Geometric Group Theory: Proceedings of a Special Research Quarter at The Ohio State University, Spring 1992 9783110810820, 9783110147438

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Ohio State University Mathematical Research Institute Publications 3 Editors: Gregory R. Baker, Walter D. Neumann, Karl Rubin

Geometric Group Theory Proceedings of a Special Research Quarter at The Ohio State University, Spring 1992

Editors Ruth Charney Michael Davis Michael Shapiro

w DE

_G_ Walter de Gruyter · Berlin · New York 1995

Editors RUTH

CHARNEY

MICHAEL

DAVIS

Department of Department of Mathematics Mathematics The Ohio State University The Ohio State University 231 West 18th Avenue 231 West 18th Avenue Columbus, Ohio 43210-1174 Columbus, Ohio 43210-1174 USA USA

MICHAEL

SHAPIRO

Department of Mathematics Cornell University Ithaca, N.Y. 14853-7901 U S A

Series Editors: Gregory R. Baker, Karl Rubin Department of Mathematics, The Ohio State University, Columbus, Ohio 43210-1174, USA Walter D. Neumann Department of Mathematics, The University of Melbourne, Parkville, VIC 3052, Australia 1991 Mathematics Subject Classification: 20-06, 51-XX, 57-XX Keywords: Hyperbolic groups, automatic groups, geometric group theory ©

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Geometric group theory : proceedings of a special research quarter at the Ohio State University, spring 1992 / editors, Ruth Charney, Michael Davis, Michael Shapiro. p. cm. - (Ohio State University Mathematical Research Institute publications ; 3) ISBN 3-11-014743-2 (acid-free paper) 1. Geometric group theory - Congresses. I. Charney, Ruth, 1 9 5 0 II. Davis, Michael, 1 9 4 9 III. Shapiro, Michael, 1948 Oct. 1 3 - . IV. Series. QA183.G455 1995 512'.2-dc20 95-12932 CIP

Die Deutsche Bibliothek - Cataloging-in-Publication Geometric group theory : proceedings of a special quarter at the Ohio State University, spring 1992 / Charney ... - Berlin ; New York : de Gruyter, 1995 (Ohio State University, Mathematical Research publications ; 3) ISBN 3-11-014743-2 NE: Charney, Ruth [Hrsg.]; International Mathematical Institute : Ohio State University

Data research ed. Ruth Institute Research ...

© Copyright 1995 by Walter de Gruyter & Co., D-10785 Berlin. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Printing: Gerike G m b H , Berlin. Binding: Lüderitz & Bauer G m b H , Berlin. Cover design: Thomas Bonnie, Hamburg.

Dedicated to the memory of Craig Squier

Preface

In the spring of 1992, a Special Research Quarter in Geometric Group Theory was held under the auspices of the International Mathematical Research Institute at the Ohio State University. This volume consists of contributions from participants of the Special Quarter. The program for the quarter included fifteen visitors for periods of one to seven weeks, and culminated with a four-day conference attended by about sixty participants. The field of geometric group theory has seen an explosion of new ideas and new faces over the past decade. An effort was made to include a large number of graduate students and recent PhD's in the Special Quarter and their enthusiastic participation was particularly gratifying. The Special Quarter in Geometric Group Theory was generously funded by the Ohio State Mathematical Reasearch Institute and the National Science Foundation to whom we are most grateful. The Research Institute. The International Mathematical Research Institute at Ohio State University was founded in 1989 to support a program of visiting research scholars in mathematics at Ohio State and to run Workshops and Special Emphasis Programs on topics of particular importance and timeliness. A Research Semester on Low Dimensional Topology was the first major program of the Institute. Since then the Institute has supported workshops on, among others, Nearly Integrable Wave Phenomena in Nonlinear Optics, Quantized Geometry, the Arithmetic of Function Fields, L -Functions Associated to Automorphic Forms, and Groups, Difference Sets, and the Monster. The Institute is currently supporting about 20-30 other research visitors (mostly short term) per year. The Institute publishes a preprint series as well as this book series, which is devoted to research monographs, lecture notes, proceedings, and other mathematical works arising from activities of the Research Institute. Acknowledgements. First and foremost, the editors thank The Ohio State University for its support of this program through the Research Institute. We also thank the staff of the Mathematics Department for their help in the organization and running of the Research Semester, particularly Marilyn Howard (administration and visas) and Marilyn Radcliff (expenses). Finally, we thank all those who took part in our program for their participation, their contributions to these proceedings, their help in refereeing, and most of all, for their patience in waiting for this volume to appear. Columbus, December 1994

Ruth Charney Michael Davis Michael Shapiro

Contents

Preface Β. .H. Bowditch Notes on Locally CAT(l) Spaces

vii 1

N. Brady Asynchronous Automatic Structures on Closed Hyperbolic Surface Groups

49

L. P. Comerford and C. C. Edmunds Genus of Powers in a Free Group

67

G. R. Conner Isoperimetric Functions for Central Extensions

73

J. Corson Groups Acting on Complexes and Complexes of Groups

79

B. Fine, F. Levine, F. Roehl and G. Rosenberger The Generalized Tetrahedron Groups

99

S. M. Gersten Finiteness Properties for Asynchronously Automatic Groups

121

B. Hu Retractions of Closed Manifolds with Nonpositive Curvature

135

P. Papasoglu On the Sub-Quadratic Isoperimetric Inequality

149

Ch. Pittet Isoperimetric Inequalities for Homogeneous Nilpotent Groups

159

J. R. Stallings Problems about Free Quotients of Groups

165

M. Shapiro and M. Stein Some Problems in Geometric Group Theory

183

Notes on Locally CAT(l) Spaces Β. H.

Bowditch

0. Introduction The term "locally CAT(l) space" refers to a complete, locally compact path-metric space satisfying a certain geometrical "comparison" axiom. This axiom is intended to capture the idea of the space having curvature bounded above by 1. Thus, for example, a riemannian manifold is locally CAT(l) if and only if all its sectional curvatures are at most 1. In this paper we give an account of various aspects of locally CAT(l) spaces, beginning with a general survey, and going on to consider spaces of loops in such spaces; finally relating these results to various geometric inequalities. Attempts to formulate curvature as a purely metric property go back to the work of Aleksandrov, Toponogov and Busemann. Thus, one treats one of the comparison theorems of riemannian geometry as axiomatic. Since curvature is a local property, we will want a formulation that allows us to pass readily from local to global. In [Gromov], Gromov introduced the term "CAT( χ ) " for a comparison axiom intended to capture the idea of a space having curvature everywhere < χ . This is defined in the context of "geodesic spaces" in which every pair of points a joined by a geodesic—a length-minimising rectifiable path. It is a metric condition on triangles formed from three geodesies edges, as we describe at the end of this chapter. In this paper, we shall restrict attention to complete locally compact path-metric spaces. These are necessarily geodesic spaces (Lemma 2.1.2). We may speak of a space being "locally" or "globally" CAT( χ ). After scaling the metric, we reduce to three qualitatively distinct cases, namely χ = —1,0, 1. If χ < 0, a locally CAT( χ ) space will be globally CAT( χ ) if and only if it is simply-connected (and hence contractible). See, for example [Pa]. To make an analogous statement in the case χ = 1, we should replace simple-connectedness (i.e. path-connectedness of the space of loops) by a related condition obtained by restricting attention to those loops which are rectifiable, and of length strictly less than 2π . Thus this new condition demands instead that the subspace of such loops be path-connected. (We explore this matter in Chapter 3.) It seems that the cases χ = — 1,0, 1, become progressively more difficult to deal with from a synthetic point of view. Much of the geometry of strictly negatively curved spaces (corresponding to χ = — 1) can even be formulated combinatorially, as can be seen from Gromov's hyperbolicity criterion [Gromov]. In the case of non-positive curvature, the appropriate combinatorial formulation has yet to be settled on. In the context of combinatorial group theory, the notions of combability and automaticity (and

2

Β. Η. Bowditch

the numerous variations thereof) are candidates (see for example, [E]). However a great deal can be done by synthetic means. Much of the basic theory, as developed in [BalGS] for example, can be carried out for CAT(O) spaces (see [BrH]). We also have the weaker notion of convexity of the distance function, as formulated by Busemann [Buseml], [Busem2]. See also [Pa] for an account of this. Unfortunately, positively curved spaces prove less amenable to synthetic argument, and much of the development has so far been confined to the riemannian category. However we can get some mileage out of the C AT( 1) axiom as we shall describe. Note that in all these cases, the CAT( χ ) axiom is intended to place only an upper bound on curvature. The curvature is thus allowed to be arbitrarily, or indeed "infinitely" negative. A synthetic means of formulating lower curvature bounds has been developed by Burago, Gromov and Perelman [BurGP], [Pe]. This theory has a somewhat different flavour, and will not directly concern us here. One recent triumph of these methods has been a version of the sphere theorem founded on purely synthetic hypotheses [GrovP]. As examples of locally CAT( χ ) spaces (besides riemannian manifolds) we can consider geometric simplicial complexes obtained by gluing together simplices of constant curvature χ . In such a complex, the link of each simplex has, itself, the structure of a geometric complex built out of spherical simplices (the case χ = 1). It turns out that the original complex will be locally CAT( χ ) if and only if the link of each vertex is globally CAT(l). (To be formally consistent, we should say that each connected component of such a link is globally CAT(l).) By induction on dimension, one can see that this is equivalent to saying that the link of each simplex should contain no closed geodesic of length strictly less than 2π . This latter observation uses the fact that a compact locally CAT(l) space is globally CAT(l) if and only if it contains no closed geodesic of length less than 2π . For a discussion of polyhedral complexes, see [Bal] or [Br]. Another context in which the CAT(l) property arises naturally concerns the realisation of 3-dimensional hyperbolic polyhedra. Given a compact convex polyhedron in hyperbolic 3-space, we can construct its dual in the de-Sitter sphere [HR]. Intrinsically, this dual is a topological 2-sphere with a singular spherical metric, i.e. it has constant curvature 1 away from a finite number of cone points. Each of the cone points has cone angle > 2π . The metric is thus locally CAT(l). Hodgson and Rivin [HR] characterise the metrics that occur in this way as precisely those which contain no closed geodesic of length < 2π . (This is slightly stronger than globally CAT(l) — a globally CAT(l) space might contain a closed geodesic of length precisely 2π .) It turns out that (up to isometry) there is a bijective correspondence between such metrics and compact hyperbolic polyhedra. A natural quantity associated to a compact locally CAT(l) space is the "systole" [ChaD]. This may be defined as the length of the shortest closed geodesic, provided this is < 2π . Otherwise, we set the systole to equal 2π . There are various equivalent ways of defining this quantity as we shall describe in Chapters 2 and 3. One can also give a definition for non-compact spaces. In all cases the space will be globally CAT(l) if and only if the systole equals 2ττ .

Notes on Locally CAT(1) Spaces

3

In Chapter 3, we introduce the notion of a "shrinkable loop" in a locally CAT(l) space. Briefly, a rectifiable loop of length less than 2π is said to be shrinkable if it can be (freely) homotoped to a point (constant loop) passing only through other rectifiable loops of length less than l i t . It turns out that if this is possible, we can always choose the homotopy so that the lengths of the intermediate loops tend monotonically to 0 (Theorem 3.1.5). A locally CAT(l) space is globally CAT(l) if and only if every loop of length less than 2π is shrinkable. (This follows from Theorem 3.1.2 and Corollary 3.1.7.) This property is the analogue of simple connectedness in CAT(O) spaces alluded to earlier. In Section 3.1, we list various other results relating to shrinkability. For example, we shall see (Corollary 3.1.4) that a closed geodesic of length less that 2π cannot be shrinkable. Most of Chapter 3 will be devoted to proving these results. (Sections 3.5 and 3.7 are digressions from the general flow of the paper.) The main technique used in these results is that of "Birkhoff curve shortening", which we describe in Section 3.3. In Section 3.7, we give an example of a compact smooth riemannian 3-manifold on which the Birkhoff process (in general) fails to converge. In Chapter 4, we mention two area inequalities from 2-dimensional riemannian geometry, to which the notion of shrinkability is relevant. Firstly shrinkability gives a convenient way of describing a dichotomy arising out of the spherical isoperimetric inequality. Secondly, it gives an hypothesis under which we can prove an area comparison theorem for triangles. Most of the material for this paper was prepared at the University of Melbourne, under an A.R.C. fellowship. It was completed at the University of Aberdeen under an S.E.R.C. research assistantship.

Definitions For reference we begin by recalling some elementary facts about path-metric spaces. We also give a formal definition of the CAT( χ ) property. In the rest of the paper, we will discuss in detail in the case χ = 1 . Suppose (X, d) is a metric space. A path (from x to y) is a map a : [0, 1] — • X (with of(0) = x and a ( l ) = y). We define the (rectifiable) length of a to be the limit, as η —• 0 of inf{£]f = i d(a(ti-\), α(ί;))}. where the infinum is taken over all subdivisions 0 = to < ti d(x, y), and it is easily seen that d' is a metric on X. Moreover, repeating this process does not give us any new

4

Β. Η. Bowditch

metrics, i.e. d" = d'. We shall say that a metric d is a path-metric if d' = d. This is equivalent to saying that for all x, y € X and e > 0, there is a rectifiable path joining χ to y of length at most d(x, y) + € . Note that the metric on a riemannian manifold is always (more or less by definition) a path-metric. The above discussion allows us to define induced path-metrics on subsets. Suppose (X, d) is a path-metric space, and Y c X is a subset with the property that any two points of Y are joined by a rectifiable path in Y. The induced path-metric on Y is defined to be d'y where dy is the metric d restricted to Y. As an example, if we take Y = S2 to be a unit sphere in euclidean 3-space, we obtain the "standard 2-sphere", whose induced path-metric is riemannian of constant curvature 1. We define a (global) geodesic from χ to y, in our path-metric space (X, d), to be a rectifiable path, α , parameterised proportionately to arc-length, which is globally length-minimizing among all rectifiable paths from χ to y . Equivalently, we can express this by saying that d(a(t), a(u)) = |i - u\d(x,y) for all t, u e [0, 1]. In general, there might be no geodesic between to given points. (Consider χ = (— 1,0), y = (1,0) in Μ2 \ {(0,0)} with the euclidean path-metric.) However, we shall see that geodesies always exist if X is complete and locally compact. Let us suppose now that (X, d) is a complete locally compact path-metric space. We can define the CAT( χ ) property as follows. Suppose that a \ , a j , «3 : [0, 1] —> X are three geodesies forming a triangle, i.e. α,(1) = α,·+ι (0) where we take subscripts mod 3. Suppose that Σ ί = ι d(a,(0), α,·(1)) < 2π/^/χ , where this condition is deemed vacuous if χ < 0. We may construct a comparison triangle in the 2-dimensional model space (Sx ,dx) of constant curvature χ , consisting of three geodesic segments a'\, a'2, a'3 ' [0, 1] —>• Sx , with d x (a'/(0), α',·(1)) = d(cti(0), α,·(1)) for ι = 1, 2, 3. This comparison triangle is well defined up to isometry in Sx . We say that X is "(globally) CAT( χ )" if for all such triangles (αϊ, Qf2, «3), for all t, u e [0,1] and for all i, j e {1, 2, 3}, we have rf(a,(i), aj(u)) < dx(u'i(t), a'j(u)). We say that X is "locally CAT( χ )" if every point has a neighbourhood which is CAT( χ ). Note that the CAT(0) property implies the following: If α, β : [0, 1] —> X are geodesies, then the map [(t, u) i-* d(a(t), ß(u))] : [0, l] 2 —> [0, 00) is convex. This property, taken as an axiom, is (equivalent to) Busemann's condition for non-positive curvature. It may be alternatively expressed in terms of bisecting edges of triangles [Buseml], [Busem2]. As already alluded to, the fundamental examples of locally CAT( χ ) spaces are riemannian manifolds of curvature at most χ . The fact that these are indeed locally CAT( χ ) is a simple deduction from Toponogov's comparison theorem (see [CheE]). We need only quote a relatively weak version here. It is well known that any riemannian manifold is covered by geodesically convex sets. Suppose that Κ is such a set, and that (αϊ, α 2,^3) is a triangle of perimeter < Ι π / ^ / χ in Κ . Then Toponogov says that the angle between α,· and α;· is less than or equal to the angle between a',· and a'j in the comparison triangle (α'ι, a'2, a'3) in the model space. To deduce CAT( χ ) from this, one just needs

Notes on Locally CAT(1) Spaces

5

a bit of spherical, euclidean or hyperbolic geometry: cf. Proposition 1.2. For more details of these matters, see for example [Br].

1. Spherical geometry In this chapter, we survey a few elementary facts of spherical geometry. We shall not write out detailed proofs here. Much of it can be dismissed as "spherical trigonometry", though most of the results can be deduced by synthetic argument without resort to computation. By "spherical geometry" we mean the geometry of the 2-sphere, S2, with the pathmetric d\, induced by embedding S2 as a unit sphere in euclidean 3-space. The diameter of (S2, d\) is thus 7Γ , and two points χ, y € S2 are antipodal if and only if d\ y) = π . If χ and y are not antipodal, they may be joined by a unique geodesic segment y] ίΞ S2. If Λ: G S2, and y, ζ e S2 \ {*}, neither antipodal to χ, then we write yxz E [0, TT] for the angle between >>] and [x, z] at χ . We say that a closed set 2 R c S is convex if R does not consist of a pair of antipodal points, and if [x, y] c R for any pair of non-antipodal points x, y e R. We say that R is strictly convex if it is convex and contains no pair of antipodal points. In the latter case, R lies inside some open hemisphere of S 2 . Moreover, if it has non-empty interior, then it is topologically a disc, bounded by a rectifiable Jordan curve, dR, of length < 2π . Any Jordan curve / c S 2 bounds two closed discs. If J is rectifiable and of length < 2π , then one of these discs, D\, is contained in an open hemisphere. (This follows from Proposition 1.3.) The other disc Dj therefore contains a closed hemisphere in its interior. We shall refer to D\ and D2 respectively as the small and large discs bounded by J . Here we shall only be concerned with polygonal Jordan curves. Definition. A (non-degenerate)polygon (or n-gon), Ρ, in S2 is a cyclically ordered set of ( n ) points (x\,x2,... ,xn) of X such that 0 < d\(xi, χ,· + ι) < π for all i, and such that Γ(Ρ) = U" =1 [;c ( , JC,+I] is a Jordan curve. (Here we are taking subscripts mod n.) We define the perimeter of Ρ to be: η perim(P) = length Γ(Ρ) =

*,·+ι). i=l

Suppose that perim(P) < 2π . Let R(P) be the small disc bounded by Ρ . We refer to R(P) as a small polygonal region. For each 1, we write ί(Ρ,χι) for the interior angle of R(P) at JC,· . Thus R(P) is convex if and only if l{P,Xi) < π for all i (so that l(P,Xi) = Xi-iXiXi+ι). We also refer to Ρ being "convex" in this case. Note that all triangles (3-gons) are convex. We make the observation that if we "open out" a triangle at a vertex, fixing the lengths of the two adjacent edges, then the length of the opposite edge increases:

6

Β. Η. Bowditch

Lemma 1.1. Suppose that the triangles Τ = (>\ x, z) and T' = (y',x',z') satisfy di(x, y) = d\ (χ', / ) and d\{x, ζ) = d\ (χ', z'). 77ie/z Z(7\ λ:) < /(Γ', λ:') if and only if di(y,z) Γ (β') obtained by sending each geodesic segment of Γ ( ß ) isometrically onto the corresponding segment of Γ ( β ' ) . Note that in a given equivalence class, there is a 1-parameter family of darts, up to isometry in S2 . This can be thought of intuitively in terms of flexing a mechanical linkage. Proposition 1.2. Suppose Q = (x, z, y, w) and Q' = (χ', ζ', yf, w') are equivalent darts. Then, the following are equivalent: (1) l(Q,x)>l(Q',x'). (2) l(Q,y) L{Q\tT) (and/or i(Q,w) > l(Q',w')). (4) di(x,y)>di(x',y'). (5) di( Z s w)>di(z',w'). (6) The natural map f : ( Γ ( β ) , PQ) —> ( Γ ( β ' ) » PQ') is distance

(7) area(Ä(j2)) > area(R(Q')).

non-increasing.

Notes on Locally CAT(1) Spaces

7

Proof. (Sketch) The equivalence of (1), (2) and (5) follows immediately from Lemma 1.1. So also does the equivalence of (3) and (4). To see that (1) and (3) are equivalent, suppose that L(Q', z') < L(Q, z). Let u' be the intersection of [ιυ, y] extended with [χ, ζ], i.e. the unique point of [λ, ζ] such that di (u,y) + d\ (y, w) = d\ (u,w). Let u' be the corresponding point on [Λ:', ζ'] , i.e. such that d\(z',u') = d\(z,u). By Lemma 1.1 applied to ( u , z , y ) , we have d\(u',y') < r d\(u, y) and so d\(wu') < d\(w', y ) + d\(y', u') < d\(w, y) + d\(y, u) = d\(w,u). Thus by Lemma 1.1 applied to (u, x, w) we see that l(Q', x') < L(Q, as required. The fact that (6) implies (4) is trivial, and that (1) and (3) together imply (6) may be seen easily from Lemma 1.1. Finally, the equivalence with (7) may be seen as follows. Suppose Q, Q' satisfy (1) and (3). Then we may erect isometric copies of R(x', y', z') and R(xyr, w') on [*, z] and [χ, ιυ], respectively, so as to lie on the same side as R(Q). Conditions (1) and (3) now tell us that these are disjoint and contained in R(Q). We thus deduce property (7).

• Note that in each equivalence class of darts there is a unique "triangular dart", Q = (x, z, y, w), for which l(Q, y) = π . In this case, R{Q) is convex, so pQ is simply the metric d\ restricted to R(Q). We have already observed that all triangles are convex. It will often be convenient to allow for "degenerate" triangles Τ = (χ, y, ζ), where the only condition is that no pair of points of {Λ:, y, z} are antipodal. Thus perim(R) < 2π . We shall need: Proposition 1.3. For all e > 0, there exists η > 0 such that if Τ = (χ, y, ζ) is a (possibly degenerate) triangle, with d\ (y, x) + d\(x, ζ) < π —€, then d\ (a, x) < j — η, where a is the midpoint of [y,z]. • As a corollary, referred to earlier, suppose that J c S2 is a Jordan curve with length J < 2π . Choose y, ζ € J so as to divide J into two subarcs, each of length less than π , and let a be the midpoint of [x, ;y]. Then, J lies in the open hemisphere centred on a . It follows easily that J bounds one large disc and one small disc.

2. Basic properties of CAT(l) spaces In this chapter, we develop some of the basic properties of a locally CAT(l) spaces. Much of what we do in this chapter can be viewed as a combination of the accounts of Ballmann [Bai] and Charney and Davis [ChaD]. In most of Section 2.1, we shall deal generally with complete locally compact spaces. In Section 2.2, we shall restrict attention to compact spaces. Nevertheless the results of that section will be relevant to the general theory developed in the rest of the paper. An important quantity to associate to a compact locally CAT(l) space X is the "systole", sys(X), of X . There are several ways one can formulate this notion. In

8

Β. Η. Bowditch

[ChaD], the authors define sys(X) to be the minimum length of a closed geodesic in X. We shall find it convenient here to demand that sys(X) < 2π , i.e. we set sys(X) = 2π if there is no closed geodesic of length < 2π . We shall see that X is globally CAT(l) if and only if sys(X) = 2π . Note that, up till now, we have not said explicitly what we mean by a "closed geodesic". There are two sensible interpretations. We could take it to mean a closed local geodesic, i.e. a non-constant map of a circle S 1 into X which is length minimising when restricted to all sufficiently small subarcs; or else we could take it to mean a closed global geodesic which is globally length minimising, i.e. the (pseudo)metric on Sl induced by the metric d is an intrinsic path-metric on S 1 . Thus a global geodesic is always embedded, whereas a local geodesic need not be. The interpretation in [ChaD] is that of a global geodesic. It turns out that both interpretations give rise to the same notion of systole. In fact, any closed local geodesic of length equal to sys(X) is necessarily a global geodesic (see Corollary 2.2.14). In Chapter 3, we give another description of the systole in terms of what we call "shrinkability" of loops. This formulation also makes sense when X is not compact.

2.1. General observations Let us get started on this road. For the moment, all we require of (X, d) is that it be a complete locally compact path-metric space. The path-metric property tells us a priori that, given any € > 0 and x j e X , there is a rectifiable path joining χ to y of length at most d(x, y) + e . In fact, all we shall need is the apparently weaker assumption that, given such χ , y and e, together with some h e [0, d(x, y)], then there is a point ζ e X satisfying d(x, z) < h + € and d(y, z) < d(x, y) — h + €. We shall see that these hypotheses imply that any two points are joined by a geodesic, so we could, in fact, take € = 0 in either of the above statements. Given χ € X and r > 0, we shall write N(x, r) = {y e X | d(x, y) < r) for the closed metric r -ball about χ. Lemma 2.1.1. For all χ € X and r > 0 , the metric ball N(x, r) is compact. Proof. Fix χ e X and suppose, for contradiction, that not all closed metric balls about χ are compact. Let r = inf{ί > 0 | N(x,t) is not compact}. By local compactness, r > 0. First, we claim that N(x, r) is compact. To see this, let (j/)/gN be any sequence of points of N(x,r). Since d is a path-metric, we can find, for any j € Ν Π [1 + oo), points yij such that d(a, yij) < r — X and d(yt, ytj) < j + X . For all such j, the ball N(a,r — j ) is compact. Thus, by a diagonal sequence argument, we can find a subsequence of natural numbers, i(k), such that for all j e Ν Π [1 + oo), the sequence (yi(k)j)keν converges as k tends to oo. Now, given k,k' e N, we have d(ym, yi(k')) ^ d(ymj, yi{W)j) + 7 + 7^) + Τ(Ρ> w h i c h c a n b e m a d e arbitrarily small.

Notes on Locally CAT(1) Spaces

9

Thus, the sequence (yi(k))keN is Cauchy, and so converges in N(x, r). Thus, N(x,r) is compact as claimed. Now, by local compactness, we can find e > 0 and some finite set A c N(x, r) such that N(x, r) c \JaeA Ν (a, e), and such that N(a, 3e) is compact for all a e A. Now, since d is a path-metric, we have N(x, r + e) c ( J a e A N(a, 3e), and we deduce that N(x,t) is compact for all t < r + €, contradicting the definition of r . • Recall the definition of a (global) geodesic given in the introduction: Definition. A (global) geodesic joining two points x, y e X is a path a : [0,1] —> X with a(0) = χ and a ( l ) = y, such that d(a(t), a(u)) = μ\ί - u\ for all t, u e [0, 1] and for some fixed μ ( = length or). Lemma 2.1.2. Given any two points x, y e X, there is a geodesic joining χ to y. Proof. Let r = ^d(x, y). Since d is a path-metric, we can find, for any i e N, a point Zi € X satisfying d(x, n) < r + j and d(y,Zi) < r + j . Applying Lemma 2.1.1, we can find a subsequence of fo) converging to a point ζ € X , satisfying d(x, z) = r and d(y, z)=r. We can now interpolate, by induction, to obtain a map, a , of the diadic rationals [0, 1] Π such that d(a(t),a(u)) = 2r\t - u\ for all t, u e [0, 1] η Z [ | ] , and with α(0) = χ and a ( l ) = y . We may now extend by continuity to a geodesic X. • α : [0, 1] Given a geodesic a : [0, 1] —> X, we write end(a) = (α(0),α(1)) for the pair of endpoints of a. Definition. A triangle in X consists of three points X\,X2,X3 £ X and three geodesic segments a \ , «2, ccj suchthat end(a,·) = (jc/_|_ ι, jC|+2) (taking subscripts mod 3). (Figure 2a)

χ

2

χ

χ Figure 2a

3

10

Β. Η. Bowditch W e write Τ = (ατι, « 2 ,

, or Τ = (cc\, (*2, ccy,x\, Χ2, χι)

if we wish to specify the

endpoints. W e write Γ ( Γ ) for the closed path αϊ U a i U a j , and write 3 p e r i m ( r ) = length Γ ( Γ ) = ^ d ( x j , jc t +i) i=1 for the perimeter

of

Τ.

Definition. Suppose Τ = (spherical) (S2,d\)

comparison

suchthat d\(x'i,

, ct2, « 3 ; χ ι , X2, X3) is a triangle with p e r i m ( T ) < 2π . A for Γ is a triangle Τ

triangle χ'i+\)

= d(xi, Xi+\)

= (α'ι,α'2,

a'y, x'\, x'2, ^'3) in

for i e { 1 , 2 , 3 } .

Such a triangle always exists, and is unique up to isometry in S2. It may happen that T ' is degenerate, i.e. Γ ( 7 " ) need not be a Jordan curve. Definition. Suppose p e r i m ( r ) < 2π . W e say that Τ is " C A T ( l ) " if di(a'i(t),

a'j{u))

for all t,u

«/(«))
π . Thus Ρ is a "dart" in the sense of Chapter 1. We can "open out" the angle at y to form an equivalent dart Q = (χ", ζ", y", w") with l(Q, y") = π . (Figure 2c)

Figure 2c Thus {χ", ζ", w") is a comparison triangle for Τ . Now Proposition 1.2(6) tells us that the natural map / : ( T ( Q ) , d \ ) — • (Γ(Ρ), ρ) is distance non-increasing, and so therefore is the composition g ο f : (T(Q),d\) —> ( Γ ( T ) , d ) . This is the CAT(l) property for Τ . • A degenerate case of a triangle is a bigott Β consisting of two geodesic arcs α, β joining the same pair of points x, y with d(x, y) < π . Thus perim(ß) = 2 d ( x , ;y). From the definition, Β is CAT(l) if and only if a = β . Definition. We say that X is "r-CAT(l)" if every triangle of perimeter strictly less than 2r is CAT(l). We say that X is "globally CAT(l)" if it is TT-CAT(I). We say that X is "locally CAT(l)" if every point has a closed neighbourhood which is (globally) CAT(l) in the induced path-metric. From the previous discussion of bigons, we see that in an r-CAT(l) space, any two points, χ and y , which are a distance less than r apart are joined by a unique geodesic, which we shall denote by [λ —»· j ] . We note that the local CAT(l) property implies local convexity in the following sense. Suppose first that (X,d) is r-CAT(l), and that 0 < s < r/2. Suppose χ e X and E N(x, Ä) . Thus d(y, z) < r, and so there is a well-defined triangle Τ = ([Λ: -»· y], [>> -»· ζ], [ζ —• x]) with vertices x,y,z. Now perim(r) < 2r, and so, applying the

12

Β. Η. Bowditch

CAT(l) property, we see that the geodesic [>> —• ζ] maps into N(x, s). We express this by saying: Lemma 2.1.4. Suppose (X,d) is r-CAT( 1). If χ € X and s < r/2, then N(x,s) convex.

is •

Now the induced path-metric on N(x, s) is just the metric d restricted to N(x,s). We see that N(x,s) is intrinsically r-CAT(l). In fact, it will follow from the results of Section 2.2 that N(x, s) must be globally CAT(l). For the moment, let us just observe that if s < r / 3 , then N(x, s) is certainly globally CAT(l), since any triangle in N(x, s) has perimeter less than 2r . It follows that any r-CAT(l) space is locally CAT(l). Suppose we just assume that (X, d) is locally CAT(l). For any χ € X, we can find some neighbourhood, Μ , which is CAT(l) in the induced path-metric, ρ . Now, there is some η > 0 such that N(x, 2η) c Μ. On N(x, η), the metrics d and ρ must agree. The argument of Lemma 2.1.4, now shows that if e < η/3, then N(x, e) is convex, and in fact itself intrinsically CAT(l). In other words, we can always take Μ to be a convex metric ball. Elaborating on these arguments, we arrive at the following alternative formulation of the local CAT(l) property: Lemma 2.1.5. The space X is locally CAT(l) if and only if for all compact subsets Κ c X, there is some e > 0 such that if Τ is a triangle in X with all its vertices in Κ, and with perim(T) < 2e, then Τ is CAT{ 1). Proof Suppose that X is locally CAT(l). In the discussion above, we saw that for each χ G X, there is some > 0 such that N(x, 2e(x)) is convex and globally CAT(l). If Κ c X is compact, there is some finite subset A c X such that Κ c Ν (χ, . We set 6 = min{e(jc) | χ e A}. Then any triangle with vertices in Κ and perimeter less than 2e lies in one of these convex balls, and is thus CAT(l). For the converse, suppose χ e X. By Lemma 2.1.1, Κ = Ν (α, π) (say) is compact. Let € be as given by the hypotheses, and choose some η < e/3. The above discussion shows that Ν(χ,η) is globally CAT(l). • Note that in particular, a compact path-metric space is locally CAT(l) if and only if it is e-CAT(l) for some € > 0. So far we have talked only about "global geodesies". We shall also want the notion of a "local geodesic". In the case of riemannian manifolds, this coincides with the usual notion of a path satisfying the geodesic differential equation. Definition. A path a : [0, 1] —> X is a local geodesic if, for some η > 0 and μ > 0 , we have d(a(t), a(u)) = μ\ί — u\ whenever t,u e [0, 1] and \t — u\ < η. Clearly, d(a(t),a(u)) < ß\t — u\ for all t,u g [0,1]. Note also that a local geodesic is a global geodesic if and only if length« = d(a(0), a ( l ) ) .

Notes on Locally CAT(1) Spaces

Lemma 2.1.6. Suppose X is r-CAT( 1), and a length at most r. Then a is a global geodesic.

[0,1]

13

X is a local geodesic of

Proof. Let i 0 = max{i e [0, 1] | length(a|[0, t]) = d(a(0), α(ί))}. Certainly tQ > 0. If α is not a geodesic, then TO < 1. In this case, choose UQ,U\ e (0, 1) so that \U\— UQ\ < η and mo < ίο < "l · Thus d(a(uo),a(u\)) = μ(μ\ — mo) = length(a|[Mo, «ι]). Let αϊ, «2 : [ 0 , 1] — • X be linear reparameterisations of a|[0, mo] a n d a | [ M o , M j ] respectively. Thus αϊ and a2 are global geodesies. (Figure 2d) α (uQ )

oc(0)

α ω

α (up

Figure 2d Since u\ > to, we have d(a(0), a(«i)) < length (a |[0, mi]) < r . Let β be the global geodesic [α(0) a(«i)] and let Τ = ( α ι , α 2 , β ) . Thus, perim(r) < 2 r , and so Τ satisfies CAT(l). Now length β < length αϊ + length α2, from which we can deduce that d ( a ( 0 ) , a ( i ) ) < length (α | [0, t]) for all t € (mo, " l ) · This contradicts the assumption that ίο < 1, and so α must be geodesic. •

2.2. Compact locally CAT(l) spaces In this section, we shall assume that (X, d) is compact and e-CAT(l). One of the main results of this section will be Theorem 2.2.11, which gives us a characterization of the systole of X . We write geod(X) for the space of all geodesies in X , with the sup-norm metric Χ χ X by end(a) = ( a ( 0 ) , a ( l ) ) . Clearly this is continuous, and Lemma 2.1.2 tells us that it is surjective. Note that length(a) = d(a(0), a ( l ) ) , and so length : geod(X) —> [0, oo) is also continuous. Given α e geod(X), we write —a for [ί ι-»· a ( l — i)] € geod(X). As observed in Section 2.1, if x, y e X with d(x,y) < €, then there is a unique geodesic joining χ to y, which we write as [x ->· y ] .

14

Β. Η. Bowditch

Lemma 2.2.2. Suppose α, β £ geod(X) with end(a) = (ζ, λ:) and end(ß) = (z, y). Suppose dsup(a, β) < € and d(x, z) + d(y, z) + d(z, χ) < 2π. Then Τ = (β, α, [χ ;y];*,;y,z) is CAT (1). Proof. We can find 0 = t\ < tj < · · · < tk = 1 such that perim 5,· < 2e and perim Ti < 2e for all i , where S, and Γ, are respectively the triangles with vertices (cc(ti),a(ti+i), ß(ti)) and (j8(/,·), ß(ti+\), a f o + i ) ) . (Figure 2e.) We can now apply Lemma 2.1.3 inductively. •

Figure 2e Corollary 2.2.3. If α, β e geod(X) with length α = length β < π , end a = end β and dsup(a, β) < €, then a = β. Proof. By Lemma 2.2.2, the bigon a U -β

is CAT(l).

•

Corollary 2.2.4. If x,y e X with d(x, y) < π, then e n d - 1 (JC, y) C geod(X) is finite. Proof By Lemma 2.2.1, and Corollary 2.2.3. In fact, | end - 1 (x, ;y)| is bounded by the number of (e/2) -balls we can pack disjointly into (geod(X), dsup). • Given r > 0, write Given (x,y) e Π ( π ) , write n(x, _Y) = | e n d - 1 (JC, 1 for all (x, y).

. Thus, by Lemma 2.1.2, n(x,y)

>

Lemma 2.2.5. The map η : Π (π) —> Ν is upper-semicontinuous. Proof Suppose (χ,·, _y,) is a sequence converging to (JC, y) G Π (π), and that η (χ ι, j,·) > m for all i e N. For each i, choose {an, · · · , aim) to be m distinct geodesies with end(a i ; ) = (JC,·, J , · ) . Since geod(X) is compact, passing to a subsequence we have otij —> aj € geod(X) for each j. Clearly end(a 7 ) = (JC, y). Since, for all i, dsup(&ij, &ik) > ^ if j φ k, we see that the a j are all distinct. Thus n(x, y) > m. • Now let Πο(ττ) = {(JC, _Y) E Π(ΤΤ) | n(x, >>) > 2}. By Lemma 2.2.5, this is a closed subset of Π(7τ). If Πο φ 0 , set I = min{d(jc, ;y) | (JC, y) e Πο(ττ)}. Otherwise, set l = π . (We may think of I as the "injectivity radius" of X. It will turn out to equal half the systole.) Given (x,y) e Π(/), there is a unique geodesic [x —• y] e geod(X) with end[jc -»• y] = ( ^ J ) ·

Notes on Locally CAT(1) Spaces

Lemma 2.2.6. The map [ ( * , Y) H-> [Λ: ->• y]] : Π(Ζ) —> geod(X) is

15

continuous.

Proof. Suppose the sequence ( X i , y , ) tends to (x,y) · >,·] must converge to [x >>]. For otherwise, since geod(X) is compact, some subsequence would converge to a geodesic β φ [x y] joining χ to y , contradicting the uniqueness of [jc —> y ] . • Proposition 2.2.7.

X is

l-CAT(1).

Proof. Suppose Τ = (α, β, γ; χ, y, ζ) has p e r i m ( r ) < 21. Thus d{z, y(t)) < I for all t ε [0, 1], so we can set 8t = [z —• y(t)]. By Lemma 2.2.6, the map [ί i-> : [0, 1] — • geod(X) is continuous. Thus, we can find 0 = t\ < ti < · · · < tk = 1 with dsuP(Sti,8ti+l) < e for all i e {0, 1, 1}. (Figure 2f) ζ

Ydj)

Y(ti+1) Figure 2f

By Lemma 2.2.2, each triangle Tt = (8ti, Stj+l, [y(f,·) Lemma 2.1.3, Τ is CAT(l).

y ( i f + i ) ] ) is CAT(l). By •

(Note that this justifies the assertion, made after Lemma 2.1.5, that if s < r / 2 , then any metric s -ball in an r-CAT(l) space is globally CAT(l). In an ^ -ball all bigons have perimeter less than 2r , so geodesies are unique, and we can take I = π .) We want to relate the "injectivity radius", I , to the "systole" of X , i.e. the length of the shortest closed geodesic. Suppose σ is a path-metric on the circle S 1 . Up to a homeomorphism of 5 1 , σ is a multiple of the standard path-metric on S 1 as the unit circle in R 2 . We can always assume that σ has this form if we wish, though we shall have no need to insist on this. Let r = \ l e n g t h ( S σ ) = d i a n ^ S 1 , σ). We say that the points t, t' e Sl are antipodal if a(t, t') = r . Definition. A closed (global) geodesic (of length 2r) is a map γ : Sl —> X such that d(y(t), γ (Μ)) = a(t, u) for all t, u e S 1 , for some path-metric σ on 5 1 .

16

Β. Η. Bowditch

Lemma 2.2.8. Suppose γ : (S\ σ) —> (X,d) is distance non-increasing, and and d(y(t), γ it')) = r whenever t, t' g 5 1 are antipodal, then γ is a global geodesic. Proof. Suppose t,u e S1 . Then d(y(t), γ(u)) < a(t, u) and d(y(t'), y(w)) < a(t', u). But a(t, u)+a(t', u) = a(t, t') = r = d(y(t), y(t')) < d(y(t), y(u))+d(y(t'), y(u)). Thus d(y(t), y(u)) = a(t, u). • Suppose α, β e geod(X) are non-constant geodesies with end(a) = end(/J). Let σ be a path-metric on S 1 of length 2r, where r = length a = length β. We divide S 1 into two subintervals I, J , each of length r , and define a map γ : S 1 —> X by letting y\I and y\J be linear reparametrizations of a and —β respectively. We write γ = a U — β. Note that y : (Sl, σ) —> (X, d) is distance non-increasing and d(y(t), yiu)) = σ(ί, u) if t, u € / or if u e / . Recall the definition of I after Lemma 2.2.5. Lemma 2.2.9. Suppose α, β e geod(X) with α φ β and end α = end β = (x,y). Suppose d(x,y) = I, and that I < π. Then γ = a U — β is a closed global geodesic. Proof. Suppose y is not a global geodesic. Then, by Lemma 2.2.8, we can find t e (0, 1) such that d(a(t), ß(\ - t)) < I. Let 2. Thus, there are at least two distinct geodesies α, β e geod(X) with end a = end β = (x, y). • In summary, we have shown (Proposition 2.2.7 and Corollary 2.2.10) that:

Notes on Locally CAT(1) Spaces

17

Theorem 2.2.11. If X is compact and locally CAT(J), then there is some I e (0, π] such that X is l-CAT(\), and either I — π, or else there is a closed global geodesic of length equal to 21. • Now, if γ is a closed global geodesic of length < In , we may divide γ into three geodesic paths, γ = αϊ U αϊ U «3 , of equal length, to form a triangle Τ = (αϊ, α2, α·$) with perim(T) = length γ . Clearly the CAT(l) property fails for Τ. It follows that the quantity I as described by Theorem 2.2.11 is uniquely determined. Definition. We define the systole, sys(X) of X to be equal to 21 where I is the quantity described by Theorem 2.2.11. Thus, either sys(X) = 2π, or else it is the length of the shortest closed global geodesic. We shall see that we can replace "global" by "local" in the above statement. Recall, from Section 2.1, the definition of a local geodesic path, α , joining two points χ and y. As with global geodesies, we shall write end (a) = (x, y). Suppose α, β : [0, 1] —> X are local geodesies with end α = end β. Provided a and β are not both constant, we can define γ = a U —ß : S 1 —> X as in the case of global geodesies. If a is the induced path-metric on Sythen γ : (Sl,a) —>· ( X , d ) is distance non-increasing and length γ = l e n g t h ( 5 σ ) = length α -I- l e n g t h . Lemma 2.2.12. Suppose α, β : [0, 1] —> X are local geodesies with αφ β, end α = end β, and length α + length/? < sys(X). Then, either length α = length/J — π, or else γ = a U — β is a closed global geodesic. (It thus follows that lengthy = length α + length β = sys(X).) Proof Let r = \ (length α + length β) < \ sys(X). If length α = length then, by Lemma 2.1.6, α, β € geod(X). Since α φ β , we must have r = 5 sys(X). Thus, either r = π - , or else, by Lemma 2.2.9, γ is a closed global geodesic. Thus, we suppose that length α > length β . Let to e [0,1) be such that (α|[0, ίο]) has length r. Let χ = α(0) = β φ) and y = α(ίο), and let αϊ : [0,1] — • X be a linear reparameterization of a|[0, 4)]. Thus, by Lemma 2.1.6, αϊ e geod(X), and so d(x,y) = r. Now β U — (α|[/ο, 1]) is a path of length r joining χ to y, and so may be reparameterized to give β\ € geod(X) with end β\ = (x, y). Note that y = a U — β = a i U —β\. Since a is locally injective near to, we must have αχ φ β \ . Now if r < π , then by Lemma 2.2.9, γ is a closed global geodesic. Thus, we are reduced to the case where r = η = ^ sys(X). We have a\,ß\ e geod(X), with end αϊ = endßi = (JC,J), and we know that a\ U —ß\ is locally geodesic at y, and hence everywhere except possibly at χ. For t e (0, 1), consider the path (α|[ί, 1]) U — (ß|[l — t, 1]). This is a local geodesic of length π, and so by Lemma 2.1.6, we have d(a(t), ß(\ — t)) = it. In other words, we have shown that d(y(u), y(u')) = π for all pairs of antipodal points u, u' G S 1 . Thus, by Lemma 2.2.8, γ is a global geodesic. •

18

Β. Η. Bowditch

Corollary 2.2.13. If a and β are distinct local geodesies with end a = end β, then sys(X) < length a + length β . • Also, in the case where β is a constant path, we obtain: Corollary 2.2.14. If a is a non-constant local geodesic with both endpoints equal, then sys(X) < length«. Moreover, if sys(X) = length a , then a is a closed global geodesic.

• Of particular interest is the case of a closed local geodesic. Definition. A map γ : Sl — X is a closed local geodesic if for some path-metric σ on Sl, and some η > 0, we have d(y(t), y(u)) = a(t, u) whenever t,u e S1 with a(t, u) < η. Thus γ : (Sl,a) —• (X,d) is distance non-increasing. As a special case of Corollary 2.2.14, we see that any closed local geodesic γ must have length at least sys(X). If length γ = sys(X), then γ is a global geodesic.

3. Spaces of loops 3.1. Introduction Throughout Chapter 3, (X, d) will denote a complete, locally compact path-metric space which is locally CAT(l). All the results refered to in this section (3.1) are valid for all such spaces. In Sections 3.3, 3.4 and 3.5, we shall be making the additional assumption that X be /-CAT(l) for some fixed I > 0 ; the general case being dealt with in Section 3.6. By a loop in X , we mean any continuous map of the circle S 1 into X . We write Ω(Χ) for the space of all loops with the sup-norm metric, dsup. Thus, dsup(a, β) = max{d(a(t), ß(t)) \ t € S 1 }. Given γ € Ω(Χ), we write lengthy e [0, oo] for the rectifiable length of γ . Thus, the map length : Ω(Χ) — • [0, oo] is lower semicontinuous. Given r e [0, oo), we write Ω ( Χ , Γ ) = {γ E Ω ( X ) I length γ < R}.

We say that γ is rectifiable if length γ < oo. (Note that we are not making any assumptions about the parameterization of a rectifiable loop.) Suppose α, β £ Ω (X, r ) . We say that a and β are r -homotopic if they lie in the same path-connected component of Ω (X, r ) . Note that there is precisely one component containing all the constant loops in X . Of particular interest is the case r — 2π . If α, β € Ω(Χ, 2ττ), we write a ~ β to mean that they are (2π·) -homotopic. We write a ~ 0, and say that a is shrinkable if it is (2π) -homotopic to a constant loop. We write α γ 0 for not a ~ 0. The property of shrinkability seems a natural one in this context. We shall show:

Notes on Locally CAT(1) Spaces

19

Theorem 3.1.1. Suppose that x,y € X, and that αϊ, aj, «3 : [0, 1] —> X are three paths joining χ to y. Let y,· = α ί + ι U —a,+2 € Ω(Χ). Suppose that lengthy, < 2π for all i G {1,2,3}. If γι ~ 0 and γι ~ 0, then y j ~ 0. Theorem 3.1.2. Suppose Τ = (α, β, γ) is a triangle in X, with perim(r) < 2π; so that Γ(Τ) = a U β U γ e Ω(Χ, 2π). If Γ (Γ) ~ 0, then Τ is CAT (I). Theorem 3.1.3. Suppose α, β are two local geodesies joining the same pair of points, with length a + length β < 2π; so that γ = a U —ß e Ω(Χ, 2π). If γ ~ 0, then a = β. Theorem 3.1.3 is somewhat analogous to Klingenberg's Lemma in the context of riemannian manifolds [GromoIKM, page 198]. An immediate corollary of Theorem 3.1.3 is: Proposition 3.1.4. If γ e Ω (Χ, 2π) is a closed local geodesic, then γ

0.

•

The definition of r-homotopy we have given is quite weak. Note that the map [γ ι—> length γ] : Ω(Χ) —> [0, oo] is not upper-semicontinuous. Thus, for example, an r-homotopy [f γ{] : [0, 1] — • Ω(Χ, r) of two loops might takes us through loops γt of length arbitrarily close to r . We define the stronger notion of monotone homotopy: Definition. Suppose α, β e Ω(Χ). We say that a is monotonically homotopic to β, and write a \ β if there is a path [t h^· j/f] : [0, 1] — • Ω(ΑΤ), such that [r h>length y,] : [0, 1] — • [0, 00] is continuous, and length y, < length α for all t . Thus if a \ β , then length β < length a . If length β < oo, we shall demand that length γ{ < oo for all t > 0. Note that it is easy to lengthen any intermediate loop y f , for example by folding it back on itself, and so there is no loss is assuming that the map [t i-^ length y f ] is monotonically decreasing, thus justifying the terminology. Most of the monotone homotopies we construct will have this property anyway. We write y \ 0 to mean that y is monotonically homotopic to a constant loop. Clearly, y \ 0 implies y ~ 0. In fact we shall show: Theorem 3.1.5. Suppose γ e Ω(Χ, 2π). Then γ ~ 0 if and only if γ \

0.

In the case where X is compact, we will have in addition: Theorem 3.1.6. Suppose X is compact, and γ e Ω(Χ). Then either γ \ γ \ a where a is a closed local geodesic.

0, or else

Putting this together with Corollary 2.2.14, we get: Corollary 3.1.7. Suppose X is compact. If γ G Ω(Χ) and lengthy < sys(X), then y ~ 0.

20

Β. Η. Bowditch

These results give us an alternative definition of the systole of X , as the minimum of 2π and the minimum length of a non-shrinkable loop in Ω(Χ, 2π). This will also serve as a definition of systole in the case where X is only locally compact (provided we replace "minimum" by "infinum"). In this case (by Theorem 3.1.3), if r = 5 sys(X) > 0, then X is r-CAT(l). In particular, X is globally CAT(l) if and only if sys(X) = 2π . This is the same as saying that Ω(Χ, 2π) is path-connected. There is some degree of arbitrariness in the formulations of r -homotopy and monotone homotopy we have chosen. For example, it would perhaps be more natural to keep track of parameterizations by demanding that all loops and homotopies be lipschitz. It turns out that this approach would lead to essentially the same notions, as we shall observe in Section 3.5. Similarly, we could restrict attention to smooth maps in the riemannian category, or to piecewise linear maps in the context of euclidean polyhedral complexes. In order to prove the results stated in this section, we shall reduce ourselves to considering polygonal loops to which we can apply the "Birkhoff curve-shortenning process". If X is /-CAT(l), and we start with a polygonal (i.e., piecewise geodesic) loop, each of whose geodesic segments has length less than I, then we can attempt to shorten it by cyclically joining the midpoints of each segment. The Birkhoff process is the iteration of this procedure. If X is compact, then some subsequence must converge, either to a point, or to a closed local geodesic. If the length of the the original loop is < 2π , then we will get convergence to a point if and only if this loop is shrinkable. It seems to be an interesting question as to when the Birkhoff process converges without having to pass to a subsequence. We give a brief discussion of this in Section 3.7. It is not relevant to the rest of the paper. In proving the results given here, it will be convenient to deal first with the case where X is /-CAT(l) for some I > 0 (i.e., "uniformly locally CAT(l)"). We describe the Birkhoff process in Section 3.3, and give complete proofs in Section 3.4. In Section 3.6, we describe how to deal with the case where X is not /-CAT(l). We begin, in Section 3.2, with a discussion of cartesian products of CAT(l) spaces, which provides a convenient means of overcoming a technical difficulty in Section 3.3.

3.2. Cartesian products Suppose ( X \ , d \ ) and (X2, d2) are complete locally compact path-metric spaces. Then so is (X, d), where X = X\ χ X2 and

Proposition 3.2.1. If (Χι, d\) and (X2, d2)

Proof. Let pt is a triangle in triangle in X,·. a ' = [m (α'ι

are

r-CAT{\),

then so is (X,

d).

: Χι χ Χι — • X, be projection to X , . Suppose Τ = (α, β, γ) X with perim(r) < 2r. Thus 7/ = (ρ,· ο a, pi ο β, pi ο γ) is a Let Γ',· = (α',, ß'i, γ'ι) be a comparison triangle in S2 for 7}. Let (t),a'2(t))] : [0, 1] —> S2 χ S2 . Similarly define β' and / , so that

Notes on Locally CAT(1) Spaces

21

Τ = (a', β', γ') is a triangle in S2 χ S2. Let ρ be the product riemannian metric on S2 χ S2 . With respect to this metric, we have perim(7") = perim(r) < 2 r , and the natural map (Γ(Γ),ί/) — • (Γ(Γ'),ρ) is distance non-increasing. Now (5 2 χ S2,p) is a riemannian manifold of curvature < 1, with no geodesies of length less that 2π , and is therefore 7r-CAT(l) (as discussed in the introduction). Thus, T' satisfies CAT(l), and so therefore does Τ . •

3.3. Polygonal curve shortening In this section, we assume that X is locally compact, and /-CAT(1) for some I > 0. We are interested in polygonal loops, which we can think of formally as cyclically ordered η -tuples of points of X . We describe the Birkhoff process of shortening such loops. We are principally interested in the question of when the Birkhoff process converges to a point. We write C(X) = Xn for the set of η -tuples x = (x\,x2,··· which we think of as cyclically ordered so that = X[. Throughout this section, η will be constant, and so will not feature in our notation. We define the maps M, L, Ε : C(X) —> [0, oo) by M(x) = max{d(xj, *,·+ι) | i = 1 , . . . , n} η L(x) = ^2d(xi,xi+1) i=1

η

E(x) =

^2d(xi,xi+1)2. 1= 1

Thus we may think of M(x), L(x) and E(x), as the "mesh", "length" and "energy" of χ respectively. Note that if Mix) < I, then we may join consecutive points and jc,'+i by a unique geodesic in X , thus justifying our interpretation of χ as a polygonal loop. Lemma 3.3.1. For all χ € C(X),

we have

E(x) < L(x)2
X to a map of the 1-skeleton g : K\(G) —> X , by mapping each edge of F\ (G) linearly to the corresponding geodesic segment in X. Thus, the boundary, 3 Τ , of each triangle Τ e F 2 (G) gets mapped to a triangle of perimeter less than h < I. Thus, applying CAT(l), we see that g\(dT, ρ) —> (X, d) is distance non-increasing. Since ρ is a path-metric on D , we obtain: Lemma 3.3.12. The map g : (K[(G), p) —>- (X, d) is distance non-increasing.

•

We now claim: Lemma 3.3.13.

(D, p) is

n-CAT(\).

Proof. First, we show that (D, p) is locally CAT(l). Secondly, we show that (D, p) contains no simple closed geodesic. The result then follows by Theorem 2.2.11. For the first part, we need to know that each interior cone-angle, l(D, υ), for υ e V \ 3 D , is at least 2π . (See the discussion of polyhedral complexes in the introduction). This is in turn equivalent to saying that each such vertex lies in the interior of a ρ -geodesic segment. Suppose ν e V \ 3 D , so that ν = v(i, j) for some (i, j) e { 2 , . . . , m) χ { 1 , . . . , η}. Let υι = v(i — 1, j) and υ2 = v(i — 1, j -I-1). Thus ui and υ2 are adjacent to υ in K\ (G), and gv is the midpoint of gv\ and g u 2 . Applying Lemma 3.3.12, we find that ρ(υι,υ2) > d(gvu gv2) = d(gvu gv) + d(gv, gv2) > p{v\, v) + p(v, v2). Thus, ν lies in the interior of the ρ-geodesic segment joining v\ to u 2 , a n d s o Z(D, υ) > 2π as required. For the second part, suppose, for contradiction that γ c D is a simple closed geodesic. Thus, γ bounds an open disc Do c. D . From the construction, it is clear by induction that V c D \ Do. It follows that γ lies inside a spherical triangular region corresponding to some Τ e F 2 (G) which is impossible. • Note that the first part of the above proof also works to show that each vertex in V ( l ) lies in the interior of a ρ -geodesic segment. We deduce: Lemma 3.3.14. / / ν € V ( l ) , then l(dD, ν) > π.

•

Notes on Locally CAT(1) Spaces

27

From the construction, if ν e V(0), then Ζ ( 3 D , ν) < π . We recall our objective of showing that E(x) — E(f(x)) is bounded below by some positive continuous function of L(x) . Set r = \L{x) and Δ = Ε (χ) - E(f(x)) > 0 . We can thus, without loss of generality, imagine Δ to be small. This implies that for each j e { 1 , . . . , n } , d(xj,xj+i) is close to 2r/n . To be more precise, set = d(xj, Xj+\) = p(v(0, j ) , υ(0, j + 1 ) ) . Then E(x)

= Σ"=ι

and E ( f ( x ) ) < Σ%ι

4 Δ and so |£ 7+ ι — Hj\ < 2Λ/~Κ for all j .

· Thus, Σ / . ι φ + ι ~ * / ) 2 < Since Σ ; = ι

=

2 r , we deduce that

I — < n^fK for all j . In particular, if Δ < r 2 / n 4 , then > r/n. Let € = π — r > 0 . Let η be the constant of Lemma 3.3.10 (given e). Let μ = min(r/2n, η) and let 8 be the constant of Lemma 3.3.11 (given η and μ ) . We have that length(3D, p) = L(x) = 2r = 2(π — e ) , and so, by Lemma 3.3.10, there is some a e D suchthat p(a, y) < π - η for all y e 3 D . Let b e 3 D be a point of 3 D furthest from a . Now, 3 D is a piecewise geodesic loop with vertices V Γ) 3D = V ( 0 ) U V ( l ) . Since (D, p) is 7r-CAT(l), we see easily that b must be the a vertex. In fact, using Lemma 3.3.14, we must have b e V ( 0 ) . In other words, b = v(0,k + 1) for some Let υι = v(l,k) and = 1). Thus v\,v2 € V ( l ) are adjacent vertices to b. Now, p(b, t>i) = p(b, v2) = £ t / 2 > r/2n > μ. Similarly, p(b, v2) > μ. Let y ι and yi be points on the geodesic segments [b, i>i] and [b, v2] respectively, such that p(b, yi) = p(b, y2) = ß- N o w c 3 D , so p(a,yi) < /o(a,b) < π - η . Also μ < 2η, and so by Lemma 3.3.11, we have p{y\, yi) < 2μ — δ. We deduce that p(v\,v2) < p(b, vi ) + p(b, v2)-S = + & + i ) - 5If we set ζ] = p(v(\, Σ"=ι

j ) , υ(1, j +1))

= d(x(l,

j),x(

1, j + 1)), we obtain E ( f ( x ) ) =

We know that ζ] < | (£,·+£,·+,) for all j , and that ft
ι

Thus

>

We derived this inequality under the assumption that Δ < r 2 / n 4 . In other words, either Δ > 8r/n or else Δ > r2/n4. So we have shown that Δ > λ = m i n ( 8 r / n , r 2 / n 4 ) . We see from Lemmas 3.3.10 and 3.3.11 that 8, and hence λ , can be assumed to vary continuously in r = L(x). We have thus proven Lemma 3.3.9 under the assumption of strict triangle inequalities (*)· To deal with the general case, the idea is to take a cartesian product with a small regular euclidean polygon. We described earlier the Birkhoff process applied to such a polygon. In particular, we see that it must satisfy ( * ) . Suppose Y is a ;r-CAT(l) space. Then Lemma 3.2.1 tells us that Χ χ Y is n CAT(l). If χ = (χι, ...,xn) € Ch(X) and y = ( y i , . . . , yn) e Cn(Y), we write (x, y) g Ch+ηίΧ x Y) for the cycle ( ( * i , > 1 ) , . . . , (xn,yn)) • We see that f i x , y) =

28

Β. Η. Bowditch

( / ( χ ) , f ( y ) ) , where /

is used to represent Birkhoff curve shortening on X , y and

Χ χ Y. In particular, if χ e C°(X) and y e Cfj(Y), then (x, y) e C%+J](X χ Now take Υ = E

2

2

to be the euclidean plane, and let y e Cn{E )

Y).

be a regular η -gon

2

of small circumradius € > 0 . Then (jc, y) e C®+1](X χ Ε ) satisfies condition ( * ) . Thus, we have that E(x, y) - E(f(x),

f ( y ) ) > k(L(x,

y)).

We now let e -»· 0 , and deduce in the limit that E(x) - E(f(x))

>

k(L(x)).

This finally concludes the proof of Lemma 3.3.9, and thus also of Proposition 3.3.8. Theorem 3.3.15. Suppose X is locally compact and l-CAT(l) h < I, then C®(X, 2N) is open and closed in C/,(X, 2JT). Proof.

for some I > 0. If

By definition, C j ( X , 2ττ) = C°h{X) Π Ch{X, 2π), so Lemma 3.3.7 tells us that

C%(X, 2tt) is open in Ch(X, 2π). Suppose, then, that the sequence xt e C®(X, 2π) converges to some χ € C/,(X, 2π). Since L : C(X) —> [0, oo) is continuous, we can assume that e C®(X, r) for some fixed r < 2π . By Proposition 3.3.8, there is some m e Ν such that L ( / m ( j c , ) ) < / / 2 for all ι . Since / and L are continuous, it follows that L(fm(x)) < 1/2 < I, and so, by Lemma 3.3.5, x e CjJ(X, 2π). This shows that CjJ(X, 2π) is closed in Ch(X, 2π).

• 3.4. Shrinkable loops As in the previous section, we suppose that X is locally compact and /-CAT(l). Under this assumption, we shall prove the results 3.1.1-3.1.5 described in Section 3.1. (We shall describe how to deal with general locally CAT(l) spaces in Section 3.6.) In the case where X is compact, we deduce 3.1.6 and 3.1.7. In Section 3.1, we defined Ω(Χ) as the space of loops Sl —> X, and Ω(Χ, r) as the subspace of rectifiable loops of length strictly less than r . We defined the equivalence relation ~ of (2zr) -homotopy on Ω (Χ, 2π), and the transitive relation \ of monotone homotopy on Ω ( X ) . In these definitions, we have made only topological hypotheses. In Section 3.5, we show that we can restrict attention to lipschitz maps and homotopies to the same effect. Note that the definition of monotone homotopy also makes sense for paths with fixed endpoints. Thus, if α, β : [0, 1] —> X both join χ to y in X , then we write a \ β if there is a homotopy [t yt] with γο = a, γ\ = β and yr(0) = χ and y/(l) = y for all t and such that [t h>· length y,] is continuous and monotonically decreasing. Lemma 3.4.1. Suppose x,y,z e X with d(x,y) Then [x —»· U [_y —»· ζ] \ [x —• z] •

< I, d(y,z)

< I and d(x,z)

< /.

Notes on Locally CAT(1) Spaces

29

Proof. Let β = [y -> z]. Since X is /-CAT(l), d(x, /8(f)) < I for all t e [0, 1]. Let yt = [λ; U 08|[ί, 1]). If u < t e [0, 1], we see that length yu < length yt, by comparison with the spherical case, using the fact that the triangle (χ, ζ, ß(t)) is CAT(l).

•

Suppose χ — ( χ ι , . . . , χη) e Ch(X), where h < I (Section 3.3). We write Γ ( * ) for the piecewise geodesic loop obtained by joining together the segments —• x,+i] for i = 1 , . . . , η. To be more precise, we divide S 1 into η equal segments /, (with respect to the standard path-metric on S 1 ) , and map /, linearly onto the image of [jc/ ->· jc(+1 ] . In this way Γ : Ch(X) —> Ω ( Χ ) is continuous. We draw the following corollaries to Lemma 3.4.1. Lemma 3.4.2. Suppose γ e Ω ( Χ ) . Suppose t\,... ,tn e Sl divide S 1 into η segments Ji,... , Jn such that length(y|7,) < I for all i e { 1 , . . . ,«}. Then γ \ Γ ( κ ( ί ι ) , . . . ,Y(tn)). Proof. Choose any homeomorphism φ : M/nZ — > S 1 such that φ(ΐ) = ti. Given r € N, let xr = (γ ο φ(ψ), γ ο φ(ψ),... ,γ ο φ(^τ)). Thus xr is a cycle of 2 rn points of X, with M(xr) < I and M(xr) 0 as r —> oo. Applying Lemma 3.4.1, we see that Γ ( χ Γ + 1 ) \ Γ(χΓ) for all r. Also, since φ([0, η] Π is dense in S 1 , we see that Γ(^ Γ ) tends to γ in Ω(Χ) as r oo. We thus split the interval [0, 1] into subintervals [^τ+Γ. ψ ] and string together these monotone homotopies so as to obtain a monotone homotopy [t h> y,] with γ\ ßr = Γ(λ γ ) and γο = γ. We conclude that Υ \ Γ ( χ 0 ) = Γ ( κ ( ί ι ) , . . . ,γ(ί„)). • Let / : Ch(X) —> Ch(X) be the Birkhoff curve shortening map. The following is immediate from Lemma 3.4.1. Lemma 3.4.3. // χ · yt] : [0, 1] — > Ω(Χ,2π) be the homotopy joining a to β. By compactness, there is some 0 , such that if u,u' e S1 with a(u,u') < 8, then d(Yt(u), yt(u')) < h for all t G [0, 1]. Thus if ui,... ,un cut S 1 into segments of length at most 8, we have that x(t) = (yt(mi), . . . , /,(«„)) e Ch(X). Also L ( x ( 0 ) < length^,) < 2π . Thus [ί x ( 0 ] gives the desired path in Ch(X, 2π). • Lemma 3.4.7. Suppose γ € Ω (Χ, 2π) and γ ~ 0. Then there is some 8 > 0 such that if u ι,... ,un G Sl divide S 1 into segments of σ -length < 8, then (y{u\),... ,γ(μη)) G C°h(X, 2;r). Proof Apply Lemma 3.4.6, with a = γ and β a constant path. We see that χ = ( y ( « i ) , . . . , γ(μη)) is connected to a constant cycle by a path in Ch(X, 2π). Now all constant paths lie in Cj}(X, 2π), and C j ( X , 2ττ) is open and closed in C/,(X, 2π) (Theorem 3.3.15). Thus χ € C j ( X , 2?r). • We can now deduce Theorem 3.1.5, namely if γ e Ω(Χ, 2π) and γ ~ 0 , then Υ

\o.

Proof of Theorem 3.1.5. Suppose γ € Ω(Χ,2π) and γ ~ 0 . Let 8 be as given by Lemma 3.4.7. Let u\,... , un e S1 divide S 1 into segments 7, such that each has σ length < 8, and such that length(y |7,·) < h for all i. Let χ = (y(uj),... , y(un)). By Lemma 3.4.7, χ e C j ( X , 2π). By Lemmas 3.4.2 and 3.4.5, we have γ \ Γ ( * ) \ 0 . • We can also give a direct proof of Proposition 3.1.4, namely if γ € Ω(Χ, 2π) is a closed local geodesic, then y / 0. Proof of Proposition 3.1.4. Suppose γ e Ω(Χ,2π) is a closed local geodesic and that γ ~ 0 . Let L(f2n(x)) (otherwise Γ(χ) is already a local geodesic). Now set € = L(x) — L(f2n(x)) and choose r e Ν Π [In, od) accordingly. Thus, by Lemma 3.4.3, we have Γ(χ) \ Γ ( f r ( x ) ) , and so Γ(*) \ Γ(>>) = a. Thus γ \ a. • Corollary 3.1.7 now follows. If X is compact, and γ e Ω (Χ), with lengthy < sys(X), then either γ \ 0 or else γ \ a with length a < length γ . But the latter case is impossible by Corollary 2.2.14. We have still to prove Theorems 3.1.1-3.1.3 for X locally compact and /-CAT(l). For these, we will need the following lemma. Lemma 3.4.8. Suppose χ e 2π). Then there is a compact n-CAT{\) space (Υ, ρ), and points yi,... ,yn £ Υ suchthat p(yi,yi+i) = d(xi,xi+i) for all i € {1,... , n}, and p(yit yj) > d(xi,xj) for all i, j € {1,... , n). Proof. In the case where χ satisfies condition (*), the construction Section 3.3 gives us a singular spherical metric ρ on the disc D, with points y i , . . . , yn e 3D satisfying the conclusion of the lemma. (Here yi = υ(0, 1) e 3D.) In the general situation, we take a product with a small regular euclidean polygon to obtain a family of metrics {p{e) | e > 0} on the disc D , with the properties d(xi,xi+1)

< p(€)(yi,yi+i)

< d(xi,xi+1)

+ e

and d(xi,xj)


0 . Thus, (Y, p) is a spherical complex. To make sense of this, note that the (D,p(€)) are all singular spherical metrics obtained from the same combinatorial complex G . Thus, each 2-cell of G is a spherical triangle. As e -»· 0, some of these 2-cells may degenerate into geodesic segments or points. The remainder converge geometrically to become the 2-cells of Υ. We may describe the 1-skeleton K\(7) as follows. If € > η > 0 , there is a natural map (K\(G), p(€)) —> (Ki(G), ρ(η)) which is linear on each edge. The metrics p(e) thus converge to a limiting pseudometric p(0) on K\(G), so that (ΑΓι(Κ),ρ) is the hausdorffification of (tfi(G), p(0)). Now, each vertex ν in the 0-skeleton Ko(Y) of Υ is obtained by collapsing some subcomplex, G(v) of G to a point. Now G(v) must be simply connected (since any short simple closed curve in (D, p(e)) bounds a small disc). We consider two cases. If G(v) Π 3D = 0 , then the link of ν in Υ is a circle. Moreover, this circle has length at least 2π . (To see this, consider the boundary of a small uniform neighbourhood of G(v) in (D, p(e)), which we can take to be a circle. By Gauß-Bonnet, this circle must have total turning at least 2ττ — 5(e), where 5(e) -> 0 as e 0.) On the other hand, if

32

Β. Η. Bowditch

G(v) Π 3Ό φ 0 , then the link of υ in Κ is a disjoint union of points and arcs. Thus, in both cases, the link is tt-CAT(I), and so (Υ, ρ) is locally CAT(l). It remains to see that (Υ, ρ) is 7r-CAT(l). By Corollary 3.1.7, it is enough to show that if γ e Ω(Κ, 2π), then γ ~ 0 in Υ. However, this follows easily, since we can approximate γ by a loop, γ' in (D,p(€)) of p(e)-length less than 2ιτ. Now y' ~ 0 in (D, p(e)), and we may use the monotone homotopy of γ' to 0 to construct a 2π -homotopy of γ to 0. • Next we prove Theorem 3.1.2 when X is l-CAT(1). Suppose that Γ is a triangle in X with perim(r) < 2π and with Γ(Γ) ~ 0. Then we claim that Τ is CAT(l). Proof of Theorem 3.1.2. Let Τ = (α, β, γ; χ, y, ζ). We choose points jcj , . . . ,X3m, cyclically ordered on ΓΧΓ) = a U β U γ, such that d(xi, < h < I for all i, and such that xm = ζ, xim = x and X3m = y · Thus χ € Ch(X, 2π) and Γ(χ) = Γ(Γ). Now Lemma 3.4.7 tells us that, provided we have subdivided Γ (Γ) finely enough, we have χ e 2π). Thus Lemma 3.4.8 gives us a 7t-CAT(1) space (7, ρ), and points yi,··· ,y3m e Υ with d(xi,xi+1) = pCy,-,y,-+i) and d(xitxj) < p(yi,yj) for all i, j G {1,... , 3m}. Now since the points x\,... ,xm lie along the geodesic a in X, we have p(y0, ym) > d(x0,xm) = Σ?= id(xi-\,*i) = Σ i l l P(yi-u yi), and so the points yo,... ,ym lie along a geodesic ö in 7 . Similarly, ym,... , y2m lie along a geodesic β and y2m, · · · . y3m lie along a geodesic γ . Let Τ be the triangle (ä, β, γ) in Υ. Now perim(f') = perim(r) < 2π , so we may construct a comparison triangle Τ = (α', β', γ') for Τ in ( S 2 , d \ ) . This is also a comparison triangle for Τ. We have a sequence of points z\,... ,Z3m around Γ(Γ') with d\(zi, Zi+\) = d(xt,xi+1) for all i. Since Υ is 7r-CAT(l), we have d\{zi,zj) < p(yi,yj) for all i,j, and so d\ (Zi,Zj) < d(xi,Xj). Since the set {jcj,... , X3m} can be chosen to include any two given points of Γ(Γ), we conclude that Τ is CAT(l). • Now suppose that α, β : [0, 1] —> X are local geodesies with the same endpoints χ and y, and with length a + length β < 2π . Thus γ = a U — β e Ω(Χ, 27γ) . We claim that if γ ~ 0, then a = β . This is Theorem 3.1.3, in the case where X is /-CAT(l). Proof of Theorem 3.1.3. For a sufficiently large natural number m, we can find points ... , xm equally spaced along a, and xm,... , xim = xo equally spaced along β ; so that if we set μο = length a/m and μι = length ß/m, we have d(xmi+j-\, xmi+j) = μι for (i, j) e {0,1} χ {1,... ,m} and d{xmi+j-\,xmi+j+\) = 2μ, for ( i j ) € {0, 1} χ {1,... , m — 1}. We assume that μο, μι < h, so that χ = (jcj ,... , X2m) € Ch(X, 2π) and Γ(χ) = γ . By Lemma 3.4.7, we can suppose that χ e C®(X, 2π). Let Y,p,yi,... ,ym be as given by Lemma 3.4.8. Thus p(ymi+j-i, ymi+j) = μ(· and p(ymi+j-i,ymi+j+i) = 2μ(· for i, j as above. Thus, the points lie along a local geodesic ö in F , and ym,... , y2m lie along a local geodesic β . Now length ä + length β = length a + length β < 2π , and Υ is 7r-CAT(l). Thus by Corollary 2.2.13, we have ä = β. It follows that μο = μι and y, = y2m-i for all i, and so Xi = X2m-i · Thus a = β . •

Notes on Locally CAT(1) Spaces

33

In Section 3.3, we defined the Birkhoff process for polygonal loops. We can also define a Birkhoff process for piecewise geodesic paths connecting two fixed points x,y e X. Suppose χ = (XQ, ... , xn) is a sequence of points of X with xo = χ, xn = y, and M(x) = max{d(Λt·, χι+i) | i = 0 , . . . , η — 1} < h < I. Let fo(x) = (χ, x'i, x'i,... , x'n,xn) where x'i = mid(;c,-_i, *,·) for i € {1,... , n]. Let f\ ο fo(x) = (j:,mid(A: / i,x / 2),mid(x / 2,x / 3),... , m i d ^ - i , x'n), y). Note that M(f\ ο /o(*)) < Μ (χ) < h. Thus we may iterate fi ο fy. Since all metric balls in X are compact (Lemma 2.1.1), we see that some subsequence ( f \ ο fo)r'(x) must converge on some y = (jo> · · · ,yn), with M(y) < h. It's not hard to see (cf. Lemma 3.3.4) that f\ ο fo(y) = y, and that the points yo,... ,yn are equally spaced along a local geodesic a joining χ to y. Now suppose that γ : [0, 1] —> X is any path joining points χ and y in X. We can choose 0 = «ο < "ο < • · · < un — 1 so that length(y |[w,_i,«,]) < h for all i = 1 , . . . , η. Let χ = (y(«o), · · · , x("n)) and let y and a be as in the previous paragraph. As in the proof of Theorem 3.1.6, we see that γ \ a. We have shown: Lemma 3.4.9. Suppose γ : [0, 1] —• X joins χ to y in X. Then there is a local geodesic a joining χ to y such that γ \ a. • We may use this to deduce Theorem 3.1.1. Suppose a \ , a 2 , «3 are paths joining χ to y, and let y, = or,+i U — 2 € Ω(Χ). Suppose γι,γζ,γ3 € Ω(Χ, 2π) and that yi ~ 0 and Y2 ~ 0. Then we claim that γι ~ 0 . Proof of Theorem 3.1.1. Let a, , y, be as above. By Lemma 3.4.9, there are local geodesies a'\, a'2, a'2 joining χ to y such that α,· \ a',·. Let y',· = α',+ι U — a'1+2 . Then y, \ y', for each i. If yi ~ 0 and γ2 ~ 0, then γ'\ ~ 0 and y'2 ~ 0 . By Theorem 3.1.3, we see that a'2 = a'3 and c/3 = α'ι. Thus a'2 = a'\ and so y' 3 = a'\ U — a'2 ~ 0. It follows that y3 ~ 0. • This concludes the proofs of the main results 3.1.1-3.1.7 of Section 3.1, in the case where X is /-CAT(l) for some I > 0. In Section 3.6, we describe how to deal with the general case.

3.5. Lipschitz maps In this section, we observe that the results of the last section go through if we restrict attention to Lipschitz maps and homotopies. Let σ be the standard path-metric on the circle S 1 = R / Z , so that the total σ length of S 1 is 1. We may define a μ-lipschitz loop as a map y : S 1 — • X such that d(y(t), y(u)) < ßa(t,u) for all t,u e 5 1 . We write Ωι(Χ) for the space of all loops which are μ-lipschitz for some μ > 0. We write r) for the subspace of loops which are μ-lipschitz for some μ < r . Thus Ω^(Χ) c Ω(Χ) and ΩΖ,(Χ,Γ) C

Ω£(Χ)ΠΩ(Χ,Γ).

34

Β. Η. Bowditch

Given α, β e , a lipschitz homotopy from a to β is a path [f yt\ : [0, 1] —> £ I l ( X ) with γ0 = a, γι = β, and dSup(y/> Yu) < λ|ί — u\ for some λ > 0. The latter condition is equivalent to saying that the map [(t,u) yf(w)] : [0, l ] x S ! —> X is lipschitz. We may now define the relations on Ωί,(Χ) and on Ω/,(Χ, 2π) by restricting to lipschitz homotopies. We make the following observations. If γ ε Ω(Χ, r ) , then we can find a degree-1 homeomorphism φ : S 1 —>· S 1 such

that γ ο φ e

SlL(X,r).

If χ e Ch(X, r), we parameterize Γ(χ) proportionately to arc-length so that Γ(χ) e Ω/,(Χ, r). If we normalise so that some fixed point of Sl gets mapped to by Γ(χ), then the map Γ : Ch(X, r) —> Ω/,(Χ, r) is continuous. If χ e Ch(X, r), then Γ(χ) \ L Γ ( / ( λ ) ) . If χ € CJ(X, 2π), then Γ(χ) \ L 0. Suppose γ € Ω^(Χ) is μ-lipschitz, and u i , . . . , un € S 1 divide S 1 into segments of σ -length at most h/μ,. Then χ = (y(uι),... , γ(ιιη)) 6 Ch(X) and γ \ ι Γ(χ). If γ G Ω^ίΧ) and γ ~ 0, then γ 0. Suppose α, β € 2π) and a ~ β . We can find points x, y e C/,(X, 2π) such that a \ Γ (λ) and β \ Γ(^) and χ and y are connected by a path in Ch(X, 2π) (Lemmas 3.4.2 and 3.4.6). LeT [i (*ι(ί), Γ.. , Jc„(i))] : [0, 1] — • Ch(X, 2π) be a such a path. Given 0 = ίο < h < · · · < tm = 1, we may approximate [11->· *,·(*)] by the piecewise geodesic path y, = U^Ui ^ - i ) *i(f/)l · Let = (yi(r), · · · > Yn(t)) · If we choose the subdivision to,... ,tm fine enough, then we have M(z(t)) < h + η, and L(z(t)) < 2π for all t, where η < I - h. Thus z ( 0 e Ch+n(X, 2π) for all t, and the map [t Γ(ζ(ί))] : [0, 1] —> Ω^ίΧ, 2ττ) gives a lipschitz homotopy from Γ(χ) to r(y). We conclude that a β. By a similar argument, we can deduce that if α, β e Ω/,(Χ) and a \ β , then

a\

L

ß.

All of the above statements follow more or less directly from the constructions of Section 3.4. We conclude that working in the category of lipschitz maps, as opposed to continuous maps, amounts to the same thing.

3.6. The general case In this section, we describe how the proofs of properties 3.1.1-3.1.5 given in Section 3.4, generalize to the case where X is locally compact and locally CAT(l). Using Lemma 2.1.5, it is enough to observe that everything of interest goes on inside some compact subset of X .

Lemma 3.6.1. Suppose ρ is α π-CAT (I) metric on the disc D, so that d D is rectifiable and length dD D

Notes on Locally CAT(1) Spaces

by f(x,t) = ax(0 where ax = [a χ] : [0, 1] — • D. By Lemma 2.2.6, / continuous. Moreover f\dD is the identity on dD. Thus, by Brouwer degree, / surjective. In other words, if y e D, there is some χ e 3D with y e 1]), and p(x, ;y) < π / 2 .

35

is is so •

Using the construction of Section 3.3, this effectively tells us that if the Birkhoff process starting with a polygonal loop of length less than 2π converges to a point, then it will do so entirely within a metric (π/2) -neighbourhood of the original polygon. More formally, suppose Κ c X is compact, and h > 0 . Write Ch(X, 2τr, K) = [x e Ch(X, r ) | jc, e Κ for all ί = 1 , . . . , η}. Fix any €q > 0 . By Lemma 2.1.1, the set K' = N ( K , f + e o ) is compact. Lemma 2.1.5 gives us some e > 0 such that any triangle with vertices in K ' and perimeter less than 2e apart may be joined by a unique geodesic in X. We can suppose that € < eo · Now suppose that h < e , and χ € Ch(X, 2π, Κ). The Birkhoff process applied to χ is well defined provided fm(x) remains in Ch(X, 2ττ, Κ'). Let C°h(X, 2ττ, Κ) = {χ € Ch(X, 2π, Κ) \ V m ( / m ( ^ ) e Ch(X, 2π, Κ')), and L(fm(x))

—»· 0}.

From Lemma 3.6.1, and the construction described in Section 3.3, we see that if χ e Cj}(X, 2π, Κ), then the image of each loop r ( f m ( x ) ) lies inside a (π/2) -neighbourhood of Γ ( χ ) , and thus inside KQ = N(K, γ + 1 ) . Now Ko is compact and lies in the interior of K ' . Thus the argument of Section 3.3 works to show that: Lemma 3.6.2.

C j ( X , 2π, Κ) is open and closed in Ch (Χ, 2π, Κ).

•

The properties 3.1.1-3.1.5 now all follow easily from the special cases proved in earlier sections. For example, to prove Theorem 3.1.5 (if γ e Ω(Χ,2π) and γ ~ 0 then Υ \ 0), we argue as follows. Let Κ be the image in X of the homotopy [0, 1] —> Ω(Χ, 2π) from γ to a constant loop. Let e > 0 be as described above, and h < e . Now as with Lemmas 3.4.2 and 3.4.6, we can find u\,... ,un e 5 1 such that γ \ Γ(χ) and χ = (y(mj), . . . , y(un)) is connected by a path to a constant cycle in Κ . Clearly any such constant cycle lies in C®(X, 2π, Κ), and so by Lemma 3.6.2, χ g C®(X, 2π, Κ). Thus, as in Lemma 3.4.5, Γ(χ) \ 0 , and so γ \ 0 . The remaining results follow by similar arguments. We just need to observe that we can decide a priori the compact subset of X in which we are interested. As observed in Section 3.1, these results allow us to define the systole of a locally CAT(l) space as sys(X) = inf({27Γ} U {length γ \ γ e Ω(Χ, 2π), γ f 0}). Thus, if I = \ sys(X) > 0, then X is /-CAT(l).

36

Β. Η. Bowditch

3.7. Convergence of the Birkhoff process We have made use of the trivial fact that some subsequence of the Birkhoff curveshortening process defined on a compact space must converge. This suffices for our applications, though it seems natural to ask when the process itself converges. We shall describe an example of a smooth riemannian 3-manifold where convergence fails for certain polygonal loops. By scaling the metric (if necessary) we can assume that the curvature is everywhere at most 1, and so this gives an example for a locally CAT( 1) space. The convergence of the Birkhoff process seems to be an open question for riemannian 2-manifolds. (See [Ga] for some discussion of the curve shortening flow in this context.) It also seems to be unknown for real-analytic riemannian manifolds. On a riemannian manifold, the Birkhoff process is closely related to curve shortening flow for smooth curves. Such a flow γ (t, u) is defined by the equation ^ = inf / J where u € Sl, r is the time parameter, and Τ = ^ J is the unit tangent to the curve [u Η» γ(ί, Μ)] . This seems a more natural construction in this context, though one has to worry about the possibility of running into singularities. We begin with some positive results. We have already observed (Lemma 3.3.6) that if the length of a polygonal loop tends to 0 under the Birkhoff process, then the process must converge to a point. By a similar argument, we shall show that the Birkhoff process converges on any compact non-positively curved manifold, indeed on any space for which the distance function is convex locally. Suppose X is a compact path-metric space. Suppose that, for some I > 0, the map [(f, u) d(a(t), ß(u))] is convex whenever α, β : [0, 1] — • X are geodesies with diam(a([0, 1]) U y3([0, 1])) < I. (For such a space, we may define sys(X) to be the length of the shortest geodesic, and we can always take I = j sys(X).) Note that all compact locally CAT(O) spaces fall into this category. We fix some η e N, and define C/,(X) as for locally CAT(l) spaces. If h < I, then we may define Birkhoff curve shortening, / : Ch(X) —> Ch(X). Given x, y e Ch(X), let >/) | i = 1 , . . . ,n}. We see that if dsupCl,^) < I, then dsup(f(x), f(y)) ( y (0) · In other words, curvature flow on fibres reduces to steepest ascent of log0 on R 2 . It is easy to see from this that the curvature flow of fibres on the 3-torus described above does not in general converge. The Birkhoff process can be thought of as a discrete analogue of the curve shortening flow, and so this approximates to steepest ascent, though the proof in this case will involve us in some messy analysis. I am indebted to Viktor Schroeder for pointing out a gap in my original argument. Suppose [t 1-»· (ß(t), z(f))] is a geodesic in R 2 χ 5 1 , so that β is a curve in R 2 . We may compute the geodesic equation to give

38

Β. Η. Bowditch

and

D ( dß\

/ s


2π as required. Now suppose that (D, p) is π-ΟΑΤ(Ι). By Lemma 4.1.3, there is some a e D such that p(a,x) < π/2 for all χ £ 3D. We can assume that α e intD. If 3D is smooth, then it is the image of a smooth embedding γ : 5 1 —> D. Fix to e Sl, and for t € Sl, let A(t) be the area of the "sector" bounded by the geodesies [a -»· γ (ίο)] and [a -» y(t)]. (More precisely, A(t) is the area of the union of the images of [a y(u)] 1 for all u in the positively oriented interval of S joining ίο to t.) Applying the Rauch comparison theorem (see [CheE]), we find that A(t) is differentiable in t, and that dA(t)/dt < \dY(t)/dt\. Integrating over S 1 we find that area(D, p) < length(3D) < 2π . The general case follows by approximating 3D by smooth curves. • We may refine the above result by quoting the following isoperimetric inequality. Suppose (D, p) is a riemannian disc of curvature < 1, and with area(D) < 4π . Then length (3D) > L(area(D)) where L(A) = «JΑ(Απ — A). Note that the extremal case is that of a spherical cap in (S2, d\) of area A, bounded by a round circle (of length equal to L(A)). This inequality, in a variety of forms, has a long history, and one could attach to it a long list of names. For an exposition of this, and many related inequalities, we refer to Osserman's articles [Ol], [02]. We may express the above inequality in terms of the "dual" problem of spanning a circle of a given length by a disc of curvature < 1. Thus, if length(3D) = L < 2it, then

42

Β. Η. Bowditch

either area(D) < A_(L) or else area(D) > A+(L), where A±(L) = 2π±^4π2 - L2 . Thus A-{L) and A+(L) are, respectively, the areas of the small and large spherical caps in the unit 2-sphere, bounded by a round circle of length L. Clearly A_(L) < 2π < A+(L), and so by Lemmas 4.1.1 and 4.1.3, we see that this dichotomy can be expressed in terms of the shrinkability of the boundary: Theorem 4.1.4. Suppose that (D, p) is a riemannian disc of curvature < 1, and that L = length (3D) < 2π. If dD ~ 0, then area (D) < A_(L), whereas if dD f 0, then area(D) > A+(L), where A±(L) = 2π ± λ/4τγ2 - L 2 . • A result related to this is the computation of the "minimal volume" of the plane by Bavard and Pansu [BavP]. See also [Bo] for a discussion of this.

4.2. Area comparison for triangles In this section give a proof that the area of a riemannian disc of curvature < 1 bounded by a shrinkable triangle is less than or equal to the area of the small region in the unit 2-sphere bounded by a comparison triangle (Proposition 4.2.7). A version of this theorem was proven by Aleksandrov [A], an account of which can also be found in [BuserK]. We begin with some observations from spherical geometry. Given two non-antipodal points x, y € S2 , we write m(x,y) for the midpoint of the geodesic joining Λ: and y. Lemma 4.2.1. There is some universal constant k > 0 such that if (x,y,z) spherical triangle, then di(m(x,y),m(x,z))

is a

1 2 < -di(y, z) + kpenm(x, y, z) .

Proof One can obtain an explicit value for k using the spherical cosine formula. Alternatively, note that the exponential map to S2 based at some point χ e S2 is analytic. Thus, if α, β : [0, 1] —> S 2 are geodesies of length < π parameterized proportionately to arc-length with α(0) = β(0) = χ, then d\{a(t), ß(t)), as a function of t, can be written in the form at (1 + tg(t)), for some constant a, and analytic function g. We can assume that length a is bounded below, so that a and g are confined to compact sets. • Given a non-degenerate spherical triangle Tq = (jq, X2, X3) in S2, we may subdivide the small region R(TQ) bounded by Γ(7o) into four smaller regions R(Tj), i = 1, 2, 3, 4, by joining the midpoints of the edges. More formally, set yi = m(jc I+ i, XI+2), and set Ti = (xi,yi+i,yi+2) for i = 1, 2, 3, and Γ4 = (y\,y2, y3). (Figure 4a) Clearly, perim(7/) < perim(7o) for all i e {1, 2, 3,4}. In fact, Lemma 4.2.2. Given e > 0, there is some μ < 1, such that if perim(7o) < 2(π — e), then perin^r,·) < juperim(7b) for i e {1,2,3,4}.

Notes on Locally CAT(1) Spaces

43

Figure 4a Proof. Since d\(xi,xj) < 2d\{yi,yj) for i, j e { 1 , 2 , 3 } , we have perim(7i) < perim(74) for i e {1, 2, 3}. Thus, it suffices to verify that perim(r4) < μ peri m( 7o). This is certainly true for small triangles (with μ, close to 5 ), so we can suppose, without loss of generality, that d\(x\, X2) and d\(χι, Λ3) are greater than some fixed constant. If perim(74) were almost equal to perim(7o), then we would have di(y2,ys) almost equal to d\ (^2, x\) +d\(x\, ys), so that the angle at x\ would be abitrarily close to π . If we now assume, in addition, that perim(7b) is bounded away from 2π , then it follows that d\ (x2, X3) is also bounded away from 0. So, by a similar argument we see that the angles at xi and X3 are also close to π . We arrive at the conclusion that perim(7b) is arbitrarily close to 2ττ , contrary to our hypothesis. • Note that the construction of the triangles Γι, Γ2, Γ3, T4 makes sense, and Lemma 4.2.2 remains valid, even if 7o is degenerate. By a "locally C A T ( l ) triangular region", we shall mean a locally C A T ( l ) path-metric Ρ on the disc D, such that 3 D consists of three geodesic segments αϊ, OC2,013 . In other words, D = R(T) and 3D = Γ(Τ), where Τ is the triangle (αι,α2,£*3). Suppose perim(r) < 2π , then by Theorem 3.1.2, and Lemma 4.1.1, the following three conditions are equivalent: ( 1 ) T h e space ( R ( T ) , P ) is t t - C A T ( I ) ,

(2) The triangle Τ i s C A T ( l ) , (3) Γ ( Γ ) ~ 0. Suppose that (R(TQ), p) is such a 7r-CAT(l) triangular region. As observed after Lemma 4.1.2, any pair of points of R{TQ) are joined by a unique geodesic. Thus, as in the case of a spherical triangle, we may join together the midpoints of the edges of 7o to obtain four well-defined triangles T\, T2, T3, T4. If Γ(Τί) happens to be an embedded curve, then it bounds a closed disc /?(7}). Since R(TI) is obtained from R(TQ) by cutting along geodesies, we see that the metric Ρ restricted to R(Ti) is already a path-metric, and is locally CAT(l). In fact, (R(Ti), p) is 7r-CAT(l), since any embedded closed geodesic in R{TI) would be an obstruction to

44

Β. Η. Bowditch

shrinking Γ(7ο) (as in the proof of Lemma 4.1.1). Thus /?(Γ,) is itself a 7r-CAT(l) triangular region. In general, there are several ways in which Γ (7o) may be degenerate, though in all cases, we obtain a 7r-CAT(l) "region" R(Tt) "bounded by" Γ(7;). Thus, Χ(7}) might consist of an arc, three arcs connected together at a common endpoint, or a (genuine) triangular region with arcs attached to one or more of the vertices. This is a somewhat tedious complication, which we shall not worry about too much. We may iterate this procedure to obtain, at the η th stage, a subdivision of R(Tq) into 4" triangular regions. (Note that triangular regions may also degenerate into points, on iteration.) More formally, we write 7\(Tq) = {Γι, Τι, Γ3, Z4}, and define inductively T„(7o) = U{Ti(7) I Τ e T„_i(7b)}. By Lemma 4.2.2, we see that max{perim(r) | Τ e Τ η (Γ 0 )} tends to 0 geometrically in η. Also, Lemma 4.2.3. The perim(7o) < 2π).

quantity

perim(r) 2

Στ€7η(Το)

is bounded

(by

some

function

of

By Lemma 4.2.2 and the CAT(l) inequality, we see that if Τ e T„(7b), then perim(r) < 2πμη . Let Τι(Γ) = {Γι, Γ2, Γ3, Γ4}. By Lemma 4.2.1 and the CAT(l) inequality, we have that for each i, 1 2 perim(7}) < - perim(r) + 3^perim(r) , Proof.

and so 4 J^perim(7;) 2 < ρβΓ^(Γ) 2 (1 + 12*perim(r) + 1=1

36k2perimir)2)

< perim(r) 2 (l + Κμη) where

Κ = 24kn

+

By induction, it follows that

I44k2n2.

Σ perim(r) 2 < (1 + Κ μ") reT„+i(7b)

perim(r) 2

^ T&7„(To)

η

i=1

Thus T„(7o) is bounded by YYiiii1 +

0 such that ηκ(δ) < e . There is some natural number η such that perim(r) < 8 for all Τ e Τ η ( Δ ) . Thus, by Lemma 4.2.6, if Γ € Τ„(Δ), I area(Ä(r» - area(/?(r))| < eperim(r) 2 . Now area(i?(A)) =

a r e a ( R ( T ) ) , and so area(R(Tf)) < €

area(/?(A)) -

Τ"€Τη(Δ)

Σ

perim(r) 2 < Cc,

7-eT„(A)

where C is constant (Lemma 4.2.3). By Corollary 4.2.5, we have ^

area(R(T')) < area(/^(Δ / )).

Te7n(A)

Thus area(Ä(A)) < area(^(A')) + Ce. The result follows by letting €

0.

•

The conclusion is independent of the lower curvature bound, —κ2 . In fact, it's not hard to see that this condition can be dropped.

Notes on Locally CAT(1) Spaces

47

The same argument can be used to show that if R (Δ) is a triangular region of curvature < 0 ( < — 1), then area(i?(A)) < area(i?(A")) where A" is a euclidean (hyperbolic) comparison triangle. Here, the bound on perim(A), and the hypothesis that Γ ( Δ ) ~ 0 are redundant. By analogy with Theorem 4.1.4, it seems reasonable to conjecture that if R(A) is a riemannian triangular region of curvature < 1, with perim(A) < 2π and with Γ ( Δ ) f 0, then we have area(Ä(A)) > 4;r - area(/?(A')).

References [A]

A.D. Aleksandrov, Metrische Räume mit einer Krümmung nicht größer als K, Der Begriff des Raumes in der Geometrie, Bericht von der Riemann Tagung (1957).

[AZ]

A.D. Aleksandrov, V.A. Zalgaller, Intrinsic geometry of surfaces, Amer. Math. Soc. Translations of Mathematical Monographs (1967).

[Bai]

W. Ballmann, Singular spaces of non-positive curvature, Chapter 10 of Sur les Groupes Hyperboliques d'aprts Mikhael Gromov (ed. E. Ghys, R de la Harpe), Progress in Maths. 83, Birkhäuser (1990).

[BalGS]

W. Ballmann, Μ. Gromov, V. Schroeder, Manifolds of Non-positive Curvature, Progress in Math. 61, Birkhäuser (1985).

[BavP]

C. Bavard, P. Pansu, Sur le volume minimal de Μ 2 , Ann. Scient. Ec. Norm. Sup. 19 (1986), 479-490.

[Bo]

B.H. Bowditch, The minimal volume of the plane, J. Austral. Math. Soc. (Series A) 55 (1993), 2 3 ^ 0 .

[Br]

M.R. Bridson, Geodesies and curvature in metric simplicial complexes, in Group Theory from a Geometric Viewpoint (ed. E. Ghys, A. Haefliger, A. Verjovsky), World Scientific (1991), 373^163.

[BrH]

M.R. Bridson, A. Haefliger, An introduction to CAT(0) spaces, in preparation.

[BurGP]

Yu. Burago, M. Gromov, G. Perelman, A. D. Alexandrov's spaces with with curvatures bounded from below, I, Usp. Mat. Nauk 47 (1992), 3-51.

[BurZ]

Yu.D. Burago, V.A. Zalgaller, Geometric Inequalities, Grundlehren der Mathematischen Wissenschaften 285, Springer-Verlag (Translation of Russian edition of 1980).

[Busemi]

Η. Busemann, Spaces with non-positive curvature, Acta Math. 80 (1948), 259-310.

[Busem2]

H. Busemann, The Geometry of Geodesies, Pure and Applied Math. 6, Academic Press (1955).

[BuserK]

P. Buser, Η. Karcher, Gromov's almost flat manifolds, Asterisque No. 81, Societe Mathematique de France (1981).

[ChaD]

R. Charney, M. Davis, Singular metrics of non-positive curvature on branched covers of riemannian manifolds, Amer. Jour, of Math. 115 (1993), 929-1009.

[CheE]

J. Cheeger, D.G. Ebin, Comparison Theorems in Riemannian Geometry, NorthHolland (1975).

[E]

D.B.A. Epstein et al, Word Processing in Groups, Jones and Bartlett (1992).

[Ga]

M.E. Gage, Curve shortening on surfaces, Annal. Scient. Ec. Norm. Sup. 23 (1990), 229-256.

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[GromolKM] D. Gromoll, W. Klingenberg, W. Meyer, Riemannsche Geometrie im Großen, Lecture Notes in Mathematics No. 55, Springer-Verlag (1968). [Gromov] [GrovP]

M. Gromov, Hyperbolic groups, in Essays in Group Theory (ed. S.M.Gersten) M.S.R.I. Publications No. 8, Springer-Verlag (1988), 75-263. K. Grove, P. Petersen, A radius sphere theorem, Invent. Math. 112 (1993), 577-583.

[HR]

C.D. Hodgson, I. Rivin, A characterization of compact convex polyhedra in hyperbolic 3-space, Invent. Math. I l l (1993), 77-111.

[01]

R. Osserman, Bonnesen-style inequalities, Amer. Math. Monthly 86 (1977), 1-29.

[02]

R. Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), 11821238.

[Pa]

F. Paulin, Constructions of hyperbolic groups via hyperbolisations of polyhedra, in Group Theory from a Geometric Viewpoint (ed. E. Ghys, A. Haefliger, A. Veijovsky), World Scientific (1991), 313-372.

[Pe]

G. Perelman, A. D. Alexandrov's spaces with with curvatures bounded from below, II, preprint.

[S]

J.J. Stoker, Geometric problems concerning polyhedra in the large, Comm. Pure and Applied Math. 21 (1968), 119-168. Faculty of Mathematical Studies, University of Southampton Highfield, Southampton S 0 9 5NH, Great Britain, email: [email protected]

Asynchronous Automatic Structures on Closed Hyperbolic Surface Groups Noel

Brady

Abstract. We examine the geometry of asynchronous automatic structures on a finitely presented group G. When G is the fundamental group of a closed hyperbolic surface we show that (up to equivalence) it has only one asynchronous automatic structure: the geodesic-language automatic structure.

Introduction One of the basic problems in the theory of automatic groups is to determine whether or not a particular finitely presented group G has an (asynchronous) automatic structure. A natural extension of this is to ask for a description of all possible (asynchronous) automatic structures on G. This latter question is interesting in its own right, as it gives us an opportunity to study the interplay of geometry and logical complexity in discrete groups. In addition, some of the techniques developed in an effort to answer this question for a given G may help us with the basic problem for other groups. In this paper we give a complete description of the set 21(G) of equivalence classes of asynchronous automatic structures on G, the fundamental group of a closed hyperbolic surface. Roughly, two such structures are equivalent if words in one asynchronously fellow travel words in the other. We give precise definitions of asynchronous automatic structure and the equivalence relation in Section 1. Here is the main result of the paper. The set S21(G) consists of equivalence classes of automatic structures on G. Theorem 1. Let G be the fundamental group of a closed hyperbolic surface. Then S2l(G) = 21(G) contains only the class of the geodesic-language automatic structure. The proof begins by describing certain geometric properties of asynchronous automatic structures, which are derived from their logical simplicity (regularity). These properties are combined with some hyperbolic geometry and a transversality argument in the plane to yield the result. This proof provides a nice example of the interplay of geometry and logical complexity which is fundamental to the theory of automatic groups. We use similar techniques in [Br] to show that lattices in the 3-dimensional Lie group Sol are not asynchronously automatic. Key words and phrases. Closed hyperbolic surface groups, asynchronous automatic structures, regular language, fellow traveller property.

50

Ν. Brady

The rest of this paper is organized as follows. In Section 1 we give the basic definitions and describe the background for the main result. In Section 2 we describe geometric properties satisfied by asynchronous automatic structures on an arbitrary group G. These are used with some topology and geometry of the hyperbolic plane to prove Theorem 1 in Section 3. We give generalizations and pose some questions in Section 4. We would like to thank the referee for the many helpful suggestions concerning the layout and content of this paper. We also thank Paul Brown who supplied the graphics for the figures.

1. Definitions and background We define the notions of (asynchronous) automatic structure, equivalence of structures, and the sets 21(G) and S2l(G). We define and list some properties of hyperbolic groups. All these definitions appear in the literature, in particular [Ε], [B], [NS] and [S]. Finally, we give the background for the main result: Theorem 1. Let G be a finitely generated group and A c G be a finite set of monoid generators. Typically, one takes A to be the union of a finite set of group generators for G and the set of their inverses. We shall assume throughout that A is closed under taking inverses. Let A* denote the free monoid on A and w w denote the natural map A* G. Call elements of A*, words over the finite alphabet A. We are interested in subsets L c A* which map onto G. We can place restrictions on the logical structure of L as follows. A finite state automaton with alphabet A, is a finite, directed graph, whose edges are labelled by elements of A such that no two edges with the same label can have the same initial vertex. We call the vertices of the graph states. One state is called the start state, and a subset of the states are called accept states. Reading the labels along a directed edgepath, with initial vertex the start state and terminal vertex an accept state, gives us an element of A* called an accepted word. A subset L c A* is called ^regular language if it is the set of accepted words of some finite state automaton with alphabet A. If L C A* —> G is a regular language and maps onto G we say it is a rational structure for G. We can also view L as a set of edgepaths in a metric space, and place geometric restrictions on it. Given a finite monoid generating set A C G (closed under taking inverses), one can construct the Cayley graph ^ ( G ) of G with respect to A. The set of vertices of T ^ G ) is just G itself. Take as the set of edges all the triples (g, a, gä) where a e A and g € G. Since A is finite, r , i ( G ) is a locally finite graph. Let each edge of TA(G) be isometric to the interval [0, 1], and define the distance between two points to be the infimum of the lengths of all paths in T^(G) joining them. This path metric makes T ^ G ) into a geodesic metric space (i.e. for all x, y € r ^ ( G ) there is a path joining χ to y of length equal to d(x,y), the distance between χ and j ) . A word w € A* gives an edgepath in r ^ ( G ) (also denoted by w ) which starts at the identity and ends at the vertex w. We parameterize these paths by arc length up to their endpoints,

Asynchronous Automatic Structures on Surface Groups

51

where they remain constant. Thus w(t) = w for all t > length(tu). Here length(u;) denotes the length of the path w, which equals the length of w as an element of A*. We say L c A* has the Κ-fellow traveller property if there exists Κ > 0 such that for all u,v e L with u~lv e Λ U {id} then d(u(t), v(t)) < K, for all t > 0. This says that if two L -paths end distance < 1 apart in r ^ G ) , then points travelling at unit speed along them (starting at the same time at the identity) stay within Κ of each other. Say that L c A* has the asynchronous Κ-fellow traveller property if there exists Κ > 0 such that for all u,v e L with u~lv e Λ U {id} there are proper monotone reparameterizations φ and ψ (depending on u and v) such that d(u((f>(t)), v(\J/(t))) < Κ for all t > 0. This is similar to the previous definition, except that we now allow points to speed up or slow down (but not backtrack) along the L -paths in order to stay within Κ of each other. We say that L C A* has a departure function D : M+ M+ if for all r > 0, for all ν e L and for all 0 < s, t < length(if) we have |i - t\ > D{r) =>• d(v(s),

v(t)) > r.

An automatic structure on G is a rational structure L —> G which satisfies the Κ -fellow traveller property. We say that G is an automatic group if it has such a structure. An asynchronous automatic structure on G is a rational structure L -> G which satisfies the asynchronous Κ-fellow traveller property, and has a departure function. This is called a boundedly asynchronous automatic structure in [E]. It is proven in [E, Theorem 3.3.4] that an automatic structure L —• G has a linear departure function. Thus an automatic structure is a special case of an asynchronous automatic structure, where the reparameterization maps are the identity and the departure function is linear. However, in [E, section 7.4] and [B, section 7] there are examples of groups which are asynchronously automatic but not automatic. Even automatic groups can admit other asynchronous automatic structures which are not automatic structures. The fundamental groups of closed hyperbolic 3-manifolds which fiber over the circle give examples of such groups. See [NS] for more details. One can always obtain new (asynchronous) automatic structures from a given structure by passing to substructures or by adding an element of A* \ L to L. In considering the set of all (asynchronous) automatic structures on a group we will not distinguish between structures formed in this way. Following [NS] we introduce an equivalence relation on these sets of structures. Define L c A* and L' c B* to be equivalent (denoted by L ~ L') if there is a Κ such that L U L ' satisfies the asynchronous Κ -fellow traveller property with respect to the metric on r^ußiCz). Denote by 21(G) the set of equivalence classes of asynchronous automatic structures on G, and by S5l(G) the subset of equivalence classes of automatic structures on G . The 'S' stands for 'synchronous' highlighting the fact that in an automatic structure the words synchronously fellow travel each other. Note that the definition of asynchronous automatic structure above, the one in [E], and the one in [NS] all give rise to the same set 21(G).

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Ν. Brady

Let L c A* be an (asynchronous) automatic structure for G and Β be another monoid generating set for G. Theorem 2.4.1 of [E] shows how to find a corresponding structure L' c B* for G. It is clear from their construction that L ~ L'. In Theorem 2.5.1 (resp. Theorem 7.3.2) of [E] it is shown that every automatic structure (resp. asynchronous automatic structure) L -»· G has a substructure L ' c L with uniqueness·, i.e. L' -»· G is an automatic structure (resp. asynchronous automatic structure), and the projection map restricted to L' is bijective. Thus we can find representatives with the uniqueness property for every class of structures by working with one specific generating set for G. This will allow us use a specific generating set in the proof of Theorem 1 in Section 3. The geometric content of the definitions of (asynchronous) automatic structure with uniqueness gives rise to the following concepts, which are the subject of intense research at present. By a synchronous combing (resp. asynchronous combing) on a finitely generated group G we mean a section σ for the projection map A* G, such that a(G) C A* satisfies the if-fellow traveller property (resp. asynchronous AT-fellow traveller property). We say that σ is a combing with departure if a(G) has a departure function. Clearly an asynchronous automatic structure on G gives rise to an asynchronous combing with departure. We shall see examples of asynchronous combings with (exponential) departure function on closed hyperbolic surface groups which are not asynchronous automatic structures. Next, we define the notions of hyperbolic group and δ-hyperbolic metric space. These were introduced by Gromov in [G]. We follow the terminology and definitions of [S]. A path connected metric space X with distance function d is called a geodesic metric space if for all points x, y in X there is an isometric map from the interval [0, d(x, j ) ] to a path in X joining χ to y. Such paths are called geodesies. If G is a finitely generated group, then the Cayley graph of G (with respect to a finite generating set A) with the usual path metric is a geodesic metric space. Given three points x, y, ζ in X one can find geodesies joining them, to form a geodesic triangle xyz. By the triangle inequality there is a comparison triangle x'y'z' in the Euclidean plane (so that d(x, y) = άε(χ', y') etc., where dg denotes the Euclidean metric). The maximum inscribed circle meets x'y'z' in points a, b, c as shown, where d£(x',b) = dE(x',c) etc. Collapse this comparison triangle down to a tripod Τ by isometrically identifying the segment [x',b] with [x', c] etc., as shown in Figure 1.1. The resulting composed map τ : xyz —• Τ is an isometry on the edges of xyz. We say that the geodesic triangle xyz is δ-thin if diam(T -1 (/?)) < 8 for all ρ e Τ. A path connected geodesic metric space X is called 8-hyperbolic if there exists 0 such that every geodesic triangle in X is δ-thin. A finitely generated group G is called hyperbolic if the Cayley graph r,i(G) is a S-hyperbolic metric space. In [S] it is shown that if Β is another finite generating set then r ^ ( G ) is 0, so the property of being a hyperbolic group is independent of the generating set.

Asynchronous Automatic Structures on Surface Groups

53

τω

Figure 1.1. 5-thin triangles The following proposition describes a property of hyperbolic groups which plays a key role in the proof of Theorem 1. It is a condensed version of two results found in [S]. First we give a definition. , Let X be a geodesic metric space with distance function d. A path w : [a, b] X, parameterized by arc length, is called a (λ, €)-quasigeodesic if there exists constants λ > 1 and e > 0 such that for all s, t e [a, b] one has \s — t\ < kd(w(s),w(t))

+ €.

One can similarly define the notion of infinite and bi-infinite (λ, e)-quasigeodesics. As the next proposition shows, these concepts are important in the theory of hyperbolic groups. Proposition 1.1. ([S, Propositions 3.2 and 3.3]) Let g be an element of infinite order in a hyperbolic group G, and let Γ denote the Cayley graph of G with respect to a finite generating set. Let a be a path in Γ from the identity vertex to g. Then the infinite path «σο = [α,

ga,...)

is a quasigeodesic. Furthermore, there exists an integer D > 0 such thatfor all points x, y on Qfoo the subsegment of α G be an asynchronous automatic structure, and suppose that there exists Ν > 0 such that every I £ L' lies in an Ν-neighborhood of the geodesic g G L with g = I. Then L ~ V. Proof Let Κ and K' be the fellow traveller constants for L and L' respectively. We need to find a K\ > max(Ä\ Κ') such that words in L U L ' asynchronously K\ -fellow travel. Suppose that I e L' and g e L satisfy d(l,g) < 1. Thus I lies in an (N + AT)-neighborhood of g. Consider I and g as paths in TA(G) parameterized by arc length. Let χ,· = I(i) so that Jto = id and xc = Ϊ where c is the length of /. Choose Zi on the geodesic g so that d ( x i , z i ) < Ν + Κ and let zo = id and zc = g- Note that d(zi,Zi±i) i(j) andd(id, Zi) > d(id, z,•(_,·))}. Since zc = g we always get the next value of i(j + l). Since d(zi,zi±\) < 2(Ν+/ίΓ) + 1, it follows that d(zt(j), z/t/'+i)) < 2(JV + £ ) + 1. Thus · Η —> G —• G/H 1 is a central extension of groups, where G is finitely presented, Η is finitely generated, and G/H is hyperbolic (in the sense of Gromov). Then G has a cubic polynomial isoperimetric function. Proof. It is a well-known fact (see [ABC] for details) that a finitely generated group is Gromov hyperbolic if and only if it has a linear isoperimetric function. Applying the previous result, we are done. • Corollary 3. A finitely generated nilpotent group of class η has a polynomial isoperimetric function of degree 2-3". A weaker result which bounds the degree in terms of the Hirsch number of the group was independently proven in [G2]. Proof. We assume corollary 1, and proceed by induction on the class of G. We let Ln(G) denote the n-th term of the lower central series for G. If G has class zero then G is abelian and thus has a quadratic isoperimetric function. Now suppose G has class n. Then, by induction, G/Ln(G) has a polynomial isoperimetric function of degree

Isoperimetric Functions for Central Extensions

75

2 · 3 n 1 . Thus, by corollary 1, G has a polynomial isoperimetric function of degree 2-3". • To prove the main result we need a slightly stronger version of a result originally proven in [Gl, Lemma 2.2]. Lemma. Let 7 be a finite presentation, Μ the maximum length of relators in 7, and let f be an isoperimetric function for 7. Then g(x) = ^(M · f ( x ) + x) is both an isoperimetric function and an isodiametric function for 7, in particular for each reduced word w of length η which represents the identity there is a van Kampen diagram for w with area and diameter each no more than g(n). The above conclusion is stronger than the statement that g is both an isoperimetric and isodiametric function for C P , since among diagrams for a particular word, those having the least area may not have the smallest diameter. Proof It suffices to consider the case where / is the Dehn function for 7 (i.e., / is minimal among all isoperimetric functions for 7 ) . We will first show that g is an isoperimetric function for 7. It is enough to check that g > / . This follows immediately from the definitions except when Μ < 1, in which case, 5(7) is a free group and CP is a presentation where each nontrivial relator is one of the generators. One can easily show that for such presentations the Dehn function is either the identity function (when Μ equals 1) or the constant zero function (when Μ equals 0). Thus f ( x ) < x, and g(;t) > f(x). Now we will show that g is an isodiametric function for 7. Let w be a reduced word of length η representing 1 in 9(7). Select a van Kampen diagram, D, for w with area no more than f(n). Choosing ρ to be a geodesic path of maximal length in we have that the diameter of D^ is the length of p. Since ρ is a geodesic, it can intersect at most half the edges of any 2-cell in D and may cross any edge (1-cell) in D^ at most once. Notice that w must cross each edge which does not lie in the boundary of any 2-cell precisely twice. Let Μ be the maximal length of a relator. Recall that i(w) denotes the length of w as a string. Then, diameter(D (1) ) = length(p) 1 < - • Μ • #{2-cells in D) 2 4- #{edges not in the boundary of any 2-cell in D} 1 t(w) 1 Thus g(x) = {(Μ • f ( x ) +

η

x) is an isodiametric function for 7.

•

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G. R. Conner

Proof of Theorem.

Let {h\,...,

hk) be a generating set for Η so that Η is the direct

product of the cyclic subgroups {hi).

Choose { g i , . . . , £*} so that

S = is a generating set for G .

...,gk,hi,

...,hk}

Let 3~(£) be the free group on the set S. Then there is a

natural homomorphism φ: 3^(5) — > G. W e proceed by choosing a finite subset R\ of J ( 5 ) so that ({R\)) = ker(φ) and the presentation

= (5; R\) of G has the following

properties: 1.

[hi,hj]e

RiVl

2. By a directed corner of σ we mean a triple a = (σι, σ, aj) where σ\ and σι are principal faces of σ . The inverse of a is given by ä = (σ2,σ, σ ι ) . Since d i m a > 2, the intersection σ\2 = σ\ Π σι is a non-empty principal face of σ\ and of σι. We sometimes identify the corner a with the loop (σι2, σι, σ, σ2, σΐ2) in the principal face graph of X. Definition. A (simplicial) complex of groups is a triple G = (X, G, φ) where 1. X is a connected simplicial complex; 2. G is a function that assigns to each non-empty simplex σ in X a group Ga and to each principal face σ' of σ an injective homomorphism G(a', σ ) : Ga -»• Ga>; 3. φ is a corner labeling function that assigns to each oriented corner a = (σι, σ, a group element φ(α) g GCTi ησ2 such that the following three axioms are satisfied: (i) φ(ά) = φ(α)~ ι for each oriented corner a . (ii) If a = (σι, σ, σ2>, then Ga ι

•

0σιησ2

/ Ga

t

\ Ga2

^

Ga\ Πσ2

is required to be a commutative diagram where the vertical arrow is the conjugation map Λ; M>φ(α)χφ(α)~χ. (iii) Given a simplex σ with dimσ > 3 and principal faces σ ι , σ ι , and a j , write Oij = ajdaj and σι 2 3 = σιΓ)σ2Γ)σ3 . Put gij = G(ai23, σί])(φ(σί,σ, σ,·)) € Gam . Then the following product in the group Gffm on the codimension 3 face is required

Groups Acting on Complexes and Complexes of Groups

81

to be trivial: 812 · ψ(σ\2, σ 2 , σ23> · £23 • φ(• Υ, a homomorphism fa\ Ga ->• # / ( σ ) for each simplex σ of X, and an element / ( r , σ) € Η/χ τ ) for each principal face τ < σ such that fx ο G(T, or) = Ad / ( τ , σ) ο tf(/(r), / ( σ ) ) ο fa (where Ad h denotes the inner automorphism χ a = (σι, σ, σ 2 ) of Χ ,

hxh~l) and such that for each corner

fal2( Η and g: Η —> F be morphisms. Then the composition g o / : G —> F is amorphismwhere g o / : X - » Ζ is the usual composition of functions; for each simplex σ in X, (go f ) „ = ga ο fa ; and for each principal face τ < σ, (g ° / ) ( * , o) = gr[/(T, or)]g(r, σ ) . Hence a category of complexes of groups is formed; the identity morphism id: G ->· G is, by definition, the identity simplicial map on the underlying complex, the identity homomorphism idcr: Ga -»• Ga on each simplex, and for each principal face r < σ , id(r, σ) = 1 € GT . Example 1.3. Consider a regular G-complex Y as in Example 1.1. Suppose we have two sections of the G-orbit map, j\, j2: cells X cells Y. For i = 1,2, and for each principal face r < σ in X , choose an element g t (r, σ) of G such that ji(j) Let G, be the complex Example 1.1. Then G j identity map on X can For each simplex σ

< gi(r,a)

• ji(a).

of groups determined using the section / and the (τ, σ) as in and G2 are isomorphic; an isomorphism / : Gi -> G2 over the be produced as follows. of X , choose an element ασ e G such that h(σ)

=ασ

· j\(a)

and define fa : Gjl (σ) -> Gj2(a) by χ H> ασ χ a~l. define / ( τ , σ) as

For each principal face r < σ ,

/ ( τ , σ) = ατ g\(r, σ) α^Γ1 g2(r,

.

One can easily check that the conditions of a morphism are satisfied. Furthermore, / has an inverse where f~l is conjugation by a~l and f~l( τ,σ)

= a~l gi(T,a)aa

gi(r, σ ) " 1 .

Therefore, altering the choices involved in forming the complex of groups produces an isomorphic one, so a regular, simplicial action determines a complex of groups uniquely up to isomorphism. We conclude this section by introducing the notion of the fundamental group of a complex of groups. We shall see in a later section that the fundamental group plays a key role in the theory of groups acting on 1-connected complexes analogous to that of the fundamental group of a graph of groups in the Bass-Serre theory of groups acting on trees. Given a complex of groups G = (X, G, φ), let F\ be the fundamental group of the graph of groups (X 1 , G | X J ) on the 1-skeleton. Then for each 2-cell σ of X , let

Groups Acting on Complexes and Complexes of Groups

83

φ(σ) denote the cyclic word obtained by reading the edges and corner labels in order with respect to some orientation around the boundary of σ . Then φ{σ) determines a conjugacy class of elements of F\; see [C2, § 2]. Definition. The fundamental group of G is defined as π\ (G) = F\/N where F\ is the fundamental group of the graph of group on the 1-skeleton X1 and Ν is the normal closure in Fi of the set of cyclic words φ{σ), for σ a 2-cell of X. Recall that the fundamental group of a graph of groups can be defined in various equivalent ways; see [Se], [SW], and [C2, § 2]. The approach most suitable for defining induced homomorphisms is to take the fundamental group to be the set of homotopy classes of words whose underlying paths are loops at the base point [Se]. Let G = (X, G, φ) and Η = (Κ, Η, ψ) be complexes of groups and / : G Η a morphism. Then there is a homomorphism /* determined on the fundamental groups of the graphs of groups on the 1-skeletons by f*\Gv = fv for each vertex ν of X and f*(e) = f(ie, e) f(e) f{ze, e)~x for each directed edge e of X. One can check that for each 2-simplex σ of I , /*( Κ satisfying two condition: 1. Constant fibers over cells. The following diagram is commutative: Ka X Dn —^

Dn

Κ

X

and ga\Ka x Int Dn is a cellular homeomorphism onto ρ~ι(σ). 2. πι-injective on fibers. For each t 6 Dn , the homomorphism 7Γ1 {Κσ X {f})

7Γ1 (P

l

(gat))

84

J. Μ. Corson

induced by restriction of ga to Κσ χ {ί} is injective. (Notice that this follows from Condition 1 for t e Int Dn as the restriction map in this case is a homeomorphism onto the fiber over ga (t).) Maps ga and ga satisfying the above conditions shall be called a characteristic pair of maps for σ . Put Kn = p~l{Xn) where Xn is the n-skeleton of X and let pn '. Kn —y Xn denote the map obtained by restricting p. If σ is an η-cell of X and ga,ga is a characteristic pair of maps for σ , then ga maps (Κσ χ Dn, Κσ χ dDn)

(Kn,

Kn~l).

Observe that Kn is the adjunction space of Kn~l and a copy of Κσ χ Dn with a map g0\Ka χ 3D n for each η-cell σ of X. The restriction pn : Kn -»· Xn is a complex of spaces on the «-skeleton of X. (Notice that in general Kn is not the n-skeleton of Κ but it contains the n-skeleton of Κ .) Lemma 2.1. Suppose X is a simplicial complex and ρ : Κ —• X is a complex of spaces over X. Let σ be a simplex of X and denote by St σ the open star of σ in X. Then the inclusion Ka c p~l(Sta) is a homotopy equivalence. (Here, and elsewhere, we identify Κσ with the fiber p~l(ba)

over the barycenter of σ .)

Proof To see this, let S = p~l (St σ) and Sn = S Π Kn . Then for η > dim σ , we see that Sn+l is homotopy equivalent to Sn as for each (n + l)-simplex r G St σ , ρ '(τ) is homeomorphic to Κτ χ r which retracts to p~l (τ) η Sn by lifting a simple homotopy collapsing r . Since S dim Ε XQY i s a covering map. A map ρ : Ε XQY ^ X is determined on the orbit spaces yielding a commutative diagram: ExY

Pr J2

° > Y

4 Ε xGY

I—

X

Now ρ : Ε XQ Υ ^ X is a complex of spaces on X. To see this, let σ be a cell of X and choose any characteristic map ga: Dn —> X for σ . Let σ be a lift of σ in Y. Since G is acting on Y without inversions, ga lifts to a

Groups Acting on Complexes and Complexes of Groups

85

characteristic map gä: Dn ->- Y for σ. There is a map ga: E/G& X Dn -» £ XG F determined such that the diagram: Dn

Ε χ

•

Ε

I x Z)n

£ / G ä

Y

I« gg

>

E x

1 Dn

χ

g

Y

i' —

X

is commutative. It is easy to verify that the maps and g a are a characteristic pair of maps for the cell σ ; the fiber over σ is E/G„ . Moreover, if Υ is 1-connected, then π\{Ε x g Y) = G. Universal covering relative to a complex of spaces. Let a complex of spaces ρ : Κ X be given. Let Κ be the universal cover of Κ . Form a space Y(p) by collapsing to a point each component of the preimage in Κ of each point of X under the composite map Κ ^ Κ Ά - Χ . Let ρ: Κ —> Υ (ρ) be the quotient map. Then Y(p) is a CW-complex whose cells correspond under ρ to the components of the preimages in Κ of the cells in X. Since the covering group G = πι (Κ) permutes these components, a cellular action of G on Y(p) is induced such that the diagram: Κ

Y ( p )

Λ Κ

G Χ

is commutative. Furthermore, ρ is a complex of spaces; characteristic pairs of maps are given as follows. Let σ be a cell of X and σ a cell of Y(p) such that ρ(σ) = σ; say dimor = η. Choose a pair of characteristic maps ga, ga for σ . Restrict ga : Κσ χ Dn -»· Κ to Κσ = Κα X {0} and consider the induced homomorphism π \ ( Κ σ ) —>· π\ ( Κ ) . Let K& be the covering space of Κσ corresponding to the kernel of this homomorphism. Then by elementary covering space theory, ga lifts uniquely to a map gä'. Kä x Dn Κ whose image corresponds to σ (under ρ). Notice that gä \K„ x Int Dn is injective since ga \ Κσ χ Int Dn is, and the composite map ρ ο gä collapses the first factor; hence it factors through the projection Proj 2 : K„ x Dn —Dn yielding a characteristic map gä'. Dn —> Y(p) for σ . The maps g„, gä are a characteristic pair for the arbitrary cell σ of Y(p); whence ρ is a complex of spaces over Y(p).

86

J. Μ. Corson

Simplicial complexes of spaces. We now restrict to simplicial complexes of spaces, that is, complexes of spaces on simplicial complexes. We show that a simplicial complex of spaces determines a complex of groups up to isomorphism. Let X be a simplicial complex. Given a complex of spaces ρ : Κ X on Χ , define Go = π\(Κσ) for each cell σ of X. Let Γ c X be the principal face graph of X (see § 1) and define a map s: Γ Κ as follows. For each vertex σ e Γ , define s(a) to be the base point of Κσ and for each edge e in Γ , say e = ( σ \ σ ) , let s(e) be a path in p~l(e) joining the base points of Κσ> and Κσ . (Recall that the edge e is identified with the edge in the subdivision X' joining the barycenters bai and ba .) We shall call such a map s a lift of the principal face graph to Κ. Now let σ' < σ be a principal face and let e be the edge joining σ' and σ in Γ . Observe that the inclusion Ka> c p~l(e) induces an isomorphism on fundamental groups (see Lemma 2.1) and Κσ c p~l(e) induces an injective homomorphism of the fundamental groups. Thus by identifying πι(Κσ') with πι(p~1(e)) and translating the base point of Κσ along the path s(e), we can regard JTI (K„) as a subgroup of π\ (Κσ>). This gives an injective homomorphism G(a',a):

Ga

Ga>.

Next let a = (σ\, σ, σ-ι) be a corner in X. Recall that we have identified a with the loop (σΐ2, σ\, σ, σι, σγ£) in Γ where σγι = σ\ Πσ2· Observe that s(a) is a loop in p _ 1 ( S t a i 2 ) based at the base point of Κση . By Lemma 2.1, the inclusion Κσ 12 c p - 1 ( S t f f i 2 ) is a homotopy equivalence. Define φ(α) to be the homotopy class in Gan = (Κσl2) corresponding to the class of the loop j ( a ) . It is easy to see that G(p) = (X, G, φ) is a complex of groups. Furthermore, G ( p ) is uniquely determined up to isomorphism of complexes of groups. For if , * Γ —> Κ are two lifts of the principal face graph and Gi and G2 are the respective complexes of groups, then there is an isomorphism / : Gi —> G2 defined as follows. On the underlying complex X, f is the identity simplicial map; for each simplex σ of X, fa is the identity homomorphism; and for each principal face r < σ, / ( τ , σ) is the homotopy class in Gx = π\ (Κτ) of the loop S2(t, σ) · 5ΐ(τ, σ ) - 1 . (Actually this loop lies in p~l(Str) and we take the corresponding class in 7Γι(ΛΤσ) under the homotopy equivalence induced by the inclusion map; see Lemma 2.1.) Definition. The complex of groups G(p) determined by a simplicial complex of spaces ρ : Κ —> X is called the associated complex of groups. Theorem 2.3. Suppose ρ: Κ —y X is a simplicial complex of spaces such that for each simplex σ of X, the inclusion induced homomorphism π\(Κσ) —> π\(Κ) is injective. Then the associated complex of groups G(p) is isomorphic to the complex of groups corresponding to the action of the universal covering group G = 7t\ (K) on Y{p); in particular, G(p) is developable.

Groups Acting on Complexes and Complexes of Groups

87

Proof. Let s: Γ - » Κ be a lift of the principal face graph and define G(p) using s . Make the choices involved in forming the complex of groups corresponding to the action of G on Y(p) as follows. Choose a maximal tree Τ c Γ containing the base point. Let s: Τ —• Κ be the unique lift of s \ Τ to the universal cover that hits the base point of Κ. Then j = ρ ο s is a continuous section of the orbit map Υ (ρ) —>- X . (Since Τ contains all the vertices of Γ , i.e., cells of X , restricting j gives a section cells X cells Y{p).) For each simplex σ of X and principal face τ < σ , define [r, σ] g π\(Κ) as the homotopy class of the loop S(YT) • Ί(Τ,Σ)

·

SO"1)

where γτ is the unique reduced path in Τ from the base point to the vertex r . Let θ: π\ (Κ) —> G be the standard isomorphism that takes the homotopy class [α] of the loop a to the unique covering transformation that maps the base point of Κ to the terminal point of the lift of a whose initial point is at the base point. Put g(τ,σ) =θ{[τ, σ]) and observe that _/'(r) < g(r, σ) · j(a) and g(r, σ) = 1 if the edge (r, σ) lies in Τ. Construct G as in Example 1.1 using the section j and connecting elements g(r, σ ) . Now s is compatible with the section j and connecting elements g(r, σ) in the following sense. Identify π\(Κσ) with a subgroup of π\{Κ) by translating the base point along the path s{ya). (This uses the assumption that Κ σ Κ is injective on the fundamental groups.) Then θ identifies π\(Κσ) with Ga (the stabilizer of 7 ( σ ) ) and there is a commutative diagram: π\(Κσ)

• G,j

[r-σ]

πχ{Κτ)

—^

Gr

where the vertical maps are conjugation by the element on the arrow. (Note that for edges in Τ , the vertical maps are inclusions.) Hence there is an isomorphism / : G(p) —> G which is the identity on X ; for each simplex σ of Χ , /σ = ΘΙπι(Κ σ ); and for each principal face τ < σ , /(τ, σ) = 1. • Now we investigate the fundamental group π\ (Κ) for a simplicial complex of spaces ρ : Κ -> X . The restriction ρ 1 : Κ 1 -> X 1 is a graph of spaces and it follows that πι(Κι) is the fundamental group of the associated graph of groups on X1; see [SW]. (Recall that Kn denotes ρ~λ(Χη) and it is not in general the n-skeleton of Κ, though it does contain it.) Observe that K 2 can be regarded as an adjunction space of K l and a copy of Κσ χ D for each 2-cell σ of X. The attaching map ga I: Κσ χ Sl ^ Kl IS 7Γ1injective and factors through Κσ χ D2. It follows from van Kampen's theorem that ΐΐ\ (Κ2) is the quotient group of π\ (AT1) obtained by adding a relation determined by the loop Sl ^ Κσ χ S{ ^ Kl for each 2-cell of X. Taking S 1 to be the subset {*} χ S 1

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where * is the base point of Κσ , we see that the relation corresponding to the 2-cell σ is the cyclic word φ(σ). Furthermore, π\(Κ) = π\ (Κ2) as the 2-skeleton of Κ is contained in K2. Hence we have the following little theorem. Theorem 2.4. Suppose ρ: Κ —>- X is a simplicial complex of spaces and let G be the associated complex of groups. Then π\(Κ) = π\ (G). • Remarks. 1. The fundamental group of a simplicial complex of spaces only depends on the associated complex of groups. In fact it only depends on the 2-complex of groups associated to ρ 2 : Κ 2 —> X 2 and furthermore, it does not even depend on the groups on the 2-cells. 2. We can define the fundamental group of a complex of groups G topologically by choosing a complex of spaces ρ : Κ X with associated complex of groups G; such complexes of spaces exist by the following theorem. Then t t i ( G ) = π\(Κ) by Theorem 2.4, independent of the choice of complex of spaces. The following result is proved in [C3] and [H2]. (In [H2], however, the definitions of complexes of groups and complexes of spaces are slightly different from ours.) Theorem 2.5. Suppose G = (X, G, φ) is a complex of groups and for each simplex a of X, suppose an Eilenberg-MacLane complex Ka of type (Ga, 1) is given. Then there exists a complex of spaces ρ : Κ —>• X with fibers Κσ such that the associated complex of groups is G.

3. Fundamental group of a complex of groups Throughout this section let G = (X, G, φ) be a fixed complex of groups. For each positive integer η, let Γ„ be the principal face graph of Xn ; see § 1. Then G determines a graph of groups (Γ„, G|T n ) on Γ„ as follows. The vertices of Γ„ are the simplices a of Xn and G assigns groups Ga to them. The edges of Γ„ are of the form e = (τ, σ) where σ is a simplex in Xn and r is a principal face of σ . Define Ge = Ga and define the monomorphisms by GT
2 bean integer. Then the fundamental group π\ (G) is isomorphic to the quotient of Fn by the normal subgroup generated by the corner relations (τ,σ\)(σι,σ)(σ2,σ)~ι(σ2,τ)~ι

=

where (σι, σ, σ2) is a corner in Xn and τ = σ\ Π σ2.

φ(σ\,σ,σ2)

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We can restate the theorem as follows. For η > 2, let Rn be the set of cyclic words over the graph of groups (Γ„, G|T n ) of the form φ(σ\, σ, σ2)~1(τ, σι)(σι, σ)(σ2, σ Γ 1 to, τ ) - 1 . Then the theorem says that { Fn | Rn ) is a generalized presentation for ττχ (G) over the graph of groups on Γ„ in the sense of [C2]. Let Rι be the set of cyclic words φ(σ), one for each 2-simplex of X. Then, by definition, tti(G) is given by the generalized presentation {F\ | R\); see §1. The theorem is proved by showing how to pass from the generalized presentation ( Fn \ Rn ) to (Fn+1 I Rn+i) by generalized Tietze transformations, which we now briefly describe; see [C2, Appendix B] for more details. Let (Γ, G) be a graph of groups and let R be a set of cyclic words over (Γ, G). Let F be the fundamental group of the given graph of groups and let Ν be the normal closure of R in F. We call (F | R) a generalized presentation of the group F/N over the graph of group (Γ, G). The passage from the generalized presentation ( F \ R) over (Γ, G) to a new generalized presentation ( F' | R') over a graph of groups (Γ", G') by one of the following three ways or its inverse will be called a generalized Tietze transformation. Type 0. Attach a tree of groups to (Γ, G) and add relators to R defining the elements of the groups on the tree: Let (Τ, Η) be a graph of groups where Γ is a tree. Identify a vertex of Τ with a vertex of Γ to form Γ' = Γ ν Γ , Define G' by G'„0 = GVQ * HVo on the identified vertex VQ , and by requiring that G' = G on Γ — {υο} and G' = Η on Τ — {uo} · Let R' consist of the cyclic words of R together with a cyclic word of the form ugYvgY~l for each vertex ν e Τ and each g in a set of generators for Ηυ, where yv is the unique reduced path in Τ joining uo to ν and ug is some word over (Γ, G) whose underlying path is a loop at i>o. Type 1. Attach a collection of redundant relators: Put Γ' = Γ , G' = G, and R' = R U S where S is a set of cyclic word over (Γ, G) that lie in the normal closure Ν of R. Type 2. Attach a collection of edges and groups to (Γ, G) and add a collection of cyclic words to R defining the new edges and groups: Attach a collection of edges {e, } to Γ and for each e,, let be a word over (Γ, G) whose underlying path is a path in Γ from r(e, ) to i(e,·) and let G'ei be a group that admits monomorphisms into G^ej) and Gr(€|.) satisfying the property: if g e G'ei and gl and gT are the images of g in Gt(C(.) and GT(ei) respectively, then wj~lgrwi(g')_1 is in N. Let G' be such an extension of G to a graph of groups on Γ' = Γ U {e,}. For each i, let r, be the cyclic word of the form ei u>i and put R' = R U {r,}. Remark. Two presentations over graphs of groups define isomorphic groups if and only if one can be transformed into the other by a finite sequence of generalized Tietze transformations; see [C2, Appendix B].

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Before giving a proof of Theorem 3.1, we first establish some notation. Form an increasing sequence of simplicial trees T\ c T2 c · · · such that Tn is a maximal tree of Γ η . Given an η-simplex σ of X (n > 2), there is a unique principal face ισ < σ such that the edge (ισ, σ) of Γ lies in the tree Tn . For if r < σ is another such principal face, then (τ, σ)(σ, ίσ) is a path in Tn . But Tn 3 Tn-\, a maximal tree for Γ π _ ι , so there is also path in Tn-\ joining τ and ισ and hence there exists a non-trivial loop in T n , a contradiction. Recall the group F(G|Γ„, Γ„) defined in [Se, § 1.5.1]. This is the group generated by the groups Ga and the elements (τ, σ), where σ is a vertex of Γ„ and (r, σ) is a directed edge of Γ„ , subject to the relations (σ, r ) = (r, σ ) - 1

and

(r, σ)χ(τ, σ)~ι = G(r, σ)(χ)

for each simplex σ in Xn , principal face r < a , a n d χ € Ga . Then Fn = π ι ( 0 | Γ η , Γ π ) is isomorphic to the quotient of F ( G | r „ , Γ Λ ) by the normal subgroup generated by the edges in Tn , [Se, Proposition 1.20]. Proof of Theorem 3.1. First we wish to pass from the generalized presentation { F\ \ R\) to { F2 I /?2) using generalized Tietze transformations. We observed above that for each 2-cell σ of X, there is a unique principal face ισ < σ such that (ισ, σ) is an edge of T2. We first do a type (0) generalized Tietze transformation: attach the edges (ί (σ), σ) to Γ ι , put the group Ga both on the edge (ι(σ),σ) and on its terminal vertex σ , and define the elements of Ga by the relations (ισ, σ)χ(ισ, σ ) - 1 = G(ia, σ)(χ),

χ e Ga.

Next use a type (2) generalized Tietze transformation to attach the remaining edges (e, σ), e φ ίσ , of Γ2 — Γ ι : put the group Ga on (e, σ), and add the defining relations (e, σ) = (ν, e)~^(a)(v,

ισ)(ισ,

σ)

where a = (e, σ, ισ) and ν = eCiia . This is a type (2) generalized Tietze transformation by condition (ii) of the definition of a complex of groups. Now if e\, ei < σ are edges of X, neither equal to ισ , then in the presence of the corner relations at the other two corners of σ , the relation (v,e\)(ei,a)(e2,oyl(v,

e2)_1 = φ(ε\,σ,

e2)

is equivalent to the relation φ(σ) = 1. Hence by using generalized Tietze transformations of type (1) to first add the redundant corner relations and then deleting the redundant relators φ ( σ ) , we arrive at the presentation ( F 2 \ R 2 ) · Now suppose η > 2. Start with the presentation ( Fn \ Rn) and form a new presentation by attaching, for each (η + l)-simplex σ of X , the edges (ισ, σ) to Γ„ , put the group Ga on the edge and on the terminal vertex, and add the defining relations (ισ, σ)χ(ισ, σ)~ι = G(ia, σ){χ), This is a generalized Tietze transformation of type (0).

χ e Ga.

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Then attach the remaining edges (r, σ ) , τ φ ισ , of Γ„ + ι — Γ„ and edge groups and add the defining relations (τ, σ) = (η, τ)~ιφ(α)(η,

ίσ)(ισ, σ)

(*)

where a = (τ, σ, ισ) and η = τΠισ . This is a type (2) generalized Hetze transformation by condition (ii) of the definition of a complex of groups. If σι, σ 2 < σ are principal faces, neither equal to ta , then the corner relation φ(σ\,σ,

σ 2 ) = (τ, σι)(σι, σ)(σ 2 , σ ) _ 1 ( τ , σ 2 ) _ 1 ,

τ=σιΠσ2,

is now a consequence of the relation in condition (iii) of the definition of a complex of groups corresponding to the principal faces σι, σ 2 , ισ < σ . Hence these redundant corner relations can be adjoined by a generalized Tietze transformation yielding the presentation ( F n + \ | /?„+i). • The generalized Tietze transformations of the above proof yield homomorphisms hn '· Fn+\ Fn determined by hn\Fn = id, hn\Ga = G(ia, σ) for each (n + 1)simplex σ of X, and hn(τ, σ) equals the RHS of (*) for each principal face τ < σ (where (ισ, σ) = 1 as it is an edge of Tn). Furthermore, hn induces an isomorphism Fn+l/(Rn+i) = Fn/(Rn). We now give two corollaries after first fixing some notation. Let F be the fundamental group of the graph of groups (Γ, G|T) determined on the full principal face graph. Let Ν is the normal closure in F of the corner relations (r, σ ι ) ( σ ι , σ ) ( σ 2 , σ ) _ 1 ( σ 2 , τ ) - 1 = φ(σ\,σ,

σ2)

where (σι, σ, σ 2 ) is a corner in X and τ = σ\ Π σ 2 . Corollary 1. The fundamental group it\(G) is isomorphic to the quotient

F/N.

Proof. Observe that Τ = U™={Tn is a maximal tree for the full principal face graph Γ . Use Τ to define F as above. Since the maximal trees Tn are nested, there are obvious inclusions F\ c F 2 c · · · and F = . It follows from the proof of Theorem 3.1 that there exists epimorphisms θη : Fn —> 7ri(G) suchthat 9n+i\Fn = 6n and ker0„ is the normal closure of Rn . Hence there is an epimorphism 9\ F —> π\ (G) defined by θ\Fn — 9n. Now ker0 is the normal closure of all the corner relations R = . For if w e F then w e Fn for some n. So w e ker θ if and only if w € ker 9n for some n . • Next consider a complex of groups coming from a simplicial complex of spaces ρ: Κ X. Recall that π\ (Κ) = it\ (G); see Theorem 2.4. Corollary 2. Suppose ρ: Κ -> X is a complex of spaces over a simplicial complex X, let s : Γ —^ Κ be a lift of the principal face graph of X, and let G be the associated complex of groups relative to s. Then there is a generalized presentation of the fundamental group Ν >—> F —» πι(Κ).

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The epimorphism F —> π\ (Κ) is determined by the inclusion induced homomorphisms Ga = 7t\{Ka) —> 7Γ[(K) (the base point of K„ is translated along the path s(ya) where γσ is the unique reduced path in Τ joining the base point of X to the vertex corresponding to σ) and by mapping each (τ, σ) to the homotopy class in π\(Κ) of the loop s(YT)s(T,a)s(Y~l). •

4. Structure of groups acting on complexes Let a group G and a connected, regular G-complex Υ be given; put X = Y/G. Then given a section j: cells X —> cells Υ and given an element g(τ, σ) e G for each principal face τ < σ in X such that j ( j ) < g(r, Υ of the orbit map. To see this, notice that G acts on the principal face graph of Υ with quotient Γ and there exists a continuous section on Τ into the principal face graph of Υ (hence into Y) of this action on a graph; see [Se, Proposition 1.14] or [SW, Proposition 4.2]. Now let r < σ be a principal face in X. If the edge (r, σ) of Γ lies in Τ, then j (τ) < j (σ) in Υ and in this case we take g(r, X by choosing a 1-connected, free G-complex Ε and using the Borel construction as in Example 2.2. Theorem 4.1. With the above notation and assumptions, the associated complex of groups G ( p ) is canonically isomorphic to G. Proof

Recall that there is a commutative diagram: ExY

κ

?r0j2

> Υ

—

χ

and E x Y is a regular covering of Κ with G acting as covering group. Observe that the proof of Theorem 2.3 did not use the assumption that Κ is the universal cover, and in fact, the argument works for any regular covering. Hence, by the proof of Theorem 2.3, it suffices to show that the inclusion induced homomorphism π\(Κσ) -»· π\(Κ) is injective for each simplex σ of X.

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As in the proof of Theorem 2.3, let θ: π\ (Κ) -> G be the standard isomorphism. Observe that the composition

identifies π\{Κσ) with Ga , and hence, π\{Κσ) σ of X , as required.

-»• π\{Κ)

is injective for each simplex •

Corollary 1. With the above notation and assumptions, there is a homomorphism F determined by Ga

G

and

G

(r, σ) ι-> g(r, σ)

for each simplex σ of X = Y/G and principal face τ < σ. Moreover, this homomorphism induces an epimorphism θ: π\ (G) —> G whose kernel is isomorphic to πι (Y).

• Remark. One now gets a presentation for a group acting on a simplicial complex from any presentation for the corresponding complex of groups it determines, for example the presentation in § 1 or one of the presentations in § 3. Compare with [Br]. Corollary 2. A complex of groups G = (X, G, φ) is developable if and only if the natural homomorphism Ga —> 7TI(G) is injective for each simplex σ of X. (The natural homomorphisms Ga π\ (G) are only determined up to conjugacy as they depend on the choice of maximal tree in the principal face graph of X.) Proof Suppose G is developable, say arising from the regular action (G, Y). Let θ: 7i\ (G) G be the homomorphism of Corollary 1. Then for each simplex σ in X , the composition Ga —» tti(G)

G

is the inclusion; whence the first map is injective. The converse follows by forming a complex of spaces with associated complex of groups G and applying Theorem 2.3. • Corollary 3. Suppose G = (X, G, ψ) is a developable complex of groups and let ρ : Κ —> X be a complex of spaces with associated complex of groups G. Then the associated complex of groups G(p) is isomorphic to the complex of groups determined by the action of TTI(G) on

Y(p).

Proof By Corollary 2, the inclusion Κσ ^ apply Theorem 2.3.

Κ is injective on fundamental groups. Now •

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5. The main theorem Recall that a group acting regularly on a simplicial complex determines a complex of groups structure, up to isomorphism, on the orbit space; see Example 1.1. A complex of groups that arises in this way from a simplicial action is called developable. All graphs of groups are developable and the fundamental theorem of Bass-Serre theory relates graphs of groups and simplicial actions without inversions on trees; see [Se, Theorem 1.13]. We now extend this result of Bass-Serre theory. Theorem 5.1. The process of obtaining a complex of groups from a regular simplicial action gives a one-to-one correspondence between the isomorphism classes of developable complexes of groups and regular simplicial actions on \-connected complexes where two simplicial actions are considered the same if they are equivariantly isomorphic. Proof Let 1-connected simplicial complexes Y and Ζ and regular actions (G, Y) and (Η, Ζ) be given. Suppose these two actions determine the same isomorphism class of complexes of groups. Then we need to show that Y and Ζ are equivariantly isomorphic simplicial complexes. That is, show the existence of an isomorphism of groups u:G Η and a simplicial isomorphism / : Y -»· Ζ such that f(g · σ) = u(g) · f (σ) for each g e G and simplex σ of Y. Put X = Y/G and let Γ be the principal face graph of X. As in §4, choose a maximal tree Τ c Γ and a continuous section j : Τ Y of the orbit map and a family of connecting elements g(r, σ) relative to j ; let G = (X, G, φ) be the corresponding complex of groups. If H' is a complex of groups determined by the action (Η, Ζ), then by assumption, there is an isomorphism G = H'. Use this isomorphism to identify the orbit spaces ZjH = X = Y/G. We shall next form an equivalent complex of groups Η = (Χ, Η, ψ) = Η' such that there exists an isomorphism / : G Η satisfying / ( τ , σ) = 1 whenever the edge (r, σ) is in the tree Τ. Lemma. There exists a continuous section k: Τ —> Ζ of the Η-orbit map and a family of connecting elements h(r, σ) such that the corresponding complex of groups Η = (Χ,Η,ψ) admits an isomorphism f : G Η satisfying / ( τ , σ) = 1 whenever the edge (τ, σ) is in the tree Τ. Proof By Zorn's Lemma, there is a maximal subtree 7o c Τ such that there exists a continuous section ko: Τ Ζ and connecting elements ho(r, σ) relative to ko such that the corresponding complex of groups Ho admits an isomorphism /o: G -»• Ho with /o(r, σ) = 1 for each edge (r, σ) in 7o. Suppose To φ Τ. Then there exists an edge (α, β) of Τ — To that has one vertex in 7o; assume α is a principal face of β in X. Let T\ = To U {(α, β)}. We get a contradiction by showing that there is a continuous section k\: Τ Ζ and connecting elements relative to this section such that the corresponding complex of groups Hi admits

Groups Acting on Complexes and Complexes of Groups

95

an isomorphism fx: G Hi with / i ( r , σ) = 1 for each edge (τ, σ) in Γι. Hence Τ = TQ and the lemma follows. Case 1. α is a vertex of TQ. Define k\ such that KX\TO = fcol^b and k\{ß) = Μα, β) • k0(ß). (Since Μα, β) stabilizes ko(a) = kx(a), indeed (&ι(α), k\{ß)) is an edge in the principal face graph of Ζ.) Thus we have defined k\ \ T\ continuously. Extend to a continuous section k\: Τ -> Ζ . (This can be done by a simple argument using Zorn's Lemma; see [Se, Proposition 1.14] or [SW, Proposition 4.2].) Then choose a family of connecting elements relative to k\ and let Hi be the corresponding complex of groups. Since ko and k\ are sections of the orbit map, there exists elements aa e Η such that k\(a) = ασ · ko(a) for each simplex σ in X , We take ασ = 1 for σ a vertex of 7o and αβ = /ο (α, β ) . Let / ' : Ho Hi be the isomorphism over the identity on X determined by the ασ ; see Example 1.3. Then / ' ( r , σ ) = 1 for each edge (τ, σ) in 7o and f'{a,ß) = fo(a, ß)~l. Hence f\= f ο / ο : G ->• Hi is an isomorphism such that, by Lemma 1.2, Μα, β) = f'a[f0(a,

ß)]f'(a,

β) = I

as f = id and f ( a , ß ) = f0(a, ß)~l. Also / i ( r , σ) = 1 for each edge (r, σ) in To. Case 2. Β is vertex of TQ . Define the section fci by the formula k\ (σ) = fo(a, β) · ko(a) for each vertex σ of Τ. Let ασ = /ο(α, β) for σ φ a and αα = 1. Then k\ (σ) = ασ • ko(a) (notice that k\(α) = ko(a)); let f : Ho —> Hi be the isomorphism determined by the α σ as in Example 1.3. Now for (r, σ) in 7o, / ' ( r , σ ) = a r / i o ( r , a)a~lh\(r,

σ)-1

= Met, β) fo(a, β)~1 = 1 and f'ia,

ß) = aah0(a,

β)α~βλ1ΐχ{α, β)~ι = Μα,

β)~ι.

(Recall that A;(r, σ) = 1 for each edge ( τ , σ ) in Γ . ) Hence, as in the other case, f\ — f ο / ο : G Hi is an isomorphism such that / i ( t , σ) = 1 for each edge (r, σ ) in Γι. • Proof of Theorem 5.1 (continued). Consider the presentations for 7TI(G) and πχ (Η) of the form given by Corollary 1 to Theorem 3.1 using the maximal tree Γ . Since / ( r , σ ) = 1 for each edge (r, σ ) in Γ , it follows that / * : π χ (G) —> πχ (Η) is determined by f*\Ga

= fa

and

/*(τ, σ) = / ( r , σ ) · (τ, σ)

for each simplex σ of X and principal face τ < σ .

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J. Μ. Corson

Define an isomorphism u : G -> Η such that the diagram: JTI(G)

—

^

ΠΙ(Η)

ι .

θι G

—=-•

Η

is commutative where the vertical arrows are the isomorphisms of Corollary 1 to Theorem 4.1. Then for each simplex σ of X, u\Ga = fa and for each principal face τ < σ , er)) = / ( τ , σ)Λ(τ, σ) . Now define / : Υ Ζ by f i g · jo)

= u(g) · k(a)

for each g G G and each simplex σ of X , This is well-defined because gΙ • j(a) g2 · j(a) implies g^gι G Ga so uig^gx) G Ησ = Hk(a). Hence u(g\) · k(a) u(g2) ·

= =

k(a).

To show that / is a simplicial map, it suffices to show that it preserves the principal face relation. Notice that the principal face pairs of simplices in Y are of the form g · ϊ(τ) < g · g(j, σ) · 7 ( σ ) where τ < σ is a principal face pair of simplices in X. Hence, given a simplex σ in X and a principal face τ < σ, it is enough to show that /O'T) is a face of / ( g ( r , σ) · ja). But / Ο ' τ ) = k{τ) < h(r, σ) · k(a) so / O ' t ) < f i r , a)h(r, = u(g(r, = f(g(r,a)

σ) · k(a)

[as / ( τ , σ) e Ητ =

Hk(T)]

σ)) · k(a) ·

ja).

Hence / is a simplicial map and it is clearly w-equivariant. Since a symmetric construction using / - 1 : Η ->• G produces a simplicial map / _ 1 : Ζ -»• y given by f~l(h-ka) = u~l(h)· j(a) for h G / / , we see that / has an inverse, whence it is an equivariant isomorphism as required. •

References [B] [Br] [CI] [C2] [C3] [HI]

G.E. Bredon, Introduction to Compact Transformation Groups, Academic Press, New York, 1972. K.S. Brown, Presentations for groups acting on simply-connected complexes, J. Pure Appl. Algebra 32 (1984) 1-10. J.M. Corson, Complexes of groups, Proc. London Math. Soc. (3) 65 (1992) 199-224. J.M. Corson, Small cancellation theory over graphs of groups, preprint. J.M. Corson, Complexes of groups II and finiteness properties of groups, preprint. A. Haefliger, Complexes of groups and orbihedra, in Group Theory from a Geometrical Viewpoint (eds. Ghys, Haefliger and Verjovsky), World Scientific, Singapore, 1991, pp. 504-542.

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97

[H2]

A. Haefliger, Extension of complexes of groups, Ann. Inst. Fourier, Grenoble 42, 1-2 (1992), 275-311.

[SW]

P. Scott and C.T.C. Wall, Topological methods in group theory, in Homological Group Theory (ed. C.T.C. Wall), London Mathematical Society Lecture Note Series 36, Cambridge University Press, 1979, 137-203.

[Se]

J.-P. Serre, Trees, Springer-Verlag, New York, 1980.

[Sp]

B. Spieler, Ph.D. thesis, Ohio State University, 1992.

University of Alabama, Box 870350, Tuscaloosa, AL 35487-0350 Email: [email protected]

The Generalized Tetrahedron Groups Benjamin Fine, Frank Levin, Frank Roehl and Gerhard Rosenberger

Abstract. We consider a class of groups which generalize the ordinary tetrahedron groups. This class was introduced by Fine and Rosenberger and independently by Vinberg. Group theoretically these groups are triangular products of generalized triangle groups. We show that these groups admit essential representations into PSL2(C) and from this deduce conditions under which they are SQ-universal,contain non-abelian free subgroups and are virtually torsion-free. Also as a consequence of the essential representations we give conditions for the generalized triangle group factors to inject into the group. Part of these conditions use the concept of a non-positively curved triangle of groups.

1. Introduction An ordinary tetrahedron group is a group with a presentation of the form

< x, y, z; xm =yk=zl

= (xy~l)p = (jz - 1 ) r = (zx~l)s = 1 > .

(1.1)

In our work on the Tits Alternative for generalized triangle groups (see [FLR1], [FLR2], [FR1], [FHR], [LR], [R3]) and on special NEC groups ([FR2]) we often came upon subgroups with presentations of the following form related to (1.1).

< ai, a2,ay, α\χ = aj =

= R™{a\,a2) = R%(ai,ai) = flf (02,^3) = 1 > (1.2)

where e, = 0 or > 2 for i = 1, 2, 3; 2 < m, p, q\ Ri (αϊ, a2) is a cyclically reduced word in the free product on a\, a2 which involves both a\ and a2, R2(a\,a3) is a cyclically reduced word in the free product on a\, 2, i = 1, 2, 3; 2 < m, p, q\ R\(a\,a2) is a cyclically reduced word in the free product on a\, a2 which involves both a\ and /?2(ö1»ö3) is a cyclically reduced word in the free product on a\, 2 for i = 1, 2, 3, p(R\) has order m, p(R2) has order p, and p(Rj) has order q. In particular the existence of any essential representation implies that the group G is non-trivial. A. generalized triangle group is a group Τ with a presentation of the form Τ =< a, b; ap = bq = R'(a, b) = I >

(2.2)

with ί > 2, p=0or p> 2, q=0 or q>2 and where R(a, b) is a cyclically reduced word in the free product on {a, b} involving both a and b. From [BMS] such groups admit essential representations into PSL2(C). A group G is a polygonal product if it can be described in the following manner. There is a polygon P. Each vertex ν corresponds to a group Gv. Each edge y corresponds to a group Gy and adjacent vertices are amalgamated via relations along the edges. The group G is then the group formed by the free product of the Gv modulo the amalgamating edge relations. This is pictured in Figure 1. Karrass, Pietrowski and Solitar have developed a subgroup theory of polygonal products which is parallel to the theory for free products with amalgamation [KPS]. If

The Generalized Tetrahedron Groups

G

101

y23 G

c

Figure 1. A polygonal product the polygon has 4 or more sides then the group G decomposes as a free product with amalgamation. For a generalized tetrahedron group G with presentation (2.1) let

Gi =< α\,α2·, a\x -

= R™{a\,a2) = 1 >

G2 =< αι,α3\ a\l = a^ =

(01,^3) = 1 >

G3 =< a2, «3; aJ2 = ae33 = R\{a2,03) = 1 > . Each of these groups is a generalized triangle group and the generalized tetrahedron group G is then a triangular product of these groups with edge amalgamations over the cyclic subgroups < a\ >, < a2 >, < «3 >. As mentioned earlier it is not clear when the vertex groups in a triangular product inject into the group. In Section 5 we give conditions for this to occur relative to the generalized tetrahedron case. We call G\, Gi, G3 the generalized triangle group factors of the generalized tetrahedron group G.

3. Essential representation We first prove the existence of essential representations and thus show that these groups are non-trivial.

Let G be a generalized tetrahedron group with presentation as in (1.2). Then G admits an essential representation into PSL2{€,).

Theorem 1.

Proof. Let G\, G2, G3 be as given above relative to G. Then G\ =< «1,02; ae\ = a,= R™(a\, a2) = 1 > is a generalized triangle group. Choose an essential representation of Gi in PSL2(C) which is possible from [BMS] and let Αι, A2 be the respective images of a\, a2. Pi qi If A, Β are projective matrices in PSL2(C) and /?j(A, B) - A B ... A ^ Ä « with all pi φ 0 if A has infinite order and 1 < ρ, < ρ if A has order ρ and all