sphere centered 2 isometry, O2 becomes the half-plane h2 ; C1  R , and the computation already a h2

performed shows that v D 22 2 , for some a2 again independent of v. The surface of the points having equal distance from O1 and from O2 is of course determined by the point in which it intersects the x3 -axis, whose height h must satisfy Z h Z h1 dx3 dx3 D H) log.h/  log.h2 / D log.h1 /  log.h/ x3 h2 x3 h p H) 2 log.h/ D log.h1 h2 / H) h D h1 h2 : a h2

2 2 a1 established above now easily imply that The relations v D 2h 2 and v D 2 1 p a1 h1 h2 D a2 is indeed independent of v, and the conclusion follows.

We can now state the result of [16]:

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Theorem 3.7 (Epstein–Penner canonical decomposition). If M is a cusped hyperbolic 3-manifold then dual to CutM .v/ .@M .v/ / there is a decomposition of M into hyperbolic ideal polyhedra whose combinatorics and hyperbolic shape of the blocks depend on M only. Once the Epstein–Penner canonical decompositions of two given cusped hyperbolic manifolds have been determined, to compare the manifolds for equality one then only needs to compare the canonical decompositions for combinatorial equivalence. Note that one does not need to check that the hyperbolic shapes of the polyhedra are the same, since combinatorial equivalence of the decompositions already ensures that the manifolds are homeomorphic to each other (whence, by rigidity, isometric to each other). The light-cone and the convex hull construction To show how one can actually construct the Epstein–Penner canonical decomposition of a given ideally triangulated n of Hn than we cusped manifold, we will exploit more of the hyperboloid model HC have done so far. We first define the (future) light-cone in the Minkoswki space R1;n with scalar product hji.1;n/ as ˚  LnC D y 2 R1;n W hyjyi.1;n/ D 0; y0 > 0 and we remark between @Hn and the projectivized  that  there is a natural identification n n light-cone P LC . Moreover for all y 2 LC one can define as follows an associated horoball ˚  n By D x 2 HC W hxjyi.1;n/  1 at and its boundary   horosphere Hy D @By . It is not hard to see that By is centered  Œy 2 P LnC D @Hn , and that all horoballs centered at some p 2 P LnC D @Hn have the form By for some y 2 LnC with Œy D p. Note that By 0  By if y 0 D y with  > 1. Turning to the effective construction of the Epstein–Penner decomposition, let us fix a cusped hyperbolic 3-manifold M , and a set of disjoint cusps in M all having one and the same volume v. These cusps lift in the universal cover of M , that we identify 3 with HC , to a family of disjoint horoballs fBy W y 2 P g for some P  L3C . Let us now establish the following crucial property of P : Lemma 3.8. P is discrete. Proof. It is of course sufficient to show that for all h > 0 the set fp 2 P W x0 .p/  hg is finite. Assuming the contrary and projecting to the disc model B 3 , we would get an infinite family of horoballs that, as Euclidean balls, have radius bounded from below. But this is impossible since the horoballs must be disjoint from each other. We now define C as the convex hull of P in R1;3 , and we note that P , and hence 3 to R1;3 . The C , are invariant under the action of 1 .M /, which extends from HC following is established in [16]:

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Proposition 3.9. C \ L3C D f  p W p 2 P ;   1g. 3 For all x 2 HC the half-line ft  x W t  0g intersects C in a half-line ft  x W t  0 .x/g for a suitable 0 .x/, and 0 .x/  x 2 @C . 3 , therefore the radial @C n L3C consists precisely of the points 0 .x/  x for x 2 HC 3 3 projection is a bijection between @C n LC and HC . @C consists of a 1 .M /-invariant family of finite-faced polyhedra that intersect L3C precisely at their vertices. 3 The polyhedra of which @C consists, projected first radially to HC and then to M under the action of 1 .M /, give the ideal decomposition of M dual to CutM .v/ .@M .v/ / as in Theorem 3.7. The tilt formula Let us suppose that M is a cusped hyperbolic manifold with a given hyperbolic ideal triangulation T . We will now describe a method, based on the results of Sakuma and Weeks [60], [53] and exploited by Weeks’ software SnapPea [61], to decide whether the Epstein–Penner canonical decomposition of M is actually T or can be obtained from T by merging together some of the tetrahedra into more complicated polyhedra. To this end we fix some v > 0 such that M contains disjoint cusps of volume v at all its ends (and note that v is easy to find using the combinatorics of T and the geometry of the hyperbolic tetrahedra that T consists of). We then concentrate on a 2-face F of T , to which two tetrahedra 1 and 2 will be incident. Let us lift 3 z 1 and  z 2 such F , 1 , 2 in HC to an ideal triangle Fz and two ideal tetrahedra  z z z that F D 1 \ 2 . The choice of v allows us to associate a point on the light-cone z 1,  z 2 , and we can consider the straight triangle F 0 L3C to each ideal vertex of Fz ,  0 0 1;3 and tetrahedra 1 , 2 in R having these points on L3C as vertices . Finally, we define #.F / as the dihedral angle in R1;3 not containing 0 formed along the plane containing F 0 by the half-hyperplanes containing 01 and 02 , and we note that #.F / is independent of the particular liftings chosen. The following is a direct consequence of Proposition 3.9: Proposition 3.10. T is the Epstein–Penner decomposition of M if and only if #.F / <  for all 2-faces F of T . More generally, the Epstein–Penner decomposition of M is obtained from T by merging together some of the tetrahedra of T if and only if #.F /   for all 2-faces F of T , and in this case the mergings to perform are those along the F ’s such that #.F / D . When T does not meet the conditions of this proposition, namely when it contains some offending 2-face F with #.F / > , a new triangulation T 0 with better chances of being a subdivision of the Epstein–Penner decomposition is obtained by performing the 2-to-3 move along F . Note however that the move can be applied only if the two tetrahedra of T incident to F are distinct. One can then start a process that searches for faces F with #.F / >  to which the 2-to-3 move can be applied, applies the move and starts over again. The process can get stuck if all F ’s with #.F / >  are incident

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to the same tetrahedron on both sides, but it is shown in [53] that if the process does not get stuck then it converges in finite time to a subdivision of the Epstein–Penner decomposition. As a matter of fact, experimentally the process always converges, and it does so very quickly. There is however one aspect of the process just described that we have not yet described how to perform algorithmically, namely the computation of the angle #.F /. This is done using the Sakuma–Weeks tilt formula, that constitutes the core of [53]. This formula associates a real number t .; E/, called the tilt, to each triangle E in R1;3 with vertices on L3C and to each tetrahedron  with vertices on L3C having the triangle as a face. To do so, the following vectors are needed: • The unit normal p to , satisfying hpjxi.1;3/ D 1 for all x 2 . • The outer unit normal m to E, satisfying hmjmi.1;3/ D 1, hmjxi.1;3/ D 0 for all x 2 E, and hmjvi.1;3/ < 0, where v is the vertex of  opposite to E. Then t .; E/ D hpjmi.1;3/ , and the knowledge of the tilts allows to apply Proposition 3.10 by means of the following: Proposition 3.11. With the notation introduced before Proposition 3.10, #.F / <  if and only if t.01 ; F 0 / C t .02 ; F 0 / < 0; and #.F / D  if and only if t .01 ; F 0 / C t .02 ; F 0 / D 0: Rather than providing a complete proof of this result, we motivate it using

an 1 1;2 0 example in one dimension less. We suppose in R that F has vertices 1 and 1

s1

0 s2

0 1 , while 01 has the further vertex and 02 has the further vertex 0 , s1 s2 0 for some s ; s > 0. Now one easily sees that #.F / D  if and only if the vectors 1 2 s1 1 s2

0 ; 0 ; 0 are aligned, which happens if s11 C s12 D 2. And more generally s1

0

s2

that #.F / <  if and only if s11 C s12 < 2. A direct computation now shows that 0 0 0 1 0 1 1 1 1 1 0 0 @ @ @ A @ A A 0 0 ; p2 D ; m1 D 0 ; m2 D 0A p1 D 1 1 1  s1 1 1 1 s2 H) t .01 ; F 0 / C t .02 ; F 0 / D hp1 jm1 i.1;3/ C hp2 jm2 i.1;3/ 1 1 D 1C 1 s1 s2 1 1 D C 2 s1 s2

and the conclusion easily follows.

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The next result, which represents the main achievement of [53], [60], shows how to effectively compute the tilts of a given hyperbolic ideal triangulation of M , knowing only the geometry of the hyperbolic tetrahedra of T and the “height” in each of them of the horospheres giving equal-volume cusps in M . Proposition 3.12. Let  2 T have vertices w0 , w1 , w2 , w3 and define Ei as the face opposite to wi . Let 0  R1;3 be the straight lifting of  with vertices on L3C determined by the choice of equal-volume cusps of M , let Ei0 be the face of 0 projecting to Ei , and set ti D t .0 ; Ei0 /. Denote by ij the dihedral angle of  along the edge joining wi and wj , and by ri the radius of the circle circumscribed to the Euclidean triangle obtained by intersecting with  the horosphere centered at wi that belongs to the system yielding in M the equal-volume cusps. Then: 0 1 0 1 0 1 1  cos 01  cos 02  cos 03 t0 r0 C Bt1 C B cos 01 C B 1  cos   cos  12 13 C Br1 C B CDB :  @t2 A @ cos 02  cos 12 1  cos 23 A @r2 A t3  cos 03  cos 13  cos 23 1 r3 Moreover ri D e di , where di is the (signed) distance between Ei and the horosphere at wi already described. We note that this result was first established in [60] in dimension 3 and then generalized in [53] for all dimensions. Giving a complete proof is beyond our scopes, 2  C the triangle but we can at least prove the last assertion in dimension 2. If in C  has vertices 0, 2, 1 and the horosphere centered at 1 cuts it at height =.z/ D h, therefore in a segment of Euclidean length h2 , then of course r1 D h1 . The point closest to 1 of the edge opposite to 1 is 1 C i , and its distance from the horosphere Rh is d1 D 1 ds D log.h/, therefore one indeed has r1 D e d1 . This concludes our s discussion of the algorithmic recognition of cusped manifolds. Existence of hyperbolic ideal triangulations Taking a subdivision of the Epstein– Penner decomposition into hyperbolic ideal polyhedra of a cusped hyperbolic manifold M , one can get a topological ideal triangulation, which in M gives a hyperbolic triangulation with some genuine and some flat tetrahedra (the four vertices are distinct but aligned). The reason is that it may not be possible to subdivide the polyhedra separately in such a way that the subdivision of the faces is matched by the gluings, therefore some flat tetrahedra may need to be inserted (see [48], [49] for a detailed discussion of this process). The paper [59] describes a sufficient condition for the existence of a subdivision of the Epstein–Penner decomposition into genuine hyperbolic ideal polyhedra, and [33] shows that up to passing to a finite cover of M one can always find a genuine hyperbolic ideal triangulation. The experimental findings of [11] strongly suggest that every cusped hyperbolic 3-manifold does possess such a triangulation, but the question is apparently open for the time being.

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One further aspect is worth mentioning. SnapPea solves the hyperbolicity equations using Newton’s method and numerical approximation, so one could view SnapPea’s finding that a certain manifold is hyperbolic merely as an informal indication. Recall however that the hyperbolicity equations are algebraic ones, and that they have at most one solution. This implies that the solution, if any, belongs to some finite extension of Q. Goodman’s software Snap [24], starting from a high-precision numerical solution, is capable of guessing what the right extension of Q is and then to check that the solution is an exact one using arithmetic, without approximation. This process has been successfully applied to the manifolds found in [11], which means that this census is immune to numerical flaws.

4 Complexity and closed manifolds This section represents a singularity in the present survey, since hyperbolic geometry only plays in it a comparatively marginal role. We will briefly discuss Matveev’s [38] complexity theory for closed manifolds, and the experimental results obtained exploiting it. To begin, we extend the notion of special spine of a closed (orientable) 3-manifold M , already discussed in Section 1, by defining a simple spine of M as a compact polyhedron P  M onto which M minus some number of points collapses, and such that the link [52] of every point of P is contained in the complete graph with four vertices. Note that special polyhedra, surfaces and graphs are simple polyhedra. For a simple polyhedron we can still define a vertex as a point appearing as in Figure 3 (right). Following Matveev we then define the complexity c.M / of a manifold M as the minimal number of vertices in a simple spine of M . To explain the reason why the “right” definition of complexity is based on the more flexible notion of simple (rather than special) spine, we need to recall that a connected sum of two manifolds M1 and M2 is a manifold M1 # M2 obtained by removing open balls from M1 and M2 , and by gluing the resulting boundary spheres. Since S 2 has only two isotopy classes of self-homeomorphisms, if M1 and M2 are connected then at most two different M1 # M2 ’s exist. Moreover if M1 and M2 are oriented (as opposed to orientable only) and one insists that the gluing homeomorphism should reverse the induced orientations, one gets a uniquely defined M1 # M2 . The 3-sphere S 3 is the identity element for the operation # of connected sum, and a manifold M is called prime if it cannot be expressed as a connected sum with both summands different from S 3 ; in addition, M is called irreducible if every 2-sphere in M bounds a 3-ball in M . Of course every irreducible manifold is prime. The following Haken–Kneser–Milnor decomposition theorem has been known for a long time [28], [29]: Theorem 4.1. The only prime non-irreducible (closed orientable) 3-manifold is S 2  S 1 . Every 3-manifold can be expressed in a unique way as a connected sum of prime ones.

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Turning back to complexity, let us define a simple spine P of M to be minimal if it has c.M / vertices and no proper subset of P is still a spine of M . We now have the following fundamental result of Matveev [38]: Theorem 4.2. (1) c.M1 # M2 / D c.M1 / C c.M2 /. (2) If M is prime then either c.M / D 0 and M 2 fS 2  S 1 ; S 3 ; P 3 .R/; L.3I 1/g; or c.M / > 0 and every minimal simple spine of M is special. Item (1) of the previous theorem, which translates into the statement that complexity is additive under connected sum, means that to know the complexity of any manifold one only needs to know that of its connected summands. And item (2) implies that, with a few easy exceptions, the complexity of a prime manifold equals the minimal number of tetrahedra required to triangulate it. Note that additivity would not be true if the definition of complexity were based on special spines, or on triangulations. Employing simple spines one has in addition the following advantage, that proves extremely useful in practice: Proposition 4.3. A special polyhedron to which one of the moves shown in Figure 11 can be applied is not a minimal spine of a prime 3-manifold of positive complexity.

1-to-0

2-to-0

Figure 11. Moves on special spines turning them into simple ones.

The proof of this result is given by Figure 11 itself, because the moves described in it preserve the property that a polyhedron be a spine of a manifold, they lead to simple polyhedra, and they reduce the number of vertices.

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According to Theorem 4.2 (2) and Proposition 4.3, given n  1, to produce the list of closed irreducible 3-manifolds of complexity n, one can proceed according to the following (partly simultaneous) steps: • Recursively construct the list of all orientable special spines with n vertices of closed manifolds (or, dually, the list of all triangulations with n tetrahedra of closed orientable manifolds). • During the construction, check whether the configurations of Figure 11 appear in incompletely constructed spines and, if so, discard automatically all their possible completions. • Once a reduced list of spines has been obtained, eliminate duplications of manifolds by repeated applications of the 2-to-3 move and its inverse, and show that the final list contains distinct manifolds using some invariants (such as homology or the Turaev–Viro invariants [56]). This strategy has been carried out by Matveev for n  8, by Martelli and the author for n D 9 (using a substantial refinement [36] of Proposition 4.3, based on a certain theory of bricks), and then independently by Martelli and Matveev (see [35] and the references quoted there) for larger values of n. We also mention that Matveev and Tarkaev [40] have written a software, based on special spines and on the idea of applying moves to simplify them, that allows to recognize any given closed manifold in a very efficient way; the web site [40] also includes very helpful electronic lists of manifolds. In addition, non-orientable versions of the census have been obtained by Amendola and Martelli [3], [4] (using results of Martelli and the author [37] on non-orientable bricks) and by Burton [9], [10]. Closed hyperbolic manifolds Matveev showed with a (complicated) theoretical argument that no closed manifold of complexity smaller than 9 can be hyperbolic. The following was proved in [36]: Theorem 4.4. There are precisely 4 closed orientable hyperbolic 3-manifolds of complexity 9, and they coincide with the 4 of smallest known volume. The four manifolds referred to in the previous statement include the Weeks manifold, now known to be the minimum-volume closed orientable hyperbolic one, thanks to a result of Milley [41] based on his joint work with Gabai and Meyerhoff on Mom’s [23]. Martelli [35] found 25 hyperbolic manifolds in complexity 10, and Martelli and Matveev found (the same!) few more in higher complexity. To conclude we mention that two completely alternative approaches to closed hyperbolic manifolds exist but will not be reviewed here. On one hand one can obtain a wealth of them doing Dehn filling on cusped manifolds (see Theorem 2.3), which SnapPea [61] allows to do very efficiently. On the other hand one can try to construct a hyperbolic structure on a given closed manifold, starting from a triangulation and using a method suggested by Casson [12] (see also Manning [34]).

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5 Geodesic boundary and graphs As many of the ideas in the realm of hyperbolic geometry, those underlying the algorithmic hyperbolization of manifolds with boundary are again due to Thurston [54], who first constructed such a structure on the complement of a certain graph in the 3-sphere. As in the case of cusped manifolds, where one starts from some hyperbolic tetrahedra and imposes matching and completeness of the structure induced on the manifold obtained by gluing them, one starts from certain parameterized building blocks and tries to solve a system of equations. To describe the building blocks we will need a model of hyperbolic space not employed so far, namely the projective n to the model, obtained by projecting radially (whence the name) the hyperboloid HC n 1;n unit disc BP of the hyperplane at height x0 D 1 in Minkowski space R . The main advantage of this model is that hyperbolic straight lines appear as Euclidean straight segments in it (but the angles are not the Euclidean ones, as it happens instead in the disc and half-space models). We define a hyperideal hyperbolic polyhedron as a polyhedron P in the space f1g  Rn containing BPn so that: • P has some hyperideal vertices, lying outside the closure of BPn , and possibly some ideal ones, lying on the boundary of BPn ; • P has some genuine edges, meeting the interior of BPn , and possibly some ideal edges, tangent at one point to the boundary of BPn ; • the ends of each ideal edge are hyperideal (i.e., not ideal) vertices of P . To such a P we will always associate the corresponding truncated hyperideal hyperbolic polyhedron Py , to define which we introduce for each hyperideal vertex v of P : • the .n  1/-sphere v  @BPn of the tangency points to @BPn of lines emanating from v; • the hyperplane ˛v in f1g  Rn  R1;n containing v ; • the closed half-space Qv in f1g  Rn bounded by ˛v and not containing v. Then we define the truncation Py of P as the intersection of P with BPn and with the Qv ’s as v varies among the hyperideal vertices of P . See Figure 12 for 2-dimensional examples of P and Py . Note that one can naturally define for a truncated hyperideal hyperbolic polyhedron the truncation faces as those lying on the ˛v ’s, and the lateral faces, coming from the original ones. The following fact is easy to show: Lemma 5.1. The truncation faces and the lateral faces of a truncated hyperideal hyperbolic polyhedron lie at right angles to each other. This implies that the hexagon in Figure 12 (left) is right-angled (even if one would not be able to tell from the picture), and that the two pentagons in Figure 12 (center and right) are right-angled at the non-ideal vertices. Turning to dimension 3, the next result follows from the previous one and will be needed below:

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Figure 12. A genuine hyperideal triangle and its truncation; a hyperideal triangle with an ideal vertex and its truncation; a hyperideal triangle with an ideal edge and its truncation.

Lemma 5.2. If v is the ideal point at which an ideal edge of a truncated hyperideal hyperbolic polyhedron Py meets @BP3 , then the intersection of Py with a sufficiently small horosphere centered at v is a Euclidean rectangle. Moduli and hyperbolicity equations Let us from now on restrict our discussion to dimension 3. Given a compact manifold M with boundary, or more generally a pair .M; A/ where A  @M is a family of closed annuli, one tries to construct on M a hyperbolic structure such that the non-toric components of @M nA are totally geodesic (so the components of A give annular cusps, whereas the tori give toric cusps). The way to do this algorithmically is again to start from an ideal triangulation T , encode by certain modules the structures of truncated hyperideal hyperbolic tetrahedra one can put on the tetrahedra of T , and try to solve certain equations on the modules that translate the fact that the structures of the tetrahedra match to give a global complete hyperbolic structure on M of the appropriate type. As for modules, the following was shown in [22] (see also [21]): Proposition 5.3. The space of modules for a hyperideal hyperbolic tetrahedron is given by the 6 dihedral angles, that vary freely subject to the following restrictions: • the sum of the angles at a hyperideal vertex is less than ; • the sum of the angles at an ideal vertex is equal to ; • the angle at an ideal edge is 0. Since this will be needed soon, we also mention that, given a choice of modules for a hyperideal hyperbolic tetrahedron , the lengths of all the edges of the corresponding y are computed by explicit formulae to be found in [21]. Note that an ideal truncated  edge always has length 0, and an edge with one or both ideal ends has length C1. Turning to the global matching of the structures on the individual tetrahedra of a triangulation, we begin from the case of a pair .M; A/ with A D ;, where one starts from an ideal triangulation T of a compact M in the usual sense. (The case A ¤ ;

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requires important variations discussed below.) In this case the matching equations express the fact that the lengths of all the truncation and lateral edges should be matched by the gluings of T . But, as a matter of fact, since a right-angled hyperbolic hexagon with finite vertices is determined by the lengths of three pairwise non-consecutive edges, the requirement that all lengths should be matched is typically redundant. As just described, the matching equations for hyperideal tetrahedra are quite different than those for ideal tetrahedra. In particular, we stress that the formulae to compute the lengths of the edges involve trigonometric and hyperbolic functions, so the equations are not algebraic ones. On the other hand the completeness equations are precisely the same: the modules give Euclidean structures up to similarity on the triangles of which each boundary torus is constituted, and completeness translates into the fact that each such torus should be Euclidean, and in turn into explicit equations in the modules along the lines of Proposition 3.4. Annular cusps For a pair .M; A/ with A  @M a family of closed annuli, the approach to hyperbolization using triangulations requires an important variation. To understand it, suppose that .M; A/ has a decomposition T into truncated hyperideal hyperbolic tetrahedra. Then an annular cusp A 2 A corresponds to an ideal edge of T . More precisely, one obtains the compact pair .M; A/ by first taking the compact manifold constructed by gluing the truncated versions of the tetrahedra in T , and then digging open cylindrical tunnels along the edges of T that come from ideal edges of the tetrahedra. This means that T itself is not an ideal triangulation of M . On the contrary, the following holds: Proposition 5.4. The ideal triangulations required to hyperbolize a pair .M; A/ are those of the form .T ; a/, where: • T is an ideal triangulation of the manifold N described next, and a is a family of edges of T ; • N is the manifold obtained from M by gluing a solid cylinder (a 2-handle) to each annulus in A; • the family of edges a, viewed in N , is precisely the family of cores of the solid cylinders glued to M to get N ; • when choosing modules for the tetrahedra in T , the ideal edges should be precisely those in a. The case with annular cusps requires the initial subtlety just described, and there is one more. The point is that there is one very special case where two “hexagons” with the same ordered lengths of the edges need not be isometric, so the matching of lengths of the edges is not sufficient to ensure consistency of the hyperbolic structure carried by a choice of modules for the tetrahedra in a triangulation, and an additional equation must be added. This occurs when the hexagon has one ideal edge and an opposite ideal vertex, so it reduces to a quadrilateral with two ideal and two finite vertices. The

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extra parameter describing the shape of such an object is described in [21] together with the method to compute it starting from the modules. Fortunately enough, after these two complications, we can show that in dealing with annular cusps no completeness issues arise: Proposition 5.5. Consider a (possibly incomplete) hyperbolic structure on .M; A/ given by a solution of the matching equations for a triangulation .T ; a/ as in the previous result. Then the structure is automatically complete at the annular cusps A. Proof. By Lemma 5.2 a horospherical cross-section at some A 2 A is obtained by gluing some rectangles, so it is a Euclidean annulus with boundary components of equal length. The double of such an annulus is a Euclidean torus (and not merely a similarity one), and the conclusion easily follows from Proposition 3.3. Canonical decomposition The recognition of hyperbolic 3-manifolds with geodesic boundary is based on an analogue for this type of manifolds of the Epstein–Penner canonical decomposition, due to Kojima [31], [32]. For a pair .M; A/ without toric cusps (but A can be non-empty, so annular cusps are allowed) the Kojima decomposition is a subdivision of M into truncated hyperideal hyperbolic polyhedra, possibly with ideal edges but without ideal vertices, and it is simply dual to the cut-locus of the boundary, as illustrated in Figure 10. This definition is of course of impractical use, but Kojima proved an analogue of Proposition 3.9 that allows the actual computation of the canonical decomposition. To state this result we need to recall more of the n geometry of the hyperboloid model HC of hyperbolic space. We define the 1-sheeted hyperboloid  n D fy 2 R1;n W hxjxi.1;n/ D C1g and, for y 2  n , n W hxjyi.1;n/  0g; Qy D fx 2 HC n noting that Qy is a geodesic half-space in HC , and that each such half-space has the n form Qy for a unique y 2  . Let us now consider a hyperbolic .M; A/ without toric cusps and recall that the z and an hyperbolic structure induces an identification between its universal cover M 3 intersection of closed geodesic half-spaces in H . Using the hyperboloid model H3C T z D fQy W y 2 P g for some family of points P   3 . As in the we then have M cusped case we now define C as the convex hull of P in R1;3 . Kojima proved the following result, stated in a rather informal way here (but carefully stated and proved in [21]):

Proposition 5.6. If .M; A/ is hyperbolic without toric cusps then the canonical decomposition of .M; A/ dual to the cut-locus of the boundary is obtained by projecting 3 radially to HC the faces of @C that meet the positive time-like half-lines.

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Turning to the case of a hyperbolic manifold with geodesic boundary .M; A/ also having toric cusps, we briefly mention that a canonical decomposition has been constructed by Kojima also in this case. The argument is somewhat more complicated, the main steps being as follows: 3 • Let T P   be the family of points such that the universal cover of .M; A/ is fQy W y 2 P g.

• Let PL  L3C be a family of points such that the family of horoballs fBy W y 2 PL g projects in .M; A/ to a family of equal-volume “sufficiently small” disjoint toric cusps. • Let P D P [ PL and define C as the convex hull in R1;3 of P . • Then the canonical decomposition of .M; A/ is obtained by projecting radially to 3 the faces of @C that meet the positive time-like half-lines, and then suitably HC subdividing those arising from vertices in PL . We only mention that how small the toric cusps should be in order for this construction to work was left implicitly determined by Kojima, and was later spelled out in a quantitative fashion in [21]. Algorithmic recognition While enumerating some class of hyperbolic manifolds with geodesic boundary, for each manifold one constructs the structure using a triangulation (which, in practice, always works for minimal triangulations if there are no topological obstructions to hyperbolicity), and then one is faced with the issue of algorithmically finding the Kojima canonical decomposition. The strategy to do so is the same as in the cusped context: starting from the given triangulation one tries to decide whether its tetrahedra represent the projections of the faces of @C , which amounts to checking whether the angles between suitable liftings of the tetrahedra, determined by the global geometry, are convex if viewed from the origin. And this can be carried out using an extension of the Sakuma–Weeks tilt formula, due to Ushijima [57] and carefully described in [21]. If some concave angle is found the combinatorics of the triangulation is changed by performing the 2-to-3 move along the offending face, until the process gets stuck (which never happens in practice) or the Kojima decomposition is reached. Experimental results The framework described above was successfully used by Frigerio, Martelli and the author [19] to list all manifolds with non-empty geodesic boundary and (possibly) toric cusps, but no annular cusp, that can be triangulated with up to four tetrahedra. The data are available online [20] and include the computation of the volume, based on results of Ushijima [58]. One of the most striking findings of [19] is that there are 56 manifolds whose canonical Kojima decomposition consists of a single hyperideal regular octahedron (with different combinatorics of the gluings). The corresponding 56 manifolds share the same volume and would be extremely difficult to distinguish from each other using

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different techniques (such as the invariants of algebraic topology or those of Turaev and Viro): it is only using hyperbolic geometry in its full power that one can tell that they are actually distinct. We also mention that this result naturally prompted the problem of enumerating all the different manifolds that can be obtained gluing the faces of the octahedron, solved by Heard, Pervova and the author in [27]. Turning to graphs, the trivalent hyperbolic ones in the most general sense (with parabolic meridians) were investigated by Heard, Hodgson, Martelli and the author in [26], where (with the restriction that the graph should have at least one trivalent vertex) all those that can be triangulated by five or less tetrahedra were enumerated and carefully analyzed. The enumeration and analysis have exploited Heard’s excellent software [25], which allows to hyperbolize and study manifolds with boundary and orbifolds in an extremely effective fashion. No systematic enumeration of hyperbolic orbifolds has been carried out so far, but the theoretical and computer methods are all in place, as described above, and the author is hoping to contribute to the topic in the future.

References [1]

C. C. Adams, Thrice-punctured spheres in hyperbolic 3-manifolds. Trans. Amer. Math. Soc. 287 (1985), 645–656.

[2]

G. Amendola, A calculus for ideal triangulations of three-manifolds with embedded arcs. Math. Nachr. 278 (2005), 975–994.

[3]

G. Amendola and B. Martelli, Non-orientable 3-manifolds of small complexity. Topology Appl. 133 (2003), 157–178.

[4]

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An introduction to asymptotic geometry Viktor Schroeder Institut für Mathematik, Universität Zürich Winterthurer Strasse 190, 8057 Zürich, Switzerland email: [email protected]

Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Motivation and examples . . . . . . . . . . . . . . . . . . . . 2.1 Basic notation . . . . . . . . . . . . . . . . . . . . . . 2.2 Motivation from geometric group theory . . . . . . . . 2.3 Stability of quasi-geodesics . . . . . . . . . . . . . . . 3 Hyperbolic spaces . . . . . . . . . . . . . . . . . . . . . . . 3.1 ı-inequality and Gromov product . . . . . . . . . . . . 3.2 Cross difference and cross difference triple . . . . . . . 4 Boundaries of hyperbolic spaces . . . . . . . . . . . . . . . . 4.1 Gromov product on the boundary . . . . . . . . . . . . 4.2 Busemann functions . . . . . . . . . . . . . . . . . . . 5 Geodesic hyperbolic spaces . . . . . . . . . . . . . . . . . . 5.1 Geodesic boundary . . . . . . . . . . . . . . . . . . . . 6 A metric structure on @1 X . . . . . . . . . . . . . . . . . . . 7 Morphisms of hyperbolic spaces . . . . . . . . . . . . . . . . 8 Möbius geometry of @1 X . . . . . . . . . . . . . . . . . . . 9 Hyperbolic approximation . . . . . . . . . . . . . . . . . . . 10 Quasimetric spaces and their Möbius geometry . . . . . . . . 10.1 Quasi-metric spaces . . . . . . . . . . . . . . . . . . . 10.2 Quasi-metrics and metrics . . . . . . . . . . . . . . . . 10.3 Complete quasi-metric spaces and the metric involution 10.4 Möbius geometry of quasi-metric spaces . . . . . . . . 10.5 Morphisms of quasi-metric spaces . . . . . . . . . . . . Appendix: Proof of Theorem 5.6 . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction These notes are an expanded version of a course on asymptotic geometry held in May 2008 at the doctoral school in Strasbourg and in a modified form at the University of Nanjing in June 2008. It is based on the book [BS1]. In these notes we want to give an introduction to some aspects of the asymptotic geometry of metric spaces. With asymptotic geometry we mean the study of the large scale aspects of metric spaces where we completely ignore the local geometry of the spaces in question. In particular the spaces can be discrete, which says in some sense that the local geometry does not exist. The main focus is the discussion of Gromov hyperbolic spaces. We think that they form a particularly nice class of spaces, where the global geometry can be studied by elementary methods: for this class of spaces one can define a boundary at infinity and it turns out that (under some natural assumptions) the whole asymptotic geometry can be decoded by the properties of the boundary at infinity. A space X is called Gromov hyperbolic if there exists a constant ı  0 such that for any quadruple of points .x; y; z; w/ 2 X 4 the two largest of the following three numbers jxyj C jzwj;

jxzj C jywj;

jxwj C jyzj

differ by at most 2ı. Here jxyj, jzwj etc. denotes the distances. It is surprising that this simple property catches some important properties of the classical hyperbolic space Hn , but is flexible enough in order to apply it for a broad class of spaces. The classical hyperbolic space has an ideal boundary @1 Hn , which is S n1 in the unit ball and Rn1 [ f1g in the upper half space model. In this classical situation there is a deep connection between the geometry of Hn and the Möbius geometry of its boundary. In particular an isometric map F W Hn ! Hn induces a Möbius map f D @1 F W @1 Hn ! @1 Hn , and vice versa, a Möbius map of the boundaries comes from an isometry of the hyperbolic spaces. To a large extent, this interplay between space and boundary can be generalized to general Gromov hyperbolic spaces. Theorem 1.1 and Theorem 1.2 are the analogous statements for Gromov hyperbolic spaces. To formulate these results we need a kind of Möbius structure on the boundary @1 X of a Gromov hyperbolic space X . Depending on a basepoint o 2 X respectively on a basepoint ! 2 @1 X , we define a quasimetric o respectively ! on @1 X. A quasi-metric satisfies a weaker kind of triangle inequality, see Section 10 for details. The quasi-metric is defined as o .x; y/ D a.xjy/o , where .j/o is the Gromov product introduced in Section 3. The discussion of the metrics o and ! in Section 4 is more involved. The quasi-metrics depend on an additional constant a > 1, which is chosen to be fixed for the purpose of this introduction. Of course the quasi-metric o depends on the basepoint o 2 X, but it turns out that for different basepoints these metrics are “essentially” Möbius equivalent. This implies that we obtain in a natural way something like a Möbius structure on the boundary @1 X.

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We want to make this point precise. We call a map f D .Z; / ! .Z 0 ; 0 / between two quasi-metric spaces bilipschitz Möbius if there is a constant L  1 such that for all x; y; z; w 2 Z with images x 0 ; y 0 ; z 0 ; w 0 2 Z 0 we have 1 Œx; y; z; w  Œx 0 ; y 0 ; z 0 ; w 0   L Œx; y; z; w; L where Œx; y; z; w is the cross ratio Œx; y; z; w D

.x; z/.y; w/ ; .x; y/.z; w/

respectively the cross ratio on .Z 0 ; 0 /. If we want to focus on the constant L, we speak about L-bilipschitz Möbius maps. We call two quasi-metric spaces .Z; / and .Z 0 ; 0 / bilipschitz Möbius equivalent if there is a bijective bilipschitz Möbius map f D Z ! Z 0 . In this case the inverse f 1 is also a bilipschitz map. Now one can show that for a Gromov hyperbolic space X there exists a constant L depending only on ı such that for any given o; o0 2 X and ! 2 @1 X , the identity maps id W .Z; o / ! .Z; o0 / and id W .Z; o / ! .Z; ! / are L-bilipschitz Möbius. This implies in particular that the cross ratio Œx; y; z; w for four points in @1 X is (up to a uniform multiplicative error) well defined and does not depend on the chosen basepoint o 2 X or basepoint ! 2 @1 X. Thus in some sense, we have a Möbius structure on @1 X. In the case that X is visual (see Definition 4.5) and geodesic, this Möbius structure of the quasi-metric space .@1 X; o / determines X up to rough isometry. To be precise: A map F W X ! X 0 between two metric spaces is called roughly isometric if there is a constant b  0 such that for all x; y 2 X and x 0 D F .x/, y 0 D F .y/, jxyj  b  jx 0 y 0 j  jxyj D b: Two metric spaces X and X 0 are called roughly isometric if there is a roughly isometric map F W X ! X 0 and in addition some constant c  0 such that to any point z 0 2 X 0 there exists x 2 X such that jF .x/z 0 j  c. To be roughly isometric is an equivalence relation among metric spaces. Theorem 1.1. Let X and X 0 be visual and geodesic Gromov hyperbolic spaces. Then X and X 0 are roughly isometric if and only if their boundaries @1 X and @1 X 0 are bilipschitz Möbius equivalent. More precisely the result states: (1) Let F W X ! X 0 be a roughly isometric map between two Gromov hyperbolic spaces, then F induces a boundary map f D @1 F W @1 X ! @1 X 0 which is bilipschitz Möbius. (2) Let X be a visual and X 0 be a geodesic hyperbolic space, and assume that f W @1 X ! @1 X 0 is a bilipschitz Möbius map. Then there exists F W X ! X 0 roughly isometric, with @1 F D f .

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The second point is a version of the classical Poincaré extension theorem, which states that a Möbius map on Rn1 [ f1g can be extended to a Möbius map (and hence isometry) of the upper half space. Theorem 1.1 essentially gives a description of a hyperbolic space up to rough isometry in terms of its boundary at infinity. However for many applications the notion of rough isometries is not flexible enough. We really aim for a description up to quasi-isometry. A map F W X ! X 0 between metric spaces is quasi-isometric (or Q-isometric) if there are c  1, b  0 such that 1 jxyj  b  jx 0 y 0 j  c jxyj C b c for all x; y 2 X, with x 0 D F .x/, y 0 D F .y/. The spaces X and X 0 are called quasi-isometric if there exists a quasi-isometric map F W X ! X 0 such that there is a constant d  0 such that for every z 0 2 X 0 there exists x 2 X with jF .x/z 0 j  d . To be quasi-isometric is an equivalence relation and comes up in many cases. Let us mention two important examples: (1) Let G be a finitely generated group, and let dS and dS 0 be the word metrics with respect to finite symmetric generating sets S respectively S 0 (compare Section 2). Then .G; dS / and .G; ds 0 / are quasi-isometric. In particular the quasi-isometry class of G is well defined. z its Riemannian universal (2) If M is a compact Riemannian manifold and M z covering, then M is quasi-isometric to 1 .M /. This is the theorem of Milnor and Švarc (see [Mi1], [Sv]). One can extend the above Theorem 1.1 to the setting of quasi-isometric maps. It turns out that in the “multiplicative” theory at infinity the Q-isometric maps correspond to the PQ-Möbius maps. A map f W Z ! Z 0 between quasi-metric spaces is called power quasi-Möbius or PQ-Möbius if there are p  1 and q  1 such that for all .x; y; z; w/ with Œx; y; z; w  1 we have 1 Œx; y; z; w1=p  Œx 0 ; y 0 ; z 0 ; w 0   q Œx; y; z; wp : q We then obtain the following correspondence: Theorem 1.2. Let X, X 0 be visual and geodesic Gromov hyperbolic spaces. (1) If F W X ! X 0 is Q-isometric, then F induces a map f D @1 F W @1 X ! @1 X 0 which is PQ-Möbius. (2) If f W @1 X ! @1 X 0 is a PQ-Möbius map. Then there exists F W X ! X 0 Q-isometric such that @1 F D f . This was proved by Paulin [Pau] in a setting for hyperbolic groups. Bonk and Schramm [BoS] proved it in the setting of PQ-symmetric maps, in [BS1] there is a proof under the additional assumption that the boundaries are uniformly perfect, in this general formulation it is due to Jordi [J]. This result gives a very nice interplay

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between the “additive” theory of hyperbolic spaces and the “multiplicative” theory of quasi-metric spaces. The additive error b of rough isometries corresponds to the multiplicative error L of bilipschitz Möbius maps. And the affine errors ct C b and 1 t  b of quasi-isometries correspond to the error functions qt p and .1=q/t 1=p in the c multiplicative setting on the boundary at infinity. This exponential (or logarithmic) relation between the metric on X and the quasimetric on @1 X is somewhat subtle. It becomes clearer after introducing the cross difference 1 hx; y; z; wi D .jxzj C jywj  jxyj  jzwj/ (1) 2 and the notation of a PQ-isometric map, which preserves the cross difference up to an affine control function. The cross difference can be rewritten in the form hx; y; z; wi D .xjy/o C .zjw/o  .xjz/o  .yjw/o ;

(2)

where o 2 X is an arbitrary basepoint. Note that the setting in (1) makes only sense for finite points x, y, z, w, while the setting in (2) also extends to points at infinity. By definition of o we see that ahx;y;z;wi D Œx; y; z; w; which gives the desired connection between the “multiplicative” cross ratio and the “additive” cross difference. We also need the additive analogon to PQ-Möbius maps. A map F W X ! X 0 between metric spaces is called power quasi-isometric or PQ-isometric if there exist c  1 and b  0 such that for x; y; z; w 2 X with hx; y; z; wi  0, 1 hx; y; z; wi  b  hx 0 ; y 0 ; z 0 ; w 0 i  c hx; y; z; wi C b: c We give a more detailed discussion of these maps in Section 7. Now one obtains Theorem 1.2 with F a PQ-isometric (instead of Q-isometric) map. To be PQ-isometric is in general much stronger than to be Q-isometric, i.e. a PQisometric map between metric spaces is automatically Q-isometric but the contrary is in general false. However we have the following important result: Theorem 1.3. Let F W X ! X 0 be a Q-isometry between geodesic metric spaces. Assume that X 0 is Gromov hyperbolic. Then (1) X is Gromov hyperbolic, (2) F is PQ-isometric. The first part of this theorem implies the fundamental fact that hyperbolicity is a Q-isometry invariant for geodesic Gromov hyperbolic spaces. We formulate this as: Corollary 1.4. Let X and X 0 be quasi-isometric geodesic metric spaces. If X is hyperbolic, then also X 0 is hyperbolic.

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This result makes it in particular possible to speak about hyperbolic groups. (For a discussion about hyperbolic groups and their boundaries, see e.g. [Gr1], [Gr2], [CDP], [G], [L], [Pau], [BeM].) A finitely generated group G is called hyperbolic if the Cayley graph .G; S/ is hyperbolic for some (and hence by the corollary any) finite symmetric generating set S. For the notion of Cayley graphs see Section 2. The second part of Theorem 1.3 is also the main step in the proof of the first part of Theorem 1.2. The proof of Theorem 1.3 relies essentially on the stability of quasigeodesics whose classical version in the hyperbolic plane is due to Morse (see [Mo1], [Mo2]). Since this stability is so fundamental we discuss the classical case briefly in Section 2. As already mentioned this article is based on [BS1]. There are many other interesting introductions to Gromov hyperbolic spaces and coarse geometry, e.g. [Ro], [Bow], [BoS], [CP], [V]. For the classical hyperbolic geometry it is still worth to look at papers of Poincaré, see e.g. [Po], and to study the survey [Mi2]. In this article we focus on the 1 W 1 correspondence between (visual and geodesic) Gromov hyperbolic spaces (up to rough isometry) and the complete quasi-metric spaces (up to bilipschitz Möbius equivalence). We have included Section 10 to discuss the fundamentals of the theory of quasi-metric spaces and their Möbius geometry. I want to thank Athanase Papadopoulos for inviting me to give the course in Strasbourg and Jianguo Cao for the invitation to Nanjing. I profited a lot from the audiences of these courses, in particular from many discussions with Julian Jordi. Finally I also thank Linus Romer for drawing the pictures.

2 Motivation and examples In this section we introduce some of the basic notation and discuss some examples which illustrate the concepts of asymptotic geometry. In particular we introduce and motivate the notion of quasi-isometry, which is fundamental in our approach. We also discuss the stability of quasi-geodesics in the hyperbolic plane. The generalization of this stability for geodesic Gromov hyperbolic spaces is an essential tool for the study of the quasi-isometric maps between such spaces.

2.1 Basic notation As usual we define a metric on a set X as a function d W X  X ! R which is (i) positive: d.x; y/  0 for each x, y 2 X and d.x; y/ D 0 if and only if x D y; (ii) symmetric: d.x; y/ D d.y; x/ for each x, y 2 X ; (iii) satisfies the triangle inequality: d.x; z/  d.x; y/ C d.y; z/ for every x, y, z 2 X. Given a metric d the value d.x; y/ is called the distance between the points x, y. We often use the notation jxyj for the distance between x, y in a given metric space X .

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A map f W X ! X 0 between metric spaces is said to be isometric if it preserves distances, i.e. jf .x/f .y/j D jxyj for all x, y 2 X . Clearly, every isometric map is injective. A map f W X ! X 0 is called a homothety if there exists a constant   0 such that jf .x/f .y/j D jxyj for all x, y 2 X. Clearly, every non-constant homothetic map is injective. A geodesic in a metric space X is a homothetic map  W I ! X , where I  R is an interval (open, closed or half-open, finite or infinite). We call  parametrized by arclength if  is isometric. The image .I / of such a map is also called a geodesic. A metric space X is said to be geodesic if any two points in X can be connected by a geodesic, i.e. if for x; y 2 X there exists a homothetic map  W Œ0; 1 ! X with .0/ D x and .1/ D y. If x; y 2 X are points of a geodesic metric space X , then we denote by xy some geodesic between x and y, even when the geodesic is not uniquely defined. If x; y; z 2 X are three points, then we denote by xyz a triangle formed by three geodesics xy, yz and zx. Again this triangle is not necessarily unique. We will often face the situation that a certain equality holds up to an error. Here the following notation is useful. Let a; b 2 R and c  0, then we use : a Dc b : as an alternative notation for jb  aj  c. Thus Dc means equality up to an (additive) error c. We also need the corresponding multiplicative notation. Let now a; b 2 .0; 1/ and let c  1, then we use a c b as a short version of error c.

1 c

a  b  c a. Thus c means equality up to a (multiplicative)

2.2 Motivation from geometric group theory We consider now an important example coming from geometric group theory. Let G be a finitely generated group and S  G a finite set of generators which is symmetric, i.e. S D S 1 . Let .G; S/ be the Cayley graph of G with respect to the generating set S . Thus the vertices of the graph are the elements of G and two vertices g; g 0 2 G are connected by an edge if g 1 g 0 2 S. We can define in a natural way a metric dS on .G; S / by deciding that every edge has length 1 and by defining dS .x; x 0 / to be the length of a shortest path between x and x 0 . Then ..G; S /; dS / is a geodesic metric space and the metric dS restricted to G  .G; S / is the word metric on G with respect to the generating set S, i.e. dS .g; g 0 / D kg 1 g 0 kS , where khkS is the minimal length of a word in the generators S representing h. The group G operates naturally by isometries on ..G; S/; dS /. In some sense the Cayley graph can be viewed as a geometric realization of the group and enables us to adapt some of the geometric

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language to the setting of group theory. For example we can speak about geodesics in the group etc. However the metric dS depends essentially on the generating set S. Here is a picture of the Cayley graphs .Z; f˙1g/ and .Z; f˙2; ˙3g/.

−2

−1

0

1

2

Consider another finite set S 0 of generators with S 0 D S 01 . Let a  1 be chosen such that khkS 0  a for all h 2 S and kh0 kS  a for all h0 2 S 0 . Then clearly 1 kgkS  kgkS 0  akgkS a for all g 2 G and it follows that the identity map id W .G; dS / ! .G; dS 0 / is bilipschitz with constant a. However there is (in general) no bilipschitz map between .G; S / and .G; S 0 / since these graphs are even topologically quite different. To compare these two metric spaces, we need the notion of a quasi-isometric map, which is a rough version of a bilipschitz map. Recall that a map f W X ! X 0 between metric spaces is bilipschitz if 1 jxyj  jf .x/f .y/j  a jxyj a for some a  1 and all x, y 2 X (in this definition, we do not require that f .X/ D X 0 ). A subset A  X 0 in a metric space X 0 is called a net, if the distances of all points 0 x 2 X 0 to A are uniformly bounded. A map f W X ! X 0 between metric spaces is quasi-isometric, if there are a  1, b  0 such that 1 jxyj  b  jf .x/f .y/j  a jxyj C b a for all x, y 2 X . In other words, a map is quasi-isometric if it is bilipschitz on large scales. If in addition, the image f .X/ is a net in X 0 , then f is called a quasi-isometry, and the spaces X and X 0 are called quasi-isometric. We also say that f is .a; b/-quasiisometric, and call a, b the quasi-isometry constants. Now we can extend the a-bilipschitz map id W .G; dS / ! .G; dS 0 /, to an .a; b/quasi-isometry ˆ W ..G; S/; dS / ! ..G; S 0 /; dS 0 /. For example map an edge connecting neighboring points g, gs in the graph .G; S / affinely onto a shortest geodesic from g to gs in the graph .G; S 0 /. This map is Lipschitz (but not bilipschitz) and an .a; b/-quasi-isometry for appropriate a  1 and b  0. The “quasi-isometry” relation is an equivalence relation on the class of metric spaces and hence .G; S/ and .G; S 0 / are in the same equivalence class. Now a geometric property of .G; S/ which is invariant under quasi-isometry is then a property which can be defined for the group G itself and which does not depend on the special generating set S.

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We will see that the hyperbolicity property (which we define in Section 3) is a quasi-isometry invariant for geodesic metric spaces. In particular we can define a finitely generated group G to be hyperbolic if .G; S / is hyperbolic for some (and hence for every) finite generating set. The quasi-isometry invariance of the notion of hyperbolicity for geodesic metric spaces is very basic for the whole subject. This quasi-isometry invariance relies on the stability of quasi-geodesics in hyperbolic spaces. Since this stability is so essential for the theory, we will give a preliminary discussion in this motivating section.

2.3 Stability of quasi-geodesics A quasi-geodesic in a metric space X is a quasi-isometric map  W I ! X where I  R is an interval. For general metric spaces a quasi-geodesic can be far from a geodesic. Consider for example the spiral  W .0; 1/ ! R2 , .t / D t .cos.ln t /; sin.ln t // in the Euclidean plane. p Since j.t /j D t and j 0 .t/j D 2 for all t , we easily see that 1 p j.t /.s/j  jt  sj  j.t /.s/j; 2 which implies that  is a quasi-geodesic. This curve is in no way close to some geodesic. In geodesic hyperbolic spaces the situation is completely different. We will show this basic phenomenon in the special case of the classical hyperbolic plane H2 . We will sketch a proof of the fact that a quasi-geodesic in H2 will stay at a uniform bounded distance from a true geodesic. This important result is due to M. Morse (see [Mo1], [Mo2]) and often called the Morse Lemma. Theorem 2.1. Let a  1, b  0. There exists H D H.a; b/ > 0 such that the image of every .a; b/-quasi-isometric map f W Œ0; L ! H2 lies in the H -neighborhood of the geodesic c W Œ0; l ! X with c.0/ D f .0/, c.l/ D f .L/, and vice versa c lies in the H -neighborhood of the image of f . We will prove here one implication. Namely that the image of f is in the H neighborhood of c. In the first step of the proof we replace f by a quasi-isometric map, which is in addition Lipschitz. Let 0 D t0 < t1 <    < tk D L be a partition of Œ0; L with jtiC1  ti j  1. We replace f by fQ W Œ0; L ! H2 . where fQ.ti / D f .ti / and fQjŒti ;ti C1  is the geodesic from f .ti / to f .tiC1 / parametrized proportionally to arclength. One easily checks that the new map fQ is still quasi-isometric and indeed a Lipschitz map and the corresponding constants can be computed from a and b. In addition the distance of f to fQ is bounded by a constant depending only on a and b. To simplify the notation we call the new map again f and the new constants again a

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and b, i.e. we assume without loss of generality that f satisfies 1 jt  t 0 j  b  jf .t/f .t 0 /j  ajt  t 0 j a for all t; t 0 2 Œ0; L. The main and essential tool of the proof is the following property of the hyperbolic plane, which can easily be deduced by writing the hyperbolic plane in Fermi coordinates with respect to the geodesic c. Lemma 2.2. Let c  H2 be a geodesic, let c W H2 ! c be the orthogonal projection, and let dc the distance to c. Assume that f W Œ˛; ˇ ! H2 is a continuous curve with dc .f .˛// D dc .f .ˇ// D r and dc .f .t//  r. Then L.f /  cosh.r/d , where L.f / is the length of f and d D d.c .f .˛//; c .f .ˇ///.

f .ˇ/

f .˛/ M0

M0 d

Assuming this lemma we prove our claim. Let M D maxfdc .f .t // j t 2 Œ0; Lg M and M 0 D 3a 2 . Let t0 2 Œ0; l such that dc .f .t0 // D M . Then there are 0 < ˛ < t0 < ˇ < L such that dc .f .˛// D M 0 D dc .f .ˇ// and dc .f .t //  M 0 for t 2 Œ˛; ˇ. Let L WD L.fjŒ˛;ˇ  /, d WD d.c .f .˛//; c .f .ˇ///. By the lemma and by the triangle inequality we have L  cosh.M 0 /d; 4 M D 4a2 M 0 : 3 Because f is a-Lipschitz and .a; b/-quasi-isometric, we compute L  2.M  M 0 / 

L  aj˛  ˇj  a2 .jf .˛/f .ˇ/j C b/  a2 .2M 0 C d C b/:

(3) (4)

(5)

From the inequalities (4) and (5) we obtain d  2M 0  b:

(6)

Let us now assume that M 0  b. Then by (6) we have d  M 0  b and hence with (5) L  4a2 d . Then by (3) we obtain cosh.M 0 /d  L  4a2 d and hence M 0  cosh1 .4a2 /. Thus we obtain M 0  maxfb; cosh1 .4a2 /g and hence M  H WD 3a2 maxfb; cosh1 .4a2 /g:

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3 Hyperbolic spaces In this section we introduce the concept of Gromov hyperbolic spaces. We define Gromov hyperbolic spaces via the ı-inequality. Later we reformulate the ı-inequality as a property of the cross difference triple.

3.1 ı-inequality and Gromov product Definition 3.1. Let ı  0. A metric space X satisfies the ı-inequality if for all points x; y; z; w 2 X we have jxyj C jzwj  maxf jxzj C jywj; jxwj C jyzj g C 2ı:

(7)

A metric space X is called Gromov-hyperbolic or simply hyperbolic if there exists a ı  0 such that X satisfies the ı-inequality. We try to give some pictorial description of this property. Assume that we are given four points x, y, z, w in X , then we have six distances which come naturally in three pairs. If we consider the three pairs (where we add distances) we obtain three numbers jxyj C jzwj;

jxzj C jywj;

jxwj C jyzj:

z

x

y

w 2ı.

The ı-inequality says that the two largest of the three numbers differ by at most



 2ı

We give two easy examples: 1. If X is a metric tree then four points look like in the following picture. In this case the three numbers are .a C b/ C .d C e/, .a C c C d / C .b C c C d /, .a C c C e/ C .b C c C d /, and hence the two largest of these numbers coincide. Indeed trees satisfy the 0-inequality.

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x

z a

d

c

b

e

y

w

2. If we consider X D R with the Euclidean metric and if we let xn ,pyn , zn , wn bep the vertices of a square of side length n, then the numbers are 2n, 2n, 2 2n. Since .2 2n  2n/ ! 1, the Euclidean space R2 does not satisfy the ı-inequality for any ı and is not hyperbolic. To give another interpretation of the ı-inequality, we introduce the following notion. 2

Definition 3.2. Let x; y; z 2 X, then 1 .jxyj C jxzj  jyzj/ 2 is called the Gromov product of y and z with respect to the point x. .yjz/x WD

There is a nice interpretation of the Gromov product in the case of geodesic metric spaces. Let X be a geodesic metric space and xyz a triangle in X formed by three geodesics xy, yz and zx. Then it is elementary to show the following: Lemma 3.3. There are uniquely determined points u 2 yz, v 2 zx and w 2 xy with the property that jxvj D jxwj, jzvj D jzuj and jywj D jyuj. We call u, v, w the equiradial points of the triangles. If we define a WD jxwj, b WD jyuj and c WD jzvj, then a D .yjz/x , b D .xjz/y and c D .yjx/z .

a

z

c v

c u

x

b a

w b

y

Thus the Gromov product .yjz/x is the formula which describes the distance of the vertex x to the equiradial points on the adjacent sides xy and xz. In the Euclidean space R2 (and also similarly in S 2 and H2 ), the equiradial points are just the touching points of the in-circle:

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z v u x w

y

In a metric tree, the three equiradial points coincide:

z c a

x

u=v=w b

y

The Gromov product is particularly useful to describe triangles in the hyperbolic plane H2 . Given three numbers a; b; c 2 .0; 1/, there exists an (up to congruence) unique triangle xyz 2 H2 with a D .xjy/z , b D .zjx/y , c D .xjy/z . Simply take the triangle xyz  H2 with jxyj D a C b, jyzj D b C c, jxzj D a C c. Thus the space .0; 1/3 parametrizes in a natural way these triangles. This parametrization works also for degenerate and ideal triangles and the space Œ0; 13 parametrizes these triangles. For example the point .a; b; 1/ with a > 0, b > 0 describes the ideal triangle with point z at infinity and equiradial points u, v, w with jxuj D jxvj D a, jyuj D jyvj D b and such that u, v are on the same horosphere based at z. In a similar way .a; 1; 1/ parametrizes the triangles with y and z at infinity and .1; 1; 1/ represents the (up to isometry) unique ideal triangle with three different points at infinity. We reformulate now the ı-inequality. We use the following notation: Definition 3.4. A ı-triple is a triple of three real numbers with the property that the two smallest of the numbers differ by at most ı. 

ı

Now the ı-inequality is equivalent to the fact that for all x; y; z; w 2 X the triple .jxyj  jzwj; jxzj  jywj; jxwj  jyzj/ is a 2ı-triple. By adding .jxwj C jywj C jzwj/ to every entry, this is equivalent to the property that .jwxj C jwyj  jxyj; jwxj C jwzj  jxzj; jwyj C jwzj  jyzj/ is a 2ı-triple. Multiplying with 12 this is equivalent to the property that for all x, y, z, w the numbers ..xjy/w ; .xjz/w ; .yjz/w / form a ı-triple.

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Since the point w here has the function of a basepoint, we reformulate the ıinequality is the following way: X satisfies the ı-inequality if for all x; y; z 2 X and for all o 2 X , ..xjy/o ; .xjz/o ; .yjz/o / is a ı-triple. Equivalently we can say that for all o; x; y; z 2 X , .xjy/o  minf.xjz/o ; .yjz/o g  ı:

(8)

We call (8) also the ı-inequality at the point o 2 X .

3.2 Cross difference and cross difference triple We can formalize the above considerations by introducing the cross difference and the cross difference triple, which are formal analogues to the cross ratio and the cross ratio triple which we consider for quasi-metric spaces in Section 10.4, where we discuss the Möbius geometry of these spaces. In some sense it is the “additive” analogue of the “multiplicative” setting of Möbius geometry. We therefore need to introduce a formal analogon to the projective space, or more precisely to the subset †  RP 2 introduced in Section 10.4. Let A2 D R3 = , where we define .a; b; c/ .a0 ; b 0 ; c 0 / if there exists a number d 2 R with .a C d; b C d; c C d / D .a0 ; b 0 ; c 0 /. We denote the equivalence class of .a; b; c/ with .a b c/. Note that we can identify A2 with f.a; b; c/ j a C b C c D 0g, and it carries a natural topology and is homeomorphic to R2 . For ı  0 the set of ı-triples Tı can be considered as a subset of A2 . For ı D 0, T0  A2 is the tripod spanned by  .2 1 1/,  .1 2 1/ and  .1 1 2/, for   0. For ı > 0 it is a thickened version of this tripod.

Tı

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We can reformulate the ı-inequality in this setting Lemma 3.5. A metric space satisfies the ı-inequality at a basepoint o 2 X if and only if for all x; y; z 2 X, ..xjy/o .yjz/o .zjx/o / 2 Tı : We define the cross difference triple as the map cdt W X 4 ! A2 , cdt.x; y; z; w/ D . .xjy/o C .zjw/o .xjz/o C .yjw/o .xjw/o C .yjz/o /; where o is an arbitrary basepoint. Note that the definition is independent from the basepoint and that we also have 1 cdt.x; y; z; w/ D  .jxyj C jzwj jxzj C jywj jxwj C jyzj/: 2 Therefore X satisfies the ı-inequality at o if and only if for all x; y; z 2 X we have cdt.x; y; z; o/ 2 Tı . Hence we can reformulate: X satisfies the ı-inequality if and only if cdt.x; y; z; w/ 2 Tı for all quadruples .x; y; z; w/ 2 X 4 . Now we cite the tetrahedron lemma (see [BS1], Section 2). Lemma 3.6. Let d12 , d13 , d14 , d23 , d24 , d34 be six numbers such that the four triples A1 D .d23 ; d24 ; d34 /, A2 D .d13 ; d14 ; d34 /, A3 D .d12 ; d14 ; d24 / and A4 D .d12 ; d13 ; d23 / are ı-triples. Then B D .d12 C d34 ; d13 C d24 ; d14 C d23 / is a 2ı-triple. Consequently, the tetrahedron lemma implies that if cdt.x; y; z; o/, cdt.x; y; w; o/, cdt.x; z; w; o/, cdt.y; z; w; o/ 2 Tı , then cdt.x; y; z; w/ 2 T2ı . Hence we have Proposition 3.7. If X satisfies the ı-inequality for some basepoint o 2 X , then X satisfies the 2ı-inequality for every other basepoint. The “additive” analogon to the cross ratio is the cross difference cd.x; y; z; w/ WD .xjy/o C .zjw/o  .xjz/o  .yjw/o ; where the basepoint is arbitrary. We also use the notation hx; y; z; wi for cd.x; y; z; w/. It is easy to see that hx; y; z; wi D

1 .jxzj C jywj  jxyj  jzwj/: 2

Furthermore we have hx; y; z; wi D .xjy/w  .xjz/w :

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4 Boundaries of hyperbolic spaces In this section we associate to every Gromov hyperbolic space X its boundary @1 X . Let X be a hyperbolic space and let o 2 X be a basepoint. A sequence .xi / is called a Gromov sequence if .xi jxj /o ! 1, i.e. if for all K  0 there exists N 2 N such that for all i; j  N we have .xi jxj /o  K. Since j.xjx 0 /o  .xjx 0 /o0 j  joo0 j this definition is independent of the chosen basepoint. Two Gromov sequences .xi / and .yi / are called equivalent, if .xi jyj /o ! 1. Remark 4.1. (1) We remark that for two Gromov sequences .xi / and .yi / we have the following equivalent statements: lim .xi jyi /o D 1 ()

i!1

lim .xi jyj /o D 1:

i;j !1

Indeed the implication ( is clear. To see ) we note that by the ı-inequality .xi jyj /o  minf.xi jyi /o ; .yi jyj /o g  ı; hence .xi jyi /o ! 1 implies .xi jyj /o ! 1, since .yi jyj /o ! 1 because .yi / is a Gromov sequence. (2) The relation “equivalent” is indeed an equivalence relation. The only nontrivial part is the transitivity. Let .xi / equivalent to .yi / and .yi / equivalent to .zi /, then .xi jyi /o D ai ! 1, .yi jzi /o D bi ! 1, hence .xi jzi /o  min.ai ; bi /  ı ! 1 and thus by part (1) .xi / and .zi / are equivalent. (3) If .xi / is a Gromov sequence and .yi / is a sequence with .xi jyi /o ! 1, then .yi / is also a Gromov sequence, which is then equivalent to .xi /. Indeed using the ı-inequality two times we have .yi jyj /o  minf.yi jxi /o ; .xi jxj /o ; .xj jyj /o g  2ı; and all terms in the minimum converge to 1. We now define @1 X as the set of equivalence classes of Gromov sequences.

4.1 Gromov product on the boundary We have seen in Section 3 that in the classical hyperbolic space one can define the Gromov product also for ideal triangles, i.e. if some vertices of the triangle are at infinity. We will generalize this now for arbitrary Gromov hyperbolic spaces. Let ;  0 2 @1 X and let .xi / 2 , .xi0 / 2  0 sequences representing these points. we would like to define .j 0 /o D lim.xi jxi0 /o . However a significant difficulty arises. This limit does not exist in general, as the following example shows.

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 y4

y4

y3

y3

y2

y2

y1

x2 x3 x4 

o

x1 x2 x3 x4

In this picture we visualize a graph, all of its edges have length 1. The sequences .xi / and .xi0 / represent the point  2 @1 X, and the sequences .yi / and .yi0 / the point  0 . Note that .xi jyi /o D 0 for all i 2 N, while .xi0 jyi0 /o D 2 for all i 2 N. To handle this difficulty, we make the following construction in order to define the desired Gromov product. For a basepoint o 2 X and points ,  0 2 @1 X , we define their Gromov product by .j 0 /o D inf lim inf .xi jxi0 /o ; i !1

where the infimum is taken over all sequences fxi g 2 , fxi0 g 2  0 . Note that .j 0 /o takes values in Œ0; 1, that .j 0 /o D 1 if and only if  D  0 , and that j.j 0 /o  .j 0 /o0 j  joo0 j for any o, o0 2 X. We obtain the following properties: Lemma 4.2. Let o 2 X and X satisfies the ı-inequality for o, and let ,  0 ,  00 2 @1 X . (1) For arbitrary sequences fyi g 2 , fyi0 g 2  0 , we have .j 0 /o  lim inf .yi jyi0 /o  lim sup .yi jyi0 /o  .j 0 /o C 2ı: i!1

i!1

(2) .j 0 /o , . 0 j 00 /o , .j 00 /o is a ı-triple. For a proof compare [BS1], Chapter 2. Similarly, the Gromov product .xj/o D inf lim inf .xjxi /o i !1

is defined for any x 2 X ,  2 @1 X, where the infimum is taken over all sequences fxi g 2 , and the ı-inequality holds for any three points from X [ @1 X . We also have that for x 2 X and a sequence zi representing  2 @1 X .xj/o  lim inf .xjzi /o  lim sup .xjzi /o  .xj/o C ı: i!1

i !1

Since .xjzi /y C .yjzi /x D jxyj this implies the estimate jxyj  2ı  .xj/y C .yj/x  jxyj:

(9)

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There are important cases, where the difficulty with the nonexistence of limit does not occur. Definition 4.3. We call a Gromov hyperbolic space X boundary continuous if the Gromov product extends continuously to the boundary in the following sense: let .xi /; .yi / sequences in X which converge to points x, N yN in X or in @1 X , then .xi jyi /o ! .xj N y/ N o for every o 2 X . Gromov hyperbolic spaces satisfying the ı-inequality with ı D 0 are boundary continuous. The classical hyperbolic space Hn is boundary continuous and more generally every proper CAT.1/-space is boundary continuous (see [BS1]). This implies that many inequalities for Gromov hyperbolic spaces become equalities in the case of CAT.1/ spaces. Sometimes it is therefore useful to trace why we have estimates which involve some ı or 2ı. Come these error terms from the non-continuity of the Gromov product on the boundary or not? We will see in the following subsection on Busemannn functions many cases of this phenomenon. Remark 4.4. In the above definition of the Gromov product we use the lim inf (and not for example the lim sup), since ı-triples are stable under the lim inf-operation: Let .ai ; bi ; ci / 2 R3 are ı-triples, i D 1; 2 : : : , then .lim inf ai ; lim inf bi ; lim inf ci / is a ı-triple. Definition 4.5. A Gromov hyperbolic space is called visual if for a given basepoint o 2 X there exists a constant D  0, such that for any point x 2 X there exists a point  2 @1 X such that joxj  .xj/o  D:

x

ÄD



o

In some sense, a space X is visual if any point x of the space can be seen by looking to the sky (the boundary at infinity) at least up to a fixed error D. We remark that this definition of visual differs from the definition of visibility, i.e. the existence of geodesics connecting pairs of points in the boundary.

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4.2 Busemann functions Let ! 2 @1 X . We define the Busemann cocycle b! W X  X ! R by b! .x; y/ WD .yj!/x  .xj!/y : : Recall that for a; b 2 R and c  0, we use a Dc b as a notation for jb  aj  c. With this notation we have: Lemma 4.6. (1) b! .x; y/ D b! .y; x/. : (2) b! .x; y/ C b! .y; z/ C b! .z; x/ D3ı 0. Proof. The first equality is obvious. To obtain the second estimate, we write the left-hand side as .yj!/x  .xj!/y C .zj!/y  .yj!/z C .xj!/z  .zj!/x : If we replace ! 2 @1 X by a finite point w 2 X, then all terms cancel in pairs and the expression is 0. Now take a Gromov sequence .wi / representing ! such that for all u; v 2 fx; y; zg, the expression .u; wi /v converges. Since .uj!/v  lim.ujwi /v  .uj!/v C ı; we have in the limit at most an error 3ı. We define two comparison functions b!u ; b!l W X  X ! R by b!u .x; y/ WD jxyj  2.xj!/y ; b!l .x; y/ WD 2.yj!/x  jxyj: We will call them the upper and lower comparison function and we will show b!l .x; y/  b! .x; y/  b!u .x; y/:

(10)

0  b!u  b!  2ı

(11)

0  b!  b!l  2ı:

(12)

More strongly we have and For boundary continuous hyperbolic spaces we have equality, i.e. all the functions b!l ; b! ; b!u coincide. To show (11) we observe b!u .x; y/  b! .x; y/ D jxyj  Œ.xj!/y C .yj!/x  2 .0; 2ı/ by equality (9). The estimate (12) follows in the same way. We define now the set B.!/ of Busemann functions based at ! 2 @1 X as the set of all functions b W X ! R such that b!l .x; y/  .b.x/  b.y//  b!u .x; y/:

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Again for boundary continuous hyperbolic spaces we here have equality. The set B.!/ has the following properties: (1) B.!/ contains all functions b!;o ./ WD b! .; o/, where o 2 X . It also contains u l the corresponding functions b!;o ./ D b!u .; o/ and b!;o ./ D b!l .; o/. (2) For every b 2 B.!/ and every constant c 2 R, the function b C c is in B.!/. (3) Any two functions b, b 0 2 B.!/ differ from each other by a constant up to an error  4ı. To see the last estimate consider the function f D b  b 0 . We show that f is constant up to error 4ı. Therefore we compute : f .x/  f .y/ D .b.x/  b.y// C .b 0 .y/  b 0 .x// D4ı b! .x; y/ C b! .y; x/ D 0: (4) B.!/ contains the functions which are constructed in the following way: let o 2 X be some basepoint and let .wi / be any sequence representing ! 2 @1 X . For x 2 X the sequence jxwi j  jowi j is bounded. Consider the bounded functions bi .x/ D jxwi j  jowi j and take an accumulation function b. Then b 2 B.!/. We show that b.x/  b.y/  b!u .x; y/, the other inequality is similar. Indeed let x; y 2 X, then there is a subsequence, which we denote for simplicity also by wi , such that b.x/ D lim.jxwi j  jowi j/ and b.y/ D lim.jywi j  jowi j/. Thus b.x/  b.y/ D lim.jxwi j  jywi j/ D lim.jxyj  2.xjwi /y /  jxyj  2.x; !/y : We introduce now the Gromov product with respect to a Busemann function. Let b 2 B.!/; then 1 .xjy/b WD .b.x/ C b.y/  jxyj/: 2 u If we take b D b!;o 2 B.!/, we obtain .xjy/b D .xjy/o  .xj!/o  .yj!/o for x; y 2 X. We denote .xjy/b also with .xjy/!;o . Note that we can extend this expression to all points x, N yN 2 @1 X n f!g. We also remark that for all x; N y; N zN 2 @1 X n f!g, .x; N y/ N !;o ; .y; N z/ N !;o ; .z; N x/ N !;o is a 2ı-triple. This will play a major role when we will consider the Möbius structure on @1 X .

5 Geodesic hyperbolic spaces We start with the definition of geodesic ı-hyperbolic spaces. Definition 5.1. A geodesic metric space is called ı-hyperbolic if for all x; y; z 2 X the following holds: if y 0 2 xy and z 0 2 xz are points with jxy 0 j D jxz 0 j  .yjz/x , then jy 0 z 0 j  ı.

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This property is visualized in the following picture, where we indicate which points have distance  ı.

ı Äı

ı

For a triangle in a ı-hyperbolic geodesic space X the equiradial points of the triangle have pairwise distance  ı. Thus in some sense every triangle is subdivided in four smaller triangles; namely three ı-thin ends which come together in a small triangle spanned by the equiradial points. Thus the triangle is ı-close to a tripod. It is not difficult to prove Lemma 5.2. Let X be a geodesic metric space. (1) If X is ı-hyperbolic, then X satisfies the ı-inequality. (2) If X satisfies the ı-inequality, then X is 4ı-hyperbolic. The property that in geodesic hyperbolic spaces triangles are thin can be used as a definition of hyperbolicity. Definition 5.3. A triangle xyz in a geodesic metric space is called -thin (here  0 is some constant) if every side of the triangle is in the neighborhood of the union of the two other sides. If X is a ı-hyperbolic geodesic space, then obviously all triangles are ı-thin. On the other hand, if all triangles are ı-thin, then the space satisfies the ı 0 -inequality for some ı 0 only depending on ı. In geodesic ı-hyperbolic spaces one can define some map Trip W X 3 ! X , which associates to a triple .x; y; z/ some equiradial point of some side of some triangle xyz. Thus in a tree, Trip.x; y; z/ is the tripod point. In general here we have to make some choices. However it follows easily from the definition that for ı-hyperbolic geodesic spaces the point Trip.x; y; z/ is uniquely defined up to error 2ı. Similarly as the Gromov product, the map Trip extends naturally to the boundary @1 X and can hence be considered as a map Trip W Xx 3 ! Xx , where Xx D X [ @1 X . One can generalize the stability of quasi-geodesics to the setting of ı-hyperbolic geodesic spaces.

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Theorem 5.4 (Stability of geodesics). Let X be a ı-hyperbolic geodesic space, a  1, b  0. There exists H D H.a; b; ı/ > 0 such that the image im.f / of every .a; b/quasi-isometric map f W Œ0; d  ! X lies in the H -neighborhood of any geodesic c W Œ0; l ! X with c.0/ D f .0/, c.l/ D f .d /, and vice versa, c lies in the H neighborhood of im.f /. Indeed it is possible to embed every hyperbolic space isometrically into some geodesic hyperbolic space. The following result is due to Bonk and Schramm [BoS]. Theorem 5.5. Let X satisfy the ı-inequality. Then X can be isometrically embedded in a complete geodesic metric space Xz such that Xz also satisfies the ı-inequality. Since we are interested in the interplay between X and the boundary @1 X we also formulate the following version. Theorem 5.6. Let X satisfy the ı-inequality. Then X can be isometrically embedded in a geodesic metric space Xy such that (1) Xy satisfies the ı-inequality; (2) @1 Xy D @1 X; (3) if X is visual, then also Xy is visual. We give a proof of this result in the Appendix. Remark 5.7. We do not know if there is a complete geodesic space such that (1), (2) and (3) hold.

5.1 Geodesic boundary For geodesic hyperbolic spaces one can define the boundary in an alternative way. Two geodesic rays ,  0 W Œa; 1/ ! X (parametrized by arclength) in a geodesic space X are called asymptotic if j.t / 0 .t/j  C < 1 for some constant C and all t  a. To be asymptotic is an equivalence relation on the set of the rays in X , and the set of classes of asymptotic rays is sometimes called the geodesic boundary of X , @g X. We then denote the class of  with .1/. (1)  

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In a geodesic hyperbolic space, asymptotic rays are at a uniformly bounded distance from each other. If a geodesic space X is Gromov hyperbolic, then obviously @g X  @1 X. Here we identify the point .1/ with the class Œ.i / 2 @1 X . There are examples where the two boundaries are different. However, there are important cases when @g X D @1 X. In particular it is not difficult to prove that for a proper geodesic hyperbolic space X we have @g X D @1 X. Given a geodesic  W Œ0; 1/ ! X we define as usual the Busemann function b W X ! R of the ray by b .x/ D lim .jx.t /j  j.t /.0/j/: t!1

We relate this to our previous definition of the Busemann cycle. It is not difficult to show that for o WD .0/ and ! WD .1/ 2 @1 X , we have : b .x/ D4ı b! .x; o/: In the case where X is boundary continuous, we have equality.

6 A metric structure on @1 X Up to now @1 X is only a set. However we can use the Gromov product to define a metric structure on the boundary. Let a > 1 and let o 2 X. Then we define  W @1 X  @1 X ! Œ0; 1/ by 0

.;  0 / WD a.j /o : The ı-inequality on @1 X implies that .;  00 /  K maxf.;  0 /; . 0 ;  00 /g for all ;  0 ;  00 2 @1 X, where K D aı . Thus  defines a K-quasi-metric on @1 X , where a K-quasi-metric on a set Z is a map  W Z  Z ! Œ0; 1 with: (i) .z; z 0 /  0 for every z, z 0 2 Z and .z; z 0 / D 0 () z D z 0 ; (ii) .z; z 0 / D .z 0 ; z/ for every z, z 0 2 Z; (iii) .z; z 00 /  K maxf.z; z 0 /; .z 0 ; z 00 /g for every z, z 0 , z 00 2 Z. We give a detailed discussion on quasi-metric spaces and its Möbius geometry in Section 10. In particular this quasi-metric defines a topology on @1 X . The quasi-metric  defined on @1 X depends on the basepoint o 2 X and the chosen parameter a > 1. If we emphasize this dependence, we write o;a . Let o, o0 2 X . Since j.j 0 /o  .j 0 /o0 j  joo0 j we compute c 1  0

o;a .;  0 /  c; o0 ;a .;  0 /

where c D ajoo j . Thus the quasi-metrics o;a and o0 ;a are bilipschitz equivalent.

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If a, a0 > 1 are different parameters then we have ˛ ; o;a0 D o;a 0

where ˛ D lnlnaa . This discussion implies in particular that the topology induced by the quasi-metric  does not depend on the basepoint o 2 X and the parameter a > 1. Indeed, one can show that the quasi-metric is complete. Proposition 6.1. If X is a hyperbolic space, then .@1 X; o;a / is a complete quasimetric space. We denote the complete quasi-metric space .@1 X; o;a / also with @o;a 1 X . We will also consider quasi-metrics coming from Busemann functions. Let ! 2 @1 X , and let b 2 B.!/. We now define for x; y 2 @1 X n f!g, b;a .x; y/ WD a.xjy/b : u with the Gromov product In particular we consider b D b!;o

.xjy/b D .xjy/o  .xj!/o  .yj!/o and obtain

!;o;a .x; y/ WD a.xjy/!;o :

This is a quasi-metric, since for all x; N y; N zN 2 @1 X n f!g, .x; N y/ N !;o ; .y; N z/ N !;o ; .z; N x/ N !;o is a 2ı-triple. We consider !;o;a as a quasi-metric on @1 X , where ! is the infinitely remote point of !;o . We should mention that in the case of CAT.1/ spaces, the quasi-metric  is indeed a metric for the choice of the constant a D e, the Euler number. This was proved by Bourdon [Bou] for the metrics o;e . For the metrics !;o;e it follows from [FS2] and is in some sense implicitly already contained in the construction by Hamenstädt [Ha]. Theorem 6.2. Let X be a CAT.1/ space, o 2 X, ! 2 @1 X . Then o;e and !;o;e are metrics on @1 X.

7 Morphisms of hyperbolic spaces We consider maps F W X ! X 0 between metric spaces which preserve certain quantities. In general we use x 0 2 X 0 as a notation for the image of x 2 X, i.e. x 0 D F .x/. We give some examples, where we formulate also equalities in terms of double inequalities. The map F is called

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• isometric if for all x; y 2 X, jxyj  jx 0 y 0 j  jxyjI • similar (or homothetic) if there exists  > 0 such that for all x; y 2 X ,  jxyj  jx 0 y 0 j   jxyjI • rough isometric if there exists b  0 such that for x; y 2 X, jxyj  b  jx 0 y 0 j  jxyj C bI • rough similar if there exists  > 0, b  0 such that for x; y 2 X ,  jxyj  b  jx 0 y 0 j   jxyj C bI • bilipschitz if there exists c  1 such that for x; y 2 X , 1 jxyj  jx 0 y 0 j  c jxyjI c • Q-isometric if there exists c  1, b  0 such that for x; y 2 X, 1 jxyj  b  jx 0 y 0 j  c jxyj C b: c In our context we also want to control other quantities. In particular we want to control the Gromov product .xjy/z and the cross difference cd.x; y; z; w/ D hx; y; z; wi, which we have defined as hx; y; z; wi WD .xjy/o C .zjw/o  .xjz/o  .yjw/o ; for an arbitrary point o 2 X. Recall that 1 .jxzj C jywj  jxyj  jzwj/ D .xjy/w  .xjz/w : 2 We have hx; y; z; wi D  hx; z; y; wi. From the cross difference we can obtain back the Gromov product and the distance. Indeed hx; y; z; zi D .xjy/z ; hx; y; z; wi D

and hx; x; y; yi D jxyj: We want to control the cross difference by special functions i , i.e. ˛ ˝

1 .hx; y; z; wi/  x 0 ; y 0 ; z 0 ; w 0  2 .hx; y; z; wi/: Affine control functions play an important role in this case.

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Definition 7.1. A map F W X ! X 0 is called power quasi-isometric or PQ-isometric if there exist a  1 and b  0 such that for all x; y; z; w 2 X with hx; y; z; wi  0, ˛ ˝ 1 hx; y; z; wi  b  x 0 ; y 0 ; z 0 ; w 0  a hx; y; z; wi C b: a If we want to emphasize the constants we speak about .a; b/-PQ-isometric maps. What is the case for quadruples with hx; y; z; wi < 0? In this case hx; z; y; wi > 0 and therefore ˛ ˝ 1 hx; z; y; wi  b  x 0 ; z 0 ; y 0 ; w 0  a hx; z; y; wi C b; a hence ˛ 1 ˝ a hx; y; z; wi  b  x 0 ; y 0 ; z 0 ; w 0  hx; y; z; wi C b: a Thus the control functions are of the form #1 .t / D at b for t  0 and #1 .t / D a1 t b for t  0, #2 .t/ D a1 t C b for t  0 and #2 .t / D at C b for t  0.

#2



b

#1

b

 Remark 7.2. The notion of PQ-isometric maps in the “additive” theory of hyperbolic spaces corresponds to the notion of PQ-Möbius maps in the “multiplicative” theory of quasi-metric spaces. For PQ-isometric maps also the Gromov product and the distance can be controlled. Indeed, if F is .a; b/-PQ-isometric, then 1 .xjy/o  b  .x 0 jy 0 /o  a .xjy/o C b a and 1 jxyj  b  jx 0 y 0 j  a jxyj C bI a in particular PQ-isometric maps are Q-isometric. Since the cross difference determines the Gromov product and hyperbolicity is described via the Gromov product one can see the following.

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Proposition 7.3. Let F W X ! X 0 be PQ-isometric. Then (1) if X 0 is hyperbolic, then X is hyperbolic; (2) if X hyperbolic, then F .X/ is hyperbolic. We sketch the proof of Theorem 1.3, which is one of the most important results of the whole theory. We restate the result: Theorem 7.4. Let F W X ! X 0 be a Q-isometry between geodesic metric spaces. Assume that X 0 is Gromov hyperbolic. Then (1) X is Gromov hyperbolic; (2) F is PQ-isometric. The proof of this theorem relies essentially on the stability of quasi-geodesics in geodesic hyperbolic spaces, which is the generalization of the geodesic stability in the classical hyperbolic space H2 discussed in Section 2 and stated in Section 5. We give a rough sketch of the idea of the proof. (1) We assume that X 0 is hyperbolic and have to prove that X is hyperbolic. Thus we have to show that triangles in X are uniformly thin. Assume the contrary, then there are triangles xi yi zi in X with a point wi 2 xi yi such that the distance of wi to the union of the sides yi zi and zi xi goes to infinity, i.e. d.wi ; yi zi [zi xi / ! 1. Since F is quasiisometric also d.F .wi /; F .yi zi / [ F .zi xi // ! 1. Now by the stability of quasigeodesics the images F .xi yi /, F .yi zi /, F .zi xi / are at uniform bounded distance from geodesics xi0 yi0 , yi0 zi0 and zi0 xi0 . This implies that d.wi0 ; yi0 zi0 [ zi0 xi0 / ! 1. Since F .xi yi / is at bounded distance from xi0 yi0 , there is a point wi00 2 xi0 yi0 with d.wi0 ; wi00 / bounded and hence also d.wi00 ; yi0 zi0 [ zi0 xi0 / ! 1. Thus xi0 yi0 is not in a bounded neighborhood to yi0 zi0 [ zi0 xi0 . This contradicts the hyperbolicity of X 0 . (2) To prove the second statement we formulate Lemma 7.5. Let F W X ! X 0 be a Q-isometric map between geodesic Gromov hyperbolic spaces. Then there exists some constant h  0 (only depending on the quasi-isometry constants and ı, ı 0 ) such that d.F .Trip.x; y; z//; Trip.x 0 ; y 0 ; z 0 //  h: Indeed the point Trip.x; y; z/ can be characterized by the property that it is the point where the three geodesics xy, yz, zx are close together. Since F .xy/, F .yz/ and F .zx/ are close to the geodesics x 0 y 0 , y 0 z 0 and z 0 x 0 , it is not difficult to see that Trip.x; y; z/ is mapped close to Trip.x 0 ; y 0 ; z 0 /. To show that a Q-isometric map F is indeed a PQ-isometric map, we have to show that the cross difference is preserved up to an affine error function. Consider now the cross difference as hx; y; z; wi D .xjy/w  .xjz/w : We visualize this in a metric tree in the following picture.

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y x

v w

u

z

Here hx; y; z; wi is the signed distance between the bifurcation points u and v, in the situation as drawn in the picture, hx; y; z; wi D juvj. If the space X is a geodesic Gromov hyperbolic space and x; y; z; w 2 X, then drawing six geodesics between these points we obtain a picture which is ı-close to the picture in a tree. We actually see that up to bounded error : hx; y; z; wi D j Trip..w; y; z/ Trip.w; x; z/j; and correspondingly ˝ 0 0 0 0˛ : x ; y ; z ; w D  j Trip..w 0 ; y 0 ; z 0 / Trip.w 0 ; x 0 ; z 0 /j: By the above lemma Trip.w; y; z/ goes to Trip.w 0 ; y 0 ; z 0 / and Trip.w; x; z/ goes to Trip.w 0 ; x 0 ; z 0 /. Thus the cross difference is distorted by a bounded additive error.

8 Möbius geometry of @1 X Up to now we have introduced the metrics o;a on @1 X and have obtained the complete quasi-metric space @o;a 1 X D .@1 X; o;a /. We now introduce a kind of Möbius structure on @1 X. The following observation about the dependence of the metric from the basepoint is essential. A map f D .Z; / ! .Z 0 ; 0 / between two quasi-metric spaces is called L-bilipschitz Möbius, where L  1, if for all x; y; z; w 2 Z with images x0; y 0; z0; w0 2 Z 0, 1 Œx; y; z; w  Œx 0 ; y 0 ; z 0 ; w 0   L Œx; y; z; w; L where Œx; y; z; w is the cross ratio Œx; y; z; w D

.x; z/.y; w/ ; .x; y/.z; w/

respectively the cross ratio on .Z 0 ; 0 /. Proposition 8.1. There exists a constant L depending only on ı and a with the followo0 ;a ing property: Let o; o0 2 X , ! 2 @1 X, then id W @o;a 1 X ! @1 X is an L-bilipschitz !;o;a X is L-bilipschitz Möbius. Möbius map and id W @o;a 1 X ! @1

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Proof. Let x; y; z; w 2 @1 X. Then Œx; y; z; wo;a D a.xjy/o C.zjw/o .xjz/o .yjw/o D ahx;y;z;wi ; where Œ  o;a is the cross ratio with respect to o;a . Now for finite points x; y; z; w 2 X the expression .xjy/o C .zjw/o  .xjz/o  .yjw/o does not depend on the basepoint. If now x; y; z; w 2 @1 X then choose sequences : .xi /, .yi /, .zi /, .wi / representing these points. We have lim sup.xi jyi /o D2ı .xjy/o and corresponding estimates for the other terms. Since .xi jyi /o C .zi jwi /o  .xi jzi /o  .yi jwi /o D .xi jyi /o0 C .zi jwi /o0  .xi jzi /o0  .yi jwi /o0 we have .xjy/o C .zjw/o  .xjz/o  .yjw/o : D8ı .xjy/o0 C .zjw/o0  .xjz/o0  .yjw/o0 : Thus Œx; y; z; wo;a L Œx; y; z; wo0 ;a for L D a8ı . Now recall that .xjy/!;o D .xjy/o  .xj!/o  .yj!/o ; and hence !;o .x; y/ D

o .x; y/ : o .x; !/o .y; !/

This is just the involution metric at the point !. For a detailed discussion compare Section 10. Therefore id W .Z; o / ! .Z; !;o / is indeed a Möbius map and Œx; y; z; w!;o;a D Œx; y; z; wo;a : In the case where the space X is boundary continuous and a > 1 is fixed, all the metrics o;a , !;o;a are Möbius equivalent metrics on @1 X . Thus we have a welldefined cross ratio Œx; y; z; w on @1 X, only depending on the parameter a but not on the chosen basepoint. For general hyperbolic spaces we have a Möbius structure up to a fixed multiplicative error. This implies that we can define what bilipschitz Möbius maps and also what PQ-Möbius maps are (compare Section 10) on @1 X without referring to a fixed metric.

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9 Hyperbolic approximation In this section we briefly sketch a construction which associates to a complete quasimetric space .Z; / a Gromov hyperbolic space X such that @1 X with some metric o;a or !;o;a is the prescribed .Z; /. There are various constructions of this type in the literature, see e.g. [TV], [BoS], [BS1]. As a classical counterpart, we consider the reconstruction of the hyperbolic space Hn (in the upper half space model) out of the boundary Rn1 [ f1g. Let .Z; / be a given complete quasi-metric space according to Definition 10.4 of completeness. In particular Z is extended if it is unbounded. We will construct a hyperbolic space X, whose boundary @1 X is canonically equal to Z [ f1g, i.e. if Z is unbounded, then @1 X D Z, and if Z is bounded, then @1 X is Z with an additional infinitely remote point. The space X will be a geodesic metric space and the geodesic boundary @g1 X will coincide with @1 X. We will consider on @1 X a metric 1;o;a defined by a Busemann function with respect to the point 1 2 @1 X . The Busemann functions belong to the geodesics which are asymptotic to 1. In addition the whole construction behaves nicely under scaling. We collect these properties in the following theorem. We note that the condition r < 1 in this theorem plays the same role as the 0 parameter a D 1=r > 1 in the definition of the metrics o;a and !;o;a . Thus r .zjz /b 0 in statement (6) is just a.zjz /b . Theorem 9.1. Let .Z; / be a complete quasi-metric space. For every 0 < r < 1 there exists a metric space Hypr .Z; / together with a function br W Hypr .Z; / ! R such that the following holds. (1) Hypr .Z; / is a geodesic Gromov hyperbolic space and br is a Busemann function. (2) @1 Hypr .Z; / D Z [ f1g canonically. (3) For all r; r 0 2 .0; 1/ we have Hypr .Z; / D we have Hypr .Z; s / D s Hypr .Z; /.

ln r 0 ln r

Hypr 0 .Z; / and for all s > 0

(4) For all z 2 Z n f1g there exists a geodesic cz W R ! X (parametrized proportionally to arclength) with cz .1/ D z, cz .1/ D 1 such that for z; z 0 2 Z n f1g there exists  2 R with cz .t/ D cz 0 .t / for all t   . (5) br is the common Busemann function for such geodesics cz with t ! 1. Thus br .x/ D lim t!1 .dr .x; c.t//dr .c.t /; c.0//. Here dr is the distance function in Hypr .Z; /. 0

(6) r .zjz /b  .z; z 0 /. In the special case where .Z; / is an ultrametric, the construction is in some sense canonical, and we consider this case first: Let 0 < r < 1 and consider the set Xy WD .Z n f1g/  R. We define the relation .z; t/ .z 0 ; s/ () t D s 

ln..z; z 0 // : ln r

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Since  is an ultrametric, this is an equivalence relation. Let Hypr .Z; / WD Xy = be the set of equivalence classes. We denote by Œz; t  the class of .z; t / and we define ²

ln..z; z 0 // dr .Œz; t; Œz ; s/ WD t C s  min t; s; ln r and br .Œz; t / D t . 0

[z, s]

³

z

[z  , t] z Thus we glue two copies fzgR and fz 0 gR of R together such that the bifurcation 0 // . Furthermore br is the Busemann function of each point is at the place  D ln..z;z ln r geodesic, with t ! 1. We obtain a tree whose ideal points are canonically Z [f1g, where 1 corresponds to cz .1/. 0 // 0 and r .zjz /b D .z; z 0 /. It is also not diffiBy construction .zjz 0 /b D  D ln..z;z ln r 0 cult to show that for r; r 0 2 .0; 1/ the space Hypr .Z; / is isometric to lnlnrr Hypr 0 .Z; / and Hypr .Z; s / is isometric to s Hypr .Z; /. Now we make the construction in the general case. It is based on the construction in [BS1] with modifications due to [J] in order that it works for general quasi-metric spaces. We construct in a combinatorial way a metric graph which is hyperbolic and has Z [ f1g as boundary. The construction is not canonical and depends on some choices. Let .Z; / be a complete K-quasi-metric space, where we now assume K > 1. Choose r D 1=.2K 3 / (in particular r < 1=K 3 ). For every k 2 Z let Vk be a maximal r k =K-separated subset of Z, where r k =K-separated means that .v; v 0 /  r k =K for different v; v 0 2 Vk . We assume in addition that the system .Vk /k2Z is hereditary, which means that Vk  VkC1 . In addition we assume that there is some “basepoint” z0 2 Z, with z0 2 Vk for all k 2 Z. Using some standard Zorn lemma argument one can see that such a system .Vk /k2Z exists. We now define V as the set of all ordered pairs .k; z/, where k 2 Z and z 2 Vk . Thus we can view V as the disjoined union of all Vk . The projection ` W V ! Z to the first coordinate is called the level function and the projection  W V ! Z onto the second coordinate is called the center function. We associate to every v 2 V the ball B.v/ WD Br `.v/ ..v// D fz 2 Z j .z; .v// < r `.v/ g: Now we define Hypr .Z; / to be the metric graph (all edges have length 1) with vertex set V , where two vertices v; w 2 V are joined by an edge when • `.v/ D `.w/ and B.v/ \ B.w/ ¤ ;, or • `.v/ D `.w/ C 1 and B.v/  B.w/. Thus two points in V can only be joined by an edge if their levels are either equal or neighbors, i.e. differ by one. An edge between vertices of the same level is called

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horizontal, an edge between points of neighboring levels is called radial. We describe geodesics in V joining points v; w 2 V as sequences of neighboring vertices. Now with the same methods as in [BS1], 6.2.10, one shows that this graph is a Gromov hyperbolic geodesic metric space. This is proved by looking at the geodesics in the graph V. If v; w 2 V , then there is a geodesic v D v0 ; v1 ; : : : ; vnC1 D w from v to w such that `.vi / < maxf`.vi1 /; `.viC1 /g for all 1  i  n. We want to indicate why this is possible: Consider a geodesic such that this property does not hold for some i , e.g. assume that `.vi1 / < `.vi / D `.viC1 /, then for the corresponding balls Bj D B.vj / we have Bi1  Bi and Bi \ BiC1 ¤ ;. Then one easily sees that Bi  B.v 0 / with `.v 0 / D `.vi /  1 and B.v 0 /  BiC1 . We can then replace the string vi 1 , vi ; vi C1 with vi1 , v 0 ; viC1 and the condition is satisfied. By making these kind of changes we finally obtain a geodesic with the desired properties. This shows that two edges v and w can be connected in V by a geodesic which has at most one horizontal edge. If such an edge occurs, it is on the lowest level. Thus schematically geodesics look as in the picture.

If we look to the geodesic on the left side, then it corresponds to a system of balls of the following type.

Let us now indicate that @1 Hypr .Z; / D Z [f1g in a canonical way. Therefore let us try to understand the geodesic boundary @g1 Hypr .Z; /. Consider a geodesic ray v0 ; v1 ; v2 ; : : : . By the behaviour of the levels of geodesics, we can assume that there exists some m 2 N such that for all i  m • either `.viC1 / D `.vi / C 1, or • `.vi C1 / D `.vi /  1. Hence the geodesic finally becomes radial (with only radial edges) where the levels are monotone increasing or monotone decreasing. In both cases the balls Bi D B.vi /

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form an inclusion chain, in the first case Bi  BiC1  BiC2     , and this chain has (since .Z; / is complete) a unique point z 2 Z as intersection point. On the other hand it is easy to see that every point z 2 Z n f1g can be realized by such a chain. Thus the endpoints of these geodesics correspond in a canonical way to the set Z n f1g. Consider on the other hand a geodesic ray v0 ; v1 ; v2 ; : : : whose levels are monotone decreasing. We assume for simplicity that `.vi / D i . Then the balls form an increasing chain B0  B1  B2     , and for any point z 2 B0 we have z 2 Bi for all i . Thus for i large enough, Bi \ B..i; z0 // ¤ ;, where .i; z0 / 2 V is our basepoint of Vi . Thus our geodesic ray is asymptotic to the ray .i; z0 /, i D 0; 1; 2; : : : , which is the canonical “ray to the point 1”. In particular we now see that for any point z 2 Z nf1g we can construct a geodesic cz as in Statement (4) of the theorem. It is now not too difficult to prove Statement (6). Namely if .z; z 0 / is (up to some multiplicative error   1) of the order r k for some k 2 Z, i.e. .z; z 0 /  r k , then the geodesics cz and cz 0 bifurcate approximately at time k, i.e. d.cz .t/; cz 0 .t// is small (smaller than 1) for t  k, and d.cz .t /; cz 0 .t // is equal to 2.t  k/ (up to some additive error only depending on r). Note also that the Busemann function b of the geodesics is equal to the level function. Thus 0 : .zjz 0 /b D k. Hence r .zjz /b  .z; z 0 / for some suitable error term only depending on k. This finishes the construction of the hyperbolic approximation. Note that in this construction we used the fact that r is sufficiently small. To obtain Hypr .Z; / for arbitrary 0 < r 0 < 1, we just scale the metric, i.e. Hypr 0 .Z; / WD ln r Hypr .Z; /. ln r 0 In the case where .Z; / is bounded, @1 Hypr .Z; / has the additional point 1. Sometimes it is useful to get rid of this point. Assume that diam.Z; / D D. Let then . Then, since Vk is r k =K-separated, it k0 be the largest integer such that k0 < lnlnD r contains only one point for k  k0 , namely the point .k; z0 /. If we throw away all levels k < k0 , then we obtain the truncated hyperbolic approximation. The boundary of the truncated approximation is in a canonical way .Z; /. Let now .Z; / and .Z 0 ; 0 / be two complete quasi-metric spaces and f W Z ! Z 0 a map. We are looking for a map F W Hypr .Z; / ! Hypr .Z 0 ; 0 / which has f as a boundary map. We have the following extension result, where we use the terminology of Section 10.5. Theorem 9.2. (1) If f is L-symmetric, then there exists a rough isometric map F W Hypr .Z; / ! Hypr .Z 0 ; 0 / with @1 F D f . (2) If f is PQ-symmetric, then there exists a PQ-isometric map F W Hypr .Z; / ! Hypr .Z 0 ; 0 / with @1 F D f . For a proof compare the similar statements in [BoS], [BS1], [J]. We give however a rough sketch without any details. If f is L-symmetric, then it sends balls in .Z; / to approximate balls in .Z 0 ; 0 /. Thus for a ball B.v/ of the hyperbolic approximation

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v 2 V , there is some v 0 2 V 0 in the hyperbolic approximation of .Z 0 ; 0 / with f .B.v// close to B.v 0 /. We indicate now why the map v 7! v 0 is a rough isometry. Let v; w 2 V be given and assume for simplicity that they are concentric, i.e. .v/ D .w/ D x, and B.v/  B.w/. Then d.v; w/ D `.v/  `.w/. Assume that there are points y near the boundary of B.v/ and z near the boundary of B.w/, i.e. .x; y/  r `.v/ and .x; z/  r `.w/ .

f y x

z

y

z

x

We note here that many of the technical difficulties in the proof come from the fact that such points y and z need not exist. Since the map is L-symmetric it turns out 0 0 that .x 0 ; y 0 /  r `.v / and .x 0 ; z 0 /  r `.w / . Taking the logarithm we obtain .`.v/  : `.w// D .`.v 0 /  `.w 0 //, which proves that distances are roughly preserved. The proof of (2) follows essentially the same reasoning with the appropriate multiplicative and additive error functions. Remark 9.3. Theorem 9.2 implies in particular that (up to rough isometry) the hyperbolic approximation Hypr .Z; / does not depend on the particular choices, for example the choice of the basepoint z0 or the choice of the system fVk gk2Z . Theorem 1.2 is formulated for L-symmetric respectively PQ-symmetric maps. We want to have these results for L-Möbius respectively PQ-Möbius maps, since we have on the boundary of a hyperbolic space a canonical Möbius structure. Now any classical Möbius map can be obtained from the composition of an involution and a symmetric map. We imitate this approach here and show first that the metric involution of a complete quasi-metric space corresponds to a rough isometric map of the hyperbolic approximations. Let therefore .Z; / be a complete quasi-metric space, let ! 2 Z n f1g and let ! .x;y/ be the quasi-metric on Z defined by involution, i.e. ! .x; y/ D .x;!/.y;!/ . Now we have the following theorem, compare [BS1], Theorem 7.3.1, or [J]. It says that metric involutions on the boundaries correspond to rough isometries in the interior. Theorem 9.4. There is a rough isometry F W Hypr .Z; / ! Hypr .Z; ! /, with @1 F D id. Here Hypr .Z; / is the truncated hyperbolic approximation in the case where Z is bounded, and @1 F denotes the induced map on Z.

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For a sketch of the proof we first assume that .Z; / is unbounded and hence 1 2 Z. In this case, we have two distinguished points in Z: the point 1, which is the infinitely remote point of the quasi-metric , but which is a finite point for ! , and the point !, which is a finite point of , but the infinitely remote point of ! . Since the hyperbolic approximation does not depend (up to rough isometry) on the particular choices, we can assume that ! is the basepoint of the hyperbolic approximation Hypr .Z; / and 1 is the basepoint of the hyperbolic approximation Hypr .Z; ! /. Let V be the vertices of the hyperbolic approximation Hypr .Z; / and V 0 be the vertices of Hypr .Z; ! /. Correspondingly, let Vk respectively Vk0 be the separated sets. For v 2 V and v 0 2 V 0 , denote the associated balls by B.v/ and B 0 .v 0 /, respectively. Now observe that for all x 2 Z, ! .x; 1/ D

1 ; .x; !/

which implies that B.k; !/ D B 0 .k; 1/. Thus these balls occur in both approximations. We can now define F .k; !/ D .k; 1/. In general, we need to define the map F only on a net of points in V . Consider the subset Vk  V with basepoint .k; !/. Next consider Vk  Vk to be the subset of all z 2 Vk with .z; !/  r s , where s  k  2. Then in particular ! is not contained in the ball B.k; z/ for z 2 Vk , and also the balls B.k; !/ and B.k; z/ are disjoint and sufficiently separated. Assume now that .v; !/  r s (up to some uniform error , i.e. an error only depending on r and not depending on k and s). Then any point u 2 B.k; z/ has distance .u; !/ 0 r s for some other uniform error 0 . Thus ! .u; z/ 00 .u; z/=r 2s , which implies that the ball Brk .z/ is approximately the ! .z/. Hence the ball B.k; z/ is approximately a ball B 0 .k  2s; z 0 / in the ball Brk2s 0 hyperbolic approximation of .Z; ! /, where z 0 2 Vk2s is a nearest point to z. Thus,  0 for z 2 Vk we define F .k; z/ D .k  2s; z /, where s is the integer part of lnr .z; !/ 0 and z 0 2 Vk2s a closest point to z. One can now show that this map is a rough isometry. One can regard the map in following way: essentially the hyperbolic approximations of .Z; / and .Z; ! / consist of the same “balls” as sets but only the radius changes. Thus the corresponding graphs of the hyperbolic approximations are really the same, but the levels of the vertices are changed. This change of levels can be understood in an easy way: There is first the distinguished geodesic .k; !/, k 2 Z, in the graph. The levels of this geodesic are just inverted. Now all other vertices lie on some geodesic cz . The geodesic cz together with the distinguished geodesic forms a tripod with tripod point u, which in the picture below is on level 2 for Hypr .Z; / and on level 2 for Hypr .Z; ! /. Now on the ray Œu; z the levels are strictly increasing, and thus they are determined by the level of u. Hence we see how the level changes.

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!

z

2

2

1 0 −1

1

!

z

−2

6

−1

0

0

−1

1

−2

2

−3

3

−4

4

5 4 3

Now we can easily generalize Theorem 9.2 to the case of L-Möbius and PQ-Möbius maps. Theorem 9.5. Let f W .Z; / ! .Z 0 ; 0 / be a map between complete quasi-metric spaces. (1) If f is an L-Möbius map, then there exists a rough isometric map F W Hypr .Z; / ! Hypr .Z 0 ; 0 / with @1 F D f . (2) If f is a PQ-Möbius map, then there exists a PQ-isometric map F W Hypr .Z; / ! Hypr .Z 0 ; 0 / with @1 F D f . Proof. Choose z 2 Z and z 0 D f .z/ 2 Z 0 . Then f D 0 B fQ B , where D id W .Z; / ! .Z; z /, fQ D f W .Z; z / ! .Z 0 ; z 0 /, 0 D id W .Z 0 ; z 0 / ! .Z; 0 /. Now and 0 extend to rough isometric maps by Theorem 9.4. The map fQ is Lsymmetric respectively PQ-symmetric by Proposition 10.10. Thus by Theorem 9.2, fQ extends as a rough isometric respectively PQ-isometric map and hence we obtain the result.

10 Quasimetric spaces and their Möbius geometry In this section we discuss quasi-metric spaces and their Möbius geometry. One important point in the discussion is the fact that in the realm of quasi-metric spaces one can always define a metric involution. This is not possible in the realm of metric spaces, since the metric involution is in general not a metric (but only a quasi-metric).

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10.1 Quasi-metric spaces Definition 10.1. A quasi-metric space is a set Z with a map  W Z  Z ! Œ0; 1 which satisfies the following conditions: (i) .z; z 0 /  0 for every z, z 0 2 Z and .z; z 0 / D 0 if and only if z D z 0 . (ii) .z; z 0 / D .z 0 ; z/ for every z, z 0 2 Z. (iii) .z; z 00 /  K maxf.z; z 0 /; .z 0 ; z 00 /g for every z, z 0 , z 00 2 Z and some fixed K  1. (iv) There exists at most one ! 2 Z such that .!; z/ D 1 for all z 2 Z n f!g. The function  is called a quasi-metric, or more specifically, a K-quasi-metric. Property (iii) is a generalized version of the ultra-metric triangle inequality (the case K D 1). If no point ! as in (iv) exists, we call Z non-extended, while Z is called extended if there is such a point !. In this case ! is called the infinitely remote point. The infinitely remote point is often also denoted by 1 2 Z. If .Z; / is a quasi-metric space, then we consider K0 D inffK  1 j Z is K-quasi-metricg. Then Z is also K0 -quasi-metric and K0 is called the optimal quasi-isometry constant. Remark 10.2. If .Z; d / is a metric space, then d is a K-quasi-metric for K D 2. In general d p is not a metric on Z for p > 1. But d p is still a 2p -quasi-metric.

10.2 Quasi-metrics and metrics In this section we introduce a natural topology on a quasi-metric space. We do that by introducing an honest metric on Z. Let .Z; / be a quasi-metric space. Since we are interested in obtaining a metric on Z, the only problem is the triangle inequality and hence the following approach is very natural. Define a map d W Z  Z ! Œ0; 1, P d.z; z 0 / D inf i .zi ; ziC1 /; where the infimum is taken over all sequences z D z0 ; : : : ; znC1 D z 0 in Z. By definition d then satisfies the triangle inequality. We call this approach to the triangle inequality the chain approach. The problem with the chain approach is that d.z; z 0 / could be 0 for points z ¤ z 0 , and so Axiom (i) is no longer satisfied for .Z; d /. Frink [Fr] realized that the chain approach works if the space .Z; / satisfies Axioms (i), (ii), (iv) above and instead of (iii) the “weak triangle” condition: (iii0 ) If .z; z 0 /  " and .z 0 ; z 00 /  " then .z; z 00 /  2". This weak triangle inequality seems to be the natural condition for success of the chain approach. Note that (iii0 ) immediately implies that the space is 2-quasi-metric. 0 Proposition 10.3 P(Frink). Let  be a 2-quasi-metric on a set Z and let for z; z 2 Z, 0 d.z; z / D inf i .zi ; ziC1 /; where the infimum is taken over all sequences z D z0 ; : : : ; znC1 D z 0 in Z. Then d is a metric on Z with 14 d    d .

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Proof. We claim that for any points z; z 0 2 Z and for any sequence z D z0 ; : : : ; znC1 D z 0 , .z; z 0 /  2.z; z1 / C 4.z1 ; z2 / C    C 4.zn1 ; zn / C 2.zn ; z 0 /:

(13)

Assume the claim is false. Then there is some value of n such that the claim does not hold. Let N be the smallest such integer. Then there are points z; z 0 ; z1 ; : : : ; zN in Z with .z; z 0 / > 2.z; z1 / C 4.z1 ; z2 / C    C 4.zN 1 ; zN / C 2.zN ; z 0 /:

(14)

Since  is a 2-quasi-metric the claim holds for n D 1 and hence N  2. It follows also that for every zr either .z; z 0 /  2.z; zr / (15) or

.z; z 0 /  2.zr ; z 0 /:

(16)

If r D 1 (15) does not hold because of (14), hence (16) does. Let s be the largest value of r for which (16) holds. Then s < N and .z; z 0 /  2.zs ; z 0 /

(17)

From the definition of s we have .z; z 0 /  2.z; zsC1 /:

(18)

Since the claim holds for k < N .zs ; z 0 /  2.zs ; zsC1 / C 4.zsC1 ; zsC2 / C    C 4.zN 1 ; zN / C 2.zN ; z 0 /; (19) .z; zsC1 /  2.z; z1 / C 4.z1 ; z2 / C    C 4.zs1 ; zs / C 2.zs ; zsC1 /:

(20)

Adding (19) and (20) and combining with (17) and (18) we obtain .z; z 0 /  2.z; z1 / C 4.z1 ; z2 / C    C 4.zN 1 ; zN / C 2.zN ; z 0 /

(21)

in contradiction to (14). The claim implies that .z; z 0 /=4  d.z; z 0 /  .z; z 0 /, hence d.z; z 0 / is positive for z ¤ z 0 . By definition d satisfies the triangle inequality and is a metric. This allows us to define a topology on Z. A set A  Z is open if for every a 2 Z n f1g there exists r > 0 such that Br .a/  A, and, in the case where 1 2 A, there exists y 2 Z n f1g and r > 0 such that A  Br .a/C . Here Br .a/ D fz 2 Z j .z; a/ < rg. Since by Frink’s result s is bilipschitz to a metric for suitable s (choose s so that s is 2-quasi-metric), the above defined topology is metrizable.

10.3 Complete quasi-metric spaces and the metric involution Definition 10.4. A quasi-metric space .Z; / is called complete if every Cauchy sequence in Z n f1g converges and, in the case where 1 2 Z, Z n f1g is unbounded.

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Recall the following terminology. Let a; b > 0 be positive numbers and let K  1. We write a K b if a 1   K: K b Then the relation K is symmetric and we clearly have transitivity in the following sense: if a K b and b L c, then a KL c. A triple .a1 ; a2 ; a3 / 2 .0; 1/3 of three positive numbers is called a multiplicative K-triple if a  K maxfaC1 ; aC2 g for D 1; 2; 3, where the indices are taken modulo 3. Thus .a1 ; a2 ; a3 / is a multiplicative K-triple if the ratio of the two largest of the three numbers is bounded by K. Property (iii) of a quasi-metric space .Z; / says that for any three distinct points z1 , z2 , z3 in Z the mutual distances ..z1 ; z2 /; .z2 ; z3 /; .z1 ; z3 // form a multiplicative K-triple. The following is a version of the tetrahedron lemma in [BS1]. Lemma 10.5. Let d12 , d13 , d14 , d23 , d24 , d34 be six numbers such that the four triples A1 D .d23 ; d24 ; d34 /, A2 D .d13 ; d14 ; d34 /, A3 D .d12 ; d14 ; d24 / and A4 D .d12 ; d13 ; d23 / are multiplicative K-triples. Then B D .d12 d34 ; d13 d24 ; d14 d23 / is a multiplicative K 2 -triple. We now discuss the metric involution. Let .Z; / be a quasi-metric space, and let ! be its infinitely remote point. Let z 2 Z n f!g be given. We put .a; b/ z .a; b/ D .z; a/.z; b/ for every a, b 2 Z, .a; b/ ¤ .z; z/, and z .!; !/ D 0. In the case b D !, this means z .a; !/ D 0 .!; a/ D 1=.1; a/, and in the case b D z, we have z .a; z/ D z .z; a/ D 1, i.e. the infinitely remote set for z is the point z. In y n , we have z .a; b/ D ..a/; .b//, where  W R yn ! R y n is the the case Z D R inversion with respect to the sphere of radius 1 centered at z 2 Rn . This justifies our terminology. Furthermore, the inversion operation is involutive in the sense that .z /! D  Lemma 10.6. Let  be a K-quasi-metric on Z with infinitely remote set Z1 D f!g. Then for every z 2 Z n !, the inversion metric z W Z  Z ! Œ0; 1 is a K 2 -quasimetric on Z with infinitely remote set fzg. Proof. Axioms (i), (ii), (iv) of a quasi-metric for z are obvious or easy to check, so we only consider Axiom (iii). Let a, b, c 2 Z be distinct points. If one of them coincides with z, then Axiom (iii) for z is obvious with any K 0  1. Thus we assume that a, b, c are different from z. Then the triple M 0 D .z .a; b/; z .a; c/; z .b; c//

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is proportional to M D ..a; b/.c; o/; .a; c/.b; o/; .a; o/.b; c//: It follows from Lemma 10.5 that M is a multiplicative K 2 -triple, i.e. Axiom (iii) for z is fulfilled with K 0 D K 2 . Remark 10.7. In general z is not a metric, even if  is a metric. In the context of metric spaces .Z; d /, the property that the corresponding dz is a metric is a special property of the space: A metric space Z is called a Ptolemy metric space if the inequality jxyj juvj  jxuj jyvj C jxvj jyuj (22) is satisfied for all x; y; u; v 2 Z. One easily shows that a space is Ptolemy if and only if the expression dz .x; y/ D jxyj defines a metric for all z 2 Z. It is proved in [FS2] that in the case of CAT.1/ jxzj jyzj spaces X, the boundary Z D @1 X is indeed a Ptolemy metric space. In particular the extended Euclidean space Rn [ f1g is a Ptolemy metric space.

10.4 Möbius geometry of quasi-metric spaces Let .Z; / be a quasi-metric space. We will always assume that Z contains at least three points. Let Dia3  Z 4 be the set of quadruples of points in Z, where one entry occurs three or four times. We call Q D Z 4 n Dia3 the set of admissible quadruples. We define the cross ratio triple as the map crt W Q ! †  RP 2 which maps admissible quadruples to points in the real projective plane defined by crt.x; y; z; w/ D ..x; y/.z; w/ W .x; z/.y; w/ W .x; w/.y; z//: Here † is the subset of points .a W b W c/ 2 RP 2 , where all entries a, b, c are nonnegative or all entries are nonpositive. Note that † can be identified with the standard 2-simplex, f.a; b; c/ j a; b; c  0; a C b C c D 1g. We use the standard conventions for calculations with 1. If 1 occurs once in Q, say w D 1, then crt.x; y; z; 1/ D ..x; y/ W .x; z/ W .y; z//. If 1 occurs twice, say z D w D 1 then crt.x; y; 1; 1/ D .0 W 1 W 1/. Like in the case of the classical cross ratio there are six possible definitions by permuting the entries and we choose the above one. We remark that Axiom (iii) of a quasi-metric space together with Lemma 10.6 says that the cross ratio triples are always multiplicative K 2 -triples, which is equivalent to the fact that the image of crt in † does not come close to the vertices .1 W 0 W 0/, .0 W 1 W 0/ and .0 W 0 W 1/. It is not difficult to check that crt W Q ! † is continuous, where Q and † carry the obvious topologies induced by X and RP 2 . Thus, if .xi ; yi ; zi ; wi / 2 Q for i 2 N and if xi ! x; : : : ; wi ! w, where .x; y; z; w/ 2 Q, then crt.xi ; yi ; zi ; wi / ! crt.x; y; z; w/.

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A map f W Z ! Z 0 between two quasi-metric spaces is called Möbius if it is injective and preserves the cross ratio triple, i.e. crt.x 0 ; y 0 ; z 0 ; w 0 / D crt.x; y; z; w/ for all admissible quadruples, where x 0 D f .x/; : : : ; w 0 D f .w/. The injectivity implies that allowed quadruples are mapped to allowed quadruples. We call two extended metric spaces Möbius equivalent if there exists a Möbius homeomorphism between them. This is equivalent to the more classical definition using the cross ratio, which is a map cr W Z 4 n Dia3 ! Œ0; 1, and which we define by cr.x; y; z; w/ D

.x; z/.y; w/ : .x; y/.z; w/

We will also use the notation Œx; y; z; w D cr.x; y; z; w/. Sometimes it is also useful to consider the so-called simple ratio triple and the simple ratio with only two arguments. Therefore consider srt W Z 3 n Dia2 ! †, srt.x; y; z/ D crt.x; y; z; 1/ D ..x; y/ W .x; z/ W .y; z//; and sr W Z 3 n Dia2 ! Œ0; 1 defined by sr.x; y; z/ D

.x; z/ ; .x; y/

where Dia2  Z 3 is the set of triples where one entry occurs two or three times. We also use the notation Œx; y; z D sr.x; y; z/. Remark 10.8. We make the assumption that Z contains at least three points in order that admissible triples and quadruples exist.

10.5 Morphisms of quasi-metric spaces We consider maps f W Z ! Z 0 between quasi-metric spaces. As above we denote by z 0 2 Z 0 the image of z 2 Z, i.e. z 0 D f .z/. A map f W Z ! Z 0 is called • snowflake if there exists s > 0 such that for all x; y 2 Z, s .x; y/  .x 0 ; y 0 /  s .x; y/I • quasi-snowflake if there are s > 0, c  1 such that for x; y 2 Z, 1 s  .x; y/  .x 0 ; y 0 /  c s .x; y/I c • symmetric if it is injective and for all admissible triples .x; y; z/ we have Œx 0 ; y 0 ; z 0  D Œx; y; zI

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• Möbius if it is injective and for all admissible quadruples Œx 0 ; y 0 ; z 0 ; w 0  D Œx; y; z; wI • quasi-symmetric or Q-symmetric if there exist continuous monotone increasing maps 1 ; 2 W Œ0; 1 ! Œ0; 1 with i .0/ D 0 and i .1/ D 1 such that for all admissible triples

1 .Œx; y; z/  Œx 0 ; y 0 ; z 0   2 .Œx; y; z/I • quasi-Möbius or (Q-Möbius) if there exist 1 ; 2 W Œ1; 1/ ! .0; 1/ as above such that for all admissible quadruples we have

1 .Œx; y; z; w/  Œx 0 ; y 0 ; z 0 ; w 0   2 .Œx; y; z; w/I • power quasi-symmetric or PQ-symmetric if there are p  1 and q  1 such that for all admissible triples with Œx; y; z  1, 1 Œx; y; z1=p  Œx 0 ; y 0 ; z 0   q Œx; y; zp I q • power quasi-Möbius or PQ-Möbius if there are p  1 and q  1 such that for all admissible quadruples with Œx; y; z; w  1, 1 Œx; y; z; w1=p  Œx 0 ; y 0 ; z 0 ; w 0   q Œx; y; z; wp I q • bilipschitz-symmetric or L-symmetric if there is L  1 such that for all admissible triples with Œx; y; z  1, 1 Œx; y; z  Œx 0 ; y 0 ; z 0   L Œx; y; zI L • bilipschitz-Möbius or L-Möbius if there is L  1 such that for all admissible quadruples with Œx; y; z; w  1, 1 Œx; y; z; w  Œx 0 ; y 0 ; z 0 ; w 0   L Œx; y; z; w: L Remark 10.9. (1) Equivalently, a map is symmetric if it preserves the simple ratio triple, i.e. srt.x 0 ; y 0 ; z 0 / D srt.x; y; z/ and Möbius if it preserves the cross ratio triple, i.e. crt.x 0 ; y 0 ; z 0 ; w 0 / D crt.x; y; z/. (2) We use the restriction Œx; y; z; Œx; y; z; w  1 in the last definitions so that the formulation of the properties is easier. Indeed one can estimate Œx 0 ; y 0 ; z 0  also in the 1 case where Œx; y; z < 1. Namely, in this case we have Œx; z; y D Œx;y;z > 1. Thus for the control functions i we have

1 .Œx; z; y/  Œx 0 ; z 0 ; y 0   2 .Œx; z; y/;

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1 1  Œx 0 ; y 0 ; z 0   :

1 .1=Œx; y; z/

2 .1=Œx; y; z/

(3) We have the obvious inclusions: symmetric implies L-symmetric implies PQsymmetric implies Q-symmetric, and the corresponding inclusions for Möbius maps. We want to discuss the relation between symmetric maps and Möbius maps. Proposition 10.10. Let .Z; /; .Z 0 ; 0 / be unbounded complete quasi-metric spaces and hence 1 2 Z and 1 2 Z 0 . (1) If f W Z ! Z 0 is Q-symmetric (resp. PQ-symmetric, L-symmetric, symmetric), then f .1/ D 1 and f is Q-Möbius (resp. PQ-Möbius, L-Möbius, Möbius). (2) If f W Z ! Z 0 is Q-Möbius (resp. PQ-Möbius, L-Möbius, Möbius) and f .1/ D 1, then f is Q-symmetric (resp. PQ-symmetric, L-symmetric, symmetric). Proof. (1) Since .Z; / is unbounded and complete, 1 2 Z. If z1 ; z2 2 Z are finite 0 .z10 ;10 / .z1 ;1/ . Since f is Q-symmetric 1 D . This implies points, then 1 D .z 0 .z10 ;z20 / 1 ;z2 / 0 1 D 1. We show that f is Q-Möbius (resp. PQ-Möbius, L-Möbius, Möbius). Consider an allowed quadruple .x; y; z; w/ 2 Z 4 . If two entries of the quadruple coincide, then Œx; y; z; w takes one of the values 0, 1, 1 and clearly Œx 0 ; y 0 ; z 0 ; w 0  has the same value. Thus we can assume that all entries are different. If 1 is one of the entries, we can express the cross ratio by a simple ratio, for example Œ1; y; z; w D Œw; z; y. If all entries are finite and distinct, then Œx; y; z; w D Œx; y; zŒw; z; y. Since f controls the simple ratio, it also controls the cross ratio. (2) Since f .1/ D 1 and Œx; y; z; 1 D Œx; y; z this is obvious. Lemma 10.11. If Z is bounded and if f W Z ! Z 0 is Q-symmetric, then f .Z/ is bounded. Proof. Choose different points x1 ; x2 2 X . If f .Z/ is unbounded, then there are .x 0 ;u0 / .x1 ;ui / u0i 2 f .Z/ with .x10 ;x 0i / ! 1. The Q-symmetry implies .x ! 1, showing 1 ;x2 / 1 2 that Z is unbounded.

Appendix: Proof of Theorem 5.6 In this section we proof Theorem 5.6. We restate the result: Theorem A.1. If X satisfies the ı-inequality, then X can be isometrically embedded into a geodesic metric space Xy such that (1) Xy satisfies the ı-inequality, (2) @1 Xy D @1 X, (3) If X is visual, then also Xy is visual.

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As a first step we prove Lemma A.2. The space X can be isometrically embedded into a metric space M1 .X / such that for every pair a; b 2 X there exists a midpoint m.a; x b/ 2 M1 .X /. Proof. We define M 0 .X/ as the set of all subsets of X containing one or two elements. Let m.a; b/ WD fa; bg 2 M 0 .X/. Note that m.a; b/ D m.b; a/. For every m D m.a; b/ 2 M 0 .X/ we define a function Hm W X ! Œ0; 1/ by 1 Hm .x/ D maxfjxaj; jxbjg  jabj: 2 One can view Hm as the distance of x to the midpoint of a and b in a tripod of side lengths jxaj, jxbj and jabj.

x

a

w b

In particular we have: (i) Hm .a/ D Hm .b/ D 12 jabj. (ii) For x 2 X there exists px 2 fa; bg such that Hm .x/ D jxpx j  12 jabj. (iii) In the case a D b, Hm .x/ D jxaj D jxbj. In the above picture we have px D b. We now define a pseudodistance on M 0 .X /. For points mi D m.ai ; bi /, i D 1; 2, let jm1 m2 j WD

sup .jx1 x2 j  H1 .x1 /  H2 .x2 //;

x1 ;x2 2X

where Hi D Hmi . Let us check the relevant points. (1) jm1 m2 j is finite. By the triangle inequality we have for all x1 ; x2 2 X that jx1 x2 j  H1 .x1 /  H2 .x2 /  jx1 px1 j C jpx1 px2 j C jpx2 x2 j  H1 .x1 /  H2 .x2 / 1 1 D ja1 b1 j C ja2 b2 j C jpx1 px2 j 2 2 1 1  ja1 b1 j C ja2 b2 j C maxfja1 a2 j; ja1 b2 j; jb1 a2 j; jb1 b2 jg: 2 2 (2) jm1 m2 j  0.

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Consider first the case H1 .b1 /  H2 .b1 /. Then let x1 D a1 and x2 D b1 and we have jm1 m2 j  ja1 b1 j  H1 .a1 /  H2 .b1 / D H1 .b1 /  H2 .b1 /  0: If H1 .b1 /  H2 .b1 /, then there is p2 2 fa2 ; b2 g with H2 .b1 / D jb1 p2 j  H2 .p2 /. Let x1 D b1 and x2 D p2 , then jm1 m2 j  jb1 p2 j  H1 .b1 /  H2 .p2 / D H2 .b1 /  H1 .b1 /  0: (3) jm1 m3 j  jm1 m2 j C jm2 m3 j: Let " > 0 be arbitrary. Then there are x1 ; x3 2 X with jm1 m3 j  "  jx1 x3 j  H1 .x1 /  H2 .x2 /: Thus for arbitrary

x20 ; x200

2 X we have

jm1 m2 j C jm2 m3 j  jm1 m3 j  "  jx1 x20 j C jx200 x3 j  jx1 x3 j  H2 .x20 /  H2 .x200 /: We show that the right-hand side (RHS) is  0 for an appropriate choice of x20 , x200 . This then proves the triangle inequality. Again we have two cases. First case: H2 .x1 / C H2 .x3 /  jx1 x3 j. Then choose x20 D x3 , x200 D x1 , and we have RHS D jx1 x3 j  H2 .x3 /  H2 .x1 /  0: Second case: H2 .x1 / C H2 .x3 /  jx1 x3 j. By (ii) there are p1 ; p3 2 fa2 ; b2 g with h2 .x1 / D jx1 p1 j  H2 .p1 / and H2 .x3 / D jx3 p3 j  H2 .p3 /. Let x20 D p1 , x200 D p3 . Then RHS D jx1 p1 j C jx3 p3 j  jx1 x3 j  H2 .p1 /  H2 .p3 / D H2 .x1 / C H2 .x3 /  jx1 x3 j  0: (4) The ı-inequality holds. Given " > 0, there are x1 ; x2 ; x3 ; x4 2 X so that jm1 m2 j C jm3 m4 j  "  jx1 x2 j C jx2 x4 j 

X

Hi .xi /

i

 max.jx1 x3 j C jx2 x4 j; jx1 x4 j C jx2 x3 j/ C 2ı 

X

Hi .xi /

i

 max.jm1 m3 j C jm2 m4 j; jm1 m4 j C jm2 m3 j/ C 2ı: (5) If m1 D m.a1 ; a1 /; m2 D m.a2 ; a2 /, then jm1 m2 j D ja1 a2 j. For all x1 ; x2 2 X we have jx1 x2 j  jx1 a1 j  jx2 a2 j  ja1 a2 j, and so jm1 m2 j  ja1 a2 j. On the other hand, let x1 D a1 , x2 D a2 . Then we see that jm1 m2 j  ja1 a2 j. (6) For m1 D m.a; a/ and m2 D m.a; b/ we have jm1 m2 j D 12 jabj.

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For all x1 ; x2 2 X we have 1 1 jx1 x2 j  H1 .x1 /  H2 .x2 /  jx1 x2 j  jx1 aj  jx2 aj C jabj  jabj: 2 2 On the other hand, choose x1 D a, x2 D b. Then 1 1 jm1 m2 j  jabj  jabj D jabj: 2 2 Thus M 0 .X/ is a pseudometric space and the corresponding metric space M1 .X / satisfies the ı-inequality by (4). The embedding a 7! Œm.a; a/ (here Œ  denotes the corresponding equivalence class) is isometric by (5) and we can consider X as a subset of M1 .X/. By (6), for two points a; b 2 X the point m.a; x b/ WD Œm.0; b/ is a midpoint in M1 .X/. We now define recursively MnC1 .X/ D M1 .Mn .X //. Then we have a chain of isometric embeddings Mn .X/ ! MnC1 .X /. WeS consider this as a chain of subsets X  M1 .X /  M2 .X/     . Let M1 .X / D Mn .X /. Then M1 .X / satisfies the midpoint property. Thus to any points x; y 2 M1 .X /, there exists a midpoint m D m.x; x y/ 2 M1 .X/ with jxmj D jmyj D 12 jxyj. To obtain a geodesic space we consider a completion of the space. This can be done in several ways. We first give an approach which proves Theorem 5.5. A complete metric space with the midpoint property is geodesic. Since the usual completion (using Cauchy sequences) does not preserve the midpoint property, we use the !-completion of M1 .X/. Here ! is a nonpricipal ultrafilter on N. Thus we choose (once and for all) a fixed nonprincipal ultrafilter !. For a metric space .Z; d /, the !-completion Z! is defined in the following way: Let Z 0 be the set of bounded sequence in .Z; d /. For two bounded sequences .xi / and .yi / in Z 0 define d! ..xi /; .yi // D !  lim d.xi ; yi /. This defines a pseudometric on Z 0 and the corresponding metric space is Z! . The canonical embedding Z ! Z! , x 7! Œ.x/, where .x/ is the constant sequence x, is an isometric embedding. Now we define Xz D .M1 .X//! . z This Then Xz is a complete geodesic space and X embeds isometrically into X. z proves Theorem 5.5. The disadvantage of this approach is that @1 X is in general larger that @1 X. Therefore we also consider the following approach. We define J1 .X/ to be the usual metric completion of M1 .X /: a point in J1 .X / is the equivalence class of a Cauchy sequence in M1 .X /. Note that X sits isometrically in J1 .X /. Furthermore for x; y 2 X there exists a geodesic joining the points in J1 .X /. Note however that J1 .X/ does not necessarily have the midpoint property. x i ; yi / is not Namely, if .xi / and .yi / are Cauchy sequences in M1 .X /, then m.x necessarily a Cauchy sequence. S We define recursively Jn .X/ D J1 .Jn1 .X //, and let Xy D J1 .X / D Jn .X /. y Then Xy is a geodesic metric space, and X is isometrically embedded in X.

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451

To obtain (2) and (3) of Theorem 5.6, we also need the following information about M1 .X /. Lemma A.3. Let o 2 X be a basepoint in X and let " > 0. Then there exists ' D '";o W M1 .X/ ! X with (i) 'jX D id, (ii) .m1 jm2 /o  "  .m1 j'.m2 //o for all m1 ; m2 2 M1 .X /. Proof. If follows from the definition of the metric on M1 .X / that for every m 2 M1 .X / and " > 0 there exists x D '.m/ 2 X such that jomj  "  joxj  Hm .x/: In the case where m 2 X  M1 .X/ we choose '.m/ D m. We show that ' satisfies the requirements. Indeed 2.m1 jm2 /o D jom1 j C jom2 j  jm1 m2 j  jom1 j C jox2 j  Hm2 .x2 / C "  sup.j˛ˇj  Hm1 .˛/  Hm2 .ˇ// ˛;ˇ

 jom1 j C jox2 j C "  sup.j˛x2 j  Hm1 .˛// ˛

D jom1 j C jox2 j  jm1 x2 j C "; where the last inequality follows from setting ˇ D x2 . Lemma A.4. Let o 2 X a basepoint in X and let " > 0. Then there exists ";o W M1 .X/ ! X with (i) jX D id, (ii) .m1 jm2 /o  "  .m1 j .m2 //o for all m1 ; m2 2 M1 .X /.

D

1 n n Proof. Define jMn .X/ WD 'o;"=2 B    B 'o;"=2 n , where 'o; W Mn .X / ! Mn1 .X / is the map of Lemma 10.5.

We play the same game with the Jn . Lemma A.5. For o 2 X and " > 0, there exists ' D '";o W J1 .X / ! X with (i) 'jX D id, (ii) .z1 jz2 /o  "  .z1 j'.z2 //o for all z1 ; z2 2 M1 .X /. Proof. Let ' D o;"=2 B" , where " W J1 .X / ! M1 .X / is a projection which moves every point by at most "=3, and hence .z1 jz2 /  "=2  .z1 j" .z2 //o , and o;"=2 is the map from the previous lemma. Finally we obtain in analogy with Lemma 10.5

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Lemma A.6. Let o 2 X be a basepoint in X and let " > 0. Then there exists D ";o W Xy D J1 .X/ ! X with (i) jX D id, (ii) .z1 jz2 /o  "  .z1 j .z2 //o for all z1 ; z2 2 Xy . y If .zi / is a Gromov We now prove that @1 Xy D @1 X. Clearly @1 X sits in @1 X. y sequence defining a point in @1 X, then .zi / is a Gromov sequence in X , which is equivalent to .zi /. This proves (2). Assume now that X is visual. This implies that there exists D 2 R such that for any x 2 X there exists a Gromov sequence .xi / with .xjxi /o  joxj  D. Let now y Then .zj .z//o  jozj  ", in particular jo .z/j  jozj  ". Since .z/ 2 X z 2 X. there is a Gromov sequence .xi / in X, with . .z/jxi /o  jo .z/j  D. Hence, .zjxi /o  minf.zj .z//o ; . .z/jxi /o g  ı  jozj  D  ı  ": This implies that Xy is also visual. This completes the proof of Theorem 5.6.

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