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*Table of contents : CoverTitle pageContentsPrefaceIntroductionCAT(0) cube complexes and groupsGeometric small cancellationLectures on proper CAT(0) spaces and their isometry groupsLectures on quasi-isometric rigidityGeometry of outer spaceSome arithmetic groups that do not act on the circleLectures on lattices and locally symmetric spacesLectures on marked length spectrum rigidityExpander graphs, property (𝜏) and approximate groupsCube complexes, subgroups of mapping class groups, and nilpotent genusBack Cover*

IAS/PARK CITY MATHEMATICS SERIES Volume 21

Geometric Group Theory Mladen Bestvina Michah Sageev Karen Vogtmann Editors

American Mathematical Society Institute for Advanced Study Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

Geometric Group Theory

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https://doi.org/10.1090//pcms/021

IAS/PARK CITY

MATHEMATICS SERIES Volume 21

Geometric Group Theory Mladen Bestvina Michah Sageev Karen Vogtmann Editors

American Mathematical Society Institute for Advanced Study

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John C. Polking, Series Editor Mladen Bestvina, Michah Sageev, Karen Vogtmann, Volume Editors IAS/Park City Mathematics Institute runs mathematics education programs that bring together high school mathematics teachers, researchers in mathematics and mathematics education, undergraduate mathematics faculty, graduate students, and undergraduates to participate in distinct but overlapping programs of research and education. This volume contains the lecture notes from the Graduate Summer School program. 2010 Mathematics Subject Classiﬁcation. Primary 20F65.

Library of Congress Cataloging-in-Publication Data Geometric group theory (American Mathematical Society) Geometric group theory / Mladen Bestvina, Michah Sageev, Karen Vogtmann, editors. pages cm. – (IAS/Park City mathematics series ; volume 21) Includes bibliographical references. ISBN 978-1-4704-1227-2 (acid-free paper) 1. Geometric group theory. I. Bestvina, Mladen, 1959– editor. II. Sageev, Michah, 1966– editor. III. Vogtmann, Karen, 1949– editor. IV. Title. QA183.G45 512.2–dc23

2014 2014029886

Copying and reprinting. Individual readers of this publication, and nonproﬁt libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the ﬁrst page of each article within proceedings volumes. c 2014 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines

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Contents

Preface

xiii

Mladen Bestvina, Michah Sageev, Karen Vogtmann Introduction

1

Michah Sageev CAT(0) Cube Complexes and Groups

7

Introduction

9

Lecture 1. CAT(0) cube complexes and pocsets 1. The basics of NPC and CAT(0) complexes 2. Hyperplanes 3. The pocset structure

11 11 16 18

Lecture 2. Cubulations: from pocsets to CAT(0) cube complexes 1. Ultraﬁlters 2. Constructing the complex from a pocset 3. Examples of cubulations 4. Cocompactness and properness 5. Roller duality

21 21 23 25 29 31

Lecture 3. Rank rigidity 1. Essential cores 2. Skewering 3. Single skewering 4. Flipping 5. Double skewering 6. Hyperplanes in sectors 7. Proving rank rigidity

35 36 37 37 38 41 41 42

Lecture 4. Special cube complexes 1. Subgroup separability 2. Warmup - Stallings’ proof of Marshall Hall’s theorem 3. Special cube complexes 4. Canonical completion and retraction 5. Application: separability of quasiconvex subgroups 6. Hyperbolic cube complexes are virtually special

45 45 45 47 48 50 51

Bibliography

53

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vi

CONTENTS

Vincent Guirardel Geometric Small Cancellation

55

Introduction

57

Lecture 1. What is small cancellation about? 1. The basic setting 2. Applications of small cancellation 3. Geometric small cancellation

59 59 59 61

Lecture 2. Applying the small cancellation theorem 1. When the theorem does not apply 2. Weak proper discontinuity 3. SQ-universality 4. Dehn ﬁllings

65 65 66 68 69

Lecture 3. Rotating families 1. Road-map of the proof of the small cancellation theorem 2. Deﬁnitions 3. Statements 4. Proof of Theorem 3.4 5. Hyperbolicity of the quotient 6. Exercises

71 71 71 72 73 76 78

Lecture 4. The cone-oﬀ 1. Presentation 2. The hyperbolic cone of a graph 3. Cone-oﬀ of a space over a family of subspaces

79 79 81 83

Bibliography

89

Pierre-Emmanuel Caprace Lectures on Proper CAT(0) Spaces and Their Isometry Groups

91

Introduction

93

Lecture 1. Leading examples 1. The basics 2. The Cartan–Hadamard theorem 3. Proper cocompact spaces 4. Symmetric spaces 5. Euclidean buildings 6. Rigidity 7. Exercises

95 95 96 97 98 99 100 101

Lecture 2. Geometric density 1. A geometric relative of Zariski density 2. The visual boundary 3. Convexity 4. A product decomposition theorem 5. Geometric density of normal subgroups 6. Exercises

103 103 103 105 106 107 108

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CONTENTS

vii

Lecture 3. The full isometry group 1. Locally compact groups 2. The isometry group of an irreducible space 3. de Rham decomposition 4. Exercises

111 111 111 113 115

Lecture 4. Lattices 1. Geometric Borel density 2. Fixed points at inﬁnity 3. Boundary points with a cocompact stabiliser 4. Back to rigidity 5. Flats and free abelian subgroups 6. Exercises

117 117 118 119 120 121 122

Bibliography

123

Michael Kapovich Lectures on Quasi-Isometric Rigidity

127

Introduction: What is Geometric Group Theory?

129

Lecture 1. Groups and spaces 1. Cayley graphs and other metric spaces 2. Quasi-isometries 3. Virtual isomorphisms and QI rigidity problem 4. Examples and non-examples of QI rigidity

131 131 133 136 137

Lecture 2. Ultralimits and Morse lemma 1. Ultralimits of sequences in topological spaces 2. Ultralimits of sequences of metric spaces 3. Ultralimits and CAT(0) metric spaces 4. Asymptotic cones 5. Quasi-isometries and asymptotic cones 6. Morse lemma

141 141 142 142 143 144 145

Lecture 3. Boundary extension and quasi-conformal maps 1. Boundary extension of QI maps of hyperbolic spaces 2. Quasi-actions 3. Conical limit points of quasi-actions 4. Quasiconformality of the boundary extension

147 147 148 149 150

Lecture 4. Quasiconformal groups and Tukia’s rigidity theorem 1. Quasiconformal groups 2. Invariant measurable conformal structure for qc groups 3. Proof of Tukia’s theorem 4. QI rigidity for surface groups

157 157 158 160 162

Appendix. 1. Hyperbolic space 2. Least volume ellipsoids 3. Diﬀerent measures of quasiconformality

165 165 166 168

Bibliography

171

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viii

CONTENTS

Mladen Bestvina Geometry of Outer Space

173

Introduction

175

Lecture 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. 1.9. 1.10.

1. Outer space and its topology Markings Metric Lengths of loops Fn -trees Topology and Action Thick part and spine Action of Out(Fn ) Rank 2 picture Contractibility Group theoretic consequences

177 177 178 178 178 178 179 179 181 181 183

Lecture 2.1. 2.2. 2.3. 2.4. 2.5.

2. Lipschitz metric, train tracks Deﬁnitions Elementary facts Example Tension graph, train track structure Folding paths

185 185 185 186 187 189

Lecture 3.1. 3.2. 3.3. 3.4. 3.5. 3.6.

3. Classiﬁcation of automorphisms Elliptic automorphisms Hyperbolic automorphisms Parabolic automorphisms Reducible automorphisms Growth Pathologies

193 193 193 196 197 197 198

Lecture 4.1. 4.2. 4.3. 4.4.

4. Hyperbolic features Complex of free factors Fn The complex Sn of free factorizations Coarse projections Idea of the proof of hyperbolicity

199 201 201 201 202

Bibliography

205

Dave Witte Morris Some Arithmetic Groups that Do Not Act on the Circle

207

Abstract

209

Lecture 1A. 1B. 1C. 1D. 1E. 1F.

1. Left-orderable groups and a proof for SL(3, Z) Introduction Examples The main conjecture Left-invariant total orders SL(3, Z) does not act on the line Comments on other arithmetic groups

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211 211 212 213 214 215 217

CONTENTS

ix

Lecture 2A. 2B. 2C. 2D.

2. Bounded generation and a proof for SL(2, Z[α]) What is bounded generation? Bounded generation of SL(2, Z[α]) Bounded orbits and a proof for SL(2, Z[α]) Implications for other arithmetic groups of higher rank

219 219 221 223 225

Lecture 3A. 3B. 3C. 3D. 3E.

3. What is an amenable group? Ponzi schemes Almost-invariant subsets Average values and invariant measures Examples of amenable groups Applications to actions on the circle

227 227 228 229 231 232

Lecture 4A. 4B. 4C.

4. Introduction to bounded cohomology Deﬁnition Application to actions on the circle Computing Hb2 (Γ; R)

235 235 237 238

Appendix. Hints for the exercises

241

Bibliography

247

Tsachik Gelander Lectures on Lattices and Locally Symmetric Spaces

249

Introduction

251

Lecture 1. A brief overview on the theory of lattices 1. Few deﬁnitions and examples 2. Lattices resemble their ambient group in many ways 3. Some basic properties of lattices 4. A theorem of Mostow about lattices in solvable groups 5. Existence of lattices 6. Arithmeticity

253 253 254 254 256 258 259

Lecture 2. On the Jordan–Zassenhaus–Kazhdan–Margulis theorem 1. Zassenhaus neighborhood 2. Jordan’s theorem 3. Approximations by ﬁnite transitive spaces 4. Margulis’ lemma 5. Crystallographic manifolds

261 261 262 262 263 263

Lecture 3. On the geometry of locally symmetric spaces and some ﬁniteness theorems 1. Hyperbolic spaces 2. The thick–thin decomposition 3. Presentations of torsion free lattices 4. General symmetric spaces 5. Number of generators of lattices

265 265 266 267 268 269

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x

CONTENTS

Lecture 4. Rigidity and applications 1. Local rigidity 2. Wang’s ﬁniteness theorem 3. Mostow’s rigidity theorem 4. Superrigidity and arithmeticity 5. Invariant random subgroups and the Nevo–Stuck–Zimmer theorem

273 273 274 276 276 277

Bibliography

281

Amie Wilkinson Lectures on Marked Length Spectrum Rigidity

283

Introduction

285

Lecture 1. Preliminaries 1. Background on negatively curved surfaces 2. A key example 3. Geodesics in negative curvature 4. The geodesic ﬂow

287 287 288 289 291

Lecture 2. Geometry and dynamics in negative curvature 1. Busemann functions and horospheres 2. The space of geodesics and the boundary at inﬁnity 3. The Liouville current, the cross ratio and the canonical contact form 4. Summary: a dictionary

293 293 297 302 304

Lecture 3. The proof, Part I: A volume preserving conjugacy 1. Otal’s Proof

305 308

Lecture 4. The proof, Part II: Volume preserving implies isometry

313

Final Comments

321

Bibliography

323

Emmanuel Breuillard Expander Graphs, Property (τ ) and Approximate Groups

325

Foreword

327

Lecture 1. Amenability and random walks A. Amenability, Folner criterion B. Isoperimetric inequality, edge expansion C. Invariant means D. Random walks on groups, the spectral radius and Kesten’s criterion E. Further facts and questions about growth of groups and random walks F. Exercise: Paradoxical decompositions, Ponzi schemes and Tarski numbers

329 329 329 330 331

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335 336

CONTENTS

xi

Lecture 2. The Tits alternative and Kazhdan’s property (T ) A. The Tits alternative B. Kazhdan’s property (T ) C. Uniformity issues in the Tits alternative, non-amenability and Kazhdan’s property (T )

339 339 341

Lecture 3. Property (τ ) and expanders A. Expander graphs B. Property (τ )

347 347 351

Lecture 4. Approximate groups and the Bourgain-Gamburd method A. Which ﬁnite groups can be turned into expanders? B. The Bourgain-Gamburd method C. Approximate groups D. Random generators and the uniformity conjecture E. Super-strong approximation

355 355 357 360 362 363

Appendix. The Brooks-Burger transfer

365

Bibliography

373

Martin R. Bridson Cube Complexes, Subgroups of Mapping Class Groups, and Nilpotent Genus

379

1. Introduction

381

2. Subgroups of mapping class groups

382

3. Fibre products and subdirect products of free groups

385

4. A new level of complication

386

5. The nilpotent genus of a group

387

6. Cubes, RAAGs and CAT(0)

389

7. Rips, ﬁbre products and 1-2-3

392

8. Examples template

394

9. Proofs from the template

395

10. The isomorphism problem for subgroups of RAAGs and Mod(S)

396

11. Dehn functions

396

Bibliography

397

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345

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Preface The IAS/Park City Mathematics Institute (PCMI) was founded in 1991 as part of the “Regional Geometry Institute” initiative of the National Science Foundation. In mid 1993 the program found an institutional home at the Institute for Advanced Study (IAS) in Princeton, New Jersey. The IAS/Park City Mathematics Institute encourages both research and education in mathematics and fosters interaction between the two. The three-week summer institute oﬀers programs for researchers and postdoctoral scholars, graduate students, undergraduate students, high school teachers, undergraduate faculty, and researchers in mathematics education. One of PCMI’s main goals is to make all of the participants aware of the total spectrum of activities that occur in mathematics education and research. We wish to involve professional mathematicians in education and to bring modern concepts in mathematics to the attention of educators. To that end, the summer institute features general sessions designed to encourage interaction among the various groups. In-year activities at the sites around the country form an integral part of the High School Teachers Program. Each summer a diﬀerent topic is chosen as the focus of the Research Program and Graduate Summer School. Activities in the Undergraduate Summer School deal with this topic as well. Lecture notes from the Graduate Summer School are being published each year in this series. The ﬁrst twenty one volumes are:

• Volume 1: Geometry and Quantum Field Theory (1991) • Volume 2: Nonlinear Partial Diﬀerential Equations in Diﬀerential Geometry (1992) • Volume 3: Complex Algebraic Geometry (1993) • Volume 4: Gauge Theory and the Topology of Four-Manifolds (1994) • Volume 5: Hyperbolic Equations and Frequency Interactions (1995) • Volume 6: Probability Theory and Applications (1996) • Volume 7: Symplectic Geometry and Topology (1997) • Volume 8: Representation Theory of Lie Groups (1998) • Volume 9: Arithmetic Algebraic Geometry (1999) • Volume 10: Computational Complexity Theory (2000) • Volume 11: Quantum Field Theory, Supersymmetry, and Enumerative Geometry (2001) • Volume 12: Automorphic Forms and their Applications (2002) • Volume 13: Geometric Combinatorics (2004) • Volume 14: Mathematical Biology (2005) • Volume 15: Low Dimensional Topology (2006) • Volume 16: Statistical Mechanics (2007) xiii Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

xiv

PREFACE

• Volume 17: Analytic and Algebraic Geometry: Common Problems, Different Methods (2008) • Volume 18: Arithmetic of L-functions (2009) • Volume 19: Mathematics in Image Processing (2010) • Volume 20: Moduli Spaces of Riemann Surfaces (2011) • Volume 21: Geometric Group Theory (2012) Volumes are in preparation for subsequent years. Some material from the Undergraduate Summer School is published as part of the Student Mathematical Library series of the American Mathematical Society. We hope to publish material from other parts of the IAS/PCMI in the future. This will include material from the High School Teachers Program and publications documenting the interactive activities that are a primary focus of the PCMI. At the summer institute late afternoons are devoted to seminars of common interest to all participants. Many deal with current issues in education: others treat mathematical topics at a level which encourages broad participation. The PCMI has also spawned interactions between universities and high schools at a local level. We hope to share these activities with a wider audience in future volumes. John C. Polking Series Editor July 2014

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https://doi.org/10.1090//pcms/021/01

Introduction Mladen Bestvina, Michah Sageev, Karen Vogtmann

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IAS/Park City Mathematics Series Volume 21, 2012

Introduction Mladen Bestvina, Michah Sageev, Karen Vogtmann Geometric group theory refers to the study of discrete groups using tools from topology, geometry, dynamics and analysis. The ﬁeld is evolving very rapidly and the present volume provides an introduction to and overview of various topics which have played critical roles in this evolution. Speciﬁcally, it contains written versions of the nine graduate lecture courses presented during the summer Park City Mathematics Institute (PCMI), held in Park City, Utah from July 1 to July 21, 2012, as well as a contribution from one of the Clay Lecturers, Martin Bridson. The Clay lecturers for this program were Martin Bridson, Alex Lubotzky and Bill Thurston. We are grateful to all of them for contributing to the energy and excitement of the program. This was the last conference in which Bill Thurston participated, and we deeply appreciate having had this opportunity to interact with him. His work changed the way we all think about and do mathematics, and in particular had enormous direct inﬂuence on the ﬁeld of geometric group theory. His visionary conjectures on the structure of 3-manifolds provided motivation for many of the directions the ﬁeld has taken. The last of these conjectures to be proved were the virtual Haken and the virtual ﬁbering conjectures, which were resolved in a spectacular manner by Ian Agol in the spring of 2012. Agol gave a series of lectures on his proof during the ﬁrst week of PCMI, which were supplemented with lectures by Piotr Przytycki and Jason Manning. His proof was the culmination of a new approach to subgroup separability developed by Dani Wise over the past ﬁfteen years. The central geometric objects in this approach are CAT(0) cube complexes, which were studied extensively by Sageev, Haglund, Wise and others. The ﬁrst lecture series in this volume, by Michah Sageev, provides an introduction to the theory of CAT(0) cube complexes and their applications to geometric group theory and topology. This includes a sketch of Caprace and Sageev’s rank rigidity theorem (which says that under mild conditions CAT(0) cube complexes are products of irreducible complexes which resemble either single lines or Gromov hyperbolic spaces) as well as an introduction to Wise’s theory of special cube complexes. Agol’s proof also employed the notion of Dehn ﬁllings of relatively hyperbolic groups. Given a group G which is hyperbolic with respect to a subgroup P , a ﬁnite subset F ⊂ P and a normal subgroup N of P which avoids F , then the Dehn ﬁlling theorem says that P/N embeds in the quotient of G by the subgroup N normally generated by N ; furthermore, G/N is relatively hyperbolic with respect to the image of P/N . The second lecture series, by Vincent Guirardel, develops a small cancellation theory, which aims in general to understand the quotient of a group by the subgroup normally generated by a given set of subgroups. Other applications c 2014 American Mathematical Society

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4

MLADEN BESTVINA, MICHAH SAGEEV, KAREN VOGTMANN, INTRODUCTION

include identifying groups which contain every countable group in some quotient as well as constructing hyperbolic groups with various types of pathological behavior. The third lecture series, by Pierre-Emmanuel Caprace, gives an introduction to the structure of CAT(0) spaces in general. While discrete group actions by isometries on CAT(0) spaces have played a central role in geometric group theory, the emphasis of these lectures is to study the complete isometry group of a locally compact CAT(0) space. This group has the structure of a locally compact topological group (which is not necessarily discrete), and many structural properties of the underlying space can be derived by combining results on locally compact groups with geometric arguments which are often quite elementary. The lectures by Misha Kapovich provide a detailed introduction to one of the central problems of geometric group theory, that of determining to what extent a group is determined by its geometric actions. Tools from logic (ultralimits) and analysis (quasiconformal maps) are developed and applied to the question of which groups are quasi-isometrically rigid, i.e. virtually determined by such actions. The lecture series by Mladen Bestvina focuses on the group Out(Fn ) of outer automorphisms of a free group and the geometry of the natural spaces on which it acts. The last few years have seen a surge of activity on this subject using the Lipschitz metric on Outer space. Bestvina’s lectures give an introduction to this metric and show how it can be used to prove the classiﬁcation theorem for elements of Out(Fn ). The ﬁnal lecture introduces related complexes, including the free factor complex and the free splitting complex, which were very recently shown to be Gromov hyperbolic, by Bestvina-Feighn and Handel-Mosher, respectively– a development which opens the way to investigate the large scale geometry of Out(Fn ). Many techniques in geometric group theory derive either directly or indirectly from the classical theory of arithmetic groups. The next two lecture series, by David Witte-Morris and Tshachik Gelander, are introductions to the modern development of this classical subject. Witte-Morris’s lectures center around the question of whether an arithmetic group can act faithfully on the circle or the real line. They begin with a concrete study of SL3 (Z), using left-orderability to show that it cannot act by homeomorphisms on the circle. They go on to introduce bounded generation, amenable groups and bounded cohomology, all in the context of this problem. Gelander discusses highlights of the theory of lattices in locally compact groups equipped with a Haar measure, elucidating many ideas behind classical theorems as well as providing an introduction to recent developments. Topics include the geometric structure of locally symmetric spaces and rigidity properties of lattices. Geometric group theory has borrowed many techniques from the theory of dynamical systems. Amie Wilkinson’s lectures introduce some of the basic ideas from this subject, including an introduction to geodesic ﬂows in nonpositive curvature, boundaries of Hadamard spaces, and the dynamics of boundary actions of isometry groups. These are all developed along the course of proving J.-P. Otal’s theorem that a negatively curved metric on a closed surface is determined by its marked length spectrum. The ninth set of lectures, by Emmanuel Breuillard, covers several analytic aspects of geometric group theory, beginning with the notions of random walks on groups, amenability and isoperimetry. The lectures include an introduction to Kazdan’s property (T) and Lubotzky’s property (τ ), as well as the recent theory of

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MLADEN BESTVINA, MICHAH SAGEEV, KAREN VOGTMANN, INTRODUCTION

5

approximate groups. The aim is to equip the reader with the knowledge and necessary background to understand recent developments regarding expander graphs and related spectral properties of groups, which are summarized in the last lecture. Finally, the article of Martin Bridson returns to the subject of cube complexes and right-angled Artin groups. He uses techniques from these theories to solve basic algorithmic problems about ﬁnitely-presented subgroups of mapping class groups of closed surfaces, and about nilpotent completions of groups. We sincerely thank all the lecturers for their beautiful lectures as well as for the write-ups they produced for this volume. Thanks also go to the Park City Mathematics Institute for giving us the opportunity to organize this program. The lectures were attended by over 80 graduate students selected after a very competitive process. We were impressed by the strength of this group and we are very optimistic about the future of the subject.

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https://doi.org/10.1090//pcms/021/02

CAT(0) Cube Complexes and Groups Michah Sageev

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IAS/Park City Mathematics Series Volume 21, 2012

CAT(0) Cube Complexes and Groups Michah Sageev Introduction CAT(0) cube complexes are a particularly nice class of CAT(0) spaces that have made their way into the foreground of Geometric Group Theory in recent years. Initially, it seemed that the class of groups acting properly and cocompactly on CAT(0) cube complexes was quite restrictive. By now, however, we know that this class is quite broad, including surface groups, hyperbolic 3-manifold groups, Coxeter groups, and small cancellation groups. Thus the study of CAT(0) cube complexes sheds light on a wide variety of groups. What sets CAT(0) cube complexes apart within the more general setting of CAT(0) spaces is a combinatorial structure that gives them the look and feel of trees. In particular, they have a rich and natural collection of separating, convex subspaces called hyperplanes from which one can deduce a host of properties that are either false or unknown for groups acting on CAT(0) spaces in general. To give just one example, while no form of the Tits Alternative is known to hold for CAT(0) spaces in general, a strong form of it is has been established for groups acting on CAT(0) cube complexes. Finally, there are quite a number of connections to other subjects, such as subgroup separability, 3-manifold theory, median algebras and spaces, and Kazhdan’s Property (T). The goal of these lectures is to give a brief introduction to the world of CAT(0) cube complexes with an eye towards giving the young geometric group theorist the tools to explore further directions of research. We assume no more than a cursory familiarity with CAT(0) spaces, with group actions on polyhedral complexes, and with covering space theory. For some of the applications, it may help to know something about 3-manifolds and hyperbolic groups, but these concepts are not necessary to understand the core material. We did not have the time in these lectures to cover the connections to median spaces and algebras or Kazhdan’s Property (T). One can learn about these topics in [13], [38], [11] and [36].

Department of Mathematics, Technion, Haifa 32000, Israel E-mail address: [email protected] c 2014 American Mathematical Society

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LECTURE 1

CAT(0) cube complexes and pocsets 1. The basics of NPC and CAT(0) complexes 1.1. Cube complexes and links We start by reviewing the basic notions of complexes and links that we will need. We will build complexes by gluing unit cubes along their faces by isometries. One starts with a disjoint union C of unit Euclidean cubes of various dimensions, together with a collection of isometries F between the faces of cubes in C. We then form the quotient space X = C/F obtained by identifying points in the domains of maps in F with their image. We will usually suppress the F notation and label edges and faces we wish to identify. We thus have a quotient map q : C → X. The complex X is called a cube complex. The “cubes of X” are the images under q of the various faces of cubes in C. Recall that the link of a vertex in X is the simplicial complex that can be realized as a “small sphere” around the vertex. We describe what a link is more precisely. Note ﬁrst that the 1-skeleton of X is a graph. It is possible that the 1-skeleton is not a simplicial graph, so that it may have loops and multiple edges. By a local edge of C we mean a subinterval of length 1/3 of an edge of a cube in C, one of whose endpoints is a vertex of a cube in C. A local edge in X is the image under q of a local edge in C. For a vertex v of X, the vertices of lk(v) are the local edges of X containing v. A collection of vertices in lk(v) span a simplex in lk(v) if and only if the corresponding local edges of X are images under q of a collection of local edges all contained in some cube in C and all of which share a vertex. See Figure 1.

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MICHAH SAGEEV, CAT(0) CUBE COMPLEXES

1.2. Nonpositively curved complexes We are now going to restrict the class of cube complexes that we will consider. Definition 1.1. A ﬂag complex is a simplicial complex with no “missing” simplices. This means that for each complete graph in the 1-skeleton of the complex, there is a simplex in the complex whose 1-skeleton is the given complete graph. Examples: (1) A simplicial graph is a ﬂag complex if and only if it has no cycle of length three. (2) Any simplicial graph is the 1-skeleton of a unique ﬂag complex, obtained by attaching a simplex to each complete subgraph. (3) The ﬁrst barycentric subdivision of any simplicial complex is a ﬂag complex. Definition 1.2. A nonpositively curved (NPC) cube complex is a cube complex whose vertex links are simplicial ﬂag complexes. A 1-connected NPC complex is called a CAT(0) cube complex. If we focus on NPC square complexes, we see that the above deﬁnition rules out the following identiﬁcations:

Figure 2. Illegal identiﬁcations in NPC square complexes. For NPC complexes of dimensions bigger than two, the ﬁrst two identiﬁcations are still not allowed, and the third identiﬁcation is allowed only if there is a 3dimensional cube containing the three given squares in its boundary. In these notes, we work with this deﬁnition. However, since there is already a well established notion of CAT(0) which has to do with thin triangles in a geodesic metric space, some remarks are in order. (1) We can deﬁne a path metric on a cube complex in the usual way as follows. We deﬁne a rectiﬁable path in X as one that can be broken into ﬁnitely

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many subpaths each of which is contained in some cube of X. If these paths are themselves rectiﬁable (in the classical sense) we can now deﬁne the length of the original path as the sum of the length of the subpaths. The distance between p and q is then deﬁned as the inﬁmum of the lengths of the rectiﬁable paths joining p and q. (2) In the event that the complex is ﬁnite dimensional, a result of Bridson [6] tells us that the above indeed deﬁnes a metric and that with this metric, the complex is a complete, geodesic metric space. (The case of locally ﬁnite complexes was treated by Moussong [33].) (3) A result of Gromov [21] then tells us that with this metric a ﬁnite dimensional NPC complex is locally CAT(0). (4) The Cartan-Hadamard theorem then tells us that if the space is 1-connected it is CAT(0) in the usual sense. (5) More recently, Leary [31], showed that all this makes sense in the inﬁnite dimensional case as well. In particular, he showed that with the above metric, an NPC cube complex (possibly inﬁnite dimensional) is a geodesic, locally CAT(0) metric space. Moreover, he showed that it is complete if and only if every ascending sequence of cubes terminates. For a treatment of general CAT(0) spaces see Bridson and Heaﬂiger’s book [7] or Caprace’s chapter in this volume. We will sometimes use the term cubed group for a group that admits a proper, cocompact action on a CAT(0) cube complex. Typically one constructs cubed groups by building compact NPC complexes. Their fundamental groups are then cubed groups. Sometimes we will be interested in a CAT(0) cube complex that admits a cocompact group action without caring too much about the group in question. Thus we use the term cocompact CAT(0) cube complex to mean a CAT(0) cube complex whose automorphism group acts cocompactly on it. Let us now look at some examples. Examples: (1) Graphs. The link of a vertex in a graph has no edges, so every graph (simplicial or not) is an NPC complex. The universal cover of a graph is a tree, which is the model CAT(0) cube complex. (2) Tori. A torus is obtained from a square by identifying opposite edges and is thus naturally a cube complex. It is easy to check that the link of the sole vertex in this complex is a cycle of length 4. Thus a torus is an NPC complex. The reader should check that a torus of every dimension is naturally an NPC complex obtained by identifying opposite faces of a single cube. What is the link of a vertex? (3) Surfaces. Consider an orientable surface of genus g > 1. Recall that it is obtained by taking a 4g-gon and identifying faces in pairs in a suitable way (see Figure 3.) We can now subdivide the 4g-gon into squares by adding the barycenters of the edges, a vertex in the center of the 4g-gon, and an edge between each new edge-barycenter and the center of the 4g-gon. The reader should check the vertex links to see that the complex obtained is indeed an NPC complex.

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MICHAH SAGEEV, CAT(0) CUBE COMPLEXES

Figure 3. Squaring a surface of genus 2. (4) Products. We ﬁrst review the notion of a join of two simplicial complexes. Recall that an abstract simplicial complex K is simply a collection of ﬁnite subsets of some underlying set S, so that this collection is closed under taking subsets. The join of two complexes K1 and K2 , with underlying sets S1 and S2 , is the complex obtained by taking as the underlying set the disjoint union S1 ∪ S2 and taking as the collection of subsets all pairwise unions of elements in K1 and K2 . For example, the join of two simplicial complexes of dimension 0 is a complete bipartite graph. The reader should now check that the join of two ﬂag complexes is a ﬂag complex. Now consider two cube complexes X and Y . Their product X × Y is naturally also a cube complex. The reader should check that if (v, w) is a vertex in X × Y , then the link of (v, w) is the join of lk(v) and lk(w). So for example, the link of a product of two trees is a complete bipartite graph. Now since the join of two ﬂag complexes is a ﬂag complex, the product of two NPC complexes is NPC. See Figure 4.

Figure 4. The local structure of a product of two trees.

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Exercise 1.1. Show that a product of two trees of valence at least three does not embed in R3 , even locally. Exercise 1.2. Prove that a simply connected square complex whose link is a complete bipartite graph is a product of two trees. (5) Irreducible lattice in a product of trees. Wise [44] and BurgerMozes [8] provide various examples of groups acting on a product of trees properly (in fact freely) and cocompactly so that the action is not a product of two proper cocompact actions. The smallest such example was produced by Janzen and Wise [29] and is described in Figure 5. It is easy to check that the link of the vertex is a complete bipartite graph, so that the universal cover is a product of two trees. It requires quite a bit more work to prove that the action is not a product of actions; the projection onto each of the factor trees is an indiscrete group. See [29]

Figure 5. The quotient of a product of trees by an irreducible lattice. (6) RAAGs. A right angled Artin group (RAAG) is a group with the following simple presentation. Start with a ﬁnite simplicial graph Γ and deﬁne A(Γ) = Γ(0) |[v, w] ⇔ (v, w) is an edge of Γ So all abelian groups and free groups are RAAGs as are all products of free groups. There is a natural complex associated to a RAAG, called the Salvetti complex R(Γ), which we can build in the following way. Start with a single vertex and add a loop for each vertex of Γ. This is the 1-skeleton of R(Γ). Now for every maximal n-clique in Γ we attach an n-torus to the complex. The n-torus can be seen as the quotient of a cube by identifying opposite faces. See Figure 6 for a simple example. The edges of the cube

Figure 6. The Salvetti complex of a RAAG. descend to a collection of n loops in the torus intersecting at a simple loop. Identify these n loops in the torus with the n-loops in the 1-skeleton of R(Γ) associated to our n-clique.

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1.3. Cubical maps and locally isometric maps We say that a map between cube complexes is a cubical map if it sends cubes to cubes and the restriction to each cube factors through a projection onto one of the factors. Notice that such a map naturally induces a simplicial map on links of vertices. We will be interested in cubical maps between NPC complexes which are π1 -injective. Here is a simple combinatorial condition to ensure this. Recall that a subcomplex L of a simplicial complex K is full if every collection of vertices in L that spans a simplex in K spans a simplex in L. Exercise 1.3. Let f : X → Y be a cubical map such that • Y is NPC. • f is locally injective. • f (lk(v)) is a full subcomplex of lk(f (v)). Show that • X is NPC. • f∗ : π1 (X) → π1 (Y ) is injective. Hint. Use the Cartan-Hadamard theorem and the fact that a map satisfying the conditions of the exercise is a local isometry. To give an intuitive idea of this condition, consider Figure 7. On the left is a local isometry and the map on the link of the central vertex to the link of the image is full. On the right, the image is not full and indeed the map is not a local isometry.

Figure 7. A local isometry and a local non-isometry. 2. Hyperplanes Hyperplanes are natural subspaces, each of which “cuts up” a CAT(0) cube complex into two halfspaces. In this section we deﬁne them and describe some of their basic properties. Definition 1.3. Let σ be an n-dimensional cube σ in X. A midcube of σ is an (n − 1) - dimensional unit cube containing the barycenter of σ and parallel to one of the faces of σ. Thus each n-cube has n midcubes, all intersecting at the barycenter. Let X be an NPC cube complex. Let denote the equivalence relation on the edges of X generated by the relation e f if and only if e and f are opposite edges of some square in X. Definition 1.4. Given an equivalence class of edges [e], the hyperplane dual to [e] is the collection of mid cubes which intersect edges in [e]. It is useful to go back to the previous examples and think about what the hyperplanes there look like:

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Figure 8. A hyperplane in a cube complex. (1) When X is a graph, the hyperplanes are simply midpoints of edges. (2) When X is a torus, each hyperplane is a simple closed curve. The universal cover of X is simply the Euclidean plane tiled by squares, and the hyperplanes there are horizontal or vertical lines. (3) When X is a higher genus surface, squared as above, it is easy to check that the hyperplanes are simple closed curves lifting to lines in the universal cover of X. (4) When X is a product, the hyperplanes are preimages (under the natural projection maps) of the hyperplanes in each of the factors. (5) Regarding Salvetti complexes, you should do the following exercise. Exercise 1.4. Show that when X is a Salvetti complex, each hyperplane is itself a Salvetti complex and describe the RAAG for which it is a Salvetti complex. Here are some more exercises. Exercise 1.5. Find an NPC square complex with a single hyperplane. Exercise 1.6. Find an inﬁnite NPC complex with two hyperplanes. Then ﬁnd one with a single hyperplane. We will sometimes be interested in a “combinatorial thickening” of a hyperˆ and denoted C(h). ˆ It is simply the union of the cubes plane, called the carrier of h ˆ of X meeting h. In an NPC complex, hyperplanes can be immersed in complicated ways. In a CAT(0) cube complex, they are much better behaved. This is captured in the following basic theorem, which we will use extensively. ˆ a hyperplane in X. Theorem 1.1. Let X be a CAT(0) cube complex and h Then the following statements hold. (1) Every hyperplane of X is embedded. (2) Every hyperplane of X separates X into precisely two components called halfspaces. (3) Every hyperplane is a CAT(0) cube complex and the inclusion of its carrier is an isometry. (4) Every collection of pairwise intersecting hyperplanes intersects. This theorem can be proved using CAT(0) geometry (see [39]) or using disk diagrams (see [40].) We will forgo providing the proof here and view this as a starting point. The following exercise is also useful. Exercise 1.7. Show that if G acts cocompactly on a CAT(0) cube complex X, then every hyperplane is acted on cocompactly by its stabilizer.

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MICHAH SAGEEV, CAT(0) CUBE COMPLEXES

Hint. Any isometry of X which sends a cell of a hyperplane to a cell of the same hyperplane preserves the entire hyperplane. 3. The pocset structure The consequence of Theorem 1.1 is that we have a natural collection of CAT(0) subcomplexes which cut the complex into halfspaces. We want to focus on the combinatorial nature of the collection of halfspaces. Let us ﬁrst set some notation. X – CAT(0) cube complex ˆ – the collection of hyperplanes of X H H – the collection of halfspaces of X h – a halfspace in H h∗ – the complementary halfspace of h ˆ – the bounding hyperplane of h h Note that H is a poset under inclusion such that • H has a natural, order-reversing, ﬁxed-point free involution h → h∗ so that h and h∗ are incomparable (this is called a pocset), • H is locally ﬁnite (meaning that there are ﬁnitely many elements between any two given elements), • there is a bound on the size of a collection of halfspaces which are not nested (follows from ﬁnite dimensionality). Following Roller [38], we are thus motivated to deﬁne the notion of a pocset (“poset with complementation”). Definition 1.5. A pocset is a poset Σ together with a ﬁxed-point free involution A → A∗ such that • A and A∗ are incomparable • A < B ⇒ B ∗ < A∗ Elements A, B ∈ Σ are nested if one of A < B, A < B ∗ , A∗ < B, A∗ < B ∗ holds. Otherwise we say that A and B are transverse. If A < B then we let [A, B] = {C|A < C < B}. This is called the interval between A and B. A pocset is said to be locally ﬁnite if every interval is ﬁnite. The width of a pocset is the size of a maximal transverse subset. We will focus on locally ﬁnite, ﬁnite width pocsets. Some of the claims will hold in somewhat broader generality, namely when every transverse subset is ﬁnite (see [22]). Example. A space with walls (S, Σ) is simply a set S together with a collection of subsets Σ closed under complementation (see Haglund and Paulin [24]). A space with walls clearly forms a pocset under inclusion. It is said to be discrete if for any two elements of a, b ∈ S the collection of subsets in Σ containing a and not containing b is ﬁnite. We will see several examples of spaces with walls later on. A simple example comes from hyperbolic surfaces. Take any ﬁnite collection of essential geodesics on a closed surface. The collection of lifts L of these curves to the universal cover H2 is a collection of lines. The space H2 − L is now naturally a discrete space with walls, where the walls are given by the lines in L; an element of Σ is a complementary region of a single line in L.

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Example. As already mentioned, a particular example of a space with walls is the pocset of halfspaces of a CAT(0) cube complex. This pocset is locally ﬁnite and the dimension of the complex is the width of the pocset. Exercise 1.8. Prove or disprove or salvage if possible. A space with walls (S, Σ) is discrete if and only if Σ is locally ﬁnite.

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LECTURE 2

Cubulations: from pocsets to CAT(0) cube complexes In this lecture, we will describe the cubulation construction described in [40]. Variants of this construction exist in various settings (Roller [38], Nica [37], ChatterjiNiblo [12] and Guralnik [22]). We will follow Roller’s treatment closely. 1. Ultraﬁlters We assume we have a locally ﬁnite pocset Σ. We wish to construct a CAT(0) cube complex X(Σ) from Σ. First, we describe the vertices of X(Σ). Definition 2.1. An ultraﬁlter α on Σ is a subset of Σ satisfying two properties: (1) Choice: for all pairs {A, A∗ } in Σ, precisely one of them is in α. (2) Consistency: A ∈ α and A < B =⇒ B ∈ α. The notion of an ultraﬁlter on a pocset is reminiscent of the classical notion of an ultraﬁlter on the natural numbers, where the pocset is the collection of subsets of N. However, as we shall see, ultraﬁlters have geometric meaning; they will be the vertices or “vertices at inﬁnity” of a CAT(0) cube complex. An ultraﬁlter α is said to satisfy the Descending Chain Condition (DCC) if every descending chain of elements of α terminates. Example. The ﬁrst example to consider is the case of a tree. The pocset Σ is the collection of halfspaces of the tree. Here a halfspace is simply a complementary region of the midpoint of an edge. Each edge has two halfspaces associated to it, and an ultraﬁlter will choose of one of these. Thus we can view the choice condition of an ultraﬁlter as a way of putting an arrow on each edge, where the arrow points towards the chosen halfspace. The consistency condition restricts the way the arrows can be oriented, as shown in Figure 1. Note that this implies that if you choose an orientation at some edge, then for all the arrows in the tail halfspace of that edge, the arrows must point towards the edge. One type of ultraﬁlter can be obtained by choosing a vertex v and having all the arrows pointing at it. This ultraﬁlter, denoted αv , is given by αv = {h|v ∈ h} So each vertex is associated with an ultraﬁlter. Note that these ultraﬁlters all satisfy DCC. Conversely, given an ultraﬁlter that satisﬁes DCC, it is not hard to see that there exists some vertex (and hence a unique vertex) such that all the arrows point at that vertex. Thus the vertices of the tree are in one-to-one correspondence with DCC ultraﬁlters. 21 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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MICHAH SAGEEV, CAT(0) CUBE COMPLEXES

Figure 1. The orientation on the right is not allowed.

Now suppose that α is an ultraﬁlter which does not satisfy DCC. For every vertex v, the consistency condition ensures that at most one edge adjacent to v is oriented away from v. Moreover, since α does not satisfy DCC, it follows that exactly one edge is oriented away from v. This means that α determines a ray emanating from v. See Figure 2.

Figure 2. An ultraﬁlter at inﬁnity. Thus, in the case of a tree, we have a picture of all the ultraﬁlters. The DCC ultraﬁlters are the the vertices of the tree and the rest correspond to boundary points of the tree. Example. Another elementary example that is good to think about is the usual squaring of the plane (Figure 3). As in the case of a tree, the DCC ultraﬁlters correspond to the vertices of the complex. The other ultraﬁlters are “at inﬁnity” as shown below. There is a line of ultraﬁlters on all four “sides” of the plane and an ultraﬁlter for each “corner”. What we see in the above two examples turns out to be the general picture as well: the collection of all ultraﬁlters forms a compactiﬁcation of the complex. We will not delve into this in these notes; for further discussion, see [22] and [34]. We just note here that for a pocset coming from the halfspaces of a ﬁnite dimensional CAT(0) cube complex, the DCC ultraﬁlters are the same as the vertices.

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Figure 3. The ultraﬁlters associated to the squaring of the Euclidean plane. Exercise 2.9. Let X be a ﬁnite dimensional CAT(0) cube complex and consider the pocset of halfspaces H. Then the DCC ultraﬁlters are precisely the ultraﬁlters associated to vertices. Namely, every DCC ultraﬁlter is of the form αv = {h|v ∈ h} 2. Constructing the complex from a pocset Given a locally ﬁnite, ﬁnite width pocset Σ and we wish to construct a CAT(0) cube complex X = X(Σ); Vertices. The vertex set X 0 of X will be the collection of DC ultraﬁlters on Σ. From now on we will use letters like v and w to denote vertices of X 0 . Edges. We join two such vertices v and w by an edge if |v w| = 2. That is, there exists A ∈ Σ such that w = (v − {A}) ∪ {A∗ }. Exercise 2.10. Let A ∈ v, then (v − {A}) ∪ {A∗ } is an ultraﬁlter if and only if A is minimal in v. Note that (v − {A}) ∪ {A∗ } is also a DCC ultraﬁlter and hence another vertex of X. Notation. When A is minimal in v, we will use the following notation: (v; A) ≡ (v − {A}) ∪ {A∗ } When B is minimal in (v; A), we will use the following notation: (v; A, B) ≡ ((v; A); B) And similarly we use the notation (v; A1 , . . . , An ) for multiple elements. Having constructed X 1 , we now need to know that it is connected. This is the content of the next exercise. Exercise 2.11. If Σ has ﬁnite width, then any two DCC ultraﬁlters are joined by a ﬁnite path.

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MICHAH SAGEEV, CAT(0) CUBE COMPLEXES

Hint. Consider two DCC ultraﬁlters v and w. One needs to show that there are ﬁnitely many A’s in Σ such that A ∈ v and A∗ ∈ w. (Why is this enough?) Suppose there are inﬁnitely many such. Then use ﬁnite width to ﬁnd a descending sequence of elements in v or w which does not terminate. Squares. We attach a square to every 4-cycle that appears in X (1) . Let us look a bit more closely at the vertices of a square. Let us say one of them is v. Then by the way we deﬁned the 1-skeleton, there are A, B ∈ v such that the two vertices of the square adjacent to v are (v; A) and (v; B). Now it is not too diﬃcult to see that the vertex diagonally opposite v in the square is simply (v; A, B). This in turn tells us that A and B are transverse. (The reader should check this.) Since ultimately X is supposed to be CAT(0), we better have a simply connected 2-skeleton. Exercise 2.12. X (2) is simply connected. Hint. Consider the shortest possible combinatorial loop γ in X 1 which is nontrivial in π1 (X 2 ). Let v be some vertex along γ. Then we will have along a sequence of vertices along γ: v, (v; A1 ), . . . , (v; A1 , . . . , An ), v(A1 , . . . , An , A∗k ) such that (1) For all i, j with 1 ≤ i < j ≤ n, we have Ai = Aj and Ai = A∗j . (2) 1 ≤ k ≤ n. If k = n, then we have backtracking along γ and it is not the shortest nontrivial loop. Now argue that An and Ak are transverse. Use this to produce a new loop which has the following sequence of vertices along it: v, (v; A1 ), . . . , (v; A1 , . . . , An−1 , A∗k ), (v; A1 , . . . , An−1 , A∗k , An ) Proceed until backtracking is produced. Higher dimensional cubes. We now construct the n-skeleton inductively. Simply add an n-cube whenever the boundary of one appears in the (n − 1)-skeleton. It is again instructive to think about what happens locally when v is a vertex of an ncube σ. The neighboring vertices in the n-cube are of the form (v; A1 ), . . . , (v; An ). Since any pair of such edges spans a square, we have that {A1 , . . . , An } is a collection of pairwise transverse elements. We then see that all the vertices of σ are of the form (v; Ai1 , . . . , Aik ) for some distinct collection of indices ij ∈ {1, . . . , n}. Conversely, a collection A1 , . . . , An of minimal, pairwise transverse elements of some vertex v, deﬁnes an n-cube. These observations allow us to establish the following. Exercise 2.13. The links of vertices of X are ﬂag complexes. Note that the dimension of X is equal to the width of the original pocset Σ. We call this construction of a cube complex from a pocset a cubulation. Exercise 2.14. Explore this construction when the pocset is not of ﬁnite width. For example, suppose the pocset is completely un-nested: no two elements are comparable. Is the complex connected? What do the components look like? Etc.

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Group actions. Let Σ be a pocset and suppose that a group G acts on Σ in an order preserving manner and without inversions (for no g ∈ G and A ∈ Σ do we have gA = A∗ ). Then we obtain an action on X(Σ)0 : gα = {gh|h ∈ α} It is easy to check that this extends to an action on X(Σ) by cellular isometries. 3. Examples of cubulations 3.1. The pocset of halfspaces Let X be a ﬁnite dimensional CAT(0) cube complex. As already noted the collection of halfspaces H(X) of X is a pocset. What is the cubulation associated to this pocset? We will come back to this question later on. 3.2. Lines in the plane A collection of lines in the plane is called discrete if there is a lower bound to the distance between two parallel lines in the collection. Exercise 2.15. Let L be a discrete collection of lines which has ﬁnitely many parallelism classes (for example, think of the plane triangulated by unit isosceles triangles). Consider the set S = R2 − L. Then the pocset associated to the space with walls (S, Ω) is a ﬁnite width, locally ﬁnite pocset. What is the cube complex associated to this pocset? 3.3. Small cancellation groups (Wise) We give here a (very) brief account of Wise’s cubulation of small cancellation groups. For a complete discussion, see [43]. Let G = S|R be a ﬁnitely presented group, where S is closed under taking inverses. We consider the presentation 2-complex K associated to this presentation. ˜ is a 2-complex, called the Cayley complex of the presentation, The universal cover K ˜ can be identiﬁed with the whose 2-cells we call relator polygons. The 1-skeleton of K ˜ is labeled by an element Cayley graph of the presentation, so that each edge of K of S. A piece of the presentation is a reduced word in S that appears as the label ˜ 1 which is contained in the boundary of more than one 2-cell of K ˜ of a path in K (see Figure 4).

Figure 4. Part of the Cayley complex and some pieces. The presentation is said to be a C (1/n) presentation for G if the length of a piece is always less than 1/nth the length of the boundary of a relator polygon in which it appears. There are many more small cancellation conditions which come up in small cancellation theory (see [32]). Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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MICHAH SAGEEV, CAT(0) CUBE COMPLEXES

Small cancellation groups are studied via disk diagrams, which we brieﬂy describe. A multi-disk is a connected, simply connected union of disks and ﬁnite trees in the plane, where the disks and trees meet along a ﬁnite set of points. A disk diagram is a multidisk in which the disk regions of the multidisk are tiled by relator polygons and the tree portions have their edges labeled by elements of S. Given a trivial word w in G, we can represent the triviality of w by a disk diagram, as seen in Figure 5. The original word w is obtained by reading the labels around the outside of the multi-disk.

Figure 5. A disk diagram and a reduction. There is a notion of reduction in a disk diagram. Suppose one has two relator polygons in a disk diagram which meet along an arc in their boundaries, and suppose that their boundaries are labeled by the same word, one read clockwise and one read counterclockwise from the same common vertex. Then one can produce a smaller disk diagram by removing the two relator polygons and identifying the remaining arcs. A disk diagram which has no available reductions is called reduced. For small cancellation groups there is a fundamental lemma [45] which gives a trichotomy regarding disk diagrams. This goes back to Greendlinger [20] and even earlier to Dehn [15]. Lemma 2.1 (Fundamental Lemma). A reduced disk diagram for a word in a C (1/6)-group is of one of the following types. (1) A single polygon. (2) A ladder: this means a diagram formed by attaching polygons and/or edges “end-to-end” in a linear fashion. The two endpoints of the ladder are either shells or spurs (see Figure 6). A shell is a polygon which is attached to the diagram along an arc whose length is less than half of the length of its boundary. A spur is simply an edge which is attached to the diagram at one endpoint and free at the other endpoint.

Figure 6. A ladder. (3) A diagram with at least 3 shells and/or spurs. A typical diagram can be seen in Figure 7.

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Figure 7. A typical diagram. From this Fundamental Lemma one can deduce that C (1/6) groups have a linear isoperimetric inequality and hence such groups are Gromov hyperbolic (see [21]). ˜ First, if necessary, subdivide the We now describe how one ﬁnds walls in K. boundary of the relator polygons so that each relator polygon has an even number of sides. ˜ 1. We build a graph Δ as follows. We have a vertex ve ∈ Δ for each edge e in K Two vertices ve and vf are joined by an edge in Δ if e and f are opposite edges in ˜ We then have a natural map some relator polygon of K. ˜ η:Δ→K , ˜ 1 , and each which maps each vertex ve of Δ to the midpoint of edge e in K edge [ve , vf ] of Δ to a straight arc in the appropriate relator polygon joining the midpoints of the edges e and f . (Notice that the C (1/6) property ensures that there is unique such polygon. ) The image of a connected component of Δ is called a wisetrack (since it is a type of track ` a la Dunwoody. See Figure 8.)

Figure 8. Some wisetracks in a small cancellation complex. Using small cancellation theory Wise then shows that each wisetrack is embedded. The idea is to use the fundamental lemma. If a wisetrack crosses itself, one sees a sequence of relator polygons as below, and this in turn gives rise to a a disk diagram without three shells or spurs, contradicting the Fundamental Lemma (see Figure 9).

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Figure 9. A self-intersection leads to a diagram which violates the Fundamental Lemma. A result of Dunwoody [16] then tells us that each of these wisetracks separate 2 ˜ K . So we take the space with walls to be the complement of the union of all ˜ and therefore acts on the cube complex. wisetracks. The original group acts on K Wise then shows that this action is proper and cocompact. We will return to this issue shortly. 3.4. Coxeter groups In a similar vain, Niblo and Reeves [35] cubulated Coxeter groups. The space with walls can be described in terms of tracks in an appropriate presentation complex. Recall that a Coxeter group has a presentation of the following form: G = S|s2i = 1, (si sj )mij Here S = {s1 , . . . , sn } is a ﬁnite set, mij = mji and 2 ≤ mij ≤ ∞. ˜ its universal cover. We let K denote the presentation 2-complex for this and K Note that the presentation complex for this presentation has relator polygons with ˜ for each pair i, j with mij < ∞, and an even number of sides. Also note that in K, mij each polygon P which reads (si sj ) along its boundary, there are a total of mij other polygons which share the same boundary as P . We can construct a quotient ˜ in which each of these “pillows” is collapsed to a single polygon. Now we build of K tracks as in the small cancellation case. Niblo and Reeves then check that the resulting complex is ﬁnite dimensional and the action on it is proper. Caprace then showed that the action on this complex is cocompact unless the original Coxeter group contains a Euclidean triangle group. For details, see [35] and [9]. 3.5. Codimension 1 subgroups A general situation where the above construction is applicable is when G is a ﬁnitely generated group and H is a subgroup that “separates” the Cayley graph of G. More precisely, the subgroup H is said to be a codimension 1 subgroup if the coset graph G/H has more than one end. In the Cayley graph, this gives rise to the following: there exists a number R, such that the R-neighborhood of H separates the Cayley graph into two deep components. If we choose one such component A ⊂ G, we see that the translates under G of A and its complement A∗ , form a collection of walls

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Σ on the set G. With a bit of work, one can show that this is a discrete collection of walls. Moreover, the action of G on the resulting cube complex has no global ﬁxed point. See [40], [36], [17] for more details. 4. Cocompactness and properness In the examples given in Sections 3.3 and 3.4, one cubulates using a collection of walls and then one would like to know that the action is proper and better yet, proper and cocompact. It turns out that cocompactness is assured by hyperbolicity and quasiconvexity, and properness by a kind of “ﬁlling” condition. Before stating some general theorems, it might be instructive to look at a simple example to highlight the ideas. 4.1. An example: curves on surfaces As discussed in Section 3, an example of a space with walls can be obtained by considering a ﬁnite collection of simple closed geodesics on a closed hyperbolic surface S. The universal cover of S is identiﬁed with the hyperbolic plane H2 and the curves on S lift to a collection of lines L in the universal cover. The space with walls is H2 − ∈L L and the walls are the halfspaces deﬁned by the lines in L. We let G = π1 (S). A collection of lines in L that pairwise intersect is called transverse. If we review the cubulation construction in which a cube complex X is constructed from this space with walls, we see that cocompactness is implied by the following two claims. Claim 2.1. For each k > 0, there are ﬁnitely many G-orbits of transverse collections of k lines in L. Claim 2.2. There is a bound on the size of a transverse collection of lines in L. Claim 2.1 ensures that in X/G, there are ﬁnitely many cubes in each dimension. Claim 2.2 ensures that the complex X is ﬁnite dimensional. To prove these claims we will need the following, which we leave as an exercise. Exercise 2.16. Let L be a transverse collection of n lines in H2 , with n > 1. Then there exists a number R = R(L) > 0 such that any line intersecting all the lines in L, intersects the ball of radius R about the origin. Proof of Claim 2.1. For k = 1, the statement is that there are ﬁnitely many conjugacy classes of lines, which is simply the fact that there are ﬁnitely many curves in the quotient of H2 under the action of G = π1 (S). For k = 2, any two transverse lines have a point of intersection, which by cocompactness, can be translated into some ﬁxed fundamental domain D for the action. Since only ﬁnitely many lines intersect D, there are only ﬁnitely many points of intersection in D. We now proceed by induction. Let L = {1 , . . . , k+1 } denote a transverse collection of lines in L. By induction, we can translate L so that the collection {1 , . . . , k } is one of ﬁnitely many transverse collections. Now by the exercise, k+1 meets the ball of radius R about the origin. By the discreteness of the pattern, there are only ﬁnitely many choices for k+1 . Exercise 2.17. Prove Claim 2.2.

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Thus, we see that the cube complex construction yields a cocompact action by G. Now let us consider properness. Each element of the group G is a hyperbolic isometry of the hyperbolic plane and hence has an axis. We say that the pattern of lines L is ﬁlling if for every g ∈ G, there exists a line ∈ L, such that the axis g of g crosses . It is not hard see that this corresponds to each complementaryregion of the union of lines in L being bounded. Each of these complementary regions corresponds to a vertex in the resulting cube complex and it is also then not hard to see that the orbit of such a vertex is unbounded. In fact, since G is torsion free, this tells us that the action of G on X is not only proper, but free. 4.2. Hyperbolicity, quasiconvex subgroups and hyperbolic 3-manifolds Now consider the more general situation in which G is a hyperbolic group and H is a quasiconvex codimension-1 subgroup. We refer the reader to any reference on hyperbolic groups ([21] or [18], for example.) We recall that a hyperbolic group G has a natural visual boundary ∂G and that H has a limit set Λ(H) ⊂ ∂G. As in Section 3.5, we obtain a pocset Σ from which we can obtain an action of G on a CAT(0) cube complex X. Regarding cocompactness, we have the following theorem [19]. Theorem 2.1 (Gitik-Mitra-Rips-S). If G is a hyperbolic and H is quasiconvex, then the action of G on X is cocompact. Remark. The proof of this theorem is a souped up version of the proof of Claim 2.1. The same proof applies if we apply the cubulation construction to a ﬁnite collection of codimension-1 subgroups. Regarding properness, thinking along the same lines as the example above leads to the following. Theorem 2.2 (Bergeron-Wise). If G is hyperbolic and H is quasiconvex, such that for every element g ∈ G, there exists a conjugate of H whose limit set Λ(H) separates the endpoints of the axis of g, then the action of G on X is proper. Bergeron and Wise [5] actually prove a more applicable result. Theorem 2.3 (Bergeron-Wise). Let G be a hyperbolic group. Suppose that for every pair of points a, b in ∂G, there exists a quasiconvex codimension-1 subgroup H whose limit set separates a and b. Then there exists a ﬁnite collection of codimension-1, quasiconvex subgroups such that the action of G on the resulting cube complex is proper. A particular application of this theorem is in the setting of 3-manifold groups, in light of the following deep result of Kahn and Markovic [30]. Theorem 2.4 (Kahn-Markovic). Let M = H3 /G be a closed hyperbolic 3-manifold. Then every great circle in S 2 = ∂H3 is a limit of quasicircles which are limit sets of quasifuchsian subgroups. In particular, every pair of points in S 2 is separated by the limit set of a quasiconvex surface subgroup. Putting the Kahn-Markovic theorem together with the above theorems on cubulations, we obtain the following. Corollary 2.1. Every hyperbolic 3-manifold group acts properly and cocompactly on a CAT(0) cube complex.

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5. Roller duality 5.1. Statement of duality We have seen two constructions in this lecture: CAT(0) cube complex X pocset of halfspaces H(X) pocset Σ cube complex X(Σ) Proposition 2.1 (Roller Duality). These constructions are dual to one another: (1) Given a ﬁnite width locally ﬁnite pocset, Σ, then H(X(Σ)) ≡ Σ. (2) Given a ﬁnite dimensional cube complex X, X(H(X)) = X. Proof. We give a proof of (2) and leave (1) as an exercise. Let Y = X(H(X)), namely the cubulation obtained from H(X). First, note that every vertex in v ∈ X 0 determines a DCC ultraﬁlter αv = {h|v ∈ h} We thus have a map Φ : X0 → Y 0 v → αv It is easy to see that Φ is injective, since any two vertices are separated by at least one hyperplane of X and hence determine diﬀerent ultraﬁlters. For any two adjacent vertices v, w ∈ X, they are separated by a unique hyperˆ transverse to the edge of which v and w are endpoints. Thus, by construcplane h tion, it follows that αv and αw will be joined by an edge in Y . We thus can extend Φ to the 1-skeleton X 1 . Once this is done, it is not hard to see that the map can be extended to the higher dimensional skeleta, since by construction, a cube is attached to Y for every 1-skeleton of a cube that appears in Y 1 . Finally, it remains to show that Φ is onto. Suppose that α is a vertex of Y , namely a DCC ultraﬁlter on the pocset H. Let αv be some vertex of Y which is in the image of Φ and let h1 , . . . , hn ∈ αv be the halfspaces so that α = (αv ; h1 , . . . , hn ) Now h1 is minimal in αv . This means that if we consider the vertex v in X, the ˆ1 is transverse to an edge adjacent to v; let [v, w] be that edge. Now hyperplane h we observe that Φ(w) = (αv ; h1 ) We continue in this manner, ﬁnding the vertices that get mapped by Φ to (αv ; h1 , . . . , hi ) for each i ≥ 1. We thus ﬁnd a vertex mapped by Φ to α, as required.

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5.2. Applications Subpocsets and collapsing. If Σ is pocset and Δ ⊂ Σ is a subpocset (i.e. a subset closed under involution), then there is a natural map ρΔ : U(Σ) → U(Δ), deﬁned by ρΔ (α) = α ∩ Δ It is elementary to check that ρΔ (α) is indeed an ultraﬁlter. It is also easy to see that this map sends DCC ultraﬁlters to DCC ultraﬁlters. Moreover it then extends to the cubes of X and we obtain a map ρΔ : X(Σ) → X(Δ). ˆ C(h), ˆ is the To see what the map ρΔ looks like, recall that the carrier of h, ˆ ˆ collection of cubes meeting a hyperplane h and is isometric to h × I. Since the pocset of halfspaces of the cube complex X(Δ) is simply Δ, the map ρΔ collapses (in the I direction) the carrier of every hyperplane associated to halfspaces which are not in Δ. We now give some examples of this construction. Orbit quotients. Suppose that a group G acts on a CAT(0) cubical complex ˆ we can look at the orbit of h ˆ under G. We then X. Then given a hyperplane h, ∗ get the pocset G(h ∪ h ), which of course is a subpocset of H. By the collapsing construction above, we get a new CAT(0) cubical complex X(G, ˆh). We call this ˆ This quotient has the property that there the orbit quotient of X associated to h. is a single orbit of hyperplanes, which is sometimes useful. Exercise 2.18. Consider Z × Z acting on the standard squaring of the plane. What are the orbit quotients? Exercise 2.19. Consider the standard description of the surface of genus two given as the quotient of the octagon whose edges are identiﬁed ababcdcd. Square the surface by putting a vertex in the middle and joining this vertex to the midpoint of each edge. Let X be the universal cover of this surface acted on by the fundamental group G of the surface. (1) What are the orbit quotients? Are they locally ﬁnite? (2) Are the actions on the orbit quotients proper? (3) G acts on the product of the orbit quotients. Is this action proper? Is it cocompact? Products. Corollary 2.2 (Recognizing Products). Let X be a CAT(0) cube complex ˆ its collection of hyperplanes. Then a decomposition of X into a product and H ˆ as a disjoint union H ˆ=H ˆ1 ∪ H ˆ2 X = X1 × X2 corresponds to a decomposition of H ˆ ˆ where every hyperplane in H1 intersects every hyperplane in H2 . Exercise 2.20. Prove this corollary. Hint. The direction that has not been discussed before is the one where a deˆ =H ˆ1 ∪ H ˆ 2 . Build the cube composition of the hyperplanes is a disjoint union H complexes X(H1 ) and X(H2 ). Show that H(X) has the same pocset structure as H(X(H1 ) × X(H2 )). A CAT(0) cube complex is called irreducible if it is not a product of two complexes. Applying Corollary 2.2 we obtain the following

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Corollary 2.3. Let X be a ﬁnite dimensional CAT(0) cube complex.Then X admits a canonical decomposition as a product of ﬁnitely many irreducible factors (up to permutation of factors). n Proof. Consider a maximal decomposition of X as a product X = i=1 Xi . Note that n is bounded by the dimension of X, so that each Xi is irreducible. Now suppose that X = m j=1 Yi is another decomposition of X into irreducibles. We then obtain transverse disjoint decompositions of the collection of hyperplanes of X: ˆ= H

n i=1

ˆi = H

m

ˆj K

j=1

ˆ i does not admit a disjoint transverse decomposition, we have Since each H ˆ ˆ ˆ j , there exists some H ˆ i such that Hi ⊂ Kj , for some j. Similarly, for each K ˆj ⊂ H ˆ i . Putting these two facts together and the fact that these are disjoint K ˆ we obtain that for each i there exists j, such that H ˆi = K ˆj, decompositions of H, and we are done.

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LECTURE 3

Rank rigidity In this lecture, we will see that under mild conditions, CAT(0) cube complexes are products of irreducible complexes that are either “line-like” or exhibit hyperbolic-like behavior. In the course of sketching a proof of this theorem, we will discuss some useful features of CAT(0) cube complexes, including the notion of an essential core and the interaction of isometries with hyperplanes. A hyperbolic isometry of a CAT(0) space is called rank one if no axis for the isometry bounds a half-ﬂat (an isometrically embedded half Euclidean plane). Example. Let X be a Gromov hyperbolic CAT(0) space. Then every isometry is rank one. The reason here is simply that there are no half-ﬂats in X. Exercise 3.21. Suppose that X is a product of two inﬁnite locally ﬁnite trees. Then no isometry of X is rank one. Is this still true when we allow locally inﬁnite trees? State and prove a generalization of this statement to products of CAT(0) spaces with extendible geodesics. Exercise 3.22. Let Γ be the pentagon graph and let X be the universal cover of the Salvetti complex associated the RAAG A(Γ). Show that there are isometries of X that are rank one and hyperbolic isometries that are not. Problem 3.1. Is there a CAT(0) space with a proper cocompact group action and an isometry with an axis that bounds a half-ﬂat, but for which the axis is not a bounded distance from an isometrically embedded ﬂat? It turns out that for symmetric spaces of higher rank and Euclidean buildings, there are no rank one elements. Together with what we saw above about products, Ballmann and Buyalo [4] were led to the following conjecture. Rank Rigidity Conjecture. Let G act properly cocompactly on a CAT(0) space X with extendible geodesics. Then one of the following three possibilities holds. (1) G contains a rank one element (2) X is a nontrivial product (3) X is a higher rank symmetric space or a Euclidean building This conjecture was originally proven in the setting of nonpositively curved manifolds by Ballmann [2] and was subsequently generalized by others (see [3] for further discussion.) The goal of this lecture is the following theorem [10]. Theorem 3.1 (Caprace-S). Let G act properly and cocompactly on an unbounded CAT(0) cube complex X. Then one of the following two possibilites holds; (1) G contains a rank one isometry (2) X contains a convex invariant subcomplex which splits as a product of two unbounded CAT(0) cube complexes 35 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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1. Essential cores When a group acts on a tree, one can reduce to an action on a minimal invariant subtree by removing edges with valence one vertices. One has a similar construction for cube complexes, which we now describe. Definition 3.1. A hyperplane is said to be essential if both of its halfspaces contain points arbitrarily far away from it. The hyperplane is said to be inessential if some neighborhood of it is the whole complex. If the hyperplane is neither essential nor inessential, we say that it is half-essential. This means that only one of the halfspaces it deﬁnes contains points arbitrarily far away from it. See Figure 1 for an example to keep in mind. Lemma 3.1. Let X be a cocompact CAT(0) cube complex. Then there exists an Aut(X)-invariant subcomplex Y such that Y decomposes as a product Y = Z × C, where Z is essential and C is ﬁnite.

Figure 1. A complex whose essential core is the real line.

Sketch of proof. We consider ﬁrst the half-essential hyperplanes. Note that if there are half-essential hyperplanes, then one exists one which is extremal, meaning that on one side of it, all the vertices are endpoints of edges transverse to the ˆ × [0, 1], hyperplane. If a hyperplane is extremal, then its carrier is of the form h ˆ where one of its boundaries, say h × {0}, is “free” in the sense that every cell meetˆ × [0, 1) from the complex ing it is contained in the carrier. Thus we can remove h and remain with a connected subcomplex. If we do this to all of the extremal hyperplanes at once, then one obtains a new complex X , invariant under Aut(X), with fewer orbits of hyperplanes. It is easy to check that X is indeed CAT(0) and convex. In fact, there is a deformation retraction from X to X . We then continue this process, eliminating orbits of half-essential hyperplanes at each stage. Since there are ﬁnitely many orbits of hyperplanes, we end up with an invariant CAT(0) subcomplex Y with no half-essential hyperplanes. Now we consider Y and observe that every essential hyperplane intersects every ˆ and ˆk are disjoint hyperplanes with h ˆ inessential hyperplane. For suppose that h ˆ ˆ essential and k inessential. Then, up to renaming the halfspaces associated to h ˆ is essential, which means there are points in and ˆk, we have that h ⊂ k. But h ˆ and these points are then arbitrarily far from ˆk. This h arbitrarily far from h contradicts the inessentiality of ˆk.

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Thus, by Corollary 2.2, we have that Y decomposes as a product Y = Z × C, where the hyperplanes associated to Z are the essential hyperplanes and the hyperplanes associated to C are the inessential ones. Since C has only inessential hyperplanes, it follows that it is ﬁnite. We already know that Y is Aut(X)-invariant. Since the notions of essential and inessential are Aut(X)-invariant, so is the decomposition Y = Z × C. The conclusion of this lemma is that whenever we have a proper, cocompact action of a group on an unbounded CAT(0) cube complex, we can pass to a proper cocompact action on an essential one, by passing to the complex Z in the lemma. One example of an essential CAT(0) cube complex is one with extendible geodesics. This means that every geodesic segment in X can be extended to a geodesic line in X. In order to simplify parts of the proof, from here on in, we will restrict to this situation: STANDING ASSUMPTION. For the rest of this lecture, we will assume that our CAT(0) cube complex has extendible geodesics. 2. Skewering In this section, we examine the ways that automorphisms behave with respect to hyperplanes. Definition 3.2. An automorphism g ∈ Aut(X) is said to skewer a halfspace h if gh ⊂ h. The term “skewer” becomes clear in the following exercise. We say that a line and a hyperplane cross if they intersect in a single point. Exercise 3.23. Let g ∈ Aut(X) and h a halfspace. ˆ (1) If g skewers h, then g is hyperbolic and any axis for g crosses h. (2) If g is hyperbolic and the axis of g crosses h, then for some n ∈ Z, we have that g n skewers h. Note that g skewers h if and only if g −1 skewers h∗ , so that it makes sense to ˆ whenever g skewers h or h∗ . speak of g skewering the hyperplane h 3. Single skewering We start by showing that hyperplanes are skewered. Proposition 3.1 (Single Skewering Lemma). Let G act cocompactly on X, then every hyperplane is skewered by some element of G. ˆ ∈ X. Let X(G, h) ˆ be the orbit quotient Proof. Consider a hyperplane h and p : X → X(G, ˆ h) the G-equivariant quotient map. There are two possibilities ˆ is bounded or not. depending on whether the diameter of X(G, h) ˆ If the diameter of X(G, h) is unbounded, then there exists a 1-skeleton geodesic ˆ 3). This means that α of length larger than the Ramsey number R(dim(X(G, h), there are three disjoint hyperplanes crossing α. Since these are all in the same orbit ˆ we can label them aˆk, bˆk, cˆk. Now if we put a transverse orientation of ˆk = p(h), on ˆk, we obtain transverse orientations on these three hyperplanes. Clearly two of them must be oriented in the same direction along α. This means that one of the elements ab−1 , bc−1 , ac−1 skewers one of the three hyperplanes (see Figure 2).

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Figure 2. The element bc−1 carries ck into bk and hence skewers cˆk. Since the action on the hyperplanes of X(G, ˆh) is transitive, it follows that the hyperplane ˆk is also skewered in X(G, ˆ h). Lifting the action to X, we see that the ˆ in the action same element that skewers ˆk in the action of G on X(G, ˆh) skewers h of G on X. ˆ is bounded. This means that The second case is that the diameter of X(G, h) there exists a ﬁxed point for the action of G on X(G, ˆh). After perhaps passing to a ﬁnite index subgroup of G, we may assume that there is a ﬁxed vertex v ∈ X(G, ˆh) for the action. Now lifting to X, we see that the collection of vertices p−1 (v) are stabilized by the action of G on X. All of these vertices lie to one side of some hyperplane in X. This contradicts the fact that X is essential and the action of G on X is cocompact. 4. Flipping An alternative to a hyperbolic element skewering a hyperplane is the following. Definition 3.3. A hyperbolic isometry g of X is said to ﬂip a halfspace h if gh ⊂ h∗ . Exercise 3.24 (Trichotomy). Let g be a hyperbolic isometry of a proper, ﬁnite dimensional CAT(0) cube complex X and let h be a halfspace. Then one of the following holds: (1) Some power of g skewers h. (2) Some power of g ﬂips h or h∗ . ˆ (3) Some power of g stabilizes h. ˆ and the last is The ﬁrst of the above possibilities is when the axis of g meets h ˆ when the axis for g lies in a bounded neighborhood of h. See Figure 3. Can the assumptions of properness or ﬁnite dimensionality be dropped? We say that a halfspace is unﬂippable if there does not exist any g ∈ G ﬂipping it. A key lemma is then the following. Lemma 3.2 (Flipping Lemma). Let G act on X properly and cocompactly. Let ˆ h be an unﬂippable halfspace. Then X decomposes as a product X = Y × R and h appears as the preimage of a point in R under the natural projection X → R. Before sketching a proof of this lemma, we recall an elementary lemma, whose proof we leave as an exercise. Exercise 3.25 (Endometry Lemma). Let X be a proper metric space with a cocompact isometry group. Let f : X → X be an isometric map. Show that f is bijective.

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LECTURE 3. RANK RIGIDITY

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ˆ ﬂips ˆk∗ and stabiFigure 3. The three possibilities. g skewers h, ˆ lizes m. Hint. Injectivity follows immediately from the fact that f is an isometry. For surjectivity, use the fact that for every R > 0 and > 0, there exists a number N such that for any ball of radius R, the size of an -separated collection of points in the ball is at most N . Sketch of proof of Flipping Lemma. The idea will be to break up the ˆ and those that do not. It will turn out hyperplanes of X into those that intersect h that this is a transverse decomposition of the collection of hyperplanes. We will thus aim to prove the following. ˆ the hyperplanes that Claim. Show that for every hyperplane ˆk disjoint from h, ˆ ˆ intersect h and k are the same. We do this in several steps. ˆ Step 1. When ˆk ⊂ h∗ , every hyperplane intersecting ˆk intersects h. ˆ as in Figˆ is a hyperplane intersecting ˆk and is disjoint from h, Suppose that m ure 4. The hyperplane ˆk is a CAT(0) cube complex and itself has extendible

ˆ Figure 4. A hyperplane meeting ˆk but not h. geodesics. By the Single Skewering Lemma 3.1 and Exercise 1.7 there exists an ˆ Now the element g has element g ∈ Stab(ˆk) which skewers the hyperplane ˆk ∩ m. ˆ ˆ an axis which is disjoint from h, so it cannot skewer h. Moreover, since g skewers a ˆ no power of g can stabilize h. ˆ So by Trichotomy hyperplane which is disjoint from h, (Exercise 3.24), it follows that some power of g ﬂips h, a contradiction. ˆ as follows. Step. 2. Step 1 yields an embedding of ˆk into h ˆ We ﬁrst deﬁne f on the vertices of ˆk. Let v be We deﬁne a map f : ˆk → h. ˆ we need to chose a halfspace ˆ meeting h, a vertex of ˆk. For every hyperplane m ˆ Simply choose the side that contains v. This gives us an ultraﬁlter bounded by m. Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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ˆ It is easy to see that this satisﬁes DCC, so we have a vertex on the halfspaces of h. f (v). The conclusion of Step 1 now tells us that the map f is injective on vertices. ˆ then m ˆ also For if v and w are two vertices of ˆk separated by a hyperplane m, ˆ intersects h and therefore separates f (v) and f (w). Finally, one checks that f extends to the cubes of ˆk. We leave this to the reader. This yields the desired embedding. ˆ ⊂ h∗ , a hyperplane intersects Step 3. For every translated hyperplane g h ˆ if and only if intersects g h. ˆ h ˆ also meets h. ˆ We already know by Step 1 that every hyperplane meeting g h ˆ ˆ Step 2 gives us an isometric embedding of g h into h. But now the Endometry Lemma (Exercise 3.25) tells us that this embedding is surjective. It then easily ˆ also meets g h. ˆ For suppose there was a follows that every hyperplane meeting h ˆ ˆ ˆ separated from ˆ meeting h and not meeting g h. Then any vertex in h hyperplane m ˆ ˆ would not be in the image of the map f deﬁned in Step 2. g h by m ˆ if and Step 4. For every hyperplane ˆk ⊂ h∗ , a hyperplane intersects h ˆ only if it intersects k. ˆ We will show Let k denote the halfspace of ˆk which contains the hyperplane h. ∗ ˆ that there exists a translate of h lying in k . By the Single Skewering Lemma, there exists g ∈ G such that gk∗ ⊂ k∗ . Since higher powers of g move points deeper and ˆ ∩ k∗ = ∅. We can also choose n such deeper into k∗ , there exists n such that g n h nˆ n ˆ = ∅. By Step 1, since g h ˆ is disjoint from h, ˆ it must also be disjoint that g h ∩ h from ˆk and therefore must be contained in k∗ . Now by Step 3, the hyperplanes ˆ are precisely those that intersect g h, ˆ so that all the hyperplanes that intersect h ˆ must intersect ˆk as well. intersecting h ˆ if and only Step 5. For every hyperplane ˆk ⊂ h, a hyperplane intersects h ˆ if it intersects k. We leave this to the reader. We proceed as in Steps 1-4, but one needs to take care in proving the last step in this case. This completes the proof of the claim. ˆ of X into a disjoint union We now break up the collection of hyperplanes H ˆ ⊥ , where ˆ=H ˆ ∪ H H ˆ = ∅} ˆ = {ˆk ∈ H| ˆ ˆk ∩ h H ˆ = ∅} ˆ ⊥ = {ˆk ∈ H| ˆ ˆk ∩ h H ˆ⊥ ˆ intersects every hyperplane in H The claim tells us that every hyperplane in H and this gives us a product decomposition X = X × X⊥ . ˆ can intersect, the space X Observe now that since no two hyperplanes in H is a tree T . There is a copy of T which appears in X as a maximal intersection ˆ The stabilizer of T acts properly and cocompactly on T of hyperplanes meeting h. with an unﬂippable hyperplane. We leave it to the reader to check that this means that T is a line.

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5. Double skewering We now seek another property which tells us more about how automorphisms of X interact with hyperplanes. Definition 3.4. An automorphism g ∈ Aut(X) is said to double skewer two nested halfspaces h ⊂ k if gk ⊂ h. Proposition 3.2 (Double Skewering Lemma). Let G act on X cocompactly, then for any two nested halfspaces h ⊂ k there exists g ∈ G double skewering h and k. Proof. We will employ the Flipping Lemma. There are two cases. First, suppose that either h,h∗ ,k or k∗ is unﬂippable. Then by the Flipping Lemma, we ˆ and have that X decomposes as a product X = Y × R and that the hyperplanes h ˆk are preimages of points p, q ∈ R. Choose g skewering k so that gk ⊂ k. Since the ˆ and ˆk are the same, it follows that gˆk ∩ h ˆ = ∅. Now hyperplanes that intersect h ˆ ˆ since there are only ﬁnitely many hyperplanes between h and k, it follows that for a suﬃciently large power of g, we have gk ⊂ h, as required. ˆ and ˆk are ﬂippable. We do two Otherwise, all the halfspaces associated to h ﬂips to obtain double skewering (see Figure 5 below).

Figure 5. Double ﬂipping leads to double skewering. More precisely, since h is ﬂippable, there exists a ∈ G such that ah∗ ⊂ h. Since h ⊂ k, we have that k∗ ⊂ h∗ , which implies that ak∗ ⊂ ah∗ ⊂ h. Now since k is ﬂippable, so is ak. This means there exists b ∈ G such that bak ⊂ ak∗ . Since h ⊂ k, we have bah ⊂ bak. So we then obtain: bah ⊂ bak ⊂ ak∗ ⊂ ah∗ ⊂ h as required.

6. Hyperplanes in sectors Consider n intersecting hyperplanes. They divide X into 2n regions which we call sectors. In this section, we will only consider sectors determined by two intersecting hyperplanes. If X were a product of two trees, then none of these regions would contain hyperplanes since every hyperplane intersects one of the original pair. However, for the CAT(0) square complex obtained by taking the universal cover of the squaring of a hyperbolic surface described in Example 3, the hyperplanes are quasigeodesics in a hyperbolic space. Thus, it is easy to see that each sector contains a hyperplane (in fact, inﬁnitely many). The following proposition tells us that this is a general phenomenon.

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Proposition 3.3 (Sector Lemma). Let X be an irreducible cocompact CAT(0) ˆ and ˆk be two intersecting hyperplanes cube complex with extendible geodesics. Let h in X. Then each of the four sectors deﬁned by X contains a hyperplane. Sketch of proof. First, we show that there exists some hyperplane disjoint ˆ and ˆk. Assume that this is not the case. We now seek a decomposition of the from h collection of hyperplanes into disjoint transverse subsets, which will contradict the ˆ denotes the collection of hyperplanes fact that X is not a product. Recall that H of X. We now focus on the following collections of hyperplanes. ˆ ˆ h = {hyperplanes disjoint from h} H ˆ ˆ Hk = {hyperplanes disjoint from k} ˆ = {hyperplanes disjoint from some hyperplane in H ˆh} H h ˆ = {hyperplanes disjoint from some hyperplane in H ˆk} H k ˆ ˆ ˆ ˆ R = H − (Hh ∪ Kk ) ˆ ∪ R ˆ is a transverse ˆ = H ˆ ∪ H We leave it to the reader to check that H h k ˆ decomposition of H. ˆ and ˆk. Say that m ˆ is such a Thus there exists some hyperplane disjoint from h hyperplane, so that m ⊂ h ∩ k. We now need to show all the other sectors contain hyperplanes as well. The Double Skewering Lemma 3.2 applied to the hyperplanes ˆ and m ˆ gives us an element g ∈ G such that gh ⊂ m. By applying a suﬃciently h high power of g, we have that g n h ⊂ m and g nˆk∩ˆk = ∅. It follows that ˆk ⊂ g n h∗ ∩g n k or ˆk ⊂ g n h∗ ∩ g n k∗ . In either case, by applying g −n we conclude that there exists a hyperplane in one of the two sectors h∗ ∩ k or h∗ ∩ k∗ . ˆ we get that there exists a hyperplane in By reversing the roles of ˆk and m one of the two sectors k∗ ∩ h or k∗ ∩ h∗ . This gives that there are hyperplanes in diagonally opposite sectors. Without loss of generality, let us assume that there is a hyperplane in h ∩ k and h∗ ∩ k∗ . ˆ as a hyperplane in h. ˆ If both of ˆ1 = ˆk ∩ h We now consider the hyperplane h ∗ ˆ the halfspaces h1 and h1 are ﬂippable as halfspaces in h, then we could apply these ﬂipping elements and obtain hyperplanes in the remaining two sectors h ∩ k∗ and ˆ is unﬂippable. h∗ ∩ k. So we can assume that one of the halfspaces h1 or h∗1 in h ˆ ∩ ˆk in ˆk is unﬂippable. Similarly, one of the hyperplanes associated to h ˆ skewering h ˆ ∩ ˆk. Let m ˆ be the We now ﬁnd a group element g ∈ Stab(h) hyperplane containing in h ∩ k. We leave it as an exercise to the reader to show ˆ into h ∩ k∗ . Similarly some power of an element in that some power of g carries m ∗ ˆ ˆ Stab(k) carries m into h ∩ k. Exercise 3.26 (The Tits Alternative). Use Proposition 3.3 to prove every group which acts cocompactly on an irreducible CAT(0) cube complex with extendible geodesics has a free subgroup of rank 2, unless the complex is a real line. Hint. Suppose that X is as in Proposition 3.3. Obtain a pattern of four disjoint halfspaces h1 , h2 , h3 , h4 such that hi ⊂ h∗j , for each i = j. Play ping-pong. 7. Proving rank rigidity We sketch now how the elements in the previous sections provide a proof of rank rigidity. Recall that we are assuming that X is an irreducible CAT(0) cube complex with extendible geodesics and that G is acting properly and cocompactly on X. We

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wish to show that there exist rank one elements in G. So we assume that no element is rank one. In the previous section, we considered sectors which were the intersection of two halfspaces. Consider a maximal collection of intersecting hyperplanes which ˆn ˆ .,h contain hyperplanes in “diagonally opposite” sectors. ˆMore precisely, let h1 , . . be a maximal collection of hyperplanes such that i hi = ∅ and the sectors i hi and i h∗i contain hyperplanes. We know such collections exist when n = 1. We will show that if X does not have rank one elements, there always exists a larger collection of hyperplanes with these properties. Since the dimension is bounded, this is a contradiction. ˆ and ˆk be the hyperplanes contained in the sectors: h ∈ hi and k∗ h∗ . Let h i i ˆ and ˆk. If ˆn+1 that intersects both h The goal will now be to ﬁnd a hyperplane h ˆn+1 will intersect the hyperplanes h ˆ1 , h ˆn and the Sector we can do this, then h ˆn+1 , h ˆ and h ˆn+1 and ˆk will tell us that there exist hyperplanes Lemma applied to h ˆ1 , . . . , h ˆn+1 }, and the proof will in diagonally opposite sectors of the collection {h be complete. By the Double Skewering Lemma, there exists g ∈ G such that gk ⊂ h. Since g is not rank one, there exists a half-ﬂat F bounding an axis for g. The intersection ˆi , ˆk and gˆk is a collection of rays meeting in points. of F with the hyperplanes h ˆi , and g −1ˆk, as in Figure 6. We We also consider the intersection of with g −1 h observe that by discreteness, there exist ﬁnitely many points of intersection of and the hyperplanes of X between any two given points of .

Figure 6. One of the hyperplanes crossing R must meet g −1ˆk or gˆk. Let R be the ray of intersection R = ˆk ∩ F . Since there are inﬁnitely many hyperplanes crossed by R and there are ﬁnitely many hyperplanes crossing between ˆ intersecting R which does not g−1ˆk ∩ and gˆk ∩ , there exists some hyperplane m −1ˆ ˆ ˆ or g m ˆ is the required hyperplane intersect between g k ∩ and g k ∩ . So either m ˆn+1 . h

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LECTURE 4

Special cube complexes In this lecture, we give a very brief introduction to special cube complexes and the notion of canonical completion and retraction. This should give you an idea of why special cube complexes have anything to do with subgroup separability. The material in this lecture follows closely some of the material in [25]. 1. Subgroup separability We ﬁrst recall some basic notions regarding subgroup separability Definition 4.1. Let G be a group and H < G. We say that H is separable if for every g ∈ G − H, there exists a ﬁnite index subgroup K < G such that H < K and g ∈ K. The group G is said to be residually ﬁnite if the trivial subgroup is separable. Exercise 4.27. Show that H < G is separable if for every g ∈ G − H, there exists a homomorphism to a ﬁnite group φ : G → F , such that φ(g) ∈ φ(H). Recall that the proﬁnite topology on G is the topology whose basic open sets are the cosets of ﬁnite index subgroups of G. If you have not seen the proﬁnite topology before, you should check that it is indeed a topology. We then have the following exercise. Exercise 4.28. A subgroup H < G is separable if and only if it is closed in the proﬁnite topology on G. Recall that a retraction φ : G → H is simply a homomorphism which is the identity on H. We say that H is a retract of G. Exercise 4.29. Let G be a residually ﬁnite group, and let H < G be a retract. Then H is separable. Hint. A retract of a Hausdorﬀ space is closed. 2. Warmup - Stallings’ proof of Marshall Hall’s theorem Marshall Hall [27] proved back in 1949 that every ﬁnitely generated subgroup of a ﬁnitely generated free group is virtually a free factor. John Stallings came up with a nice, very elementary graph-theoretic proof of this fact [42]. Roughly speaking, the idea is to represent the subgroup H < G as an immersion of graphs Δ → Γ where Γ has fundamental group G. One then constructs a ﬁnite cover in which Δ lifts to an embedding. This then tells you that H is a virtual retract and a virtual free factor. By Exercise 4.29, we then have that H is separable. We review this more closely. 45 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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Theorem 4.1. Every ﬁnitely generated subgroup of a ﬁnitely generated free group is a virtual retract. Proof. This will be a proof by example. Refer to Figures 1, 2, and 3 below. Consider the free group G = a, b and consider the subgroup H = ab2 a−1 , abab2 . First we represent H < G as a map between bouquets of circles Δ → Γ.

Figure 1. Representing a subgroup by a map between bouquets of circles.

On the level of fundamental group, this map represents our subgroup H. Note that the map is not necessarily an immersion (i.e., a local embedding). In this case, for example, there is more than one edge on the left graph labeled a and pointing out of the central vertex. To remedy this, we “fold” two such edges in the graph on the left as shown in the diagram below to obtain a new graph with fewer edges. The map still represents the same subgroup, and the map is closer to an immersion. We continue to fold, reducing the number of edges each time, until we obtain an immersion.

Figure 2. Folding the source graph. Note that on the graph on the right (in Figure 2) there is no more folding that can be done, so that we now have an immersion of graphs representing our subgroup. We now replace Δ by this new folded graph and we have a new map Δ → Γ which is an immersion. Now that we have an immersion, the next thing to do is to complete. There are many ways to do this. We describe what Haglund and Wise call “canonical completion”. The idea is to add edges to the graph Δ until the map becomes a covering space. Consider the a-loop in Γ. The fact that the map Δ → Γ is an immersion tells us that each component of the preimage of the a-loop in Γ is either a cycle of a’s, an arc of a’s, or a single vertex. If it is a loop of a’s we do nothing. If it is an arc of a’s we add an edge labeled with an a to complete the arc of a’s to a loop of a’s. Finally, if it is a vertex, then it means that we have a vertex with only b-edges adjacent to it. In this case, we simply attach an a-loop at that vertex. See Figure 3.

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We then do the same for the b’s. It is now easy to check that the resulting map from the completed graph Γ to Γ is a covering space. Now note that not only do we have a ﬁnite covering space Γ of the original bouquet of circles Γ, The original map Δ → Γ factors through an embedding Δ → Γ and there is a natural retraction map Γ → Δ. Simply map each added aedge to the arc of a-edges it completes and each a-loop to the vertex it is attached to; and do the same for the b’s.

Figure 3. Completing an immersion of graphs. The dotted edges are the edges we add to complete. We thus obtain that the subgroup H is a retract. It is also easy to see from this picture that H is virtually a free factor. Exercise 4.30. Use the proof of the above theorem to show that a ﬁnitely generated free group is residually ﬁnite. Since the free group is residually ﬁnite, Exercise 4.29 and Theorem 4.1 now tell us that every ﬁnitely generated subgroup of a ﬁnitely generated free group is separable. 3. Special cube complexes One of Wise’s main goals was to seek a more general setting in which the technique of the previous section for graphs can be made to work. This led to the notion of special cube complexes introduced by Haglund and Wise [26], which we now describe. ˆ and ˆk in an NPC complex First we describe “osculation”. Two hyperplanes h are said to osculate at a vertex v if there exist edges e and f with endpoint v such that e and f are not on the boundary of a square and e and f are transverse to the ˆ and ˆk. If h ˆ = ˆk we say that h ˆ self-osculates. If h ˆ = ˆk and h ˆ and ˆk also hyperplanes h ˆ ˆ intersect, we say that h and k interosculate. See Figure 4. A hyperplane is said to be 2-sided if it separates its carrier. This is the same as saying that the carrier is a product of the hyperplane with an interval. Definition 4.2. Let X be an NPC cube complex. We say that X is special if every hyperplane is embedded, 2-sided, and does not self-osculate, and no two hyperplanes interosculate.

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Figure 4. Self-osculation and interosculation. A group is said to be special if it is the fundamental group of a compact special cube complex. The group is said to be virtually special if it has a ﬁnite index subgroup which is special. Exercise 4.31. Examine the examples of NPC cube complexes discussed in Lecture 1 and decide which is special. Exercise 4.32. Let X → Y be a locally isometric map between NPC cube complexes and suppose that Y is special. Then X is special. Recall now Example 6 of Lecture 1, namely the Salvetti complex associated to a right-angled Artin group. We call such a complex a RAAG complex for short. It is easy to see that these cube complexes are indeed special (the reader should check this). The reason that these examples are so important is because of the following proposition. Proposition 4.1. Let X be a compact NPC cube complex. Then X is special if and only if there exists a locally isometric embedding to a RAAG complex. In particular, the fundamental group of a compact special cube complex is a subgroup of a right-angled Artin group. Sketch of proof. If there exists a locally isometric embedding X → R, then we know by Exercise 4.32 that since R is special, so is X. On the other hand, suppose that X is special. We let Γ be the graph whose vertices correspond to hyperplanes of X and where two vertices are joined by an edge if and only if the corresponding hyperplanes in X intersect. We consider the RAAG-complex R = R(Γ). We now construct a map X (1) → R in the natural way: vertices of X get mapped to the unique vertex of R, and edges of X get sent to edges that cross the corresponding hyperplane in R. Now one checks that this map can be extended over the all cubes of X to a cubical map and that the map is a local isometry. We will focus on compact special cube complexes, although it is possible to discuss matters in the context of complexes with ﬁnitely many hyperplanes. A ﬁnal remark is that by a result proved independently by Davis-Januszkiewicz [14] and Hsu-Wise [28], RAAGs are linear. Thus, we know immediately that every virtually special group is residually ﬁnite. Therefore, in order to show that a subgroup of a virtually special group is separable, we just need to show that it is a virtual retract. 4. Canonical completion and retraction We now wish to generalize the canonical completion and retraction construction from the world of graphs to special cube complexes. First, we consider an immersion X → R from an NPC complex X to a RAAG complex R. We will build a covering space of R which we call C(X, R). An instructive example is seen in Figure 5.

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Figure 5. Canonical completion and retraction for a map to a RAAG complex.

First we focus on the 1-skeleton of X as a map to the 1-skeleton of R, which is a bouquet of circles. We canonically complete as in the case of graphs. The result is the 1-skeleton of C(X, R). Now one checks that one can add squares and higher dimensional cubes wherever they “should be”. For example, on the top left side of X, there is an annulus. We added an arc with a double arrow to complete the original double arrow arc to a loop. Now one simply glues in another annulus so that the original annulus is completed to a torus. The original square on the lower left of X has four extra arcs attached and four extra squares. To make this work requires a bit of thought and checking some cases. We leave it to the reader. We have then built a covering space of X and there is indeed a retraction C(X, R) → X extending the one described earlier for graphs. Generally, for a locally isometric immersion between two special cube complexes X → Y , where Y is not necessarily a RAAG complex, one uses a ﬁber product construction. Given two cubical maps X → Y and Z → Y between cube complexes, one can construct a complex denoted X ⊗Y Z called the ﬁber product of X and Z over Y , which is a subspace of X × Y . It consists of an n-cube for each pair of n-cubes σ ∈ X and τ ∈ Z that get mapped to the same n-cube in Z. If the original maps X → Y and Z → Y were covering spaces, this corresponds to the usual common covering space. If you are not familiar with this notion, you should ﬁrst draw some simple examples of ﬁber products using graphs. Figure 6 shows one such example. Note that the ﬁber product need not be connected, but in the situation we will be looking at there will be a natural component to focus on. So now given a local isometric embedding between special cube complexes X → Y , we know there exists a RAAG complex R and a locally isometric embedding Y → R. The composition X → Y → R gives a map X → R, and we can form the canonical completion C(X, R) for this map. We also have a natural covering map C(X, R) → R. One then deﬁnes C(X, Y ) as a ﬁber product: C(X, Y ) ≡ Y ⊗R C(X, R)

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Figure 6. A simple ﬁber product. We have a retraction which comes from the composition C(X, Y ) → C(X, R) → X, where the second map is the retraction produced before. The canonical completion C(X, Y ) becomes diﬃcult to draw for complicated examples (even the one in Figure 5), but Figure 7 displays a simple example.

Figure 7. A general completion and retraction. This particular example is connected, but you should be aware that they need not be. However there is a natural embedding of X in C(X, Y ) and one usually focuses on this component. 5. Application: separability of quasiconvex subgroups In the previous section, we saw that special cube complexes have something to do with subgroup separability. In particular, any subgroup of a special cube complex group which can be represented as an immersion is separable. A particular application of this is the following result of Haglund and Wise. Theorem 4.2. If G is virtually special and Gromov hyperbolic, then every quasiconvex subgroup is separable. We ﬁrst remark that we may assume that G is itself special. For if we prove that the quasiconvex subgroups of a ﬁnite index subgroup of G are separable, then the quasiconvex subgroups of G are separable as well. Secondly, we remark that since G is virtually special, it is linear. To prove the theorem, we ﬁrst need to discuss the construction of a combinatorial convex core. We give the construction in the form of an exercise. For more details see Haglund [23].

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Suppose that G is Gromov hyperbolic and the fundamental group of a compact NPC cube complex Y . Then we have G acting on the universal cover X, which is a Gromov hyperbolic CAT(0) cube complex. Now consider the orbit of some vertex under a quasiconvex subgroup H < G. Since H is ﬁnitely generated, there exists some neighborhood N of this orbit which is connected. Now using geometry of Gromov hyperbolic spaces, one can prove the following lemma, which we leave as an exercise. Exercise 4.33. There exists some constant C such that if σ is a cube distance at least C from N , one of the hyperplanes meeting σ is disjoint from N . One now builds a convex hull for H as follows. For each halfspace h let C(h) ˆ This is a convex subcomplex of X. We denote the union of h with the carrier of h. then set C(h) Hull(H) = N ⊂h

Some thought and Exercise 4.33 will then tell you that Hull(H) is contained in some neighborhood of H. In particular this is a convex subcomplex of X on which H acts cocompactly. We call Hull(H) a combinatorial convex hull for H. Observe that Hull(H)/H embeds naturally into X/H, and X/H → X/G = Y is a covering space and hence a locally isometric embedding. We thus have a locally isometric embedding of NPC complexes Hull(H)/H → Y . We now apply the canonical completion and retraction construction to this local isometric embedding to obtain a ﬁnite covering Y of Y and a retraction Y → Hull(H)/H. We thus obtain a retraction G → H and, so that by Exercise 4.29, H is separable. 6. Hyperbolic cube complexes are virtually special This lecture was a very cursory introduction to special cube complexes. Wise, together with Haglund and others, extensively developed the theory of special cube complexes and proved several very deep and diﬃcult theorems about them. For a treatment of much of this theory the reader should consult Wise’s upcoming book [45]. Agol [1] combined these theorems with an ingenius coloring argument to obtain the following theorem, which was Wise’s ultimate goal. Theorem 4.3. Every hyperbolic group which acts properly and cocompactly on a CAT(0) cube complex is virtually special. This is a startling theorem which has far reaching implications. For example, it settles the long-standing virtual Haken conjecture by telling us that every hyperbolic manifold has a ﬁnite cover which contains an embedded π1 -injective surface. To see this, ﬁrst note that Corollary 2.1 tells us that the fundamental group G of a hyperbolic 3-manifold is the fundamental group of a compact NPC cube complex. Since G is Gromov hyperbolic, we then know by Theorem 4.3 that G is virtually special. In particular it follows that the quasiconvex surface subgroups produced by Kahn and Markovic are separable. But now a theorem of Scott [41] tells us that a π1 -injective immersion of a surface, whose corresponding subgroup in the 3-manifold group is separable, homotopes to an embedding in a ﬁnite cover. This provides the desired Haken ﬁnite cover.

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Bibliography 1. Ian Agol, The virtual haken conjecture, Preprint, 2012. With an appendix by I. Agol, D. Groves and J.Manning. 2. Werner Ballmann, Nonpositively curved manifolds of higher rank, Ann. of Math. (2) 122 (1985), no. 3, 597–609. , Lectures on spaces of nonpositive curvature, DMV Seminar, vol. 25, Birkh¨ auser Ver3. lag, Basel, 1995, With an appendix by Misha Brin. 4. Werner Ballmann and Sergei Buyalo, Periodic rank one geodesics in Hadamard spaces, Geometric and probabilistic structures in dynamics, Contemp. Math., vol. 469, Amer. Math. Soc., Providence, RI, 2008, pp. 19–27. 5. Nicolas Bergeron and Daniel T. Wise, A boundary criterion for cubulation, Amer. J. of Math. 134 (2023), no. 3, 843–859. 6. Martin Bridson, Geodesics and curvature in metric simplicial complexes, Group Theory from a Geometrical Viewpoint, Proc. ICTP Trieste 1990 (A. Haeﬂiger E. Ghys and A. Verjovsky, eds.), World Scientiﬁc, Singapore, 1991, pp. 373–464. 7. Martin R. Bridson and Andr´e Haeﬂiger, Metric spaces of non-positive curvature, SpringerVerlag, Berlin, 1999. MR1744 486 ´ 8. Marc Burger and Shahar Mozes, Lattices in product of trees, Inst. Hautes Etudes Sci. Publ. Math. (2000), no. 92, 151–194 (2001). MR2002i:20042 9. Pierre-Emmanuel Caprace, Conjugacy of 2-spherical subgroups of Coxeter groups and parallel walls, Algebr. Geom. Topol. 6 (2006), 1987–2029 (electronic). MR2263057 10. Pierre-Emmanuel Caprace and Michah Sageev, Rank rigidity for cat(0) cube complexes, Geometric And Functional Analysis 21 (2011), 851–891, 10.1007/s00039-011-0126-7. 11. Indira Chatterji, Cornelia Drutu, and Frederic Haglund, Kazhdan and haagerup properties from the median viewpoint, Adv. Math. 225 (2010), 882–921. 12. Indira Chatterji and Graham Niblo, From wall spaces to CAT(0) cube complexes, Internat. J. Algebra Comput. 15 (2005), no. 5-6, 875–885. MR2197811 (2006m:20064) 13. Victor Chepoi, Graphs of some CAT(0) complexes, Adv. in Appl. Math. 24 (2000), no. 2, 125–179. MR1748966 (2001a:57004) 14. Michael W. Davis and Tadeusz Januszkiewicz, Right-angled Artin groups are commensurable with right-angled Coxeter groups, J. Pure Appl. Algebra 153 (2000), no. 3, 229–235. 15. Max Dehn, Papers on group theory and topology, Springer-Verlag, New York-Berlin, 1987, Translated from the German and with introductions and an appendix by John Stillwell, With an appendix by Otto Schreier. 16. M. J. Dunwoody, The accessibility of ﬁnitely presented groups, Invent. Math. 81 (1985), no. 3, 449–457. MR87d:20037 17. V. N. Gerasimov, Semi-splittings of groups and actions on cubings, Algebra, geometry, analysis and mathematical physics (Russian) (Novosibirsk, 1996), Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk, 1997, pp. 91–109, 190. MR99c:20049 ´ Ghys and P. de la Harpe (eds.), Sur les groupes hyperboliques d’apr` 18. E. es Mikhael Gromov, Progress in Mathematics, vol. 83, Birkh¨ auser Boston Inc., Boston, MA, 1990, Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. MR92f:53050 19. Rita Gitik, Mahan Mitra, Eliyahu Rips, and Michah Sageev, Widths of subgroups, Trans. Amer. Math. Soc. 350 (1998), no. 1, 321–329. 20. Martin Greendlinger, On Dehn’s algorithms for the conjugacy and word problems, with applications, Comm. Pure Appl. Math. 13 (1960), 641–677. MR23#A2327 21. M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. 53 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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22. D. Guralnik, Coarse decompositions of boundaries for CAT(0) groups, Preprint. arXiv:math/0611006 [math.GR], 2006. 23. Frederic Haglund, Finite index subgroups of graph products, Geom. Dedicata 135 (2008), 167–209. 24. Fr´ ed´ eric Haglund and Fr´ed´ eric Paulin, Simplicit´ e de groupes d’automorphismes d’espaces a ` courbure n´ egative, The Epstein birthday schrift, Geom. Topol., Coventry, 1998, pp. 181–248 (electronic). MR2000b:20034 25. Fr´ ed´ eric Haglund and Daniel T. Wise, A combination theorem for special cube complexes, Submitted. 26. Fr´ ed´ eric Haglund and Daniel T. Wise, Special cube complexes, Geom. Funct. Anal. 17 (2008), no. 5, 1 551–1620. MR2377497 27. Marshall Hall, Jr., Coset representations in free groups, Trans. Amer. Math. Soc. 67 (1949), 421–432. 28. Tim Hsu and Daniel T. Wise, Separating quasiconvex subgroups of right-angled Artin groups, Math. Z. 240 (2002), no. 3, 521–548. MR1924 020 29. David Janzen and Daniel T. Wise, A smallest irreducible lattice in the product of trees, Alg. Geom. Top. 9 (2009), 2191–2201. 30. Jeremy Kahn and Vladimir Markovic, Immersing almost geodesic surfaces in a closed hyperbolic three manifold, Ann. of Math. 175 (2012), no. 3, 1127–1190. 31. I.J. Leary, A metric Kan–Thurston theorem, Preprint. arXiv:1009.1540, to appear in Journal of Topology. 32. Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Springer-Verlag, Berlin, 1977, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. 33. G´ abor Moussong, Hyperbolic coxeter groups, Ph.D. thesis, Ohio State University, 1988. 34. Amso Nevo and Michah Sageev, Poisson boundaries of CAT(0) cube complexes, to appear in Groups Geom. Dyn. 35. G. A. Niblo and L. D. Reeves, Coxeter groups act on CAT(0) cube complexes, J. Group Theory 6 (2003), no. 3, 399–413. MR2004e:20072 36. Graham A. Niblo and Martin A. Roller, Groups acting on cubes and Kazhdan’s property (T), Proc. Amer. Math. Soc. 126 (1998), no. 3, 693–699. MR98k:20058 37. Bogdan Nica, Cubulating spaces with walls, Algebr. Geom. Topol. 4 (2004), 297–309 (electronic). MR2059 193 38. M.A. Roller, Poc sets, median algebras and group actions. An extended study of Dunwoody’s construction and Sageev’s theorem, Preprint, Univ. of Southampton, 1998. 39. Pascal Rolli, Notes on cat(0) cube complexes, Preprint, 2012. 40. Michah Sageev, Ends of group pairs and non-positively curved cube complexes, Proc. London Math. Soc. (3) 71 (1995), no. 3, 585–617. MR97a:20062 41. Peter Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. (2) 17 (1978), no. 3, 555–565. 42. John R. Stallings, Topology of ﬁnite graphs, Invent. Math. 71 (1983), no. 3, 551–565. 43. Daniel T. Wise, Cubulating small cancellation groups, GAFA, Geom. Funct. Anal. 14 (2004), no. 1, 150–214. 44. Daniel T. Wise, Complete square complexes, Comment. Math. Helv. 82 (2007), no. 4, 683–724. MR2341837 (2009c:20078) , From riches to raags, Regional Conference Series in Mathematics, vol. 117, Amer. 45. Math. Soc., Providence, 2012.

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https://doi.org/10.1090//pcms/021/03

Geometric Small Cancellation Vincent Guirardel

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IAS/Park City Mathematics Series Volume 21, 2012

Geometric Small Cancellation Vincent Guirardel Introduction The aim of these lectures is to present geometric small cancellation, also known as very small cancellation, introduced by Gromov in [Gro01b, Gro01a], and further developed by Gromov-Delzant, Arzhantseva-Delzant, Coulon, and DahmaniGuirardel-Osin [DG08, AD, Cou11, DGO]. Starting from a group G acting on a hyperbolic space, together with a family of subgroups satisfying a small cancellation condition, this theory studies the quotient of G by the normal subgroup generated by the given subgroups. Applications of the theory include the construction of monsters (i.e. groups with pathological properties), by taking iterated small cancellation quotients. The Dehn ﬁlling theory of relatively hyperbolic groups can also be understood from this framework. Beyond hyperbolic and relatively hyperbolic groups, this small cancellation theory has applications concerning groups having nice actions on hyperbolic spaces such as the mapping class group of a surface, the outer automorphism group of a free group, or the group of birational transformations of the projective plane. In the ﬁrst lecture, we start by discussing a classical small cancellation condition, applications of small cancellation, and then state the geometric small cancellation theorem. In the second lecture, we discuss weak proper discontinuity as a way to produce small cancellation subgroups, and in particular, we present an application to SQuniversality. The next two lectures are devoted to the proof of the geometric small cancellation theorem. In this proof, one produces a suitable hyperbolic space by a cone-oﬀ construction, and one describes the normal subgroup generated by a small cancellation family via its action on this cone-oﬀ space. Lecture three is about this description of the normal subgroup via the theory of very rotating families on the hyperbolic cone-oﬀ. Lecture four describes the construction and the properties of the hyperbolic cone-oﬀ.

Institut de Recherche Mathématique de Rennes et Institut Universitaire de France, Université de Rennes 1 et CNRS (UMR 6625), 263 avenue du Général Leclerc, CS 74205, F-35042 RENNES Cédex, France. http://perso.univ-rennes1.fr/vincent.guirardel E-mail address: [email protected] I would like to warmly thank Rémi Coulon. First for his help preparing the lectures (the material about the cone-oﬀ follows his own work), and then for his very useful comments on the text that helped a lot improve the exposition. I acknowledge support from the Institut Universitaire de France and ANR grant ANR-11-BS01-013 c 2014 American Mathematical Society

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LECTURE 1

What is small cancellation about? 1. The basic setting The basic problem tackled by small cancellation theory is the following one. Problem. Let G be a group, and R1 , . . . , Rn some subgroups of G. Give conditions under which you understand the normal subgroup R1 , . . . , Rn G and the quotient G/R1 , . . . , Rn . In combinatorial group theory, there are various notions of small cancellation conditions for a ﬁnite presentation S|r1 , . . . , rk . In this case, G is the free group S, and Ri is the cyclic group ri . Essentially, these conditions ask that any common subword between two relators has to be short compared to the length of the relators. More precisely, a piece is a word u such that there exist cyclic conjugates r˜1 , r˜2 of relators ri1 , ri2 (i1 = i2 is allowed) such that r˜i = ubi (as concatenation of words) with b1 = b2 . Then the C (1/6) small cancellation condition asks that in this situation, |u| < 16 |r1 | and |u| < 16 |r2 |. One can replace 16 by any λ < 1 to deﬁne the C (λ) condition. Then small cancellation theory says, among other things, that the group S|r1 , . . . , rk is a hyperbolic group, that it is torsion-free if no relator is a proper power. Moreover, when G is torsion-free, the 2-complex deﬁned by the presentation is aspherical (meaning in some sense that there are no relations among relations), and in particular G is 2-dimensional. There are many variants and generalizations of this condition. Building on Max Dehn’s work on surface groups, this started in the 50’s with the work of Tartakovskii, Greendlinger, and continued with Lyndon, Schupp, Rips, Olshanskii, and many others [Tar49, Gre60, LS01, Ol’91a, Rip82]. Small cancellation theory was generalized to hyperbolic and relatively hyperbolic groups by Olshanskii, Delzant, Champetier, and Osin [Ol’91b, Del96, Cha94, Osi10]. An important variant is Gromov’s graphical small cancellation condition, where the presentation is given by killing the loops of a labelled graph, and one asks for pieces in this graph to be small [Gro03]. This lecture will be about geometric small cancellation (or very small cancellation) introduced by Gromov in [Gro01a], and further developed by Gromov-Delzant, Arzhantseva-Delzant, Coulon, and Dahmani-Guirardel-Osin [DG08, AD, Cou11, DGO]. There are other very interesting small cancellation theories, in particular, Wise’s small cancellation theory for special cube complex [Wis11]. 2. Applications of small cancellation Small cancellation is a large source of examples of groups (the following list is very far from being exhaustive !). 59 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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VINCENT GUIRARDEL, GEOMETRIC SMALL CANCELLATION

Interesting hyperbolic groups The Rips construction allows us to produce hyperbolic groups (in fact small cancellation groups) that map onto any given ﬁnitely presented group with ﬁnitely generated kernel. This allows us to encode many pathologies of ﬁnitely presented groups into hyperbolic groups. For instance, there are hyperbolic groups having a ﬁnitely generated subgroup whose membership problem is not solvable [Rip82]. There are many useful variants of this elegant construction, see for instance [BO08, BW05, OW07, Wis03]. Dehn ﬁllings Given a relatively hyperbolic group with respect to P , and N P a normal subgroup, then if N is deep enough (i.e. avoids a ﬁnite subset F ⊂ P \ {1} given in advance), then P/N embeds in G/N , and G/N is relatively hyperbolic with respect to P/N [GM08, Osi07]. Normal subgroups Small cancellation allows us to understand the structure of the corresponding normal subgroup. For instance, Delzant shows that for any hyperbolic group G there exists n such that for any hyperbolic element h ∈ G, the normal subgroup generated by hn is free [Del96]. This is because hn is a subgroup satisfying a small cancellation condition (see below). The same idea shows that if h ∈ M CG is a pseudo-Anosov element of the mapping class group (or a fully irreducible automorphism of a free group), then for some n ≥ 1, the normal subgroup generated by hn is free and purely pseudo-Anosov [DGO]. This uses the fact that M CG acts on the curve complex, which is a hyperbolic space [MM99], and that hn is a small cancellation subgroup when acting on the curve complex. Similar arguments work in the outer automorphism group of a free group Out(Fr ) and in the Cremona group Bir(P2 C) because they have a nice action on hyperbolic space [BF10, Can11, CL]. Many quotients Small cancellation theory allows us to produce many quotients of any non-elementary hyperbolic group G: it is SQ-universal [Del96, Ol 95]. This means that for any countable group A there exists a quotient G Q in which A embeds (in particular, G has uncountably many non-isomorphic quotients). Small cancellation theory also allows us to prove SQ universality of Mapping Class Groups, Out(Fn ), and the Cremona group Bir(P2 ) [DGO]. More generally, this applies to groups with hyperbolically embedded subgroups [DGO] (we will not discuss this notion in this lecture, only the existence of hyperbolic elements with the WPD property, see Section 2). Abundance of quotients makes it diﬃcult for a group with few quotients to embed in such a group. This idea can be used to prove that lattices in higher rank Lie groups don’t embed in mapping class groups, or Out(Fn ) [DGO, BW11], the original proof for mapping class group is due to Kaimanovich-Masur [KM96]. Monsters The following monsters are (or can be) produced as limits of inﬁnite chains of small cancellation quotients: (1) Inﬁnite Burnside groups. For n large enough, r ≥ 2, the free Burnside group B(r, n) = s1 , . . . , sr |∀w, wn = 1 is inﬁnite [NA68, Iva94, Lys96, Ol 82, DG08], see also the notes by Rémi Coulon [Cou].

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LECTURE 1. WHAT IS SMALL CANCELLATION ABOUT?

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(2) Tarski monster. For each prime p large enough, there is an inﬁnite, ﬁnitely generated group all whose proper subgroups are cyclic of order p [Ol 80]. (3) Osin’s monster. There is a ﬁnitely generated group not isomorphic to Z/2Z, such that all its non-trivial elements are conjugate [Osi10]. (4) Gromov’s monster. This is a ﬁnitely generated group that contains a uniformly embedded expander, and which therefore does not uniformly embed in a Hilbert space [Gro03, AD]. This gives a counterexample to the strong form of the Baum-Connes conjecture [HLS02]. 3. Geometric small cancellation The goal of this lecture is to describe geometric small cancellation, introduced by Gromov in [Gro01b, Gro01a]. We give some preliminary deﬁnitions before stating the results. 3.1. Preliminaries and notations A metric space X is δ-hyperbolic if it is geodesic, and if it satisﬁes the δ-hyperbolic 4-point inequality: for all x, y, z, t ∈ X, d(x, y) + d(z, t) ≤ max{d(x, z) + d(y, t), d(x, t) + d(y, z)} + 2δ. This implies that for any geodesic triangle, any side is contained in the 4δ-neighbourhood of the two other sides. We denote by [x, y] a geodesic between x and y; although there is no uniqueness of geodesics, this usually does not lead to confusion. An R-tree is a 0-hyperbolic space. We denote by δH2 the hyperbolicity constant of the hyperbolic plane H2 . A geodesic metric space is CAT (−1) if its triangles are thinner than comparison triangles in H2 (see [BH99] for details). Such a space is δH2 -hyperbolic. Given a subset Q of a hyperbolic space X and r ≥ 0, we denote by Q+r its r-neighbourhood. We say that Q is almost convex if for all x, y ∈ Q, there exist x , y ∈ Q and geodesics [x, x ], [x , y ], [y , y] such that d(x, x ) ≤ 8δ, d(y, y ) ≤ 8δ, and [x, x ] ∪ [x , y ] ∪ [y , y] ⊂ Q. It follows that the path metric dQ on Q induced by the metric dX of X is close to dX : for all x, y ∈ Q, dX (x, y) ≤ dQ (x, y) ≤ dX (x, y) + 32δ. Recall that Q ⊂ X is K-quasiconvex if for all x, y ∈ Q, any geodesic [x, y] is contained in Q+K . This notion is weaker as it does not say anything about dQ (Q might even be disconnected). However, if Q is K-quasiconvex, then for all r ≥ K, Q+r is almost convex. Also note that δ-hyperbolicity implies that an almost convex subset is 8δ-quasiconvex. 3.2. Moving families and the geometric small cancellation Let X be a δ-hyperbolic space, and G be a group acting on X by isometries. Consider Q = (Qi )i∈I a family of almost convex subspaces of X, and R = (Ri )i∈I a corresponding family of subgroups such that Ri is a normal subgroup of the stabilizer of Qi . This data should be G-invariant: G acts on I so that Qgi = gQi , and Rgi = gRi g −1 . Let us call such data a moving family F. We now deﬁne the injectivity radius and the fellow traveling constant of a moving family. The small cancellation hypothesis deﬁned below will ask for a large injectivity radius and a small fellow traveling constant.

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The injectivity radius measures the minimal displacement of all non-trivial elements of all Ri ’s: inj(F) = inf{d(x, gx)|i ∈ I, x ∈ Qi , g ∈ Ri \ {1}}. Here the inﬁmum is taken only over x ∈ Qi , but since Qi is almost convex and Ri -invariant, the deﬁnition would not change much if we took the inﬁmum over all x ∈ X. The fellow traveling constant between two subspaces Qi , Qj measures how long they remain at a bounded distance from each other. Technically, Δ(Qi , Qj ) = diam Q+20δ . ∩ Q+20δ i j Because Qi , Qj are almost convex in a hyperbolic space, any point of Qi that is far from Q+20δ ∩ Q+20δ is far from Qj , so this really measures what we want. The i j fellow traveling constant of F is deﬁned by Δ(F) = sup Δ(Qi , Qj ). i=j

Definition 1.1. Assume that X is δ-hyperbolic, with δ > 0. The moving family F satisﬁes the (A, λ)-small cancellation condition if it satisﬁes (1) large injectivity radius: inj(F) ≥ Aδ, and (2) small fellow traveling compared to injectivity radius: Δ(F) ≤ λ inj(F). Remark 1.2. • It is convenient to say that some subgroup R < G satisﬁes the (A, λ)-small cancellation condition if the family R of all conjugates of R together with a suitable family of subspaces of X, makes a small cancellation moving family. • The (A, λ)-small cancellation hypothesis (for A large enough) implies that each Ri is torsion-free, because every element of Ri \ {1} is hyperbolic. • It is often convenient to take I = Q, and to view R as a group attached to each subspace in Q: R = (RQ )Q∈Q , or conversely, to take I = R and to view Q as a space attached to each group in R: Q = (QH )H∈R . • We don’t assume any properness on X, and no ﬁniteness on I/G. Relation with classical small cancellation The small cancellation hypothesis (almost) covers the classical small cancellation condition C (λ) in the following way. The group G is the free group, acting on its Cayley graph X, (Ri )i∈I is the family of cyclic groups generated by the conjugates of the relators, and (Qi )i∈I is the family of their axes. In this context, the injectivity radius is the length of the smallest relation, and the fellow traveling constant is the length of the largest piece between relators. The large injectivity radius assumption is empty because the Cayley graph of the free group is δ-hyperbolic for any δ > 0. The small fellow traveling constant assumption is (a strengthening of) the C (λ) small cancellation assumption. However, contrary to classical small cancellation, the constants A0 , λ0 in the small cancellation theorem below are not explicit and far from optimal. Graphical small cancellation also ﬁts in this context. In this case, the groups Ri ’s need not be cyclic any more, they are conjugates of the subgroups of G deﬁned by some labelled subgraphs.

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LECTURE 1. WHAT IS SMALL CANCELLATION ABOUT?

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The small cancellation theorem Theorem 1.3 (Small cancellation theorem). There exist A0 , λ0 such that if F satisﬁes the (A0 ,λ0 )-small cancellation hypothesis then (1) Ri |i ∈ I is a free product of a subfamily of the Ri ’s, (2) Stab(Qi )/Ri embeds in G/Ri |i ∈ I (3) small elements survive: for every C > 0, there exists AC , λc such that if F satisﬁes the (AC , λc )-small cancellation condition, then any non-trivial element whose translation length is at most Cδ is not killed in G/Ri . (4) G/Ri |i ∈ I acts on a suitable hyperbolic space. Remarks. In the setting of C (1/6) small cancellation, the groups Ri are conjugates of the cyclic groups generated by relators. Thus, if Qi is the axis of some conjugate r of a relator, then Stab(Qi ) is the maximal cyclic subgroup containing r. In particular, Stab(Qi )/Ri is trivial if r is not a proper power, and Stab(Qi )/Ri Z/kZ if r = uk for some u that is not a proper power. In (3), one can even prove that elements of translation length at most inj(F)(1− max{C1 λ, CA2 }) are not killed. It is diﬃcult to state right now the properties of the suitable hyperbolic space X in (4). One of the main goals of these lectures is to describe this space X. One can still say that one of its main properties is that X has a controlled geometry, including a controlled hyperbolicity constant. However, one can say more assuming that our initial space X is proper, and that the action of G is proper and cocompact (so that G is a hyperbolic group). If each Stab(Qi )/Ri is ﬁnite, and I/G is ﬁnite, then X is also proper with a proper cocompact action of G/Ri |i ∈ I so G/Ri |i ∈ I is also a hyperbolic group.

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LECTURE 2

Applying the small cancellation theorem Assume that we have a group G acting on a space X. We are going to see how to produce small cancellation moving families, and how to use them. 1. When the theorem does not apply Given a group G acting on a hyperbolic space, small cancellation families may very well not exist, except for trivial ones. A ﬁrst type of silly example is the solvable Baumslag-Solitar group BS(1, n) = a, t|tat−1 = an , n > 1. This group acts on the Bass-Serre tree of the underlying HNN extension, but there is no small cancellation family. Exercise 2.1. Prove this assertion. Note that any two hyperbolic elements of BS(1, n) share a half axis. Since we think of small cancellation families as a way to produce quotients, one major obstruction to the existence of such families occurs if G has very few quotients, for instance if it is simple. This is the case for the simple group G = Isom+ (Hn ) for example. If we restrict ourselves to ﬁnitely generated groups, an irreducible lattice in Isom+ (H2 ) × Isom+ (H2 ) acts on H2 (in two ways), but any nontrivial quotient is ﬁnite by the Margulis normal subgroup theorem [Mar91]. Similar, but more sophisticated examples include Burger-Mozes simple group [BM00], a lattice in the product of two trees viewed as a group acting on one of these two trees, or some Kac-Moody groups when the twin buildings are hyperbolic [CR09]. Exercise 2.2. What are trivial small cancellation families? Here are examples: (1) The empty family. (2) Take Q = {X} consisting of the single subspace X, and R = {N } consists of a single normal subgroup of G, (including the case N = {1} and N = G). (3) Another way is to take Q a G-invariant family of subspaces that satisfy the fellow traveling condition (for instance bounded subspaces), and take (RQ )Q∈Q a copy of the trivial group for each subspace. More generally, a trivial small cancellation family is a family such that Ri = {1} except for at most one index i. Prove that if G is simple, then there exists A, λ such that any (A, λ)-small cancellation moving family is trivial in the above sense. Hint: Consider a small cancellation moving family (Qi )i∈I , (Ri )i∈I . Since G is simple, Ri = Stab(Qi ) by the small cancellation Theorem. If h1 ∈ Ri1 \ {1}, h2 ∈ N Ri2 for i1 = i2 , prove that hN 1 h2 satisﬁes the WPD property below, contradicting that G is simple. 65 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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2. Weak proper discontinuity In hyperbolic groups, the easiest small cancellation family consists of the conjugates of a suitable power of a hyperbolic element. The proof is based on the properness of the action. In fact, a weaker notion, due to Bestvina-Fujiwara is suﬃcient. Preliminaries about quasi-axes Here we discuss the notion quasi-axes for hyperbolic elements g ∈ G. This is a g-invariant almost convex subset of X, that is quasi-isometric to R, with constants depending only on δ. One could deﬁne such a quasi-axis in terms of the boundary at inﬁnity of X, but because we don’t assume properness of X, we prefer avoiding this. To make many statements simpler, we will always assume that X is a metric graph, all of whose edges have the same length. Deﬁne [g] = inf{d(x, gx)|x ∈ X} the translation length of g. Recall that g is hyperbolic if the orbit map Z → X deﬁned by i → g i x is a quasi-isometric embedding (for some x, equivalently for any x). This occurs if and only the stable norm of g, deﬁned as g = limi→∞ 1i d(x, g i x) is not zero (the limit exists by subadditivity, and does not depend on x). These are closely related as [g] − 16δ ≤ g ≤ [g] [CDP90, 10.6.4]. In particular, if [g] > 16δ then g is hyperbolic. Consider a hyperbolic element g. Deﬁne the characteristic set of g as Cg = {x|d(x, gx) = [g]} (a non-empty set since X is a graph). We want to say that if [g] is large enough, Cg is close to being a bi-inﬁnite line (with constants independent of g). Given x ∈ Cg , consider the bi-inﬁnite path l = lx,g = ∪i∈Z [g i x, g i+1 x]. One easily checks that if y ∈ l, then d(y, gy) = [g], so l is contained in Cg . Moreover, l a local geodesic: any subsegment of length [g] is geodesic. By stability of 100δ-local geodesics [BH99, Th 1.13 p.405], there exists a constant C depending only on δ such that if [g] ≥ 100δ, lx,g and ly,g are at Hausdorﬀ distance at most C. Similar arguments show that if [g] ≥ 100δ, for any k, Cg and Cgk are at Hausdorﬀ distance at most C for some constant C depending only on δ. In this sense, if [g] ≥ 100δ, Cg is a good quasi-axis for g. If g is hyperbolic with [g] ≤ 100δ, then there is k such [g k ] ≥ 100δ, and a better quasi-axis for g would be Cgk (note that it is g-invariant). Finally, we want the quasi-axis to be almost convex. One easily checks that Cgk is 2C + 4δ-quasiconvex. Thus, we deﬁne the quasi-axis of g as Ag = Cg+2C+4δ where k is the smallest power of g such that k [g k ] ≥ 100δ. Lemma 2.3. There exists a constant C such that for all hyperbolic isometry g, for all x ∈ Ag and all i ∈ Z, ig ≤ d(x, g i x) ≤ ig + C. This follows from the fact that the quasi-axes Ag and Agi are at bounded Hausdorﬀ distance, and from the inequality [g i ] − 16δ ≤ g i = ig ≤ [g i ]. Weak proper discontinuity Definition 2.4. We say that g ∈ G, acting hyperbolically on X, satisﬁes the WPD property (for weak proper discontinuity) if there exists r0 such that for every pair of points x, y ∈ Ag at distance at least r0 , the set of all elements a ∈ G that move both x and y by at most 100δ is ﬁnite: #{a ∈ G|d(x, ax) ≤ 100δ, d(y, ay) ≤ 100δ} < ∞.

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LECTURE 2. APPLYING THE SMALL CANCELLATION THEOREM

67

Obviously, if the action of G on X is proper, then any hyperbolic element g satisﬁes the WPD property. In particular, any element of inﬁnite order in a hyperbolic group satisﬁes the WPD property. Here is an equivalent deﬁnition: Definition 2.5. g satisﬁes the WPD property if for all l, there exists rl such that for every pair of points x, y ∈ Ag at distance at least rl , the set of all elements a ∈ G that move both x and y by at most l is ﬁnite: #{a ∈ G|d(x, ax) ≤ l, d(y, ay) ≤ l} < ∞. Exercise 2.6. Prove that the deﬁnitions are equivalent. A lot of interesting groups have such elements. Example 2.7. (1) If G is hyperbolic or relatively hyperbolic, then any hyperbolic element satisﬁes the WPD property for the action of G on its Cayley graph if G is hyperbolic, or Bowditch’s space with horoballs. (2) If G is a non-cyclic right-angled Artin group that is not a direct product, then G acts on a tree in which there is an element satisfying the WPD property (see [DGO, cor. 6.50]). (3) If G is the mapping class group of a surface (of large enough complexity) acting on its curve complex, any pseudo-anosov element is a hyperbolic element satisfying the WPD property [BF07]. (4) If G = Out(Fn ), or G is the Cremona group Bir(P2 ), then G acts on a hyperbolic space with an element satisfying the WPD property [BF10, CL]. (5) If G acts properly on a proper CAT (0) space Y , and if g is a rank one hyperbolic element (its axis does not bound a half plane), there is an element satisfying the WPD property for some action of G on some hyperbolic space (see [Sis11], based on [BBF10]). Proposition 2.8. Assume that g satisﬁes the WPD property. Then for all A, λ, there exists N such that the moving family consisting of the conjugates of g N , together with their quasi-axes, satisﬁes the (A, λ)-small cancellation condition. Corollary 2.9. If G contains a hyperbolic element with the WPD property, then G is not simple. Exercise 2.10. Prove the proposition. Hints: First prove that there exists a constant Δ such that if Ag fellow travels with Ahgh−1 = hAg on a distance at least Δ, then hAg is at ﬁnite Hausdorﬀ distance from Ag . For this, show that if the fellow-traveling distance Δ(Ag , Ahgh−1 ) is large, there is a large portion of Ag that is moved by bounded amount by g i .hg ±i h−1 for many i’s. Then apply the WPD property to deduce that h commutes with some power of g, hence maps Ag at ﬁnite Hausdorﬀ distance. To conclude that this gives a small cancellation family, prove that the subgroup E(g) = {h ∈ G|dH (h.Ag , Ag ) < ∞} is virtually cyclic (where dH denotes the Hausdorﬀ distance). Now take N such that [g N ] is large compared to Δ(Ag , Ahgh−1 ), and such that g N E(g) (recall that the deﬁnition of a moving family requires the group Ri to be normal in the stabilizer of the corresponding space Qi ).

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Remark 2.11 (Remark about torsion). If G is not torsion-free, choosing N such that [g N ] is large compared to Δ(Ag , Ahgh−1 ) is not suﬃcient, as is shown by the exercise below. Exercise 2.12. Let F be a ﬁnite group, ϕ : F → F a non-trivial automorphism, of order d. Let G = (Z ϕ F ) ∗ Z =< a, b, F |∀f ∈ F, af a−1 = ϕ(f ) >. Let X be a Cayley graph of this group. Show that the family of conjugates of ak does not satisfy any small cancellation condition if k is not a multiple of d. 3. SQ-universality We will greatly strengthen Corollary 2.9 saying that G is not simple if it contains a hyperbolic element with the WPD property. Definition 2.13. A group G is SQ-universal1 if for any countable group A, there exists a quotient of G in which A embeds. Since there are uncountably many 2-generated groups, and since a given ﬁnitely generated group has only countably many 2-generated subgroups, a SQ-universal group has uncountably many non-isomorphic quotients. Theorem 2.14. If G is not virtually cyclic, acts on a hyperbolic space X, and contains a hyperbolic element satisfying the WPD property, then G is SQ-universal. The ﬁrst step in the proof consists in producing a free subgroup satisfying the small cancellation condition. Proposition 2.15. Assume that G is not virtually cyclic, acts on a hyperbolic space X, and contains a hyperbolic element h satisfying the WPD property. Then for all (A, λ), there exists H < G a free group of rank 2 and QH ⊂ X an H-invariant almost convex subset, so that (1) the conjugates of H and the corresponding translates of QH form a moving family satisfying the (A, λ)-small cancellation condition, and (2) the stabilizer of QH is H × F for some ﬁnite subgroup F . Exercise 2.16. Prove the proposition if G is torsion-free. Hint: prove that there is some conjugate k of h such that Δ(Ah , Ak ) is ﬁnite. Replace h, k by large powers so that their translation length is large compared to Δ(Ah , Ak ). The consider something like a = h1000 k1000 h1001 k1001 . . . h1999 k1999 , and b = h2000 k2000 h2001 k2001 . . . h2999 k2999 , and H = a, b. Note that such H might fail to satisfy the small cancellation condition in presence of torsion. Indeed, there may be some element of ﬁnite order that almost ﬁxes only half of the axis of a, so that Δ(Aa , Acac−1 ) might be large. The proof sketched in the exercise works if E(h) = Z × F for some ﬁnite subgroup F , and if F < E(k) for all conjugate k of h. To prove the proposition in full generality, one constructs h such that this holds, see [DGO, Section 6.2]. Proof of the Theorem. It is a classical result that every countable group embeds in a two generated group [LS01]. Thus it is enough to prove that any twogenerated group A embeds in some quotient of G. Let F2 → A be an epimorphism, 1 SQ

stands for subquotient

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LECTURE 2. APPLYING THE SMALL CANCELLATION THEOREM

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and N be its kernel. Let H < G be a free group of rank 2 satisfying the small cancellation hypothesis as in the proposition, and let QH ⊂ X be the corresponding subspace in the moving family. View N as a normal subgroup of H. We claim that N also satisﬁes the (A, λ)-small cancellation. Indeed, we assign the group gN g −1 to the subspace g.QH . For this to be consistent, we need N to be normal in Stab(QH ). This is true because Stab(QH ) = H × F . Applying the small cancellation theorem, we see that Stab(QH )/N embeds in G/N . It follows that A H/N embeds in G/N . 4. Dehn ﬁllings Let G be a relatively hyperbolic group with respect to a subgroup P (we assume that there is one parabolic group only for notational simplicity). By deﬁnition, this means that G acts properly on a proper hyperbolic space X with the following properties: there is a G-invariant family Q of disjoint, almost-convex horoballs in X; all the horoballs in Q are in the same orbit, and their stabilizers are the conjugates of P ; and G acts cocompactly on the complement of these horoballs. In fact, we can additionally assume that the distance between any two distinct horoballs is as large as we want, in particular, greater than 40δ. This means that the fellow traveling constant for Q is zero! Given R0 P , the family R of conjugates of R0 deﬁnes a moving family F = (R, Q). Now for the small cancellation theorem to apply, we need the injectivity radius to be large. This clearly fails since elements of R0 are parabolic, so their translation length is small. However, the following variant of the small cancellation theorem holds. In the small cancellation hypothesis, replace the large injectivity radius (asking that all points of Qi are moved a lot by each g ∈ Ri \ {1}), by the following one asking this only on the boundary of Qi : Theorem 2.17. Consider a moving family on a hyperbolic space with the notations above. There exists A0 a universal constant such that the following holds. Assume that Δ(Q) = 0 (the Qi ’s don’t come close to each other), and that ∀i ∈ I, ∀g ∈ Ri \ {1}, ∀x ∈ ∂Qi , d(x, gx) > A0 δ. Then the conclusion of the small cancellation theorem still holds, where Assertion (3) is modiﬁed as follows: any non-trivial element whose translation length is at most Cδ, and which is not contained in a conjugate of some Ri is not killed in G/Ri . Let ∗ be a base point on the horosphere ∂Q preserved by P . Since P acts cocompactly on ∂Q, consider r > 0 such that the P -orbit of B(∗, r) contains ∂Q. Now if R0 avoids the ﬁnite set S ⊂ P of all elements g ∈ P \ {1} such that d(∗, g∗) ≤ 2r + A then R0 satisﬁes this new assumption. We thus get the Dehn ﬁlling theorem: Theorem 2.18 ([Osi07, GM08]). Let G be hyperbolic relative to P . Then there exists a ﬁnite set S ⊂ P \ {1} such that for all R0 P avoiding S, • P/R0 embeds in G/R0 • G/R0 is hyperbolic relative to P/R0 . In particular, if R0 has ﬁnite index in P , then G/R0 is hyperbolic.

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In fact, the proof allows us to control the hyperbolicity constant of the hyperbolic space on which the quotient group acts. This can be a very useful property.

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LECTURE 3

Rotating families 1. Road-map of the proof of the small cancellation theorem The goal of the remaining two lectures is to prove the geometric small cancellation theorem. There are essentially two main steps in the proof, each step involving only one of the two main hypotheses. (1) Construct from the space X and the subspaces Qi a cone-oﬀ X˙ by coning all the subspaces Qi , and prove its hyperbolicity. This step does not involve the groups Ri , so this is independent of the large injectivity radius hypothesis. (2) Because the spaces Qi have been coned, each subgroup Ri ﬁxes a point in ˙ and thus looks like a rotation. Our moving family becomes a rotating X, family. One studies the normal group N = (Ri )i∈I via its action on the cone-oﬀ. This is where the large injectivity radius assumption is used: it translates into a so-called very rotating assumption saying somehow that every non-trivial element of Ri rotates by a large angle. The group ˙ G/N naturally acts on the quotient space X/N , and the hyperbolicity of ˙ the quotient space X/N is then easy to deduce. In this lecture, we discuss the second step which involves the study of rotating families. 2. Deﬁnitions Consider a group G acting on a δ-hyperbolic space X. Definition 3.1. A rotating family is a collection {Rc , c ∈ C} of subgroups of G indexed by a subset C ⊂ X such that • Rc ﬁxes c for all c ∈ C • C is G-invariant • and ∀g ∈ G, ∀c ∈ C, Rgc = gRc g −1 . One says that the rotating family is ρ-separated if any two distinct points in C are at distance at least ρ. The set C is called the set of apices of the family, and the groups Rc are called the rotation subgroups of the family. Note that this deﬁnition implies that Rc is a normal subgroup of the stabilizer Stab(c) of c ∈ C. Let us reformulate this deﬁnition. Start with a group G, and consider R1 ,. . . ,Rk some subgroups of G. The goal is to understand the quotient G/R1 , ..., Rk . We assume that every Ri ﬁxes a point ci such that Ri is a normal subgroup of the stabilizer of ci , and that ci is not in the G-orbit of cj for i = j. Then one gets 71 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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a rotating family by putting C = G.{c1 , . . . , ck }, and by deﬁning {Rc , c ∈ C} by Rgci = gRi g −1 . The fact that there no ambiguity in this deﬁnition is a consequence of the fact that Ri is normal in the stabilizer of ci . The following deﬁnition formalizes the fact that every non-trivial element of Ri rotates by a large angle. Definition 3.2 (Very rotating condition: local version). We say that the rotating family is very rotating if the following holds. Consider c ∈ C, g ∈ Rc \ {1}, and x, y ∈ B(c, 40δ)\B(c, 20δ). If d(x, y) ≤ d(x, c)+d(c, y)−10δ, then any geodesic between x and gy contains c. Intuitively, the very rotating condition says that if x, c, y make a small angle at c, and g ∈ Rc \ {1}, then x, c, gy makes a large angle at c. This somehow means that g rotates by a large angle. This is for instance the case if X is CAT (−1), and if for all x ∈ X \ {c}, the geodesics [c, x], [c, gx] make an angle of at least 2π (see Lemma 4.8). For exposition reasons, the deﬁnition above is slightly diﬀerent from the one in [DGO], but this does not change the nature of the results. The very rotating condition is local around an apex. It implies the following global condition. This shows in particular that Rc acts freely and discretely on X \ B(c, 20δ). Lemma 3.3 (Very rotating condition: global version). Consider c ∈ C, and x, y ∈ X at distance at least 20δ from c such that d(x, y) ≤ d(x, c) + d(c, y) − 22δ. Then for any g ∈ Rc \ {1}, any geodesic between x and gy contains c. In particular, for any choice of geodesics [x, c], [c, gy], their concatenation [x, c]∪[c, gy] is geodesic. Proof. To unify notations, write x1 = x, x2 = y. For i ∈ {1, 2}, let pi , qi ∈ [c, xi ] be such that d(pi , c) = 20δ, and d(qi , c) = 11δ. By thinness of a triangle with vertices c, x1 , x2 , d(q1 , q2 ) ≤ 4δ. In particular, d(p1 , p2 ) ≤ d(p1 , q1 )+4δ +d(q2 , p2 ) = d(p1 , c) + d(c, p2 ) − 18δ. The local very rotating hypothesis says that d(p1 , c) + d(c, gp2 ) = d(p1 , gp2 ). Consider any geodesic [x1 , gx2 ], and let p1 , q1 , gq2 , gp2 ∈ [x1 , gx2 ] be such that d(pi , xi ) = d(pi , xi ) and d(qi , xi ) = d(qi , xi ). By the triangle inequality, d(pi , c) ≥ d(pi , c) = 20δ. By thinness of the triangle x1 , c, gx2 , d(qi , qi ) ≤ 4δ, so d(q1 , q2 ) ≤ 12δ, and d(p1 , p2 ) ≤ d(p1 , q1 ) + 12δ + d(p2 , q2 ) = 30δ. Thus, the local very rotating condition applies to the points p1 , p2 , and we get that [p1 , gp2 ] contains c, and so does [x1 , gx2 ]. 3. Statements Now we state some results describing the structure of the normal subgroup generated by the rotating family. Theorem 3.4. Let (Rc )c∈C be a ρ-separated very rotating family, with ρ large enough compared to the hyperbolicity constant δ. Let N = Rc |c ∈ C. Then (1) Stab(c)/Rc embeds in G/N . More generally, if [g] < ρ and g ∈ N then g ∈ Rc for some c ∈ C, (2) there exists a subset S ⊂ C such that N is the free product of the collection of (Rc )c∈S , and (3) X/N is hyperbolic. Remark 3.5. ρ ≥ 120δ is enough for the ﬁrst two assertions. For the last one, we need it to be large enough to apply the Cartan-Hadamard theorem (Theorem 3.11 below), see Proposition 3.12 below.

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LECTURE 3. ROTATING FAMILIES

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The ﬁrst assertion follows from the following form of the Greendlinger Lemma which we are going to prove together with the theorem. The classical Greendlinger Lemma says that if a cyclically reduced word w represents the trivial element in a small cancellation group, then it has a subword u that is a subword of a relator r with |u| > |r|/2, thus w can be shortened by replacing u by the inverse of the rest of the relator. This is the basis of Dehn’s algorithm for solving the word problem in a small cancellation group. Theorem 3.6. (Greendlinger Lemma) Every element g in N that does not lie in any Rc is loxodromic in X, it has a g-invariant geodesic line l, this line contains a point c ∈ C such that there is a shortening element at c in l (as deﬁned below). Definition 3.7. Let l be a geodesic, and c ∈ l. A shortening element at c in l is an element r ∈ Rc \ {1} such that if q1 , q2 are the two points in l at distance 20δ from c, then d(q1 , rq2 ) ≤ 10δ. Assume that l is a g-invariant geodesic line, and that there is a shortening element r at c ∈ l. Then up to exchanging the roles of q1 and q2 , we can assume that q1 , q2 , gq1 are aligned in this order in l. Since d(q1 , gq1 ) = [g], we get [gr] ≤ d(r −1 q1 , gq1 ) ≤ d(r −1 q1 , q2 ) + d(q2 , gq1 ) ≤ 10δ + d(q1 , gq1 ) − d(q1 , q2 ) = [g] − 30δ, so [gr] ≤ [g] − 30δ. Thus, Greendlingers’s Lemma gives a form of (relative) linear isoperimetric inequality: every element g of N is the product of at most [g]/30δ elements of the rotation subgroups. 4. Proof of Theorem 3.4 The proof is by an iterative process, described by Gromov in [Gro01b], see [DGO]. To perform it, we construct inductively a sequence of subsets called windmills with a set of properties that remain true inductively (see Deﬁnition 3.8 and Figure 1). To each windmill W ⊂ X, we associate the group GW generated by {Rc |c ∈ W }. As the windmills we construct are going to exhaust X (see Proposition 3.9), the groups GW will exhaust the normal subgroup N = Rc |c ∈ C. Definition 3.8 (Windmill). A windmill is a subset W ⊂ X satisfying the following axioms. (1) (2) (3) (4)

W is almost convex, W +40δ ∩ C = W ∩ C = ∅, The group GW generated by c∈W ∩C Rc preserves W , There exists a subset SW ⊂ W ∩ C such that GW is the free product ∗c∈SW Rc . (5) (Greendlinger) Every elliptic element of GW lies in some Rc , c ∈ W ∩ C, other elements of Rc have an invariant geodesic line l such that l ∩ C contains a point at which there is a shortening element (as in Deﬁnition 3.7).

To initiate this inductive process, we choose c ∈ C, and we take as initial windmill W0 = {c} (we could also choose for instance W0 = B(c, r) with r ≤ ρ − 50δ)).

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W

c3

c2

c1

W1

W2

Figure 1. A windmill. The iterative step starts with a windmill W , constructs W1 , W2 , and the end of the iterative step, the new windmill W is a thickening of W2 . Proposition 3.9 (Inductive procedure). For any windmill W , there exists a windmill W containing W +10δ and W +60δ ∩C, and such that GW = GW ∗(∗x∈S Rc ) for some S ⊂ C ∩ (W \ W ). Proof of Theorem 3.4 from Proposition 3.9. Starting from W0 = {c}, deﬁne Wi+1 from Wi by applying the proposition. Since ∪i Wi = X, ∪i GWi = N . The Greendlinger Lemma follows, and so does the fact that N is a free product. Proof of Proposition 3.9. If W +60δ does not intersect C, we just inﬂate W by taking W = W +10δ . Otherwise, we construct W in several steps. Step 1. Let C1 = C ∩ (W +60δ \ W ). For each c ∈ C1 choose a projection pc of c on W , and a geodesic [c, pc ]. This choice can be done GW -equivariantly because GW acts freely on C1 (by Greendlinger hypothesis, and the very rotating assumption). Deﬁne W1 = W ∪ c∈C1 [c, pc ]. Almost convexity of W easily implies that W1 is 12δ-quasiconvex. Note for future use that for any c ∈ C1 , W1 \ [c, pc ] is also 12δ-quasiconvex for the same reason. Since C is ρ-separated with ρ > 112δ, any point in C \ {c} is at distance at least 52δ from [c, pc ]. This implies that W1+52δ ∩ C = W1 ∩ C. Step 2. The group G = Rc |c ∈ W1 is the group generated by GW and by {Rc |c ∈ C1 }. We deﬁne W2 = G .W1 . Abstract nonsense shows that G = Rc |c ∈ W2 . Step 3. We take W = W2+12δ = G .(W1+12δ ). Let us check that W satisﬁes Axiom 2 of a windmill. We have W +40δ = +40δ +52δ W2 . Since W1+52δ ∩ C = W1 ∩ C, we get W ∩ C = W2 ∩ C ⊂ W ∩ C. Axiom 2 follows. It also follows that G = GW , and that W is GW -invariant so Axiom 3 follows.

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LECTURE 3. ROTATING FAMILIES

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To prove that W satisﬁes the other axioms, we ﬁrst look at how W1 is rotated around some c ∈ C1 . So take h ∈ Rc \ {1}, and look at W1 ∪ hW1 . Consider x ∈ W1 \ [c, pc ] and y ∈ h(W1 \ [c, pc ]). Consider qx ∈ [c, x] and qx ∈ [c, pc ], both at distance 20δ from c. Deﬁne qy ∈ [c, y] and qy ∈ [c, hpc ] similarly. By thinness of the triangle c, x, pc , d(qx , qx ) ≤ 4δ: otherwise, there would be some qx ∈ [pc , x] such that d(qx , qx ) ≤ 4δ, and by 12δ-quasiconvexity of W1 \ [c, pc ], d(qx , W1 \ [c, pc ]) ≤ 16δ, so d(c, W1 \ [c, pc ]) ≤ 36δ, a contradiction. Similarly, d(qy , qy ) ≤ 4δ, and since h−1 qy = qx , d(qx , h−1 qy ) ≤ 8δ. The global very rotating condition implies that any geodesic from x to y contains c. We note that h is a shortening element of [x, y] at c. We have proved: Lemma 3.10 (Key lemma). Fix c ∈ C1 , h ∈ Rc \ 1, x ∈ W1 \ [c, pc ] and y ∈ h(W1 \ [c, pc ]). Then any geodesic from x to y contains c and h is a shortening element of [x, y] at c. By 12δ-quasiconvexity of W1 and hW1 , we get that for any x, y as in the lemma, [x, y] is in the 12δ-neighbourhood of W1 ∪ hW1 . If x lies in [c, pc ] or y lies in h[c, pc ], we get similarly that [x, y] lies in the 12δ-neighbourhood of W1 ∪ hW1 so W1 ∪ hW1 is 12δ-quasiconvex. Now we prove that W2 has a tree-like structure (see Figure 1). Recall that W2 = G .W1 . Let Γ be the graph with vertex set V = VC (Γ) VW (Γ), where VC (Γ) is the set of apices in G .C1 , and VW (Γ) is the set of translates of W under G . We put an edge between gW and gc for any g ∈ G and c ∈ C1 . Rc1 Rc2 GW

Rc3 .. .

Rc4 .. .

Figure 2. The graph of groups deﬁning the tree T , where c1 , c2 , . . . is an enumeration of S1 Since C1 is GW -invariant, we consider S1 ⊂ C1 a set of representatives of ˆ = GW ∗ C1 /GW (note that S1 may be inﬁnite). We deﬁne the free product G (∗c∈S1 Rc ), viewed as a tree of groups with trivial edge groups as in the Figure 2. Let T be the corresponding Bass-Serre tree. We denote by uW ∈ T the vertex stabilized by W , and for each c ∈ S1 , we denote by uc ∈ T the vertex stabilized by Rc . Denote by VW (T ) the set of vertices of T in the orbit of uW , and VC (T ) the set of other vertices of T . Any edge of T has an endpoint in VW (T ) and the other endpoint in VC (T ). The inclusions GW ⊂ G and Rc ⊂ G induce an epimorphism ˆ → G , and there is a natural ϕ-equivariant map f : T → Γ sending uW to ϕ:G W , and uc to c for all c ∈ S1 . We prove that ϕ and f are isomorphisms. We ﬁrst note that the key lemma implies that f does not identify any pair of vertices in VW (T ) at distance 2 from each other. Next, we claim that f is injective on the set of vertices adjacent to uW . Otherwise, there are c, c ∈ S1 and g, g ∈ GW such that f (guc ) = f (g uc ), i.e. gc = g c .

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Since S1 is a set of representatives for the orbits of GW , this implies that c = c , and the element g −1 g ﬁxes c. As noted above, the Greendlinger Axiom 5 for W implies that GW acts freely on C1 so g −1 g = 1. This proves the claim, and shows more generally that f does not identify any pair of vertices in VC (T ) at distance 2 from each other. We now claim that f does not identify any pair of points u = u ∈ VC (T ). Consider the segment [u, u ]T ⊂ T , and let u = u1 , u2 , . . . , un = u be the points in [u, u ]T ∩ VC (T ). Let ci ∈ C be the image of ui under f . Consider the path γ in X deﬁned as a concatenation of geodesics [c1 , c2 ]X , [c2 , c3 ]X , ..., [cn−1 , cn ]X . The key lemma (applied around c2 ) shows that any geodesic from c1 to c3 contains c2 so in particular, γ3 = [c1 , c2 ] ∪ [c2 , c3 ] is a geodesic and there is a shortening element at c2 for [c1 , c2 ] ∪ [c2 , c3 ]. Similarly, [c2 , c3 ] ∪ [c3 , c4 ] is geodesic, and there is a shortening element at c3 for [c2 , c3 ] ∪ [c3 , c4 ]. Then the global very rotating condition at c3 applies to γ3 ∪ [c3 , c4 ] and shows that γ3 ∪ [c3 , c4 ] is geodesic. By induction, we get that γ is geodesic so c1 = cn hence f (u) = f (u ), which proves our claim. Finally, a similar argument shows that f is injective in restriction to VW (T ). Indeed, if gW = g W ∈ VW (T ), consider a path of the form [x, c1 ]X .[c1 , c2 ]X . . . [cn , y]X where x ∈ gW , y ∈ g W and {c1 , . . . , cn } = [gW, g W ]T ∩ VC (T ). The argument above shows that this path is geodesic. It follows that f is injective. Injectivity of ϕ follows since an element of ker ϕ has to ﬁx T pointwise, and is therefore trivial. Since f and ϕ are obviously onto, they are isomorphisms. This proves that G can be written as a free product as in the Proposition, and that W satisﬁes Axiom 4. The paths [x, c1 ]X .[c1 , c2 ]X . . . [cn , y]X considered above also have shortening pairs at ci . The very rotating condition implies that any geodesic segment between x and y has to contain ci and is therefore of this form. Since W1 is 12δ-quasiconvex, it follows that so is W2 . It follows that W is almost convex, and Axiom 1 holds. The Greendlinger Axiom is similar: if g ∈ GW is elliptic in the tree T , there is nothing to prove because W is a windmill. If g is hyperbolic in T , its axis contains a vertex in u ∈ VC (T ). Let u = u1 , u2 , . . . , un = gu be the points in [u, gu]T ∩ VC (T ), and let ci ∈ C be the image of ui under f . Then the g-translates of [c1 , c2 ]X .[c2 , c3 ]X . . . [cn−1 , cn ]X form a g-invariant bi-inﬁnite geodesic, and there is a shortening element in at each ci . 5. Hyperbolicity of the quotient The goal of this section is to prove the hyperbolicity of the quotient space X/N . We will prove local hyperbolicity, and use the Cartan-Hadamard Theorem. 5.1. The Cartan Hadamard Theorem The Cartan-Hadamard theorem allows us to deduce global hyperbolicity from local hyperbolicity, see [DG08], and more detailed account in Coulon’s notes [Cou], (see also [OOS09, Th 8.3]). Theorem 3.11 (Cartan-Hadamard Theorem). There exist universal constants C1 , C2 such that the following holds. Consider a geodesic space Y and some δ > 0 such that • Y is C1 δ-locally δ-hyperbolic, and

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LECTURE 3. ROTATING FAMILIES

77

• Y is 32δ-simply connected. Then Y is (globally) C2 δ-hyperbolic. We will denote by RCH (δ) = C1 δ and δCH (δ) = C2 δ. The assumption that Y is locally δ-hyperbolic asks that for any subset {a, b, c, d} ⊂ Y whose diameter is at most RCH (δ), the 4-point inequality holds: d(a, b) + d(c, d) ≤ max{d(a, c) + d(b, d), d(a, d) + d(b, c)} + 2δ. The assumption that Y is 32δ-simply connected means that the fundamental group of Y is normally generated by free homotopy classes of loops of diameter at most 32δ. Equivalently, one may ask that the Rips complex P32δ (Y ) is simply connected. We will apply this Theorem to Y = X/N . Note that since X is δ-hyperbolic, X is 4δ-simply connected [CDP90, Section 5, prop. 1.1]. Since N is generated by isometries ﬁxing a point, X/N is also 4δ-simply connected. Indeed, let γ be a loop in X/N . Lift it to γ in X, joining x to gx with g ∈ N . Write g = gn ...g1 with gi ﬁxing a point. One can homotope γ rel endpoints to ensure that γ contains a ﬁxed point c of gn (just insert a path and its inverse). Then γ = γ1 .γ2 where the endpoint of γ1 and the initial point of γ2 are c. Downstairs, this is gives a homotopy. Now change γ2 to gn−1 γ2 . Downstairs, this does not change the path. The new path γ1 .gn−1 γ2 joins x to gn−1 . . . g1 x.Repeating, we can assume g = 1 where pi is a path with so that γ is a loop in X. By hypothesis, γ = i pi li p−1 i origin at x, and li is a loop of diameter at most 4δ. Projecting downstairs, we get the same property for the projection. Thus, in view of the Cartan-Hadamard Theorem, it is enough to prove local hyperbolicity of the quotient. 5.2. Proof of local hyperbolicity We will only prove the proposition in the particular case where X is a cone-oﬀ of radius ρ (see Corollary 4.3 in the next lecture). The main simpliﬁcation is that in this case, the neighbourhood of an apex is a hyperbolic cone over a graph, and so is its quotient. Thus we can apply Proposition 4.6 saying that such a hyperbolic cone is locally 2δH2 -hyperbolic, where δH2 is the hyperbolicity constant of H2 . Proposition 3.12. Under the assumptions of 3.4, assume that X and the rotating family are obtained by coning-oﬀ a small cancellation moving family, as described in the next section, where ρ is the radius of the cone-oﬀ. We denote by δ the hyperbolicity constant of X, N be the normal group generated by the rotating family, X = X/N , and C the image of C in X/N . Let δ = max{δ, δH2 }, and assume that ρ ≥ 10 max{RCH (δ )}. Then (1) for each apex c ∈ C, the 9ρ/10-neighbourhood c in X is ρ/10-locally 2δH2 hyperbolic, (2) the complement of the 8ρ/10-neighbourhood C in X is ρ/10-locally δhyperbolic. In fact, any subset of diameter at most ρ/10 in X \ C +8ρ/10 isometrically embeds in X/N , and (3) X/N is δCH (δ )-hyperbolic. In the case where our rotating family is not obtained by coning-oﬀ, one can also prove that X/N is locally hyperbolic with worse constants, see [DGO]. Proof. The ﬁrst assertion is a direct consequence of the fact that the hyperbolic cone over a graph is 2δH2 -hyperbolic (Proposition 4.6).

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For the second assertion, let E ⊂ X \ C +8ρ/10 be a subset of diameter at most ρ/10, and let E be its ρ/10-neighbourhood. We claim that E injects into X, so that E isometrically embeds in X. Now assume on the contrary that there are x, y ∈ E and g ∈ N \ {1} such that y = gx. In particular [g] < ρ, so by Assertion 1 of Theorem 3.4, g ∈ Rc for some c ∈ C. Then the very rotating condition implies that any geodesic [x, y] contains c, so d(x, c) ≤ 3ρ/10, a contradiction. To conclude, we have shown that X/N is ρ/10-locally δ -hyperbolic with δ = max{δ, δH2 }. Since X/N is 4δ-simply connected, and since ρ > 10RCH (δ ), the Cartan-Hadamard Theorem says that X/N is globally δCH (δ )-hyperbolic. 6. Exercises Exercise 3.13. Assume that ρ δ. Let E ⊂ X be an almost convex subset, and assume that E does not intersect the ρ/10-neighbourhood of C. Prove that E isometrically embeds in X/N . Hint: prove that any subset of E of diameter ρ/100 isometrically injects in X/N . Then say that a ρ/100-local geodesic in X/N is close to a global geodesic. Exercise 3.14. Assume that G is torsion-free, and that for all c, Stab(c)/Rc is torsion-free. Prove that G/N is torsion-free. Hint: use the fact that an elliptic isometry of a δ-hyperbolic space has an orbit of diameter at most 16δ. Then given g ∈ G/N of ﬁnite order, look for a lift in G with smallest translation length.

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LECTURE 4

The cone-oﬀ 1. Presentation The goal of this section is, given a hyperbolic space X and a family Q of almost convex subspaces, to perform a coning construction of these subspaces, thus obtaining a new hyperbolic space X˙ called the cone-oﬀ space. The eﬀect of this operation is to transform a small cancellation moving family on X into a very rotating family ˙ on this new space X. This construction has been introduced by Gromov in [Gro01a, Gro03], and further developed by Gromov, Delzant and Coulon [DG08, Cou11, Cou]. A construction of this type was introduced before by Bowditch in the context of relatively hyperbolic groups (with cone points at inﬁnity). See also Farb’s and Groves-Manning’s constructions [Far98, GM08]. We follow [Cou11], with minor modiﬁcations and simpliﬁcations. For simplicity we assume that X is a metric graph, all of whose edges have the same length. This is no loss of generality: if X is a length space, the graph Y with vertex set X where one connects x to y by an edge of length l if d(x, y) ≤ l satisﬁes ∀x, y ∈ X, dX (x, y) ≤ dY (x, y) ≤ dX (x, y) + l. We don’t assume that X is locally compact. Topologically, we are going to cone a family Q of subgraphs (see Figure 1), and to put a geometry by identifying the added triangles with sectors of H2 of ﬁxed radius ρ (see Section 2 for details). Thus ρ is a parameter of this construction, to be chosen.

X˙

X Qi

Figure 1. The cone-oﬀ. The assumptions will be that X is δ-hyperbolic with δ very small, and that we have a family Q of almost convex subspaces having a small fellow traveling length. The features of the resulting space will be as follows: (1) X˙ is a hyperbolic space, whose hyperbolicity constant is good (meaning: a universal constant; in particular, ρ can be chosen to be very large compared to this hyperbolicity constant). 79 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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VINCENT GUIRARDEL, GEOMETRIC SMALL CANCELLATION

(2) If a group G acts on X, preserving Q, and if to each Q ∈ Q corresponds a group RQ (in an equivariant way) preserving Q and with suﬃciently large ˙ injectivity radius, then (RQ )Q∈Q is a very rotating family on X. With quantiﬁers, the main result of this section will be: Theorem 4.1. There exist constants δc , Δc , ρ0 , δU as follows. If X is δc hyperbolic, and Q is a family of almost convex subspaces such that Δ(Q) ≤ Δc , then for all ρ ≥ ρ0 , the corresponding cone-oﬀ X˙ satisﬁes: (1) X˙ is R-locally (2δH2 )-hyperbolic (where δH2 is the hyperbolicity constant of H2 , and R = RCH (2δH2 ) is the constant required by the Cartan-Hadamard theorem). (2) It is globally δU -hyperbolic (with δU = δCH (2δH2 )). (3) If F = (RQ )Q∈Q is a moving family whose injectivity radius is at least ˙ 2π sinh(ρ), then (RQ )Q∈Q is a 2ρ-separated very rotating family on X. Note that the hyperbolicity constant δU of X˙ does not depend on X or ρ. In fact, the geometry of the cone-oﬀ is even nicer than this δU -hyperbolicity. Indeed, this space is CAT (−1, ε), meaning in a precise sense “almost CAT (−1)”. This property introduced in [Gro01a] implies hyperbolicity with a hyperbolicity constant close to δH2 , but gives in particular a much better control of bigons than in a standard δH2 -hyperbolic space (at least when ε is small enough). We will not discuss this property here. ˙ It is important that ρ is large compared to the hyperbolicity constant δU of X, in particular to apply the theorem about rotating families. We have the freedom to do so in Theorem 4.1 since δU is independent of ρ. The hypotheses on X in the theorem can be achieved by rescaling the metric if the fellow traveling constant Δ(Q) of Q is ﬁnite. However, if F = (RQ )Q∈Q is a moving family, this rescaling scales down the injectivity radius accordingly. In order to get the very rotating condition on the cone-oﬀ, Assertion 3 of Theorem 4.1 requires RQ to have a large injectivity radius after rescaling. To achieve this, the initial injectivity radius has to be large compared to the initial hyperbolicity constant and the initial fellow traveling constant. This is exactly what the small cancellation hypothesis asks for. Corollary 4.2. For any ρ ≥ ρ0 , there exists Aρ , λρ > 0 such that if (RQ )Q∈Q is an (Aρ , λρ )-small cancellation moving family on X, then (RQ )Q∈Q is a 2ρseparated very rotating family on X˙ α , the cone-oﬀ of radius ρ of Xα , where Xα is the rescaling of X by a factor α > 0. Proof. One can take Aρ =

2π sinh(ρ) δc

and λρ =

Δc 2π sinh(ρ) .

Indeed, if Xα is

the space X where the metric is multiplied by the factor α = min{ δδc , ΔΔc }, then Theorem 4.1 applies to Xα . Moreover, if (RQ )Q∈Q satisﬁes the (Aρ , λρ )-small cancellation condition, it acts on Xα with injectivity radius at least 2π sinh ρ. Assertion 3 of Theorem 4.1 implies that (RQ )Q∈Q acts on the cone-oﬀ X˙ α of Xα as a 2ρ-separated very rotating family.

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LECTURE 4. THE CONE-OFF

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Let ρU ≥ ρ0 be such that Theorem 3.4 about rotating families applies to any 2ρU -separated very-rotating family on a δU -hyperbolic space (where δU is the hyperbolicity constant of the cone-oﬀ given by Theorem 4.1). Applying Corollary 4.2 to this value of ρ, we get: Corollary 4.3. There exists A0 , λ0 > 0 such that if F = (RQ )Q∈Q is an (A0 , λ0 )-small cancellation moving family on X, then (RQ )Q∈Q is a very rotating family on X˙ α , the cone-oﬀ of radius ρU of a rescaled version of X. It is now easy to deduce the small cancellation Theorem 1.3. Proof of the small cancellation Theorem 1.3. All the assertions follow immediately from Corollary 4.3 and Theorem 3.4, except maybe Assertion (3) saying that given C > 0, elements with a translation length at most Cδ survive in the quotients if the small cancellation constants are good enough. Greendlinger’s Lemma says that any element g ∈ G \ {1} with translation length less than 2ρ in X˙ α survives, except if contained in some group of our small cancellation moving family F. But if we take A > C, the small cancellation assumption says that elements of Ri \ {1} act on X with translation length greater that Aδ ≥ Cδ. Now given C > 0, consider ρ such that Cδc < 2ρ. Assume that our moving family F satisﬁes the (Aρ , λρ )-small cancellation condition with Aρ , λρ as in Corollary 4.2. Let X˙ α be the space given by this corollary. Then any element acting on X with translation length at most Cδ, acts on the spaces Xα and X˙ α with translation length at most Cδc < 2ρ, and is therefore not killed in the quotient. 2. The hyperbolic cone of a graph Given ρ > 0, and α ∈ (0, π), consider a hyperbolic sector of radius ρ and angle α in H2 . The arclength of its boundary arc of circle is l = α sinh ρ. If α ≥ π, one can still deﬁne a hyperbolic sector of angle α by gluing several sectors of angle less than π. ρ α

l = α sinh ρ

ρ

If Q is a metric graph, all whose edges have length l, the hyperbolic cone over Q is the triangular 2-complex C(Q) = ([0, ρ] × Q)/∼ where ∼ is the equivalence relation that collapses Q × {0} to a point. The cone point c = Q × {0} is also called the apex of C(Q). We deﬁne a metric on each 2-cell of C(Q) by identifying it with the hyperbolic sector of radius ρ and arclength l. We identify Q with Q × {ρ}, but we distinguish the original metric dQ from the new metric dC(Q) . If we want to emphasize the dependance in ρ, we will denote the cone by Cρ (Q). For t ∈ [0, ρ], x ∈ Q we denote by tx the image of (t, x) in C(Q). There is an explicit formula for the distance in C(Q) [BH99, Def 5.6 p.59]: dQ (x, x ) cosh d(tx, t x ) = cosh t cosh t − sinh t sinh t cos min π, . sinh(ρ)

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VINCENT GUIRARDEL, GEOMETRIC SMALL CANCELLATION

This formula allows one to deﬁne the hyperbolic cone over any metric space. We shall not use it directly (except in the proof of Fact 4.14). Instead, we will use the following basic facts. Proposition 4.4 ([BH99, Chap I.5, Prop. 5.10]). (1) For each x ∈ Q, the radial segment {tx| t ∈ [0, ρ]} is the only geodesic joining c to x; (2) For each x, y ∈ Q such that dQ (x, y) ≥ π sinh ρ, then for any t, s ∈ [0, ρ] the only geodesic joining tx to sy is the concatenation of the two radial segments [tx, c] ∪ [c, sy]. (3) For each x, y ∈ Q such that dQ (x, y) < π sinh ρ, and all s, t ∈ (0, ρ], there is a bijection between the set of geodesics between x and y in Q and the set of geodesics between tx and sy in C(Q). None of these geodesics go through c. The map C(Q) \ {c} → Q deﬁned by tx → x is called the radial projection. Exercise 4.5. Prove that the radial projection is locally Lipschitz. Note that it is not globally Lipschitz in general. Prove that the local Lipschitz constant tends to 1 as one gets closer to Q: for each ε > 0, there exists t0 < ρ such that for each tx ∈ C(Q), and with t ≥ t0 , there exists a neighbourhood U of tx in C(Q) such that the restriction of the radial projection on U is (1 + ε)-Lipschitz. The hyperbolic cone on a tripod is CAT (−1) because it is obtained by gluing CAT (−1) spaces over a convex subset. It follows that the cone over a tree is CAT (−1) since by Proposition 4.4, any geodesic triangle is contained in the cone over a tripod. This extends to the hyperbolic cone over an R-tree which can be deﬁned by writing the R-tree as an increasing union of metric trees (with edges of varying lengths), or by the distance formula above. One can also view this fact as a particular case of Beretosvkii’s theorem saying that, writing κ = π sinh ρ, the hyperbolic cone of radius ρ over any CAT (κ)-space, is CAT (−1) [BH99, Chap I.5, Th 3.14]. In particular, the hyperbolic cone over an R-tree is δH2 -hyperbolic. Proposition 4.6. The hyperbolic cone of any radius, over any graph, is 2δH2 hyperbolic. This is analogous to the hyperbolicity of a Groves-Manning combinatorial horoballs [GM08]. Remark 4.7. We don’t want to assume local compactness of Q. The fact that Q is a graph whose edges have the same length is used to ensure that Q and C(Q), (and later the cone-oﬀ) are geodesic spaces. Indeed, a theorem by Bridson shows that any connected simplicial complex whose cells are isometric to ﬁnitely many convex simplices in Hn , and glued along their faces using isometries, is a geodesic space [BH99, Th 7.19]. This can be easily adapted to our situation where 2-cells are all isometric to the same 2-dimensional sector. The following very simple proof is due to Coulon. Proof. Let C be such a cone, and c its apex. One checks the hyperbolic 4point inequality: given x, y, z, t ∈ C, we want to prove that one of the following inequalities holds L : xy + zt ≤ xz + yt + 4δH2

R : xy + zt ≤ xt + yz + 4δH2

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LECTURE 4. THE CONE-OFF

83

(where we use the notation xy = d(x, y)). Since any 3-point set is isometric to a subset of a tree, and since the cone over a tree is CAT (−1), for any 3 points u, v, w ∈ C, we know that u, v, w, c satisfy the δH2 -hyperbolic 4-point inequality. Consider the inequalities Lx : cy + zt ≤ cz + yt + 2δH2 , Rx : cy + zt ≤ ct + yz + 2δH2 Ly : cx + zt ≤ ct + xz + 2δH2 , Ry : cx + zt ≤ cz + xt + 2δH2 Lz : ct + xy ≤ cx + yt + 2δH2 , Rz : ct + xy ≤ cy + xt + 2δH2 Lt : cz + xy ≤ cy + xz + 2δH2 , Rt : cz + xy ≤ cx + yz + 2δH2 . We know that for each u ∈ {x, y, z, t}, either Lu or Ru holds. If Lx and Lt hold, then summing, we see that L holds. Similarly, assuming that neither L nor R holds, we get Lx ⇒ ¬Lt ⇒ Rt ⇒ ¬Ry ⇒ Ly ⇒ ¬Lz ⇒ Rz . Up to exchanging the role of z and t, we may assume that Lx holds, and therefore that so do Rt , Ly and Rz . Summing up Lx + Rt + Ly + Rz , we get that L + R holds, so either L or R holds. The deﬁnition of the hyperbolic cone generalizes naturally to ρ = ∞, where one glues on each edge a sector of horoball with arclength l (explicitly, each triangle is isometric to [0, l]×[1, ∞) in the upper half-plane model of H2 ). The same argument shows that the a horospheric cone over any graph is also 2δH2 -hyperbolic. Lemma 4.8 (Very rotating condition). Recall that c is the apex of C(Q). Assume that some group R acts on Q, and that dQ (y, gy) ≥ 2π sinh ρ for all y ∈ Q, g ∈ R \ {1}. Then for all x1 , x2 ∈ C(Q) such that d(x1 , gx2 ) < d(x1 , c) + d(x2 , c), then any geodesic from x1 to x2 in C(Q) contains c. In particular, R satisﬁes the very rotating condition on C(Q). Proof. For i = 1, 2, denote xi = ti yi with yi ∈ Q. To prove that any geodesic [x1 , x2 ] contains the apex c, we have to check that dQ (y1 , y2 ) ≥ π sinh ρ. By the triangle inequality, no geodesic [x1 , gx2 ] contains c so dQ (y1 , gy2 ) ≤ π sinh ρ. By hypothesis on g, dQ (y1 , y2 ) ≥ dQ (y2 , gy2 ) − dQ (gy2 , y1 ) ≥ 2π sinh ρ − π sinh ρ ≥ π sinh ρ. 3. Cone-oﬀ of a space over a family of subspaces Let X be a δ-hyperbolic metric graph, whose edges all have the same length. Let Q be a family of almost convex subgraphs. We ﬁx some radius ρ > 0. For every Q ∈ Q, C(Q) is the hyperbolic cone of radius ρ over Q. We denote its apex by cQ . Later, we will also consider a moving family F = (RQ )Q∈Q . Definition 4.9. The hyperbolic cone-oﬀ of X over Q, of radius ρ, is the 2complex ⎛ ⎞ X˙ = ⎝X (C(Q))⎠ ∼ Q∈Q

where ∼ is the equivalence relation that identiﬁes for each Q ∈ Q, the subset of X deﬁned by Q, and its image in C(Q).

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VINCENT GUIRARDEL, GEOMETRIC SMALL CANCELLATION

The metric on X˙ is the corresponding path metric. ˙ for any R > 0, Although C(Q) may fail to be isometrically embedded in X, ˙ any subset of B(cQ , ρ − R) of diameter at most R is isometrically embedded in X. X

X˙

Qi

Recall that we assume that every Q ∈ Q is almost convex in the following sense: for all x, y ∈ Q, there exist x , y ∈ Q such that d(x, x ) ≤ 8δ, d(y, y ) ≤ 8δ and all geodesics [x, x ], [x , y ], [y , y] are contained in Q. In particular, for all x, y ∈ Q, dX (x, y) ≤ dQ (x, y) ≤ dX (x, y) + 32δ. Once hyperbolicity of X˙ is established, Assertion 3 of Theorem 4.1 is immediate from Lemma 4.8. Indeed if x ∈ Q, and injX (F) ≥ 2π sinh ρ, then dQ (x, gx) ≥ dX (x, gx) ≥ 2π sinh ρ and Lemma 4.8 concludes that the very rotating property ˙ holds in C(Q). Since the ball of radius ρ/2 in C(Q) isometrically embeds in X, and since the very rotating condition happens in the ball of radius 40δ around an apex, Assertion 3 of Theorem 4.1 holds as long as we take ρ0 ≥ 80δU . We note that if X is δ-hyperbolic, then it is 4δ-simply connected, hence so is X˙ by the Van Kampen theorem (each Q ∈ Q is connected because it is almost convex). Thus, by the Cartan-Hadamard Theorem, to prove the hyperbolicity of ˙ it is enough to prove that X˙ is R-locally 2δH2 -hyperbolic, with R = RCH (2δH2 ). X, In other words, Assertion 1 of Theorem 4.1 implies Assertion 2. Thus Theorem 4.1 follows from the following result. Theorem 4.10. Fix R = RCH (2δH2 ) as above. There exists δc , Δc > 0 such that for all δc -hyperbolic metric graph X whose edges have the same length, and for all Δc fellow-traveling family Q of almost convex subgraphs of X, and all ρ > 7R, the hyperbolic cone-oﬀ of radius ρ of X over Q is R-locally (2δH2 )-hyperbolic. The limit case of the theorem is as follows. Lemma 4.11. Let T be an R-tree, Q be a family of closed subtrees of T , any two of which intersect in at most one point. Then the cone-oﬀ T˙ of T over Q is δH2 -hyperbolic (in fact CAT (−1)). Remark 4.12. We only deﬁned the cone-oﬀ of a graph over a family of subgraphs, but the deﬁnition extends immediately to the setting of the lemma. Proof of the lemma. If Q is ﬁnite and T is a ﬁnite metric tree, then T˙ is δH2 -hyperbolic. For instance, this follows by induction on #Q using the fact that the space obtained by gluing two δH2 -hyperbolic spaces over a point is δH2 -hyperbolic (see also [BH99, Th II.11.1]). For the general case, consider x1 , x2 , x3 , x4 ∈ T˙ , and write T˙ as an increasing union of cone-oﬀs S˙ n of ﬁnite trees, with {x1 , x2 , x3 , x4 } ⊂ S˙ n for all n, such that dS˙ n (xi , xj ) → dT˙ (xi , xj ). The 4-point inequality of S˙ n thus implies the 4-point inequality for T˙ .

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3.1. Ultralimits to prove local hyperbolicity Let ω : 2N → {0, 1} be a non-principal ultraﬁlter. By deﬁnition, this is a ﬁnitely additive “measure” deﬁned on all subsets of N, such that ω(N) = 1, and ω(F ) = 0 for every ﬁnite subset F ⊂ N. Zorn’s Lemma shows that for any inﬁnite subset E, there is a non-principal ultraﬁlter such that ω(E) = 1. Given a sequence of properties Pi depending on i ∈ N, we say that Pi holds for ω-almost every i if ω({i|Pi true}) = 1. Since ω takes values in {0, 1}, if Pi does not hold for ω-almost every i, its negation holds for ω-almost every i. If (ti )i∈N is any sequence of real numbers, one can always deﬁne limω ti ∈ [−∞, ∞]: this is the only l ∈ [−∞, ∞] such that for any neighborhood U of l, ti ∈ U for ω-almost every i. Thus, the ultraﬁlter ω selects an accumulation point of the sequence. Let (Xi , ∗i )i∈N be a sequence of pointed metric spaces. Let B ⊂ i Xi be the set of all sequences of points (xi )i∈N such that d(xi , ∗i ) is bounded ω-almost everywhere, i.e. on a subset of ω-measure 1 (equivalently limω d(xi , ∗i ) < ∞). By deﬁnition, the ultralimit of (Xi , ∗i ) for ω is the metric space X∞ = B/∼ where (xi )i∈N ∼ (yi )i∈N if limω d(xi , yi ) = 0, and where the distance between (xi )i∈N and (yi )i∈N is deﬁned as limω d(xi , yi ). If xi ∈ Xi is a sequence of points such that d(xi , ∗i ) is bounded ω-almost everywhere, we deﬁne the ultralimit of xi as the image of (xi )i∈N in X∞ . We will use ultralimits in the following fashion. Note that we do not rescale our metric spaces, contrary to what one does in the construction of asymptotic cones. Assume that (Xi )i∈N is a sequence of metric spaces such that any ultralimit of Xi is δ-hyperbolic (for any ultraﬁlter, and any base point ∗i ). Then for all R, ε > 0, Xi is R-locally (δ + ε)-hyperbolic for i large enough. Indeed, if this does not hold, then there is a subsequence Xik and a subset {xik , yik , zik , tik } ⊂ Xik of diameter at most R that contradicts the 4-point (δ + ε)-hyperbolicity condition. Taking ∗i = xi as a base point, and taking ω a non-principal ultraﬁlter such that ω({ik }k∈N ) = 1, we get an ultralimit X∞ in which the ultralimit of the points {xik , yik , zik , tik } contradicts δ-hyperbolicity. 3.2. Proof of the local hyperbolicity of the cone-oﬀ Proof of Theorem 4.10. Let R be given. We need to prove that any 4-point set {x, y, z, t} of diameter at most R satisﬁes the 2δH2 -hyperbolic inequality. We use that any subset of B(cQ , ρ − R) of diameter at most R is isometrically embedded ˙ Since C(Q) is 2δH2 -hyperbolic, we are done if {x, y, z, t} is contained in in X. B(cQ , ρ − R). There remains to check that there exist Δc , δc such that the 2R-neighborhood of X in X˙ is R-locally 2δH2 -hyperbolic. If not, there are two sequences δi , Δi converging to 0, such that for each i ∈ N, one can ﬁnd a counterexample as follows: there are • a δi -hyperbolic space Xi , • a family Qi of almost convex subsets of Xi with Δ(Qi ) ≤ Δi , and • a radius ρi > 7R, so that the cone-oﬀ X˙ i of radius ρi of Xi over Qi contains a subset {xi , yi , zi , ti } ⊂ X˙ i of diameter at most R for which the 4-point 2δH2 -hyperbolicity inequality fails. Let ∗i ∈ Xi be a point at distance at most 2R from xi . We note that {xi , yi , zi , ti } ⊂ B(∗i , 3R).

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VINCENT GUIRARDEL, GEOMETRIC SMALL CANCELLATION

Let ω be a non-principal ultraﬁlter, and X˙ ∞ the ultralimit of X˙ i pointed at ∗i . We denote by ∗ ∈ X˙ ∞ the ultralimit of the sequence (∗i )i∈N . Let x, y, z, t ∈ X˙ ∞ be the ultralimit of the points xi , yi , zi and ti . Since 2δH2 > δH2 , to get a contradiction, it is enough to prove that x, y, z, t satisfy the 4-point δH2 -hyperbolicity inequality. We want to compare X˙ ∞ with the cone-oﬀ on an R-tree. Let T be the ultralimit of Xi pointed at ∗i (this is an R-tree). We denote by ∗T ∈ T the ultralimit of (∗i )i∈N . To deﬁne a cone-oﬀ of T , we need to deﬁne a family of subtrees Q of T . Given a sequence of subspaces Q = (Qi )i∈N ∈ i∈N Qi (each Qi ∈ Qi is a subset of Xi ,), we say that this sequence is non-escaping if there exists qi ∈ Qi such that d(qi , ∗i ) is bounded ω-almost everywhere, Let Q∞ ⊂ ( i∈N Qi )/ ∼ω be the set of non-escaping sequences up to equality ω-almost everywhere. Given Q = (Qi )i∈N a non-escaping sequence, let Qω be the ultralimit of (Qi )i∈N based at qi . Note that this ultralimit does not depend on the choice of qi as long as d(qi , ∗i ) is bounded ω-almost everywhere. There is a natural map Qω → T induced by the inclusions Qi → Xi . This map is an isometry because the inclusion Qji → Xi is an isometry up to an additive constant bounded by 32δi , and δi converges to 0. Thus we identify Qω with its image in T . Then we deﬁne the collection of all possible such subsets Qω by Q = (Qω )Q∈Q∞ , and we consider T˙ the corresponding cone-oﬀ with radius ρ = limω ρi (note that ρ might be inﬁnite, in which case we construct the corresponding horospheric cone-oﬀ). Lemma 4.13. (1) For Q = Q ∈ Q∞ , Qω ∩ Qω contains at most one point. In particular T˙ is δH2 -hyperbolic. (2) There is a natural 1-Lipschitz map ψ˙ : T˙ → X˙ ∞ that maps isometrically BT˙ (∗T , 3R) to BX˙ ∞ (∗, 3R). The lemma allows us to conclude the proof: {x, y, z, t} ⊂ BX˙ ∞ (∗, 3R), which is isometric to a subset of the δH2 -hyperbolic space T˙ , so x, y, z, t satisfy the 4-point δH2 -hyperbolicity inequality. Proof of Lemma 4.13. For Assertion 1, consider Q = (Qi )i∈N , Q = (Qi )i∈N with Qi = Qi for ω-almost every i. Given x ∈ Qω ∩Qω , there are sequences (xi )i∈N , (xi )i∈N representing x such that xi ∈ Qi , xi ∈ Qi . In particular limω d(xi , xi ) = 0. If y ∈ Qω ∩ Qω is another point, there exist similarly, yi ∈ Qi , yi ∈ Qi representing y, so that in particular, limω d(yi , yi ) = 0. If x = y, then d(x, y) > 0, so d(xi , yi ) and d(xi , yi ) are bounded below by d(x, y)/2 for ω-almost every i. By almost convexity, we see that Qi fellow travels Qi by at least d(x, y)/4 for ω-almost every i. Since Δi tends to 0, we get Qi = Qi for almost every i, so Q = Q , a contradiction. This proves Assertion 1. Now we deﬁne the map ψ˙ : T˙ → X˙ ∞ . Inclusions ϕXi : Xi → X˙ i are 1Lipschitz and deﬁne naturally a 1-Lipschitz map ψ : T → X˙ ∞ . Similarly, given Q = (Qi )i∈N ∈ Q∞ , the inclusions ϕC(Qi ) : Cρi (Qi ) → X˙ i induce a 1-Lipschitz map ψC(Qω ) : Cρ (Qω ) → X˙ ∞ . Since for each Q ∈ Q∞ , ψ coincides with ψCρ (Qω ) in restriction to Qω , these maps induce a 1-Lipschitz map ψ˙ : T˙ → X∞ . Note that in general, ψ˙ may be not onto. ˙ Given x ∈ B ˙ (∗, 3R), To prove Assertion 2, we deﬁne a partial inverse ψ of ψ. X∞ represent x by a sequence xi ∈ X˙ i with dX˙ i (xi , ∗i ) ≤ 3R.

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LECTURE 4. THE CONE-OFF

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If xi lies in Xi (i.e. not in the interior of a cone) for ω-almost every i, we want to deﬁne ψ (x) as the ultralimit in T of xi . For this ultralimit to exist, we have to prove that dXi (xi , ∗i ) is bounded ω-almost everywhere. But since ρi > 3R, any geodesic [∗i , xi ] avoids the ρi − 3R neighbourhood of any apex. Now there exists M such that the radial projection is locally M -Lipschitz (independently of ρi , see Exercise 4.15). It follows that the radial projection of this geodesic has length bounded by 3RM , so the ultralimit of xi in T exists. Similarly, if xi lies in a cone for ω-almost every i, write xi = si yi for some si < ρi , and yi ∈ Xi . The argument above shows that dXi (∗i , yi ) is bounded, so the ultralimit of yi in T exists, we denote it by y. Moreover, the sequence Q of cones Qi containing xi is non-escaping, so we can deﬁne ψ (x) as sy in the cone Qω , s = limω si . ˙ There It is clear from the deﬁnition that ψ (x) is a preimage of x under ψ. remains to show that ψ is 1-Lipschitz. It is based on the following technical fact, proved below. Fact 4.14. For any ρ0 , ε, D0 > 0, there exists n ∈ N such that the following holds. Consider a graph X, and a cone-oﬀ X˙ of radius ρ ≥ ρ0 . Then for any pair of points x, y ∈ X˙ with d(x, y) ≤ D0 , there is a path p joining x to y in X˙ such that • the length of p is at most dX˙ (x, y) + ε • p is a concatenation of at most n paths, each of which is either contained in X or in a cone C(Q). To conclude, take ρ0 = 7R, ρ > 7R, ε > 0, and let D0 = 6R + 3ε. We assume that ε is small enough so that D0 < 7R. Consider n given by the fact. Consider x, y ∈ BX˙ ∞ (∗, 3R), write x and y as an ultralimit of sequences xi , yi ∈ BX˙ i (∗i , 3R + ε). Consider pi a path joining xi , yi of length at most dX˙ i (xi , yi ) + ε ≤ D0 and which is a concatenation of at most n sub-paths as in the fact. We can assume that p is a concatenation of exactly n subpaths: pi = p1i · p2i · · · pni . Because D0 < 7R < ρ, pi stays at distance at least 7R − D0 from the cone point. Fix k ∈ {1, . . . , n}. If pki ⊂ Xi for ω-almost every i, then the ultralimit of pki deﬁnes a path pk∞ ⊂ T . Otherwise, pki is contained in a cone for ω-almost every i. Since the distance from pki to the cone point is bounded below by 7R − D0 > 0, its radial projection has bounded length by Exercise 4.15. It follows that the radial projections of pki converge to a path of ﬁnite length in T . Thus, writing pki as a map pki : [0, 1] → Qi,k × [0, ρi ], and taking an ultralimit deﬁnes a path pk∞ in T˙ . We thus get a path p∞ = p1∞ · · · pn∞ joining ψ (x) to ψ (y), and whose length is at most limω dX˙ i (xi , yi ) + ε = dX˙ (x, y) + ε. Proof of Fact 4.14. Consider p ⊂ C(Q) a geodesic path avoiding the apex and whose endpoints are in Q. Denote by l its length and L the length of its radial projection. We claim that Ll goes to 1 as l tends to 0 independently of ρ. More precisely, we claim that for any λ > 1, there exists η > 0 such that if l ≤ η, then L ≤ λl where η does not depend on ρ as long as ρ ≥ R0 . To prove the claim, we use that l and L are related by the relation cosh l = cosh ρ − sinh ρ cos 2

2

L sinh ρ

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which can be rewritten as sinh l/2 = sinh ρ sin

L 2 sinh ρ

.

Since sin(x) ≥ x − 16 x3 and argsh(x) ≥ x − 16 x3 for all x ≥ 0, one can deduce 1 (1 + sinh1 2 ρ )L3 . Since ρ ≥ ρ0 > 0, we get Ll ≥ the estimate L ≥ l ≥ L − 24 1 1 − 24 (1 + sinh12 ρ0 )L2 and the claim follows. To prove the fact, consider a path p in X˙ joining x to y, of length d(x, y) + ε/2. We can assume that p is a concatenation of paths p1 , . . . , pk where each pi is either contained in a cone, or contained in X. If two consecutive paths are contained in the same cone or are both contained in X, we can replace them by their concatenation to decrease k. Thus for each i ∈ {2, . . . , k − 1}, if pi is contained in a cone C(Q), d(x,y)+ε 0 +ε then the endpoints of pi are in Q. Let λ = RR0 +ε/2 ≤ d(x,y)+ε/2 , and consider η as in the claim above. For each i ∈ {2, . . . , k − 1} such that pi is contained in a cone and has length at most η, we replace it by its radial projection pi , and we deﬁne pi = pi for all other indices i. The length of the obtained new path p is at most λ(d(x, y) + ε/2) ≤ d(x, y) + ε. Since each pi that is not contained in X has length at least η, there are at most n0 = (R0 + ε)/η such sub-paths. By concatenation of consecutive paths contained in X, we get that p is a concatenation of at most 2n0 + 3 paths, each of which is either contained in a cone, or contained in X. Exercise 4.15. Given ρ > 0 denote by pρ : BH2 (0, ρ) \ 0 → S(0, ρ) the radial projection on S(0, ρ), the circle of radius ρ. Prove that given r, ρ0 > 0, there is a constant M such that for any ρ ∈ [ρ0 + r, ∞), the restriction of the radial projection ˚ ρ − r) is locally M -Lipschitz. to of B(0, ρ) \ B(0, Hint: Since the closest point projection H2 → B(0, ρ−r) is distance decreasing, it is enough to bound the Lipschitz constant of the restriction of pρ to the circle of radius ρ − r. Using polar coordinates, prove that this follows from the fact that sinh ρ sinh(ρ−r) decreases with ρ.

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[Gro01a] Misha Gromov. Mesoscopic curvature and hyperbolicity. In Global diﬀerential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000), volume 288 of Contemp. Math., pages 58–69. Amer. Math. Soc., Providence, RI, 2001. MR1871000 (2003a:53052) [Gro01b] Misha Gromov. CAT(κ)-spaces: construction and concentration. Zap. Nauchn. Sem. S.Petersburg. Otdel. Mat. Inst. Steklov. (POMI), 280(Geom. i Topol. 7):100–140, 299–300, 2001. MR1879258 (2002j:53045) [Gro03] Misha Gromov. Random walk in random groups. Geom. Funct. Anal., 13(1):73–146, 2003. MR1978492 (2004j:20088a) [GM08] Daniel Groves and Jason Fox Manning. Dehn ﬁlling in relatively hyperbolic groups. Israel J. Math., 168:317–429, 2008. MR2448064 (2009h:57030) [HLS02] N. Higson, V. Laﬀorgue, and G. Skandalis. Counterexamples to the Baum-Connes conjecture. Geom. Funct. Anal., 12(2):330–354, 2002. MR1911663 (2003g:19007) [Iva94] Sergei V. Ivanov. The free Burnside groups of suﬃciently large exponents. Internat. J. Algebra Comput., 4(1-2):ii+308, 1994. MR1283947 (95h:20051) [KM96] Vadim A. Kaimanovich and Howard Masur. The Poisson boundary of the mapping class group. Invent. Math., 125(2):221–264, 1996. MR1395719 (97m:32033) [LS01] Roger C. Lyndon and Paul E. Schupp. Combinatorial group theory. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1977 edition. MR1812024 (2001i:20064) [Lys96] I. G. Lysënok. Inﬁnite Burnside groups of even period. Izv. Ross. Akad. Nauk Ser. Mat., 60(3):3–224, 1996. MR1405529 (97j:20037) [Mar91] G. A. Margulis. Discrete subgroups of semisimple Lie groups, volume 17 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1991. MR1090825 (92h:22021) [MM99] Howard A. Masur and Yair N. Minsky. Geometry of the complex of curves. I. Hyperbolicity. Invent. Math., 138(1):103–149, 1999. MR1714338 (2000i:57027) [NA68] P. S. Novikov and S. I. Adjan. Inﬁnite periodic groups. I. Izv. Akad. Nauk SSSR Ser. Mat., 32:212–244, 1968. MR0240178 (39:1532a) [OW07] Yann Ollivier and Daniel T. Wise. Kazhdan groups with inﬁnite outer automorphism group. Trans. Amer. Math. Soc., 359(5):1959–1976, 2007. MR2276608 (2008a:20049) [Ol 80] A. Ju. Ol šanski˘ı. An inﬁnite group with subgroups of prime orders. Izv. Akad. Nauk SSSR Ser. Mat., 44(2):309–321, 479, 1980. MR571100 (82a:20035) [Ol 82] A. Yu. Ol shanski˘ı. The Novikov-Adyan theorem. Mat. Sb. (N.S.), 118(160)(2):203–235, 287, 1982. MR658789 (83m:20058) [Ol’91a] A. Yu. Ol’shanski˘ı. Geometry of deﬁning relations in groups, volume 70 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1991. Translated from the 1989 Russian original by Yu. A. Bakhturin. MR1191619 (93g:20071) [Ol’91b] A. Yu. Ol’shanski˘ı. Periodic quotient groups of hyperbolic groups. Mat. Sb., 182(4):543– 567, 1991. MR1119008 (92d:20050) [Ol 95] A. Yu. Ol shanski˘ı. SQ-universality of hyperbolic groups. Mat. Sb., 186(8):119–132, 1995. MR1357360 (97b:20057) [OOS09] Alexander Yu. Ol shanskii, Denis V. Osin, and Mark V. Sapir. Lacunary hyperbolic groups. Geom. Topol., 13(4):2051–2140, 2009. With an appendix by Michael Kapovich and Bruce Kleiner. MR2507115 (2010i:20045) [Osi10] Denis Osin. Small cancellations over relatively hyperbolic groups and embedding theorems. Ann. of Math. (2), 172(1):1–39, 2010. MR2680416 (2012a:20068) [Osi07] Denis V. Osin. Peripheral ﬁllings of relatively hyperbolic groups. Invent. Math., 167(2):295–326, 2007. MR2270456 (2008d:20080) [Rip82] E. Rips. Subgroups of small cancellation groups. Bull. London Math. Soc., 14(1):45–47, 1982. MR642423 (83c:20049) [Sis11] Alessandro Sisto. Contracting elements and random walks, 2011. [Tar49] V. A. Tartakovski˘ı. Solution of the word problem for groups with a k-reduced basis for k > 6. Izvestiya Akad. Nauk SSSR. Ser. Mat., 13:483–494, 1949. MR0033816 (11:493c) [Wis03] Daniel T. Wise. A residually ﬁnite version of Rips’s construction. Bull. London Math. Soc., 35(1):23–29, 2003. MR1934427 (2003g:20047) [Wis11] Daniel T. Wise. The structure of groups with quasiconvex hierarchy, 2011. available at http://www.math.mcgill.ca/wise/papers.html.

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https://doi.org/10.1090//pcms/021/04

Lectures on Proper CAT(0) Spaces and Their Isometry Groups Pierre-Emmanuel Caprace

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IAS/Park City Mathematics Series Volume 21, 2012

Lectures on Proper CAT(0) Spaces and Their Isometry Groups Pierre-Emmanuel Caprace Introduction CAT(0) spaces, introduced by Alexandrov in the 1950’s, were given prominence by M. Gromov, who showed that a great deal of the theory of manifolds of non-positive sectional curvature could be developed without using much more than the CAT(0) condition (see [BGS85]). Since then, CAT(0) spaces have played a central role in geometric group theory, opening a gateway to a form of generalized diﬀerential geometry encompassing non-positively curved manifolds as well as large families of singular spaces such as trees, Euclidean or non-Euclidean buildings, and many other cell complexes of non-positive curvature. Excellent introductions on CAT(0) spaces may be found in the literature, e.g. in the books [Bal95] and [BH99]. The goal of these lectures is to present some material not covered by those references. While rigidity of (usually discrete) group actions on non-positively curved space is a standard theme of study in geometric group theory, the main idea we would like to convey is that, in the locally compact case, the spaces themselves turn out to be much more rigid than one might expect as soon as they admit a reasonable amount of isometries. This phenomenon will be highlighted by placing a special emphasis on the full isometry group of a proper CAT(0) space. Taking into account the fact the this isometry group is naturally endowed with a locally compact group topology which is possibly non-discrete, many structural (and especially rigidity) properties of the underlying space can be derived by combining results on locally compact groups with (mostly elementary) geometric arguments. A number of results obtained with this approach are presented in this course. In the ﬁnal lecture, we will come back to discrete groups and present some results whose proof relies heavily on the preceding study of non-discrete group actions. Although some of the very basics on CAT(0) spaces will be recalled, a familiarity with the aforementioned standard references is recommended. We have chosen to present the results not always in their most general form, but rather in a way that makes their statement simpler and hopefully more enlightening. More general statements, detailed arguments and further results may be found in the papers [CM09a, CM09b, CM13, CZ13]. All the original results presented here have been obtained in collaboration with Nicolas Monod or with Gaˇsper Zadnik. UCLouvain – IRMP, Chemin du Cyclotron 2, box L7.01.02, 1348 Louvain-la-Neuve, Belgium E-mail address: [email protected] P.-E. C. is an F.R.S.-FNRS research associate at UCLouvain, Belgium, supported in part by FNRS grant F.4520.11 and by the ERC grant #278469. c 2014 American Mathematical Society

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Acknowledgements Special thanks are due to the Park City Mathematical Institute for its hospitality, and to the organisers of the 2012 programme. I am very grateful to Gaˇsper Zadnik fur useful comments on an earlier version of this document, for suggesting several exercises, and for supervising the problem sessions concerning these lectures at PCMI.

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LECTURE 1

Leading examples 1. The basics Let (X, d) be a metric space. A geodesic map is an isometric map ρ : I → X of a convex subset I ⊆ R to X, where the real line R is endowed with the Euclidean distance. The map ρ is called a geodesic segment (resp. ray, line) if I is a closed interval (resp. I is a half-line, I = R). It should be noted that the notion of geodesic introduced here is a global one, as opposed to the corresponding notion in diﬀerential geometry. A geodesic metric space is a metric space (X, d) in which any two points are joined by a geodesic segment. Examples 1.1. • The Euclidean space (Rn , dEucl ) is a geodesic metric space. • More generally, a Riemannian manifold, viewed as a metric space with its canonical distance function, is a geodesic metric space provided it is complete. An incomplete Riemannian manifold need not be a geodesic metric space. • A metric graph, with all edges of length one, is a geodesic metric space. Let (X, d) be a geodesic metric space. Given a triple (x, y, z) ∈ X 3 , a Euclidean comparison triangle for (x, y, z) is a triple (ˆ x, yˆ, zˆ) of points of the Eux, yˆ), d(y, z) = dEucl (ˆ y , zˆ) and d(z, x) = clidean plane R2 such that d(x, y) = dEucl (ˆ z, x ˆ). Notice that any triple in X admits some Euclidean comparison triangle. dEucl (ˆ A CAT(0) space is a geodesic metric space all of whose triple of points (x, y, z) ∈ X 3 satisfy the following condition: given a Euclidean comparison triangle (ˆ x, yˆ, zˆ) in R2 , any point p ∈ X which belongs to some geodesic segment joining y to z in X satisﬁes the inequality d(x, p) ≤ dEucl (ˆ x, pˆ), where pˆ ∈ R2 is the unique point of R2 such that d(y, p) = dEucl (ˆ y , pˆ) and d(p, z) = dEucl (ˆ p, zˆ). The following fundamental properties of CAT(0) spaces are straightforward to deduce from the deﬁnition. Proposition 1.2. Let (X, d) be a CAT(0) space. Then: (i) (X, d) is uniquely geodesic, i.e. any two points are joined by a unique geodesic segment. (ii) X is contractible. 95 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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Examples 1.3. • The Euclidean space (Rn , dEucl ) is a CAT(0) space. So is any pre-Hilbert space. • A complete, simply connected, Riemannian manifold M , endowed with its canonical distance function, is a CAT(0) space if and only if M has non-positive sectional curvature. See [BH99, Theorem 1.A.6]. So is in particular the real hyperbolic space Hn . • A metric graph X is a CAT(0) space if and only if X is a tree. This short list of examples already illustrates that the category of CAT(0) spaces encompasses both smooth and singular objects. The singular character expresses itself by the fact that geodesics may branch, i.e. two distinct geodesic segments may share a common sub-segment of positive length. There are several ways to construct new examples of CAT(0) spaces from known ones. A subset Y of a CAT(0) space (X, d) is called convex if the geodesic segment joining any two points of Y is entirely contained in Y . Clearly, a convex subset of a CAT(0) space is itself a CAT(0) space when endowed with the induced metric. Another key feature of the CAT(0) condition is its stability under Cartesian products. The proof is left as an exercise. Proposition 1.4. Let (X1 , d1 ) and (X2 , d2 ) be CAT(0) spaces. Then the Cartesian product X = X1 ×X2 , endowed with the metric d deﬁned by d2 = d21 +d22 , is a CAT(0) space. Throughout these notes, a Cartesian product of CAT(0) spaces will always be endowed with the CAT(0) metric as in Proposition 1.4. The product X = X1 × X2 is called a CAT(0) product space. Various more exotic constructions, like gluing two CAT(0) spaces along an isometric convex subset, also preserve the CAT(0) condition. We close this section with the following noteworthy facts, for which we refer to Cor. II.3.10 and II.3.11 in [BH99]. Proposition 1.5. (i) The Cauchy completion of a CAT(0) space is itself CAT(0). (ii) An ultraproduct of CAT(0) spaces is itself CAT(0). In particular, the asymptotic cones of a CAT(0) space are CAT(0). 2. The Cartan–Hadamard theorem A fundamental feature of the CAT(0) condition is that it is a local condition, as is the condition of being non-positively curved in the realm of Riemannian geometry. This matter of fact is made precise by the following basic result, for which we refer to [Bal95, Theorem I.4.5] and [BH99, Theorem II.4.1]. Theorem 1.6 (Cartan–Hadamard). Let (X, d) be a complete connected metric space. If every point of X admits some neighbourhood which is CAT(0) when endowed with the appropriate restriction of d (we then say that (X, d) is locally CAT(0)), such that folthen there is a unique distance function d˜ on the universal cover X lowing two conditions hold:

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→ X is a local isometry; • the covering map X ˜ is a CAT(0) space. d) • (X, The metric d˜ coincides with the length metric (also called inner metric) We refer to [Bal95, §1.1] and [BH99, §I.3] for detailed treatinduced by d on X. ments of those notions. At this point, let us just observe that a non-convex subset of a CAT(0) space may very well be CAT(0) provided it is endowed with the induced length metric. The Cartan–Hadamard theorem yields a wealth of further examples of CAT(0) spaces constructed as universal covers of compact metric spaces that are locally CAT(0). A typical situation is that of a ﬁnite piecewise Euclidean cell complex X, endowed with the length metric d induced by the Euclidean metric on each cell. The Cartan–Hadamard theorem ensures that the universal covering cell complex is naturally a CAT(0) space provided (X, d) is locally CAT(0). Verifying that X a given ﬁnite piecewise Euclidean cell complex is locally CAT(0) is usually highly non-trivial (although, in theory, it can be done algorithmically, see [EM04]). There are only two special cases where this question can be decided by means of an easy combinatorial criterion, as described in the following (see [BH99, §II.5] and the lectures by M. Sageev). Theorem 1.7. Let X be a connected piecewise Euclidean cell complex endowed with the length metric d induced by the Euclidean metric on each cell. If any of ˜ is a d) the following conditions holds, then (X, d) is locally CAT(0), and hence (X, CAT(0) space: (i) X is two-dimensional, and for each vertex v ∈ X (0) and each sequence (σ1 , σ1 , . . . , σn ) of pairwise distinct 2-faces such that σi ∩ σi+1 is an edge containing v for all i ∈ Z/nZ, the sum over all i of the interior angles of the faces σi at the vertex v is at least 2π. (ii) Each cell in X is a Euclidean cube with edge length one, and the link of every vertex is a ﬂag complex. The CAT(0) spaces constructed as in Theorem 1.7(ii), which are called CAT(0) cube complexes, are endowed with a rich combinatorial structure which provides an important additional tool in their study. This explains why results known about CAT(0) cube complexes are usually much ﬁner than those describing more general classes of CAT(0) spaces. Nevertheless, it turns out that CAT(0) cube complexes are much more ubiquitous that one might think at a ﬁrst sight. We refer to the lectures by M. Sageev for more information. From now on, a metric space (X, d) will simply be denoted by its underlying set of points X, the distance function being by default denoted by the letter d, unless explicitly mentioned otherwise. 3. Proper cocompact spaces The class of all CAT(0) spaces is vast and wild; it is not a realistic goal to understand it exhaustively. In the rest of the course, we shall frequently impose that the spaces under consideration satisfy (some of) the following conditions: • Properness. A metric space is called proper if all of its closed balls are compact. In particular such a space is locally compact.

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• Cocompactness. A metric space X is called cocompact if its full isometry group Is(X) acts cocompactly, i.e. if the orbit space Is(X)\X is compact. • Geodesic completeness. A geodesic metric space X is called geodesically complete (one also says that X has extendible geodesics) if every geodesic segment can be prolonged to a (potentially non-unique) bi-inﬁnite geodesic line. As in Riemannian geometry, the notions of properness, completeness and geodesic completeness are related in the case of locally compact spaces: Theorem 1.8 (Hopf–Rinow). Let X be a locally compact CAT(0) space. (i) X is proper if and only if it is complete. (ii) If X is geodesically complete, then it is proper. Proof. See [Bal95, Theorem I.2.4] and [BH99, Proposition I.3.7].

Among all proper cocompact CAT(0) spaces, there are two leading families of examples, namely symmetric spaces and Euclidean buildings. Those are the spaces naturally associated with semi-simple Lie groups or semi-simple linear algebraic groups over local ﬁelds. We shall now brieﬂy recall the basic deﬁnitions. 4. Symmetric spaces A symmetric space is a Riemannian manifold M such that the geodesic symmetry σx centered at each point x ∈ M is a global isometry. Equivalently, for each x ∈ M there is an isometry σx ∈ Is(M ) ﬁxing x, whose diﬀerential is the central symmetry of Tx M . Basic examples are provided by the sphere Sn , the Euclidean space Rn and the real hyperbolic space Hn . As the example of the sphere shows, a symmetric space can be positively curved. A symmetric space is said to be of non-compact type if it has non-positive sectional curvature and no non-trivial Euclidean factor. Any such space M is thus a CAT(0) space which is proper, cocompact (in fact homogeneous!) and geodesically complete. It can be constructed as a coset space M = G/K, where G is a non-compact, connected semi-simple Lie group and K < G is a maximal compact subgroup. The metric on G/K comes from the Killing form of the Lie algebra of G. A prominent example is provided by the case G = SLn (R) and K = SO(n). The coset space M = G/K can be identiﬁed with the collection of scalar products on Rn for which the unit ball has the same volume as the unit ball with respect to the standard Euclidean metric dEucl . A description of the symmetric space M may be consulted in [BH99, §II.10]; the discussion below provides an alternative approach. The distance function on M can be deﬁned as follows. Given two scalar products x1 = (·, ·)1 and x2 = (·, ·)2 on Rn , it is a standard fact (see e.g. [HJ90, Th. 7.6.4]) that there exists some basis of Rn with respect to which both products are represented by a diagonal Gram matrix, say diag(λ1 , . . . , λn ) and diag(μ1 , . . . , μn ). The distance from x1 to x2 is then deﬁned by n 2 λi (4.1) d(x1 , x2 ) = . log μi i=1

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It then turns out that (M, d) is a CAT(0) space (this is however non-trivial to verify, see Exercise 1.4). The following key feature of that space is easy to deduce from the deﬁnition given above. Proposition 1.9. Let (M, d) be the symmetric space associated with SLn (R). Then any two points of M are contained in a common ﬂat of dimension n − 1. A ﬂat of dimension k in a CAT(0) space is a subset isometric to the Euclidean space Rk . The rank of a symmetric space is the maximal dimension of a ﬂat. The above property is a special instance of a general property: in a symmetric space of rank r, any two points are contained in a common r-ﬂat. 5. Euclidean buildings Let W ≤ Is(Rn ) be a discrete reﬂection group, i.e. a discrete subgroup generated by orthogonal reﬂections through hyperplanes. The discreteness of W implies that the collection H of all hyperplanes associated with reﬂections in W is locally ﬁnite, i.e. every ball meets only ﬁnitely many hyperplanes in H. In fact, the pattern determined by H deﬁnes a cellular decomposition of Rn , which is called a Euclidean Coxeter complex. A chamber in that complex is deﬁned as a connected component of the space Rn − H∈H H. The group W acts sharply transitively on the set of chambers. The top-dimensional cells in a Coxeter complex coincide with the closures of the chambers, which may be non-compact. Any lower dimension cell is the intersection of a closed chamber with a set of hyperplanes in H. A Euclidean building is a cell complex Δ satisfying the following two conditions: (1) Any two cells are contained in a common subcomplex, called an apartment, which is (combinatorially) isomorphic to a Euclidean Coxeter complex. (2) Given any two apartments A1 and A2 in Δ, there is an isomorphism ϕ : A1 → A2 ﬁxing the intersection A1 ∩ A2 pointwise. A Euclidean building is thus primarily a combinatorial object. It always possesses a CAT(0) metric realization: Proposition 1.10. Let Δ be a Euclidean building. Then Δ has a metric realisation (|Δ|, d) such that for each apartment A ⊂ Δ, the restriction of d to |A| is the Euclidean metric. The metric space (|Δ|, d) is a complete CAT(0) space. Proof. Axiom (2) implies that all apartments are combinatorially isomorphic. Fix a Euclidean metric on one of them, and transport this metric to all the others via the isomorphisms provided by (2). The axioms imply that this yields a geometric realisation |Δ| endowed with a well deﬁned map d : |Δ|×|Δ| → R+ whose restriction to each apartment is the Euclidean metric. One may then verify that (|Δ|, d) is a metric space which satisﬁes the CAT(0) condition. The fact that it is geodesic is immediate from (1). See [AB08, Theorem 1.16] for details. The simplest example of a building is when W is the inﬁnite dihedral group acting properly on the real line. In that case, the corresponding Coxeter complex is the simplicial line, and a Euclidean building having that Coxeter complex as type of apartments is a simplicial tree without vertex of valency one. Conversely any

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simplicial tree without vertex of valency one is a Euclidean building. Likewise, if W is a product of n copies of the inﬁnite dihedral groups acting properly on Rn , the corresponding buildings are products of n trees. Euclidean buildings are natural ‘discrete’ analogues of symmetric spaces. In fact, to any semi-simple linear algebraic group over a local ﬁeld (e.g. SLn (Qp )), one may associate a Euclidean building on which the group acts isometrically, transitively on the chambers. This is part of the Bruhat–Tits theory [BT72]. Let us merely mention here that the key feature of symmetric spaces pointed out in Proposition 1.9 is shared by Euclidean buildings: Proposition 1.11. In the CAT(0) realization of a Euclidean building of dimension n, any two points are contained in a common n-ﬂat. The fact that the rank coincides with the dimension is of course peculiar to buildings; the symmetric space associated with SLn (R) has rank n − 1 and di. In fact, one has the following characterization of Euclidean mension (n−1)(n+2) 2 buildings among locally compact CAT(0) spaces, a proof of which can be found in [BL06, Cor. 1.7]: Theorem 1.12 (Kleiner). Let X be a locally compact CAT(0) space of geometric dimension n. If any two points are contained in a common n-ﬂat, then X is the metric realization of a Euclidean building. The notion of geometric dimension was introduced by B. Kleiner [Kle99]. It can be deﬁned as the supremum over all compact subsets K ⊂ X of the topological dimension of K. If X is a piecewise Euclidean cell complex, the geometric dimension coincides with the maximal dimension of a cell. For further information and alternative characterizations, see [Kle99]. A more detailed introduction on Euclidean buildings can be found in [Bro89]. See also [AB08] for a comprehensive account. The Euclidean buildings deﬁned above are sometimes called discrete Euclidean buildings, in order to distinguish them within a more general class of objects, called R-buildings (or non-discrete Euclidean buildings). Those generalize discrete buildings in the same way as Rtrees generalize simplicial trees; they appear naturally in the Bruhat–Tits theory of reductive groups over ﬁelds with a non-discrete valuation. They also pop up as asymptotic cones of symmetric spaces of non-compact type, as proved by Kleiner and Leeb (see [KL97], as well as the lectures by M. Kapovich). 6. Rigidity Symmetric spaces and Euclidean buildings should be considered as leading examples of CAT(0) spaces. This is not only justiﬁed by the fact that their features serve as a basis for the intuition in the study of more general CAT(0) spaces, but also because these spaces (especially in rank > 1) seem to be the most rigid among all proper CAT(0) spaces. We ﬁnish this ﬁrst lecture by mentioning some instances of this matter of fact. Products of symmetric spaces and Euclidean buildings arise naturally in the study of arithmetic groups (see the lectures by T. Gelander and by D. Morris). The 1 ]), with p1 , . . . , pr distinct primes. Indeed prototypical example is Γ = SLn (Z[ p1 ...p r the diagonal embedding of Γ in G = SLn (R) × SLn (Qp1 ) × · · · × SLn (Qpr ) is a

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LECTURE 1. LEADING EXAMPLES

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lattice embedding. In particular the discrete group Γ acts properly on the model CAT(0) space of G, which is the product X = M × Δp1 × · · · × Δpr of the symmetric space M of SLn (R) with the Euclidean buildings Δpi of SLn (Qpi ). The following result highlights a strong rigidity property of the arithmetic group Γ. 1 ]), with Theorem 1.13 ([CM09a, Th. 1.14 and 1.15]). Let Γ = SLn (Z[ p1 ...p r n ≥ 3 and r ≥ 0, act by isometries on a proper, cocompact, geodesically complete CAT(0) space Y . Assume that Γ acts minimally in the sense that it does not preserve any non-empty closed convex subset Z Y . Then Y is a product of symmetric spaces and Euclidean buildings, which is a subproduct of the model space X.

Notice that no properness assumption is made on the action of Γ on Y . The theorem shows that Γ admits only few minimal actions on proper cocompact CAT(0) spaces: all of them occur as projections of the Γ-action on the model space X on a subproduct. In particular, when the model space has only one factor, i.e. when Γ = SLn (Z), it follows that any minimal action of Γ on a proper cocompact CAT(0) space is either trivial, or proper and coincides with the standard Γ-action on the symmetric space SLn (R)/SO(n). Theorem 1.13 can be viewed as a rigidity property of the arithmetic group 1 ]). The following result should rather be interpreted as a rigidity SLn (Z[ p1 ...p r property of its model space X. We recall the isometries of a CAT(0) fall into three families, called elliptic, hyperbolic and parabolic respectively. Elliptic isometries are those which ﬁx points. Hyperbolic isometries are those which preserve some geodesic line and act non-trivially along it. Parabolic isometries are all the others; they can have translation length zero or not, and should be viewed as the wilder type of isometries, especially when the ambient space is not locally compact. See [Bal95, §II.3] and [BH99, §II.6]. Theorem 1.14 ([CM09b, Th. 1.5]). Let X be a locally compact, geodesically complete, cocompact CAT(0) space. Assume that the full isometry group Is(X) contains a lattice Γ which is ﬁnitely generated, residually ﬁnite, and indecomposable in the sense that it does not split non-trivially as a direct product, even virtually. If X admits some parabolic isometry, then X is a product of symmetric spaces and Euclidean buildings. The hypothesis that Γ < Is(X) be a ﬁnitely generated lattice is automatically satisﬁed if Γ is a discrete group acting properly and cocompactly on X. The content of this course includes some of the main ingredients coming into the proofs of Theorems 1.13 and 1.14. 7. Exercises Exercise 1.1. Let X be a proper metric space and let G ≤ Is(X). Show that the orbit space G\X is compact if and only if there is a ball in X which meets every G-orbit.

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Exercise 1.2. Let (X1 , d1 ) and (X2 , d2 ) be CAT(0) spaces. Given p ∈ [1, ∞), let dp be the metric on the cartesian product X = X1 × X2 deﬁned by dp = dp1 + dp2 . Show that (X, dp ) is a CAT(0) space if and only if p = 2. Exercise 1.3. (i) Let X = X1 × X2 be a CAT(0) product space. Show that X is geodesically complete if and only if X1 and X2 are both so. (ii) Show that every CAT(0) space embeds as a convex subset in some geodesically complete CAT(0) space. (iii) Show that Theorem 1.8(ii) fails for spaces that are not locally compact. Exercise 1.4. Let M = SLn (R)/SO(n) and d : M × M → R be the map deﬁned by (4.1). (i) Show that d is well deﬁned, i.e. it does not depend on the choice of a diagonalizing basis. (ii) Given a positive deﬁnite n × n matrix A, we denote by λ(A) the vector formed by its eigenvalues put in non-increasing order. A result by Lidskii (see [Bha97]) asserts that for A, B positive deﬁnite, one has log(λ(AB)) ≺ log λ(A) + log λ(B). The expression (x1 , . . . , xn ) ≺ (y1 , . . . , yn ) for two nonincreasing sequences means that the latter sequence majorizes the former,

k

k

n

n i.e. i=1 xi ≤ i=1 yi for all k, and i=1 xi = i=1 yi . Use Lidskii’s result to show that (M, d) is a metric space. (iii) (Open problem) Find a direct proof that (M, d) is a CAT(0) space, without using diﬀerential geometry.

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LECTURE 2

Geometric density 1. A geometric relative of Zariski density Let X be a CAT(0) space and G < Is(X) be a group of isometries. The G-action is called minimal if G does not preserve any non-empty closed convex subset X X. The group G is called geometrically dense if G acts minimally and without a ﬁxed point at inﬁnity of X. (For a brief recap on points at inﬁnity, see §2 below.) This notion can be viewed as coarsely related to Zariski density in the case of linear groups. Indeed, if X is a symmetric space of non-compact type, then any geometrically dense subgroup G < Is(X) is Zariski dense. This can be deduced from the Karpelevic–Mostow theorem. If in addition X is irreducible of rank ≥ 2, the converse holds by a theorem of Kleiner and Leeb [KL06]. In rank one symmetric spaces, there exist Zariski dense subgroups which do not act minimally (see Exercise 2.10). A CAT(0) space is called irreducible if it does not split as a CAT(0) product space in a non-trivial way. The symmetric space associated with a simple Lie group is always irreducible, as is the Euclidean building associated with a simple algebraic group over a local ﬁeld. The following property of the full isometry group of a proper CAT(0) space could be viewed as some very weak form of ‘simplicity’. Theorem 2.1 ([CM09a, Th. 1.10]). Let X be a proper CAT(0) space which is irreducible, not isometric to the real line, and has ﬁnite-dimensional visual boundary ∂X. Given a geometrically dense subgroup G < Is(X), any normal subgroup N G is either trivial or geometrically dense. If X = R is the real line, a non-trivial normal subgroup N G still acts minimally on X, but may obviously ﬁx the two elements of ∂X. The notion of dimension referred to in the theorem and appearing frequently in the rest of these notes, is Kleiner’s geometric dimension deﬁned in the previous lecture. The condition that X has ﬁnite-dimension visual boundary ∂X is automatic if X is cocompact (see [Kle99, Th. C]), or if X itself is ﬁnite-dimensional (see [CL10, Prop. 2.1]). 2. The visual boundary The visual boundary of X is the set of asymptotic classes of geodesic rays. It is denoted by ∂X. The visual boundary ∂X comes equipped with two diﬀerent natural topologies, which are both preserved by Is(X): • The cone topology, which is deﬁned by viewing X ∪ ∂X as the space of all geodesic segments and rays issuing from some ﬁxed base point, endowed with the topology of uniform convergence on bounded subsets. 103 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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That topology is independent of the choice of the base point. Moreover, when X is proper, the space X ∪ ∂X is compact by the Arzela–Ascoli theorem, the subset X is open and dense, and ∂X is closed, hence compact. In particular X ∪ ∂X with the cone topology is a compactiﬁcation of X, usually called the visual compactiﬁcation. • The topology induced by the angular metric. The angle between two points ξ, η ∈ ∂X is deﬁned by ∠(ξ, η) = sup ∠x (ξ, η), x∈X

where ∠x (ξ, η) denotes the Alexandrov angle at x between the unique geodesic rays issuing from x and pointing to ξ and η respectively. The angular metric is indeed a metric, and the associated topology is ﬁner (and often strictly ﬁner) than the cone topology. For a detailed treatment of the visual boundary, see [BGS85, §3–4], [Bal95, § II.1–II.4] and [BH99, §II.8–II.9]. A fundamental fact is that if X is complete, then the metric space (∂X, ∠) is a complete CAT(1) space, see [BH99, Th. II.9.20]. CAT(1) spaces are metric spaces with an upper curvature bound: they are deﬁned in a similar way as CAT(0) spaces, except that comparison triangles are chosen in the unit 2-sphere (and moreover the deﬁning inequality is only requested for geodesic triangles of perimeter less than 2π). Loosely speaking, CAT(1) spaces are to spherical geometry what CAT(0) spaces are to Euclidean geometry. More information on CAT(1) geometry can be found in the aforementioned references; here we shall content ourselves with mentioning a few facts needed in the sequel. It is well known that a bounded subset Z of a complete CAT(0) space X admits a unique circumcenter, i.e. a unique point c such that Z is contained in the closed ball B(c, R) of radius R around c, where R is deﬁned as R = inf x∈X {r ∈ R | Z ⊂ B(x, r)}. The number R is called the circumradius of Z. In CAT(1) geometry, a similar statement holds provided the subset Z is assumed to have circumradius < π/2, see [BH99, Prop. II.2.7]. The following important result, due to Balser and Lytchak, shows that this can be extended to sets of circumradius ≤ π/2 provided the set Z is convex and ﬁnite-dimensional: Theorem 2.2 ([BL05, Prop. 1.4]). Let Z be a ﬁnite-dimensional complete CAT(1) space. If Z has circumradius π/2, then the set of circumcenters of Z has circumradius < π/2. In particular the full isometry group Is(Z) ﬁxes a point in Z. We emphasize that Theorem 2.2 fails without the ﬁnite-dimensionality assumption, see Exercise 2.3. The typical situation in which we shall apply Theorem 2.2 is the following: the space Z will be a closed convex subset of the visual boundary ∂X of a proper CAT(0) space X. As mentioned above, the ﬁnite-dimensionality hypothesis is automatically satisﬁed if X is cocompact, since the full visual boundary ∂X is then ﬁnite-dimensional. It should be noted that the circumradius of Z as a subset of X may be smaller than the intrinsic circumradius of Z, deﬁned by inf z∈Z {r ∈ R | Z ⊂ B(z, r)}. It is of course the intrinsic circumradius that has to be used when applying Theorem 2.2 to a closed convex subset Z ⊆ ∂X. In the situations we shall encounter, the upper bound of π/2 on the circumradius will be deduced from the following observation.

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Proposition 2.3. Let X be a proper CAT(0) space, and (Yi )i∈I be a descending chain of closed convex subsets. If i∈I Yi is empty, then i∈I ∂Yi is a non-empty closed convex subset of ∂X, whose circumradius is at most π/2. Proof. Pick x ∈ X and let yi be its orthogonal projection to Yi . If the set (yi )i∈I is bounded, then i∈I Yi is non-empty. Assume that this is not the case. We can then extract a countable chain (Yi(n) )n≥0 such that the sequence (yi(n) ) converges to to the cone topology. In some boundary point ξ ∈∂X with respect particular n Yi(n) is empty and Z = n ∂Yi(n) = i Yi . Notice moreover that ξ belongs to Z. It remains to show that for each η ∈ Z, we have ∠(ξ, η) ≤ π/2. To this end, observe that there is a sequence yn ∈ Yi(n) converging to η in the cone topology. We have π/2 ≤ ∠yi(n) (x, yn ) by the properties of the projection [BH99, Prop. II.2.4], and ∠yi(n) (x, yn ) ≤ ∠yi(n) (x, yn ) by the CAT(0) condition, where ∠ denotes the angle in a Euclidean comparison triangle. It follows that ∠x (yi(n) , yn ) ≤ π/2. By [BH99, Prop. II.9.16], this implies that ∠(ξ, η) ≤ π/2, as desired. 3. Convexity A map f : X → R is called convex if for each geodesic ρ : I → X, the composed map f ◦ ρ : I → R is convex. In that case, sublevel sets of f are convex subsets of X. Here are a few examples: • Given a point p ∈ X, the distance to p, namely dp : X → R : x → d(x, p) is convex: this follows right away from the CAT(0) condition. Its sublevel sets are nothing but balls around p. • Given a complete convex subset Y ⊂ X, the distance to Y , namely dY : X → R : x → d(x, Y ) = inf d(x, y) y∈Y

is convex, see [BH99, Cor. II.2.5]. Its sublevel sets are called tubular neighbourhoods of Y and denote by Nr (Y ) = f −1 ([0, r]). • Given a geodesic ray ρ : [0, ∞) → X, the function bρ : X → R deﬁned by bρ (x) = lim d(x, ρ(t)) − t t→∞

is well deﬁned, convex and 1-Lipschitz, see Exercise 2.7. It is called the Busemann function associated with ρ. Its sublevel sets are called horoballs centered at the endpoint ξ = ρ(∞). If ρ is another geodesic ray having ξ as endpoint, then the Busemann functions bρ and bρ diﬀer by a constant, so that the collection of horoballs centered at ξ does not depend on the choice of a geodesic ray pointing to ξ. • Given an isometry g ∈ Is(X), its displacement function dg : X → R deﬁned by dg (x) = d(x, g.x) is convex and 2-Lipschitz, see Exercise 2.5. The inﬁmum of the displacement function is called the translation length, and is denoted by |g|. The sublevel set f −1 ([0, |g|]) = f −1 (|g|), which is thus closed and convex, is denoted by Min(g). It is non-empty if and only if g is not parabolic.

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The existence of isometries with a constant displacement function witnesses the presence of a Euclidean factor: Proposition 2.4 ([BH99, Th. II.6.5]). A CAT(0) space X admits a nontrivial isometry with constant displacement function if and only if X splits as a product X ∼ = R × X . We record the following consequence: Corollary 2.5. Let X be a CAT(0) space without non-trivial Euclidean factor. For any group G < Is(X) acting minimally on X, the centraliser ZIs(X) (G) is trivial. In particular so is the center Z (G). Proof. Let g ∈ ZIs(X) (G). Then the displacement function dg is G-invariant in the sense that dg is constant on each G-orbit. In particular G preserves all sublevel sets of dg , which are closed and convex. By minimality, it follows that dg has no non-trivial sublevel set; in other words dg is constant, and Proposition 2.4 concludes the proof. Here is another straightforward application of convexity. Lemma 2.6. Let X be a complete CAT(0) space. Given G < Is(X) and two points y, z ∈ X, we have ∂Conv(G.y) = ∂Conv(G.z), where Conv(Y ) denotes the convex hull of Y , and G.y the G-orbit of y. Proof. Set Y = Conv(G.y) and Z = Conv(G.z). Then Y and Z are both G-invariant. Setting r = d(Y, z), we obtain G.z ⊆ Nr (Y ). Since Nr (Y ) is closed and convex, this yields Z ⊆ Nr (Y ). Similarly Y ⊆ Nr (Z), and hence Y and Z are a bounded Hausdorﬀ distance apart. Therefore ∂Y = ∂Z, see Exercise 2.1. Lemma 2.6 allows one to associate a canonical subset ΔG ⊆ ∂X to the group G, deﬁned as the visual boundary of the closed convex hull of some (arbitrarily chosen) orbit. We call ΔG the convex limit set of G. It contains (generally as a proper subset) the usual limit set ΛG, which is deﬁned as the intersection with the visual boundary ∂X of the closure of some G-orbit in the union X ∪ ∂X, endowed with the cone topology. Notice that the convex limit set is deﬁned as the visual boundary of a complete CAT(0) subspace of X, and is thus a complete CAT(1) space. In other words, it is closed and convex in ∂X. 4. A product decomposition theorem Let X be a complete CAT(0) space and be a geodesic line in X. It is then a standard fact (see [BH99, Th. II.2.14]) that the union P( ) of all geodesic lines having the same endpoints as in the visual boundary, is a closed convex subset of X, which splits as a CAT(0) product P( ) ∼ = R × C. This fact is actually the key point in the proof of Proposition 2.4. Our next task is to extend that statement to more general subspaces than lines. To this end, we need an additional piece of terminology. A closed convex subset Y ⊆ X is called boundary-minimal if for every closed convex subset Z Y , we have ∂Z Y . For instance, a geodesic line is boundaryminimal while a geodesic ray is not. Theorem 2.7 ([CM09a, Prop. 3.6]). Let X be a proper CAT(0) space and let Δ ⊆ ∂X. Set CΔ = {Y ⊆ X | Y is boundary-minimal and ∂Y = Δ}. Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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Then the union CΔ is a closed convex subset which splits as a CAT(0) product

CΔ ∼ = Y × C. Moreover CΔ coincides with the set of ﬁbers Y × {c} | c ∈ C . Proof. We shall only prove the key point, namely the fact that for any two sets Z1 , Z2 ∈ CΔ , the distance function to Z1 , denoted by dZ1 , is constant on Z2 . Let Z2 ⊆ Z2 be a non-empty sublevel set of the restriction of dZ1 to Z2 . Thus there is some r > 0 such that Z2 = {z ∈ Z2 | dZ1 (z) ≤ r}, and Z2 is closed and convex. Let then ξ ∈ Δ and pick any p ∈ Z2 . Let also ρ : [0, ∞) → X be the geodesic ray issuing from p and pointing to ξ. Since ξ ∈ Δ = ∂Z2 and since Z2 is closed and convex, it follows that the point ρ(t) belongs to Z2 for all t. Since ξ also belongs to ∂Z1 , the ray ρ([0, ∞)) is entirely contained in a tubular neighbourhood of Z1 . It follows that the map t → dZ1 (ρ(t)) is bounded convex function. It must therefore be non-increasing. Since ρ(0) = p ∈ Z2 , it follows that ρ(t) ∈ Z2 for all t. In particular ξ belongs to ∂Z2 . This proves that ∂Z2 = ∂Z2 . Since Z2 is boundaryminimal, we deduce that Z2 = Z2 which proves that the function dZ1 is constant on Z2 , as claimed. The rest of the proof of the theorem uses the Sandwich Lemma [BH99, Ex. II.2.12], and is similar to the special case of the parallel set of a geodesic line mentioned above. Further details are provided in [CM09a, Prop. 3.6]. Remark that the set CΔ is potentially empty. 5. Geometric density of normal subgroups We are now in a position to complete the proof of geometric density for normal subgroups. Proof of Theorem 2.1. Let N G be a non-trivial normal subgroup and Δ = ΔN be its convex limit set. If Δ is empty, then the N -orbits are bounded and the ﬁxed point set X N of N is thus a non-empty closed convex G-invariant subset. By minimality, we must have X N = X, whence N is trivial. We assume henceforth that Δ is non-empty and consider the set CΔ . Suppose now that CΔ is empty. Then by Zorn’s lemma, there exists a chain of closed convex subspaces (Yi )i∈I such that ∂Yi = Δ for all i, and i Yi = ∅. By Proposition 2.3, it follows that Δ has intrinsic circumradius at most π/2. Since Δ is G-invariant, Theorem 2.2 implies that G ﬁxes a point in ∂X, a contradiction. Thus CΔ is non-empty. By Theorem 2.7, the union CΔ is then a non-empty closed convex subset splitting as a product of the form Y × C with all ﬁbers Y × {c} belonging to CΔ . Since CΔ is G-invariant and since the G-action on X is minimal, it follows that X = CΔ . Since X is irreducible, the product decomposition X∼ = Y ×C must be trivial. Thus either Y or C is reduced to a singleton. The former case is impossible, since it would mean that the elements of CΔ are singletons, which is absurd since they have a non-empty visual boundary. Thus X ∼ = Y × {c}, which implies that X belongs to CΔ . Thus X is boundary-minimal. It follows that N acts minimally on X. Indeed, given a non-empty closed convex N -invariant subset Z X, we have ΔN ⊆ ∂Z ⊆ ∂X. Since ΔN = ∂X, we have ∂Z = ∂X, whence Z = X since X is boundary-minimal. This proves that any non-trivial normal subgroup N G acts minimally on X. It remains to show that N does not ﬁx any point at inﬁnity. Suppose on

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the contrary that N ﬁxes some ξ ∈ ∂X. Then the commutator subgroup [N, N ] annihilates the Busemann character centered at ξ (see Exercise 2.7) and therefore stabilises each horoball around ξ. In particular it does not act minimally on X. But N being normal in G, its commutator subgroup [N, N ] is also normal in G, and is thus trivial by the ﬁrst part of the proof. Thus N is abelian. This is absurd, since a group acting minimally on CAT(0) space without Euclidean factor must be center-free by Corollary 2.5. Remark that the ﬁnite-dimensionality of ∂X was only used through the application of Theorem 2.2. It is an interesting question to determine whether Theorem 2.1 holds if X is a proper CAT(0) space with inﬁnite-dimensional visual boundary. Clearly Theorem 2.1 can be bootstrapped, thereby giving information on subnormal subgroups: Corollary 2.8. Let X be a proper cocompact CAT(0) space which is irreducible, and not isometric to the real line. Let G < Is(X) be a geometrically dense subgroup and H < G be a non-trivial subnormal subgroup. Then H is still geometrically dense; in particular: (i) ZG (H) = 1, (ii) H does not split non-trivially as a direct product, (iii) H is not soluble, (iv) H does not have ﬁxed points in X. Proof. That H is geometrically dense is immediate from an iterated application of Theorem 2.1, and (iv) follows right away. Part (i) is a consequence of Corollary 2.5, Part (ii) follows from (i). Part (i) also implies that a subnormal subgroup cannot be abelian, which implies (iii). 6. Exercises Exercise 2.1. Let X be a CAT(0) space and Y, Z ⊆ X be two convex subsets. Show that if Y and Z are a bounded Hausdorﬀ distance apart, then ∂Y = ∂Z. The converse does not hold in general. Exercise 2.2. Construct an example of a proper cocompact CAT(0) space X whose full isometry group is minimal, but not geometrically dense. Exercise 2.3. Show that Theorem 2.2 fails if Z is inﬁnite-dimensional. (Hint: a counterexample may be constructed as a closed convex subset of the unit sphere in a Hilbert space). Exercise 2.4. Let X be a proper CAT(0) space and let G < Is(X). (i) Show that if G does not ﬁx any point in ∂X, then G stabilises a non-empty closed convex subset X ⊆ X on which its action is minimal. This minimal G-invariant subspace X need not be unique, even if G acts without ﬁxed point in X. (ii) Show that if G acts cocompactly on X, then the same conclusions hold. (iii) Show that if X is geodesically complete and G acts cocompactly, then G acts minimally. Exercise 2.5. Show that the displacement function of an isometry of a CAT(0) space is convex and 2-Lipschitz.

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Exercise 2.6. Let X be a metric space and G < Is(X). A function f : X → R is called G-invariant if f is constant on G-orbits, namely f (g.x) = f (x) for all x ∈ X and g ∈ G. A function f : X → R is called G-quasi-invariant if for all g ∈ G, the map X → R : x → f (g.x) − f (x) is constant. Assuming this is the case, we denote the diﬀerence by c(g). Show that the map G → R : g → c(g) is a homomorphism. Exercise 2.7. Let X be a CAT(0) space. (i) Show that Busemann functions associated with geodesic rays in X are well deﬁned, convex and 1-Lipschitz. (ii) Show that any Busemann function associated with a geodesic ray pointing to ξ ∈ X is quasi-invariant under the stabiliser Gξ of ξ in the full isometry group G = Is(X). (iii) Show that the corresponding homomorphism Gξ → R deﬁned as in Exercise 2.6 depends only on ξ. This homomorphism is called the Busemann character at ξ. Exercise 2.8. Let X be a complete CAT(0) space and G < Is(X). (i) Show that if X is geodesically complete, then every bounded convex function is constant. (ii) Show that if X is boundary-minimal, then every bounded convex function is constant. (iii) Show that G acts minimally on X if and only if every continuous G-invariant convex function is constant. (iv) Show that G is geometrically dense if and only if every continuous G-quasiinvariant convex function is constant. Exercise 2.9. An action of a group G on a topological space Z by homeomorphism is called (topologically) minimal1 if G does not preserve any non-empty closed subset Z Z. Equivalently, the G-action is minimal if and only if every G-orbit is dense in Z. Let M denote the symmetric space of G = SLn (R). Show that the G-action on the visual boundary ∂M is minimal if and only if n = 2. Exercise 2.10. Let G = SL2 (R). (i) Show that a subgroup Γ < G is Zariski dense if and only if Γ is not virtually soluble. (ii) Show that G contains Zariski dense subgroups that are not geometrically dense as isometry groups of the hyperbolic plane H2 . Exercise 2.11. Let X be a proper CAT(0) space. (i) Show that if X is boundary-minimal, then ∂X has circumradius > π/2. (ii) Show that if X has ﬁnite-dimensional boundary and if Is(X) acts minimally, then X is boundary-minimal.

1 This

standard notion of minimality in topological dynamics should not be confused with the notion of minimality introduced above in the realm of CAT(0) geometry.

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LECTURE 3

The full isometry group 1. Locally compact groups Our strategy in studying the full isometry group of a proper CAT(0) space is to combine geometric arguments with information arising from the structure theory of locally compact groups. The following classical fact shows that locally compact groups pop up naturally in our setting: Theorem 3.1. Let X be a proper metric space. Then the full isometry group Is(X), endowed with the compact open topology, is a locally compact (Hausdorﬀ ) topological group, and the natural action of Is(X) on X is continuous and proper.

Proof. See Exercise 3.1.

The continuity of the action of G = Is(X) on X means that the map G×X → X is continuous. The properness of the action means that for each ball B in X, the set {g ∈ G | g.B ∩ B = ∅} has compact closure in G. A deep result in the theory of locally compact groups which we shall invoke is the following: Theorem 3.2 (Gleason–Yamabe [MZ55, Th. IV.4.6]). Let G be a connected locally compact group. Then any identity neighbourhood in G contains a compact normal subgroup K G such that G/K is a Lie group. 2. The isometry group of an irreducible space Combining the results obtained thus far yields the following. Corollary 3.3. Let X be a proper CAT(0) space with ﬁnite-dimensional boundary, such that X is irreducible and Is(X) is geometrically dense. Then Is(X) is either a virtually connected simple Lie group, or Is(X) is totally disconnected (potentially discrete). Proof. Let G = Is(X). Thus G is a locally compact group by Theorem 3.1. The connected component of the identity G◦ is a closed normal subgroup of G. By Theorem 3.1, any compact subgroup of G has a bounded orbit, hence a ﬁxed point in X. Corollary 2.8(iv) thus ensures that G the only compact subnormal subgroup of G is trivial. In particular G◦ has no non-trivial compact normal subgroup, and must thus be a connected Lie group by Theorem 3.2. Corollary 2.8(iii) implies that the solvable radical, as well as the center, of G◦ is trivial, hence G◦ is a center-free semi-simple Lie group. It is thus a product of simple groups, which can have at most one non-trivial factor by Corollary 2.8(ii). This shows that G◦ is a centerfree simple Lie group. 111 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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A consequence of the classiﬁcation of simple Lie groups is that the outer automorphism group Out(G◦ ) is ﬁnite. The conjugation action of G on G◦ yields a continuous map ϕ : G → Out(G◦ ), whose kernel is thus a closed normal subgroup of G of ﬁnite index. Notice that Ker(ϕ) = G◦ · ZG (G◦ ). By Corollary 2.8(i), either G◦ or its centraliser must be trivial. In the former case, the group G is totally disconnected. In the latter case, the identity component G◦ has ﬁnite index in G, so that G is virtually a connected simple Lie group. It is not surprising that, in the Lie group case of Corollary 3.3, much ﬁner information on X can be extracted from the structure theory of simple Lie groups. Each maximal compact subgroup K < G = Is(X) ﬁxes a point in X, and we thus get an equivariant embedding of the symmetric space M = G/K into X. Notice however that this embedding need not be isometric, even up to scaling. Explicit examples of this phenomenon have recently been constructed by Monod and Py [MP12] with G = SO(n, 1) acting cocompactly on a proper CAT(0) space X, containing no isometric (and even homothetic) copy of the hyperbolic space Hn . Of course, the cocompactness of the action implies that X is quasi-isometric to the symmetric space of G. That X is genuinely isometric to the symmetric space is however true if one imposes in addition that X be geodesically complete: Theorem 3.4 ([CM09a, Th. 7.4]). Let X be a locally compact geodesically complete CAT(0) space and G be a virtually connected semi-simple Lie group acting continuously, properly and cocompactly on X by isometries. Then X is equivariantly isometric to the symmetric space of G (up to an appropriate scaling of each irreducible factor). The same conclusion holds under the slightly weaker hypotheses that the action is minimal with full limit set, and that the boundary of X is ﬁnite-dimensional. One should next analyze the totally disconnected case of Corollary 3.3. Since that case includes the situation that Is(X) be discrete, conclusions in the same vein as those of Theorem 3.4 cannot be expected. The following useful facts can however be derived under the hypothesis of geodesic completeness: Theorem 3.5 ([CM09a, §6]). Let X be a locally compact geodesically complete CAT(0) space and G be a totally disconnected locally compact group acting continuously and properly on X by isometries. Then: (i) The action is smooth in the sense that the pointwise stabiliser of every open set is open in G. (ii) Every G-orbit is discrete. (iii) If the G-action is cocompact, then G does not contain parabolic isometries. (iv) If the G-action is cocompact, then X admits a locally ﬁnite G-equivariant decomposition into convex pieces, such that the piece σ(x) supporting a point x ∈ X is deﬁned as the ﬁxed-point-set of the stabiliser Gx . Remark that if X is geodesically complete, any group acting cocompactly automatically acts minimally (see Exercise 2.4). Therefore, the following dichotomy follows immediately by combining the previous three results. Corollary 3.6. Let X be a locally compact, geodesically complete, irreducible CAT(0) space such that Is(X) acts cocompactly without a ﬁxed point at inﬁnity. Then either Is(X) acts transitively on X, or Is(X) has discrete orbits.

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It should be emphasized that this dichotomy no longer holds without the hypothesis that Is(X) has no ﬁxed point at inﬁnity. Concrete examples illustrating this matter of fact are provided by the millefeuille spaces constructed in [CCMT12, §7]. 3. de Rham decomposition In the previous section, we focused on irreducible CAT(0) spaces. One should next show that the general case reduces to the irreducible one. This will require to impose suitable assumptions, since a ‘de Rham decomposition theorem’ cannot be expected in full generality for CAT(0) spaces, due to the possible presence of inﬁnite-dimensional pieces. This happens even for locally compact spaces: a CAT(0) space can very well be compact and inﬁnite-dimensional, as is easily seen by considering compact convex subsets of a Hilbert space. The following remarkable result, due to Foertsch and Lytchak, shows that inﬁnite-dimensionality is the only obstruction to a ‘de Rham decomposition’ at a very broad level of generality: Theorem 3.7 (Foertsch–Lytchak [FL08]). Let X be a ﬁnite-dimensional geodesic metric space. Then X admits a canonical product decomposition X∼ = Rn × X1 × · · · × Xp , where n, p ≥ 0, and where each factor Xi is irreducible, and neither reduced to a singleton, nor isometric to the real line (the right-hand side is given the 2 -metric). Every isometry of X preserves the decomposition, up to a permutation of possibly isometric factors among the Xi . In particular Is(Rn ) × Is(X1 ) × · · · × Is(Xp ) is a ﬁnite-index normal subgroup of Is(X). In the case of CAT(0) spaces, we have the following analogue: Theorem 3.8 ([CM09a, Cor. 5.3]). Let X be a proper CAT(0) space with ﬁnite-dimensional visual boundary ∂X, and such that Is(X) acts minimally. Then X admits a canonical CAT(0) product decomposition, with the same properties as in Theorem 3.7. The latter statement cannot be deduced directly from Theorem 3.7, since the hypotheses do not imply in general that X itself be ﬁnite-dimensional. A detailed proof of Theorem may be found in [CM09a, §5.A]. An alternative approach in case X is cocompact can be taken using the following. Proposition 3.9. Let Z be a ﬁnite-dimensional, complete CAT(1) space. Then Z admits a canonical decomposition as a join Z∼ = Sn ◦ Z1 ◦ · · · ◦ Zp , where Sn is the Euclidean n-sphere and each Zi is not a sphere and does not decompose non-trivially as a join for all i. Every isometry of Z preserves the decomposition, up to a permutation of possibly isometric factors among the Zi . Proof. Let X be the Euclidean cone over Z, deﬁned as in [BH99, Def. I.5.6]. By Berestovskii’s theorem [BH99, Th. II.3.14], the space X is a CAT(0) space, which is ﬁnite-dimensional since Z is so. (However X is not locally compact in general.) Every isometry of Z extends to an isometry of the cone X. The conclusion now follows by applying Theorem 3.7 to X.

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One may now conclude the proof of Theorem 3.8 as follows. Proof of Theorem 3.8. By Exercise 2.11, the space X is boundary-minimal. It follows that for every product decomposition X ∼ = Y1 × · · · × Yq , each factor Yi is unbounded (see Exercise 2.1) and thus has a non-empty visual boundary ∂Yi . In other words, every product decomposition of X determines a join decomposition of the visual boundary ∂X, the factors in both decompositions being canonically in one-to-one correspondence. From Proposition 3.9, it follows that X admits at least one product decomposition X ∼ = Rn × X1 × · · · × Xq with a maximal Euclidean n factor R and ﬁnitely many irreducible non ﬂat factors. At this point, we know that X admits at least one decomposition as a product of ﬂat and irreducible factors, and that each such decomposition corresponds to some regrouping of factors in the canonical join decomposition of ∂X aﬀorded by Proposition 3.9. The desired result now follows from Proposition 3.10 below, which implies that the visual boundary of an irreducible factor of X does not admit any non-trivial join decomposition. Indeed, this shows that the various factors in the maximal decompositions of X and ∂X are canonically in one-to-one correspondence, so that the canonicity of the decomposition of X follows from that of ∂X. Proposition 3.10. Let X be a proper CAT(0) space such that Is(X) acts cocompactly and minimally. Then X admits a non-trivial product decomposition X = X1 × X2 if and only if ∂X admits a join decomposition ∂X = Δ1 ◦ Δ2 with Δi = ∂Xi . Proof. (See [BH99, Th. II.9.24] for the case when X is geodesically complete.) The ‘only if’ part is clear. We assume henceforth that ∂X = Δ1 ◦ Δ2 . Since X is cocompact, it follows from [GO07] that every point ξ in ∂X admits some opposite ξ , i.e. ξ is such that ξ and ξ are the endpoints of some geodesic line. Moreover, given ξ ∈ Δi , any point ξ opposite ξ also belongs to Δi . By intersecting two horoballs respectively centered at ξ and ξ , one constructs closed convex subsets of X whose visual boundary is exactly Δ3−i (see [CM09a, Lem. 3.5]). From Exercise 2.11, we infer that X is boundary-minimal, and hence that each factor Δi must have radius > π/2. Therefore, the set CΔi from Theorem 2.7 is non-empty by Proposition 2.3. Theorem 2.7 then provides a product decomposition X = X1 × X2 such that ∂Xi = Δi , as desired. The possibility that Is(X) may ﬁx a point at inﬁnity is not excluded in Theorem 3.8, and does indeed occur sometimes (see Exercise 2.2). However, assuming that the full isometry group is geometrically dense, the results obtained thus far assemble to yield the following, which already sheds some light on the conclusions of Theorem 1.14. Corollary 3.11. Let X be a locally compact geodesically complete CAT(0) space. Assume that Is(X) acts cocompactly without a ﬁxed point at inﬁnity. Then X admits a canonical product decomposition X∼ = M1 × · · · × Mp × Rn × Y1 × · · · × Yq , which is preserved by all isometries upon permutations of isometric factors, where Mi is an irreducible symmetric space of non-compact type, and Yj has a totally disconnected isometry group, which acts smoothly and does not contain any parabolic isometry. Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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Proof. Since X is geodesically complete, any cocompact group action is minimal (see Exercise 2.4). Theorem 3.8 provides a canonical product decomposition for X, and the various properties of the irreducible non Euclidean factors were established in Corollary 3.3 and Theorems 3.4 and 3.5. 4. Exercises Exercise 3.1. Let X be a proper metric space and let Is(X)p.o. denote the full isometry group of X endowed with the point-open topology. Let also ϕ : Is(X) → X X be the natural embedding of Is(X) in the space X X of all maps from X to X, endowed with the product topology. (i) Show that ϕ(Is(X)) is closed in X X . (ii) Show that ϕ : Is(X)p.o. → X X is a homeomorphism onto its image. (iii) Deduce from (i) and (ii) that Is(X)p.o. is locally compact. (iv) Show that the point-open and the compact-open topology on Is(X) coincide. (v) Conclude the proof of Theorem 3.1. Exercise 3.2. Let G be a locally compact group acting by isometries on a proper metric space X. (i) Show that the following conditions are equivalent: (a) the G-action is continuous, (b) the orbit maps G → X : g → g.x are continuous for all x ∈ X, (c) the homomorphism α : G → Is(X) induced by the action is continuous. (ii) Assuming that the G-action is continuous, show that the following conditions are equivalent: (a) the G-action is proper, (b) the homomorphism α : G → Is(X) induced by the action is continuous is proper. If in addition G is separable, then those conditions are also equivalent to: (c) Ker(α) is compact and α(G) is closed in Is(X). Exercise 3.3. Let X be a locally compact geodesically complete CAT(0) space. (i) Prove the assertions (ii) and (iii) in Theorem 3.5 using point (i). (ii) Show that if a non-discrete totally disconnected locally compact group G acts continuously and properly on X, then some geodesic in X must branch. (iii) Show that if Is(X) is geometrically dense and every geodesic can be prolonged into a unique bi-inﬁnite geodesic line, then Is(X) is a Lie group.

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LECTURE 4

Lattices 1. Geometric Borel density The phenomenon of geometric density of normal subgroups has been discussed in Theorem 2.1. We shall now present a related statement for lattices. In the light of the analogy between geometric density and Zariski density, this could be interpreted as a geometric version of the Borel density theorem (in fact, the classical statement can indeed be deduced from the geometric version, see [CM09b, Prop. 2.8]). Theorem 4.1 ([CM09b, Th. 2.4]). Let X be a proper CAT(0) space without a non-trivial Euclidean factor. Let G be a locally compact group and ϕ : G → Is(X) be a continuous homomorphism. If ϕ(G) is geometrically dense, then so is ϕ(Γ) for each lattice Γ < G (and, more generally, for each closed subgroup of ﬁnite covolume). The proof consists in two parts: the ﬁrst is to show the absence of Γ-ﬁxed points at inﬁnity, which is established by adapting an argument of Adams and Ballmann [AB98]; the second is to show that the Γ-action is minimal. Since some technicalities can be avoided when Γ is assumed cocompact, we will content ourselves with a discussion of the second part of the proof in that special case. Proof of Theorem 4.1. For simplicity, we assume that Γ < G is cocompact and only discuss the proof of Γ-minimality; for a complete proof in the general case, the reader should consult [CM09b]. Hence we admit that the ﬁrst part of the proof has already been accomplished, namely that ϕ(Γ) does not ﬁx any point in ∂X. It follows that there is a nonempty Γ-invariant closed convex subsets Y ⊆ X on which Γ acts minimally (see Exercise 2.4). We need to show that Y = X. To this end, consider the distance function dY to Y . Since Γ is cocompact in G, it follows that for each x ∈ X, the map G → R : g → dgY (x) is continuous and bounded. In particular the function dgY (x)dg f : X → R : x → G/Γ

is well deﬁned. Moreover it is convex and 2-Lipschitz since dY is so. By construction, it is G-invariant. Since G acts minimally on X, the map f must be constant. It follows that for almost all gΓ ∈ G/Γ, the map dgY is aﬃne, i.e. it is both convex and concave (see Exercises 4.1 and 4.2). We have seen that there exists g ∈ G such that dgY is aﬃne. Since dhY = dY ◦ h−1 for all h ∈ G, we infer that dhY is aﬃne for all h; in particular so is dY . It follows from Exercise 4.1(ii) that the level sets of dY are all convex. Let Y be a level set of dY . Thus Y is closed, convex and Γ-invariant. Since Y is Γ-invariant and Γ-minimal, the restriction of dY to Y is also constant. Using the 117 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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Sandwich Lemma [BH99, Ex. II.2.12], one may conclude that Y is equivariantly isometric to Y via the orthogonal projection. This implies that Y is Γ-minimal, and that X decomposes as a product X ∼ = Y × C, so that the ﬁbers Y × {c} are precisely the minimal Γ-invariant closed convex subsets. If C contains two distinct points, we deﬁne m as their midpoint and consider the ﬁber Y = Y × {m}. Since it is a minimal Γ-invariant subspace, the arguments above show that dY is aﬃne. This is impossible by Exercise 4.1(iii). Thus C is reduced to a singleton, and hence Γ acts minimally. As in the case of normal subgroups, Theorem 4.1 yields algebraic restrictions on lattices: Corollary 4.2. Let X be a proper CAT(0) space without a non-trivial Euclidean factor, G < Is(X) be a closed subgroup which is geometrically dense and Γ be a lattice in G. Then: (i) ZG (Γ) = 1. (ii) If Γ is ﬁnitely generated, then NG (Γ) is a lattice (containing Γ as a ﬁnite index subgroup). Proof. (i) follows from Theorem 4.1 and Corollary 2.5. For (ii), observe that the ﬁnite generation assumption implies that Aut(Γ) is countable. Hence so is NG (Γ) by (i). Since the normaliser of a closed subgroup is closed, it follows that NG (Γ) is a countable locally compact group, and must thus be discrete by Baire’s category theorem. A discrete subgroup containing a lattice is itself a lattice, whence the conclusion. 2. Fixed points at inﬁnity Most results obtained so far used the condition that Is(X) be geometrically dense as a hypothesis. If a group G acts cocompactly, or without a ﬁxed at inﬁnity, on X, then there always exists some non-empty G-invariant closed convex subset Y ⊆ X on which G acts minimally (see Exercise 2.4). So the hypothesis that Is(X) acts minimally is inessential. On the other hand, one cannot expect that the full isometry group Is(X) of a proper CAT(0) space be always geometrically dense on some minimal invariant subspace Y ⊆ X (see Exercise 2.2 and the remark following Corollary 3.6). The next result shows that this is indeed the case provided the full isometry group contains a lattice. Theorem 4.3 ([CM13, Th. L]). Let X be a proper cocompact CAT(0) space and assume that Is(X) acts minimally. Let Γ < Is(X) be a lattice (e.g. a discrete group acting properly cocompactly on X). Then the only points in the visual boundary ﬁxed by Γ lie in the boundary of the maximal Euclidean factor of X. Since a locally compact group containing a lattice is unimodular, Theorem 4.3 follows by combining the following result with the geometric Borel density from the previous section: Theorem 4.4 ([CM13, Th. M]). Let X be a proper cocompact CAT(0) space and assume that Is(X) acts minimally. If Is(X) is unimodular, then Is(X) has no ﬁxed point at inﬁnity.

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Notice that the minimality assumption in both theorems is harmless: indeed, since the action is assumed cocompact, we may simply replace X by some minimal Is(X)-invariant subspace Y ⊆ X. One should however be aware that Y may admit isometries that do not extend to X. It is thus conceivable (and it indeed happens, see Exercise 4.3) that Is(X) ﬁxes points at inﬁnity, while Is(Y ) never does by Theorem 4.4. The proof of Theorem 4.4 requires further geometric preliminaries and is thus postponed to the next section. A weaker version of Theorem 4.3 was ﬁrst proved in [CM09b, Th. 3.14] under the additional hypothesis that Γ be ﬁnitely generated. At this point, let us merely present the simplest version of the argument, due to Burger–Schroeder [BS87], under the stronger assumption that Γ is cocompact; it can be proved directly, without invoking Theorem 4.4: Lemma 4.5. Let X be a proper CAT(0) space and Γ be a discrete group acting properly cocompactly on X. If a ﬁnitely generated subgroup Λ < Γ ﬁxes some ξ ∈ ∂X, then Λ ﬁxes some ξ which is opposite ξ in the sense that {ξ, ξ } are the endpoints of a geodesic line. Applying the lemma to the whole group Λ = Γ, which is ﬁnitely generated since it is cocompact, we ﬁnd an opposite pair of Γ-ﬁxed points. Assuming in addition that the Γ-action is minimal, the Product Decomposition Theorem (see Theorem 2.7) then yields a Γ-invariant splitting X ∼ = R × X such that ξ and ξ are the endpoints of the line factor. Thus the conclusion of Theorem 4.3 holds in case Γ is cocompact. 3. Boundary points with a cocompact stabiliser Let X be a proper CAT(0) space and ξ ∈ ∂X be a boundary point. We deﬁne the following subgroup of the stabiliser Is(X)ξ of ξ: Is(X)uξ = g ∈ Is(X)ξ : lim d g · r(t), r(t) = 0 ∀ r with r(∞) = ξ , t→∞

where r is a geodesic ray. One veriﬁes that Is(X)uξ is a closed normal subgroup of Is(X)ξ (see Exercise 4.4). Notice that Is(X)uξ is contained in the kernel of the Busemann character βξ centered at ξ (see Exercise 2.7). In fact, the subgroup Is(X)uξ may be view as the intersection of Ker(βξ ) with the kernel of the action of Is(X)ξ on some other CAT(0) space denoted Xξ and called the transverse space at ξ. It is deﬁned as a completed quotient of the space of all geodesic rays pointing to ξ. We refer to [CM13] for details. The subgroup Is(X)uξ can be interpreted as a unipotent radical of the stabiliser Is(X)ξ , which justiﬁes the choice of notation. This interpretation is motivated by a version of the Levi decomposition theorem for parabolic subgroups of semi-simple Lie or algebraic groups, which can be established for CAT(0) spaces as soon as the stabiliser Is(X)ξ acts cocompactly on X (see [CM13, Th. J and Th. 3.12]). In the present notes, we will only use the following fact, which can be deduced from the aforementioned Levi decomposition: Proposition 4.6. Let X be a proper CAT(0) space and G < Is(X) be a closed subgroup. Let ξ ∈ ∂X be such that the stabiliser Gξ acts cocompactly on X. Then the group Guξ = G ∩ Is(X)uξ acts transitively on the set Opp(ξ), which is non-empty.

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It is not diﬃcult to show that if Gξ acts cocompactly on X, then the set Opp(ξ) is non-empty, and the group Gξ acts transitively on it. Proposition 4.6 ensures that the smaller subgroup Guξ remains transitive on Opp(ξ). This fact plays a crucial role in excluding ﬁxed points at inﬁnity for cocompact actions of unimodular groups, as we shall now see it. Proof of Theorem 4.4. The isometry group Is(Rn ) of the Euclidean space is unimodular and acts without a ﬁxed point at inﬁnity. By Theorem 3.8, there is thus no loss of generality in assuming that X has no non-trivial Euclidean factor. Assume for a contradiction that G = Is(X) ﬁxes some point ξ ∈ ∂X. Since G acts cocompactly on X by hypothesis, the set of opposites Opp(ξ) is non-empty (see Exercise 4.4), and the subgroup Guξ acts transitively on it by Proposition 4.6. We claim that Guξ is compact. Since G = Gξ acts minimally, this implies that u Gξ , which is normal in Gξ , must be trivial. Therefore the set Opp(ξ) is reduced to a singleton, say {ξ }. In particular ξ is ﬁxed by G, and Theorem 2.7 applied to the pair {ξ, ξ } yields a product decomposition of X with a line factor, contradicting that the maximal Euclidean factor of X is trivial. In order to prove the claim, we proceed as follows. Let : R → X be a geodesic line such that (−∞) = ξ and (+∞) = ξ . By cocompactness, there is a sequence (gn ) in G such that d(gn . (0), (n)) is bounded. Since G ﬁxes ξ it follows that for each individual element g ∈ G, the sequence of conjugates (gn ggn−1 ) is bounded (i.e. relatively compact) in G. By an application of the Baire category theorem, one deduces that for each compact subset U ⊂ G, the union n gn U gn−1 has compact closure. Consider now an element g ∈ Guξ . This implies that any limit point of the sequence of conjugate (gn ggn−1 ) ﬁxes pointwise the line . Choosing some compact neighbourhood Q of the pointwise stabiliser of in G, we infer that gn ggn−1 belongs to Q for all suﬃciently large n. This holds for any individual element g ∈ Guξ , and another application of the Baire category theorem implies that for each compact subset V ⊂ Guξ , one has gn V gn−1 ⊂ Q for all suﬃciently large n. We now ﬁx some compact identity neighborhood U in X. Thus 0 < vol(U ) < ∞, where vol denotes a left Haar measure on G. We have seen that the set P = −1 n gn U gn is compact, and thus has ﬁnite volume. Now, for each compact subset u V ⊂ Gξ , we ﬁnd gn U V gn−1 = gn U gn−1 gn V gn−1 ⊂ P Q for all suﬃciently large n. Since P Q is compact, it has ﬁnite volume. The unimodularity of G implies that the Haar measure is conjugacy invariant. Thus vol(U V ) < vol(P Q) < ∞. This holds for every compact subset V ⊂ Guξ . Thus vol U Guξ < ∞, from which it follows that Guξ is compact, as claimed. 4. Back to rigidity We ﬁnally come back to Theorem 1.14 and describe the main steps of its proof: (1) Since Is(X) is cocompact and X geodesically complete, the Is(X)-action is minimal (Exercise 2.4). (2) The existence of a lattice in Is(X) implies that Is(X) is geometrically dense by Theorem 4.4. (3) We are then in a position to invoke Corollary 3.11, which yields a canonical decomposition X ∼ = M1 × · · · × Mp × Rn × Y1 × · · · × Yq , where Mi is an irreducible

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(4)

(5)

(6)

(7)

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symmetric space of non-compact type, and Yj has a totally disconnected isometry group, which acts smoothly and does not contain any parabolic isometry. The hypothesis that X has some parabolic isometry can now be re-interpreted: it simply means that X has at least one non-trivial symmetric space factor. At this point, if the space X is irreducible, we are done. Otherwise we may assume that X has several non-trivial factors. This implies that Γ may be viewed as a lattice in a product of locally compact groups; this gives access to superrigidity results, that are available for lattices in product groups at a high level of generality, notably through works by Burger [Bur95], Shalom [Sha00], Monod [Mon06], Gelander–Karlsson–Margulis [GKM08]. The residual ﬁniteness assumption, combined with the indecomposability of Γ is then used is an essential way: it is shown to imply that the Γ-action on each irreducible factor of X. The connection between residual ﬁniteness of the lattice and the faithfulness of its action on the factors was ﬁrst discovered by Burger and Mozes in their work on lattices in products of trees [BM00]. It was extended to lattices in products of CAT(0) spaces in [CM09b, Th. 4.10] (see also [CM12, Prop. 2.4]). Here, we deduce that the Γ-action on the non-trivial symmetric space factor yields in particular a faithful linear representation of Γ. The rest of the proof consists in using this linear representation combined with superrigidity tools to establish that Γ is an S-arithmetic group; this step closely follows the way in which Margulis deduced his arithmeticity theorems from superrigidity (see [Mar91] and the lectures by T. Gelander). Finally, once Γ has been identiﬁed as an S-arithmetic group, further applications of superrigidity imply that the closure of the image of Γ in the isometry group of each irreducible factor Yj of X is a semi-simple algebraic group. That Yj must be the model space (symmetric space or Euclidean building) for the semi-simple group in question is ﬁnally established, using the geodesic completeness hypothesis.

5. Flats and free abelian subgroups Some key problems on proper CAT(0) spaces, groups and lattices, remain open; the most famous among them are perhaps the Rank Rigidity Conjecture, the Tits alternative, the existence of inﬁnite torsion subgroups in CAT(0) groups, or the Flat Closing conjecture. It goes beyond the scope of these lectures to discuss all those problems, or to present the state of the art in each case. We shall content ourselves with a brief discussion of the latter. We start with the following well known open (and notoriously diﬃcult) problem. Question 4.7. Let X be a proper CAT(0) space and Γ be a discrete group acting properly and cocompactly on X. Is it true that Γ is Gromov hyperbolic if and only if Γ does not contain any subgroup isomorphic to Z2 ? The ‘only if’ direction is true by a well known property of hyperbolic groups (independent of CAT(0) geometry). Evidence for the reverse implication is provided by the following result. Theorem 4.8 ([BH99, Th. III.H.1.5]). Let X be a proper CAT(0) space and Γ be a discrete group acting properly and cocompactly on X. Then Γ is Gromov hyperbolic if and only if X does not contain any 2-ﬂat.

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In view of that theorem, answering Question 4.7 amounts to deciding whether the existence of a 2-ﬂat in X implies the existence of a Z2 subgroup in Γ. More generally, one can ask the following. Question 4.9 (Flat Closing conjecture, see [Gro93, 6.B3 ]). Let X be a proper CAT(0) space and Γ be a discrete group acting properly and cocompactly on X. Given n > 0, does the existence of an n-ﬂat in X imply the existence of a Zn subgroup in Γ? For n = 1, the answer is known to be positive, due to E. Swenson [Swe99, Th. 11]. It is generally believed that the answer for n = 2 should be negative, and that examples should be found among CAT(0) square complexes. For general n, the answer is positive when X is a symmetric space, see the lectures by T. Gelander in this volume. When X is a Hadamard manifold (not necessarily symmetric), the answer is also positive for all n, due to Bangert–Schroeder [BS91], but much more delicate to establish. We ﬁnish by mentioning a general result for CAT(0) groups in this direction. Theorem 4.10 ([CZ13, Cor. 1]). Let X be a locally compact geodesically complete CAT(0) space and Γ a discrete group acting properly and cocompactly on X. If X splits non-trivially as a CAT(0) product of n factors, then Γ contains a copy of Zn . Remark that if X is a non-trivial product of n factors, then each factor is unbounded and X contains some n-ﬂat (because X is geodesically complete). Thus the hypotheses in Theorem 4.10 are (strictly) stronger than the existence of some n-ﬂat in X. The proof of Theorem 4.10 is indirect: it relies on the fundamental decomposition provided by Corollary 3.11, and then treats separately the case of symmetric spaces (where the result is well known, as mentioned above) and the case of spaces with a totally disconnected isometry group. 6. Exercises Exercise 4.1. Let X be a CAT(0) space. A map f : X → R is called aﬃne if for each geodesic ρ : I → X, the composed map f ◦ ρ : I → R is aﬃne. (i) Show that f is aﬃne if and only if f and −f are both convex. (ii) Show that the level sets of an aﬃne map are convex. (iii) Suppose that X splits as a CAT(0) product X = Y × [0, 1]. Show that the distance function to the ﬁber Y × {0} is aﬃne, while the distance to Y × { 12 } is not. Exercise 4.2. Let X be a complete CAT(0) space. Let also (Ω, μ) be a measure space and (fω )ω∈Ω be a family of convex functions on X such that the map ω → fω (x) is integrable for all x ∈ X. Show that if the map f : x → Ω fω (x)dμ(ω) is constant, then fω is aﬃne for μ-almost all ω. Exercise 4.3. Show that Theorem 4.4 can fail if Is(X) does not act minimally. Exercise 4.4. Let X be a proper CAT(0) space and ξ ∈ ∂X be a boundary point. (i) Show that Is(X)uξ is a closed normal subgroup of Is(X)ξ . (ii) Assume that Is(X)ξ is cocompact on X. Show that Opp(ξ) is non-empty, and that Is(X)ξ acts transitively on it.

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[AB98]

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[BGS85]

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[Bro89] [BT72] [Bur95]

[BM00] [BS87]

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[CCMT12] Pierre-Emmanuel Caprace, Yves de Cornulier, Nicolas Monod, and Romain Tessera, Amenable hyperbolic groups, to appear in J. Europ. Math. Soc. (2012). Preprint. [CL10] Pierre-Emmanuel Caprace and Alexander Lytchak, At inﬁnity of ﬁnite-dimensional CAT(0) spaces, Math. Ann. 346 (2010), no. 1, 1–21, DOI 10.1007/s00208-009-0381-1. MR2558883 (2011d:53075) [CM09a] Pierre-Emmanuel Caprace and Nicolas Monod, Isometry groups of non-positively curved spaces: structure theory, J. Topol. 2 (2009), no. 4, 661–700, DOI 10.1112/jtopol/jtp026. MR2574740 (2011i:53051) , Isometry groups of non-positively curved spaces: discrete subgroups, J. Topol. [CM09b] 2 (2009), no. 4, 701–746, DOI 10.1112/jtopol/jtp027. MR2574741 (2011i:53052) , Fixed points and amenability in non-positive curvature, Math. Ann. 356 [CM13] (2013), no. 4, 1303–1337, DOI 10.1007/s00208-012-0879-9. MR3072802 , A lattice in more than two Kac-Moody groups is arithmetic, Israel J. Math. [CM12] 190 (2012), 413–444, DOI 10.1007/s11856-012-0006-3. MR2956249 [CS11] Pierre-Emmanuel Caprace and Michah Sageev, Rank rigidity for CAT(0) cube complexes, Geom. Funct. Anal. 21 (2011), no. 4, 851–891, DOI 10.1007/s00039-011-0126-7. MR2827012 (2012i:20049) [CZ13] Pierre-Emmanuel Caprace and Gaˇsper Zadnik, Regular elements in CAT(0) groups, Groups Geom. Dyn. 7 (2013), no. 3, 535–541, DOI 10.4171/GGD/195. MR3095707 [EM04] Murray Elder and Jon McCammond, CAT(0) is an algorithmic property, Geom. Dedicata 107 (2004), 25–46, DOI 10.1023/B:GEOM.0000049096.63639.e3. MR2110752 (2005k:20100) [FL08] Thomas Foertsch and Alexander Lytchak, The de Rham decomposition theorem for metric spaces, Geom. Funct. Anal. 18 (2008), no. 1, 120–143, DOI 10.1007/s00039008-0652-0. MR2399098 (2010c:53061) [GKM08] Tsachik Gelander, Anders Karlsson, and Gregory A. Margulis, Superrigidity, generalized harmonic maps and uniformly convex spaces, Geom. Funct. Anal. 17 (2008), no. 5, 1524–1550, DOI 10.1007/s00039-007-0639-2. MR2377496 (2009a:53074) [GO07] Ross Geoghegan and Pedro Ontaneda, Boundaries of cocompact proper CAT(0) spaces, Topology 46 (2007), no. 2, 129–137, DOI 10.1016/j.top.2006.12.002. MR2313068 (2008c:57004) [Gro93] M. Gromov, Asymptotic invariants of inﬁnite groups, Geometric group theory, Vol. 2 (Sussex, 1991), London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1–295. MR1253544 (95m:20041) [HJ90] Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1990. Corrected reprint of the 1985 original. MR1084815 (91i:15001) [Kle99] Bruce Kleiner, The local structure of length spaces with curvature bounded above, Math. Z. 231 (1999), no. 3, 409–456, DOI 10.1007/PL00004738. MR1704987 (2000m:53053) [KL97] Bruce Kleiner and Bernhard Leeb, Rigidity of quasi-isometries for symmetric spaces ´ and Euclidean buildings, Inst. Hautes Etudes Sci. Publ. Math. 86 (1997), 115–197 (1998). MR1608566 (98m:53068) , Rigidity of invariant convex sets in symmetric spaces, Invent. Math. 163 [KL06] (2006), no. 3, 657–676, DOI 10.1007/s00222-005-0471-y. MR2207236 (2006k:53064) [Lee00] Bernhard Leeb, A characterization of irreducible symmetric spaces and Euclidean buildings of higher rank by their asymptotic geometry, Bonner Mathematische Schriften [Bonn Mathematical Publications], 326, Universit¨ at Bonn, Mathematisches Institut, Bonn, 2000. MR1934160 (2004b:53060) [Mar91] G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991. MR1090825 (92h:22021) [Mon06] Nicolas Monod, Superrigidity for irreducible lattices and geometric splitting, J. Amer. Math. Soc. 19 (2006), no. 4, 781–814, DOI 10.1090/S0894-0347-06-00525-X. MR2219304 (2007b:22025) [MP12] Nicolas Monod and Pierre Py, An equivariant deformation of hyperbolic spaces, to appear in Amer. J. Math. (2012). Preprint. [MZ55] Deane Montgomery and Leo Zippin, Topological transformation groups, Interscience Publishers, New York-London, 1955. MR0073104 (17,383b)

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4. BIBLIOGRAPHY

[Sha00] [Swe99]

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Yehuda Shalom, Rigidity of commensurators and irreducible lattices, Invent. Math. 141 (2000), no. 1, 1–54, DOI 10.1007/s002220000064. MR1767270 (2001k:22022) Eric L. Swenson, A cut point theorem for CAT(0) groups, J. Diﬀerential Geom. 53 (1999), no. 2, 327–358. MR1802725 (2001i:20083)

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https://doi.org/10.1090//pcms/021/05

Lectures on Quasi-Isometric Rigidity Michael Kapovich

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IAS/Park City Mathematics Series Volume 21, 2012

Lectures on Quasi-Isometric Rigidity Michael Kapovich Introduction: What is Geometric Group Theory? Historically (in the 19th century), groups appeared as automorphism groups of certain structures: • Polynomials (ﬁeld extensions) — Galois groups. • Vector spaces, possibly equipped with a bilinear form — Matrix groups. • Complex analysis, complex ODEs — Monodromy groups. • Partial diﬀerential equations — Lie groups (possibly inﬁnite-dimensional ones) • Various geometries — Isometry groups of metric spaces, both discrete and nondiscrete. A goal of Geometric Group Theory (which I will abbreviate as GGT) is to study ﬁnitely-generated groups G as automorphism groups (symmetry groups) of metric spaces X. Accordingly, the central question of GGT is: How are the algebraic properties of a group G reﬂected in the geometric properties of a metric space X and, conversely, how is the geometry of X reﬂected in the algebraic structure of G? This interaction between groups and geometry is a fruitful two-way road. An inspiration for this viewpoint is the following (essentially) bijective correspondence (established by E. Cartan): Simple noncompact connected Lie groups ←→ Irreducible symmetric spaces of noncompact type.

Here the correspondence is between algebraic objects (Lie groups of a certain type) and geometric objects (certain symmetric spaces). Namely, given a Lie group G one constructs a symmetric space X = G/K (K is a maximal compact subgroup of G) and, conversely, every symmetric space corresponds to a Lie group G (its isometry group) and this group is unique. Imitating this correspondence is an (unreachable) goal of GGT.

Department of Mathematics, University of California, Davis, California 95616 E-mail address: [email protected] The author was partially supported by NSF grants DMS-09-05802 and DMS-12-05312. The author is also grateful to Dustin Mayeda and to an anonymous reader for remarks and corrections. c 2014 American Mathematical Society

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LECTURE 1

Groups and spaces Convention. Throughout these lectures, I will be working in ZFC: Zermelo– Fraenkel Axioms of set theory + Axiom of Choice. 1. Cayley graphs and other metric spaces Recall that we are looking for a correspondence: Groups ←→ Metric Spaces The ﬁrst step is to associate with a ﬁnitely-generated group G a metric space X. Let G be a group with a ﬁnite generating set S = {s1 , ..., sk }. Then we construct a graph X, whose vertex set V (X) is the group G itself and whose edges are [g, gsi ], si ∈ S, g ∈ G. (If gsi = gsj , i.e., si = sj , then we treat [g, gsi ], [g, gsj ] as distinct edges, but this is not very important.) We do not orient edges. The resulting graph X = ΓG,S is called a Cayley graph of the group G with respect to the generating set S. Then the group G acts (by multiplication on the left) on X: Every g ∈ G deﬁnes a map g(x) = gx,

x ∈ V (X) = G.

Clearly, edges are preserved by this action. Since S is a generating set of G, the graph X is connected. We now deﬁne a metric on the graph X = ΓG,S . If X is any connected graph, then we declare every edge of X to have unit length. Then we have a well-deﬁned notion of length of a path in X. The distance between vertices in X is the length of the shortest edge-path in X connecting these points. Exercise 1.1. Shortest edge-paths always exist. One can also think of the graph X as a cell complex, which we then conﬂate with its geometric realization (a topological space). Then, one can talk about points in X which lie in the interiors of edges. We then identify each edge with the unit interval and extend the above metric to all of X. As we will see, later, this distinction between the metric on V (X) and the metric on X is not very important. The metric on G = V (X) is called a word-metric on G. Here is why: Example 1.2. Let X be a Cayley graph of a group G. The distance d(1, g) from 1 ∈ G to g ∈ G is the same thing as the “norm” |g| of g, the minimal number m of symbols in the decomposition (a “word in the alphabet S ∪ S −1 ”) ±1 ±1 g = s±1 i1 si2 ...sim

of g as a product of generators and their inverses. Note: If g = 1 then m := 0. 131 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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Thus, we have a correspondence: Groups −→ Metric spaces, Cayley : G → X = A Cayley graph of G. Is this the only correspondence? Is this map “Cayley” well deﬁned? We will see that both questions have negative answers and our ﬁrst goal will be to deal with this issue. (Note that inﬁnite ﬁnitely-generated groups G have inﬁnitely many distinct ﬁnite generating sets.) Definition 1.3. Let X be a metric space and G be a group acting on X. The action G X is called geometric if: 1. G acts isometrically on X. 2. G acts properly discontinuously on X, i.e., ∀ compact C ⊂ X, the set {g ∈ G : gC ∩ C = ∅} is ﬁnite. 3. G acts cocompactly on X: X/G is compact. Informally, a group G is a group of (discrete) symmetries of X if G acts geometrically on X. (Note that in some natural situations in GGT one considers non-geometric actions of groups on metric spaces, but we will not address this in these lectures.) Example 1.4. Suppose G is a ﬁnitely-generated group and X is its Cayley graph. Then the action of G on X is geometric. Question: What is the quotient graph X/G? Other metric spaces which appear naturally in GGT are connected Riemannian manifolds (M, ds2 ). In this case, the distance between points is T d(x, y) = inf{ ds = |p (t)|dt} p

0

where the inﬁmum is taken over all paths p connecting x to y. When dealing with connected Riemannian manifolds we will always implicitly assume that they are equipped with the above distance function. Example 1.5. Suppose that M is a compact connected Riemannian manifold ˜ is the universal cover of M (with with the fundamental group π = π1 (M ), X = M lifted Riemannian metric), π acts on X as the group of covering transformations for the covering X → M . Then π X is a geometric action. More generally, let φ : π → G be an epimorphism, X → M be the covering corresponding to Ker(φ). Then the group of covering transformations of X → M is isomorphic to G and, thus, G acts geometrically on X. Note: For every ﬁnitely-generated group G there exists a compact Riemannian manifold M (of every dimension ≥ 2) with an epimorphism π1 (M ) → G. Thus, we get another correspondence Groups −→ Metric Spaces: Riemann : G → X = a covering space of some M as above. Thus, we have a problem on our hands: We have too many candidates for the correspondence Groups → Spaces and these correspondences are not well-deﬁned. What do diﬀerent spaces on which G acts geometrically have in common?

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2. Quasi-isometries Definition 1.6. a. Let X, X be metric spaces. A map f : X → X is called an (L, A)-quasi-isometry if: 1. f is (L, A)-coarse Lipschitz: d(f (x), f (y)) ≤ Ld(x, y) + A. 2. There exists an (L, A)-coarse Lipschitz map f¯ : X → X, which is “quasiinverse” to f : d(f¯f (x), x) ≤ A, d(f f¯(x ), x ) ≤ A. When the constants L, A are not important, we will simply say that f is a quasiisometry. b. Spaces X, X are quasi-isometric to each other if there exists a quasi-isometry X → X . Note that every (L, 0)-quasi-isometry f is a bilipschitz homeomorphism; if, in addition, L = 1, then f is an isometry. Example 1.7. 1. Every bounded metric space is QI to a point. 2. R is QI to Z. 3. Every metric space is QI to its metric completion. Here and in what follows I will abbreviate “quasi-isometry” and “quasi-isometric” to QI. Exercise 1.8. • Every quasi-isometry f : X → X is “quasi-surjective”: ∃C < ∞|∀x ∈ X , ∃x ∈ X|d(x , f (x)) ≤ C. • Show that a map f : X → X is a quasi-isometry iﬀ it is quasi-surjective and is a “quasi-isometric embedding”: ∃L, ∃A so that ∀x, y ∈ X: 1 d(x, y) − A ≤ d(f (x), f (y)) ≤ Ld(x, y) + A. L • Composition of quasi-isometries is again a quasi-isometry. • Quasi-isometry of metric spaces is an equivalence relation. Exercise 1.9. 1. Let S, S be two ﬁnite generating sets for a group G and d, d be the corresponding word metrics. Then the identity map (G, d) → (G, d ) is an (L, 0)-quasi-isometry for some L. 2. G is QI to its Cayley graph X. The map G → X is the identity. What is the quasi-inverse? Given a metric space X, we thus have a semigroup QI(X) consisting of quasiisometries X → X. This semigroup, of course, is not a group, since quasi-isometries need not be invertible. However, one can form a group using QI(X) as follows. We deﬁne the equivalence relation ∼ on QI(X) by f ∼ g ⇐⇒ dist(f, g) = sup{d(f (x), g(x)) : x ∈ X} < ∞. Then the quotient QI(X) = QI(X)/∼ is a group: If f¯ is quasi-inverse to f , then [f ]−1 = [f¯] where [h] denotes the projection of h to QI(X).

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Definition 1.10. 1. A geodesic in a metric space X is a distance-preserving map γ of an interval I ⊂ R to X. A geodesic ray is a geodesic whose domain in a half-line in R. If I = [a, b] then we will use the notation pq, p = γ(a), q = γ(b), to denote a geodesic connecting p to q. We will frequently conﬂate geodesics and their images. 2. A metric space X is called geodesic if for every pair of points x, y ∈ X, there exists a geodesic γ : [0, T ] → X, connecting x to y. 3. A metric space X is proper if every closed metric ball in X is compact. A subset N of a metric space X is called an -separated R-net if: (1) For all x = y ∈ N , d(x, y) ≥ . (2) For every x ∈ X there exists y ∈ N so that d(x, y) ≤ R. Here > 0, R < ∞. Exercise 1.11. 1. Let X be a Cayley graph of a group G. Then G is a separated net in X. 2. Every metric space X admits a separated net. (You need Zorn’s lemma to prove this.) Definition 1.12. Suppose that X is a proper metric space. A sequence (fi ) of maps X → Y is said to coarsely uniformly converge to a map f : X → Y on compacts, if: There exists a number R < ∞ so that for every compact K ⊂ X, there exists an iK so that for all i > iK , ∀x ∈ K,

d(fi (x), f (x)) ≤ R.

To simplify the notation, we will say that limci→∞ fi = f . Note that the usual uniform convergence on compacts implies coarse convergence. Proposition 1.13 (Arzela–Ascoli theorem for coarsely Lipschitz maps). Fix real numbers L, A and D and let X, Y be proper metric spaces so that X admits a separated R-net. Let fi : X → Y be a sequence of (L1 , A1 )-Lipschitz maps, so that for some points x0 ∈ X, y0 ∈ Y we have d(f (x0 ), y0 ) ≤ D. Then there exists a subsequence (fik ), and a (L2 , A2 )–Lipschitz map f : X → Y , so that limck→∞ fi = f . Furthermore, if the maps fi are (L1 , A1 ) quasi-isometries, then f is also an (L3 , A3 )–quasi-isometry for some L3 , A3 . Proof. Let N ⊂ X be a separated net. We can assume that x0 ∈ N . Then the restrictions fi |N are L -Lipschitz maps and, by the usual Arzela–Ascoli theorem, the sequence (fi |N ) subconverges (uniformly on compacts) to an L -Lipschitz map f : N → Y . We extend f to X by the rule: For x ∈ X pick x ∈ N so that d(x, x ) ≤ R and set f (x) := f (x ). Then f : X → Y is an (L2 , A2 )–Lipschitz map. For a metric ball B(x0 , r) ⊂ X, r ≥ R, there exists ir so that for all i ≥ ir and all x ∈ N ∩ B(x0 , r), we have d(fi (x), f (x)) ≤ 1. For arbitrary x ∈ K, we ﬁnd x ∈ N ∩ B(x0 , r + R) so that d(x , x) ≤ R. Then d(fi (x), f (x)) ≤ d(fi (x ), f (x )) ≤ L1 (R + 1) + A. This proves coarse convergence. The argument for quasi-isometries is similar.

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Theorem 1.14 (Milnor–Schwarz lemma). Suppose that G acts geometrically on a proper geodesic metric space X. Then G is ﬁnitely generated and (∀x ∈ X) the orbit map f : g → g(x), f : G → X, is a quasi-isometry, where G is equipped with a word-metric. ¯R (x0 ) be the Proof. Our proof follows [15, Proposition 10.9]. Let B = B closed R-ball of radius R in X centered at x0 , so that BR−1 (x0 ) projects onto X/G. Since the action of G is properly discontinuous, there are only ﬁnitely many elements si ∈ G such that B ∩ si B = ∅. Let S be the subset of G which consists of belongs to S iﬀ si does). Let the above elements si (it is clear that s−1 i r := inf{d(B, g(B)), g ∈ G \ S}. Since B is compact and B ∩ g(B) = ∅ for g ∈ / S, r > 0. We claim that S is a generating set of G and that for each g ∈ G (1.1)

|g| ≤ d(x0 , g(x0 ))/r + 1

where |·| is the word length on G (with respect to the generating set S). Pick g ∈ G and connect x0 to g(x0 ) by a shortest geodesic γ in X. Let m be the smallest integer so that d(x0 , g(x0 )) ≤ mr + R. Choose points x1 , ..., xm+1 = g(x0 ) ∈ γ, so that x1 ∈ B, d(xj , xj+1 ) < r, 1 ≤ j ≤ m. Then each xj belongs to gj (B) for some gj ∈ G. Let 1 ≤ j ≤ m, then gj−1 (xj ) ∈ B and d(gj−1 (gj+1 (B)), B) ≤ d(gj−1 (xj ), gj−1 (xj+1 )) < r. Thus the balls B, gj−1 (gj+1 (B)) intersect, which means that gj+1 = gj si(j) for some si(j) ∈ S. Therefore g = si(1) si(2) ....si(m) . We conclude that S is indeed a generating set for the group G. Moreover, |g| ≤ m ≤ (d(x0 , g(x0 )) − R)/r + 1 ≤ d(x0 , g(x0 ))/r + 1. The word metric on the Cayley graph ΓG,S of the group G is left-invariant, thus for each h ∈ G we have: 1 1 d(h, hg) = d(1, g) ≤ d(x0 , g(x0 ))/r + 1 = d(h(x0 ), hg(x0 )) + 1. r r Hence for any g1 , g2 ∈ G 1 d(g1 , g2 ) ≤ d(f (g1 ), f (g2 )) + 1. r On the other hand, the triangle inequality implies that d(x0 , g(x0 )) ≤ t|g| where d(x0 , s(x0 )) ≤ t ≤ 2R for all s ∈ S. Thus 1 d(f (g1 ), f (g2 )) ≤ d(g1 , g2 ). t We conclude that the map f : G → X is a quasi-isometric embedding. Since f (G) is R-dense in X, it follows that f is a quasi-isometry. Thus, if instead of isometry classes of metric spaces we use their QI classes then both Cayley and Riemann correspondences are well-deﬁned and are equal to each other! Now, we have a well-deﬁned map geo : ﬁnitely-generated groups −→ QI equivalence classes of metric spaces.

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Problem: This map is very far from being 1-1, so our challenge is to “estimate” the ﬁbers of this map. Exercise 1.15. Show that the half-line is not QI to any Cayley graph. Prove ﬁrst that every unbounded Cayley graph contains an isometrically embedded copy of R (hint: use Arzela-Ascoli theorem). Then show that there is no QI embedding f : R → R+ . Hint: Replace f with a continuous (actually, piecewise-linear) QI embedding h so that d(f, h) ≤ C and then use the intermediate value theorem to get a contradiction. Example 1.16. Every ﬁnite group is QI to the trivial group. In particular, from the QI viewpoint, the entire theory of ﬁnite groups (with its 150 year-old history culminating in the classiﬁcation of ﬁnite simple groups) becomes trivial. Is this good news or is this bad news? This does not sound too good if we were to recover a group from its geometry (up to an isomorphism). Is there a natural algebraic equivalence relation on groups which can help us here? 3. Virtual isomorphisms and QI rigidity problem In view of Milnor-Schwarz lemma, the following provide examples of quasi-isometric groups: 1. If G < G is a ﬁnite-index subgroups then G is QI to G . (G acts on a Cayley graph of G isometrically and faithfully so that the quotient is a ﬁnite graph.) 2. If G = G/F , where F is a ﬁnite group, then G is QI to G . (G acts isometrically and transitively on a Cayley graph of G so that the action has ﬁnite kernel.) Combining these two examples we obtain Definition 1.17. 1. G1 is virtually isomorphic (which will be abbreviated as VI) to G2 if there exist ﬁnite index subgroups Hi ⊂ Gi and ﬁnite normal subgroups Fi Hi , i = 1, 2, so that the quotients H1 /F1 and H2 /F2 are isomorphic. 2. A group G is said to be virtually cyclic if it is VI to a cyclic group. Similarly, one deﬁnes virtually abelian groups, virtually free groups, etc. Exercise 1.18. VI is an equivalence relation. To summarize: By the Milnor–Schwarz lemma, V I ⇒ QI. Thus, if we are to recover a group from its geometry (treated up to QI), then the best we can hope for is to recover the group up to VI. This is bad news for people in ﬁnite group theory, but good news for the rest of us. Remark 1.19. There are some deep and interesting connections between the theory of ﬁnite groups and GGT, but quasi-isometries do not see these. Informally, quasi-isometric rigidity is the situation when the arrow V I ⇒ QI can be reversed. Definition 1.20. 1. We say that a group G is QI rigid if every group G which is QI to G, is in fact VI to G.

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2. We say that a class C of groups is QI rigid if for every group G which is QI to some G ∈ C, there exists G ∈ C so that G is VI to G . 3. A property P of groups is said to be “geometric” or “QI invariant” whenever the class of groups satisfying P is QI rigid. Note that studying QI rigidity and QI invariants is by no means the only topic of GGT, but this will be the topic of my lectures. 4. Examples and non-examples of QI rigidity At ﬁrst glance, any time QI rigidity holds (in any form), it is a minor miracle: How on earth are we supposed to recover precise algebraic information from something as sloppy as a quasi-isometry? Nevertheless, instances of QI rigidity abound. We refer the reader to [6] for some of the proofs and further references: Examples of QI rigid groups/classes/properties (all my groups are ﬁnitelygenerated, of course): • • • • •

• • • • •

•

• • •

Free groups. (J. Stallings; see [6].) Free abelian groups. (M. Gromov; P. Pansu; see [6].) Class of nilpotent groups. (M. Gromov; see [6].) Class of fundamental groups of closed (compact, without boundary) surfaces. (This is a combination of work of P. Tukia; D. Gabai; A. Casson and D. Jungreis, see [34], [14], [5].) Class of fundamental groups of closed (compact, without boundary) 3dimensional manifolds. (This is a combination of work of R. Schwartz [28]; M. Kapovich and B. Leeb [19]; A. Eskin, D. Fisher and K. Whyte [11], and, most importantly, the solution of the geometrization conjecture by G. Perelman.) Class of ﬁnitely-presentable groups. (M. Gromov; see [6].) Class of hyperbolic groups. (M. Gromov; see [6].) Class of amenable groups. Class of fundamental groups of closed n-dimensional hyperbolic manifolds. For n ≥ 3 this result, due to P. Tukia [33], will be the central theorem of my lectures. Class of uniform lattices in each connected simple noncompact Lie group G (i.e., discrete cocompact subgroups Γ in G). (This is a combination of work of P. Pansu [23]; P. Tukia [33]; R. Chow [4]; B. Kleiner and B. Leeb [20]; A. Eskin and B. Farb [10].) Every non-uniform lattice in a connected simple Lie group (i.e., a discrete subgroup Γ in a simple noncompact Lie group G so that G/Γ has ﬁnite volume but noncompact). For instance, every group which is QI to SL(n, Z) is in fact VI to SL(n, Z). (This is a combination of work of R. Schwartz [28]; B. Farb and R. Schwartz [13]; A. Eskin [9]. We also refer to Farb’s paper [12] for a summary in the uniform and nonuniform case.) Solvability of the word problem (say, for ﬁnitely-presented groups). Cohomological dimension over Q. (R. Sauer, [26].) Admitting a “geometric” action on a contractible CW-complex (i.e., an action which is cocompact on each skeleton and is properly discontinuous). (See [6].)

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• Admitting an amalgam decomposition (amalgamated free product or HNN decomposition) over a ﬁnite subgroup. (J. Stallings [30]; see also [6].) • Admitting an amalgam decomposition over a virtually cyclic subgroup. (P.Papasoglou, [24].) Rule of thumb: The closer a group (or a class of groups) is to a Lie group, the higher are the odds of QI rigidity. Examples of failure of QI rigidity: • Suppose that S is a closed oriented surface of genus ≥ 2 and π = π1 (S). Then Z × π is QI to any Γ which appears in any central extension 1 → Z → Γ → π → 1. The same holds if π is replaced by a nonelementary hyperbolic group. (This was independently observed by D.B.A. Epstein, S. Gersten, G. Mess; see [6].) For instance, the fundamental group Γ of the unit tangent bundle of S is realized this way. • In particular, the property of being the fundamental group of a compact nonpositively curved Riemannian manifold with convex boundary is not QI invariant. • There are countably many VI classes of groups which act geometrically on hyperbolic 3-space. All these groups are QI to each other by MilnorSchwarz lemma. Same for all irreducible nonpositively curved symmetric spaces of dimension ≥ 3. • Class of solvable groups is not QI rigid. (A. Erschler, [7].) • Class of simple groups is not QI rigid: F2 × F2 is QI to a simple group. (M. Burger and S. Mozes, [2].) • Class of residually-ﬁnite groups is not QI rigid. (M. Burger and S. Mozes, [2].) • Property (T) is not QI invariant. (S. Gersten and M. Ramachandran; see [6].) A few open problems: • Is the class of fundamental groups of closed aspherical n-dimensional orbifolds QI rigid? • Is the class of polycyclic groups QI rigid? (Conjecturally, yes.) • Is the class of elementary amenable groups QI rigid? • Prove QI rigidity for various classes of Right-Angled Artin Groups (RAAGs): It is known that some of these classes are QI rigid but some are not (e.g., F2 × F2 ). • Are random ﬁnitely-presented groups QI rigid? • Construct examples of QI rigid hyperbolic groups whose boundaries are homeomorphic to the Menger curve. • Classify up to a quasi-isometry fundamental groups of compact 3-dimensional manifolds. • Verify QI invariance of JSJ decomposition (in the sense of Leeb ad Scott [22]) of closed nonpositively curved Riemannian manifolds of dimension ≥ 4. (Note that the 3-dimensional case was done in [19].) • Is the Haagerup property (see [3] for the deﬁnition) QI invariant? • Is the Rapid Decay property (see e.g. [21] for the deﬁnition) QI invariant? • Is the property of having uniform exponential growth QI invariant?

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• Is the class of hyperbolic free-by-cyclic groups Fn Z QI rigid (n ≥ 3)? • Is every group QI to a torsion-free group? • Prove QI rigidity for classes of lattices (including reducible ones) in every semisimple Lie group G. Open cases include products of nonuniform lattices in rank 1 and higher rank Lie groups, e.g. SL(2, Z)×SL(n, Z), n ≥ 3. Where do the tools of GGT come from? From almost everywhere! Here are some examples: • Group theory (of course) • Geometry (of course) • Topology (point-set topology, geometric topology, algebraic topology) • Lie theory • Analysis (including PDEs, functional analysis, real analysis, complex analysis, etc.) • Probability • Logic • Dynamical systems • Homological algebra • Combinatorics In these lectures, I will introduce two tools of QI rigidity: Ultralimits (coming from logic) and quasiconformal maps (whose origin is in geometric analysis and complex analysis).

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LECTURE 2

Ultralimits and Morse lemma Motivation: Quasi-isometries are not nice maps, they need not be continuous, etc. We will use ultralimits of metric spaces to convert quasi-isometries into homeomorphisms. Also, in many cases, ultralimits of sequences of metric spaces are simpler than the original spaces. We will use this to prove stability of geodesics in hyperbolic space (Morse Lemma).

1. Ultralimits of sequences in topological spaces. Definition 2.1. An ultraﬁlter on the set N of natural numbers is a ﬁnitelyadditive measure ω deﬁned for all subsets of N and taking only the values 0 and 1. In other words, ω : 2N → {0, 1} is: • Finitely-additive: ω(A ∪ B) = ω(A) + ω(B) − ω(A ∩ B). • ω(∅) = 0. We will say that a subset E of N is ω-large if ω(E) = 1. Similarly, we will say that a property P (n) holds for ω-all natural numbers if ω({n : P (n) is true }) = 1. Trivial examples of ultraﬁlters are those for which ω({n}) = 1 for some n ∈ N (such ultraﬁlters are called principal). I will always assume that ω vanishes on all ﬁnite sets, in other words, I will consider only nonprincipal ultraﬁlters. Existence of ultraﬁlters does not follow from the Zermelo-Fraenkel (ZF) axioms of set theory, but it follows from ZFC. We will use ultraﬁlters to deﬁne limits of sequences: Definition 2.2. Let X be a Hausdorﬀ topological space and ω an ultraﬁlter (on N). Then, for a sequence (xn ) of points xn ∈ X, we deﬁne the ω-limit (ultralimit), limω xn , to be a point a ∈ X so that: For every neighborhood U of a, the set {n ∈ N : xn ∈ U } is ω-large. In other words, xn ∈ U for ω-all n. As X is assumed to be Hausdorﬀ, limω xn is unique (if it exists). Exercise 2.3. If lim xn = a (in the usual sense) then limω xn = a for every ω. I will ﬁx an ultraﬁlter ω once and for all. Exercise 2.4. If X is compact then every sequence in X has an ultralimit. Hint: Use a proof by contradiction. In particular, every sequence tn ∈ R+ has ultralimit in [0, ∞]. Exercise 2.5. What is the ultralimit of the sequence (−1)n in [−1, 1]? 141 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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2. Ultralimits of sequences of metric spaces Our next goal is to deﬁne ultralimit for a sequence of metric spaces (Xn , dn ). The deﬁnition is similar to the Cauchy completion of a metric space: Elements of the ultralimit will be equivalence classes of sequences xn ∈ Xn . For every two sequences xn ∈ Xn , yn ∈ Xn we deﬁne dω ((xn ), (yn )) := lim dn (xn , yn ) ∈ [0, ∞]. ω

Exercise 2.6. Verify that dω is a pseudo-metric. (Use the usual convention ∞ + a = ∞, for every a ∈ R ∪ {∞}.) Of course, some sequences will be within zero distance from each other. As in the deﬁnition of Cauchy completion, we will identify such sequences (this is our equivalence relation). After that, dω is “almost” a metric: The minor problem is that sometimes dω is inﬁnite. To handle this problem, we introduce a sequence of “observers”, points pn ∈ Xn . Then, we deﬁne limω Xn = Xω , the ultralimit of the sequence of pointed metric spaces (Xn , pn ) to be the set of equivalence classes of sequences xn ∈ Xn so that dω ((xn ), (pn )) < ∞. Informally, Xω consists of equivalence classes of sequences which the “observers” can see. In case (Xn , dn ) = (X, d), we will refer to limω Xn as a constant ultralimit. Exercise 2.7. • If X is compact then the constant ultralimit limω X is isometric to X (for any sequence of observers). • Suppose that X admits a geometric group action. Then the constant ultralimit limω X does not depend on the choice of the observers. • Suppose that X is a proper metric space. Then for every bounded sequence pn ∈ X the constant ultralimit limω X is isometric to X. • Show that limω Rk is isometric to Rk . Let (Xn , pn ), (Yn , qn ) be pointed metric spaces and fn : Xn → Yn a sequence of isometries, so that lim dYn (fn (pn ), qn ) < ∞. ω

Then the sequence (fn ) deﬁnes a map fω : Xω → Yω ,

fω (xω ) = ((fn (xn ))).

It is immediate that the map fω is well-deﬁned and is an isometry. In particular, the ultralimit of a sequence of geodesic metric spaces is again a geodesic metric space. 3. Ultralimits and CAT(0) metric spaces Recall that a CAT(0) metric space is a geodesic metric space where triangles are “thinner” than triangles in the plane. One can express this property as a 4-point condition: Definition 2.8. A geodesic metric space X is said to be CAT(0) if the following holds. Let x, y, z, m ∈ X be points such that d(x, m) + d(m, y) = d(x, y). Let x , y , z , m ∈ R2 be their “comparison” points, i.e.: d(x, m) = d(x , m ), d(m, y) =

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d(m , y ), d(x, y) = d(x , y ), d(y, z) = d(y , z ), d(z, x) = d(z , x ). (Thus, the triangle with vertices x, y, m is degenerate.) Then d(z, m) ≤ d(z , m ). For instance, hyperbolic spaces Hn are CAT(0). The important property of CAT(0) spaces is that they are uniquely geodesic, i.e., for any pair of points x, y ∈ X there is a unique geodesic connecting x to y. Exercise 2.9. Ultralimits of sequences of CAT(0) spaces are again CAT(0). Hint: Start with a 4-point conﬁguration xω , yω , zω , mω ∈ Xω with a degenerate triangle with vertices xω , yω , mω . Represent the points xω , yω , zω by sequences xn , yn , zn ∈ Xn . Use the CAT(0) property to ﬁnd mn ∈ Xn representing mω so that the triangle spanned by xn , yn , mn is degenerate. 4. Asymptotic cones The ultralimits that we will be using are not constant. We start with a metric space (X, d) and a sequence of positive scale factors λn so that limω λn = 0. Then set dn := λn d. Hence, the sequence (X, dn ) consists of rescaled copies of (X, d). Definition 2.10. An asymptotic cone of X, denoted Cone(X), is the ultralimit of the sequence of pointed metric spaces: Cone(X) = limω (Xn , λn d, pn ). Note that, in general, the asymptotic cone depends on the choices of ω, (λn ) and (pn ), so the notation Cone(X) is ambiguous, but it will be always implicitly understood that ω, (λn ) and (pn ) are chosen in the deﬁnition of Cone(X). Exercise 2.11. Let G = Zk be the free abelian group with its standard set of generators. Let X = G with the word metric. Then Cone(X) is isometric to Rk with the 1 -metric corresponding to the norm (x1 , ..., xk ) = |x1 | + ... + |xk |. Definition 2.12. A (geodesic) triangle T in a metric space X is a concatenation of three geodesic segments in X: xy, yz, zx, where pq denotes a geodesic segment connecting p to q. We will use the notation T = [x, y, z] to indicate that x, y, z are the vertices of T . A triangle T is called δ-thin if every side of T is contained within distance ≤ δ from the union of the two other sides. A geodesic metric space X is called δ-hyperbolic if every geodesic triangle in X is δ-thin. When we do not want to specify δ, we will simply say that X is Gromov-hyperbolic. Lemma 2.13. Suppose that X is the hyperbolic space Hk , k ≥ 2. Then every asymptotic cone Xω = Cone(X) is a tree. (Note that this tree branches at every point and has inﬁnite (continual) degree of branching at every point xω : the cardinality of the number of components of Xω − {xω } is the cardinality of the continuum.) Proof. We need to verify that every geodesic triangle Tω = [xω , yω , zω ] ⊂ Xω is 0-thin, i.e., every side is contained in the union of two other sides. First of all, we know, that Xω is CAT(0) and, hence, uniquely geodesic. Thus, the triangle Tω appears as an ultralimit of a sequence of geodesic triangles Tn = [xn , yn , zn ] in Xk = (X, λk dX ). Each triangle Tn in (X, dX ) is δ-thin, where δ ≤ 1 (see Appendix 1). Therefore, the triangle Tn , regarded as a triangle in Xk , is λk δ-thin. Since limω λk δ = 0, we conclude that Tω is 0-thin. Exercise 2.14. Show that every closed geodesic m-gon [x1 , ..., xm ] in a tree T is 0-thin, i.e., the side [xm , x1 ] is contained in the union of the other sides.

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Lemma 2.15. Suppose that α is a simple topological arc in a tree T . Then α, after a reparameterization, is a geodesic arc. Proof. Let α : [0, 1] → T be a continuous injective map (a simple topological arc), x = α(0), y = α(1). Let α∗ = [x, y] be the geodesic connecting x to y. I claim that the image of α contains the image of α∗ . Indeed, we can approximate α by piecewise-geodesic (nonembedded!) arcs αn = [x0 , x1 ] ∪ ... ∪ [xn−1 , xn ],

x0 = x, xn = y.

Then the above exercise shows that αn contains the image of α∗ for every n. Therefore, the image of α also contains the image of α∗ . Considering the map α−1 ◦ α∗ and applying the intermediate value theorem, we see that the images of α and α∗ are equal. 5. Quasi-isometries and asymptotic cones Suppose that f : X → X is an (L, A)-quasi-isometric embedding: 1 d(x, y) − A ≤ d(f (x), f (y)) ≤ Ld(x, y) + A. L Pick a sequence of scale factors λn , a sequence of observers pn ∈ X and their images qn := f (xn ). Then,

Let dXn

λn d(x, y) − λn A ≤ λn d(f (x), f (y)) ≤ Lλn d(x, y) + λn A. L = λn dX , dXn = λn dX . Hence:

1 dX (x, y) − λn A ≤ dXn (f (x), f (y)) ≤ LdXn (x, y) + λn A. L n Thus, after taking the ultralimit: fω : Xω → Xω ,

fω ((xn )) = (f (xn )),

we get: 1 dω (x, y) ≤ dω (fω (x), fω (y)) ≤ Ldω (x, y) L for all x, y ∈ Xω . Therefore, fω is a bilipschitz embedding, since the additive constant A is gone! Even better, if f was quasi-surjective then fω is surjective. Thus, fω : Xω → Xω is a homeomorphism! The same observation applies to sequences of quasi-isometric embeddings/quasiisometries as long as the constants L, A are ﬁxed. Exercise 2.16. Rn is QI to Rm iﬀ n = m. Exercise 2.17. Suppose that Rn → Rn is a QI embedding. Then f is quasisurjective. Hint: If not, then, taking an appropriate sequence of scaling factors and observers, and passing to asymptotic cones, we get fω : Rn → Rn , a bilipschitz embedding which is not onto. This map has to be open by the invariance of domain theorem (since the dimensions of the domain and range are the same), it is also proper since fω is bilipschitz. Thus, fω is also closed. It follows that fω is onto. Unfortunately, we cannot distinguish Hn from Hm (these are real-hyperbolic spaces of dimensions n ≥ 2, m ≥ 2 respectively) using asymptotic cones since all of their asymptotic cones are isometric to the same tree [8]!

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LECTURE 2. ULTRALIMITS AND MORSE LEMMA

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6. Morse lemma Let X = Hn be real-hyperbolic n-space. A quasi-geodesic in X is a QI embedding f : J → X, where J is an interval in R (either ﬁnite or inﬁnite). Theorem 2.18 (Morse Lemma1 ). There exists a function D(L, A) so that every (L, A)-quasi-geodesic α in X is D-Hausdorﬀ close to a geodesic α∗ . Proof. I will ﬁrst prove the Morse Lemma in the case of ﬁnite quasi-geodesics. Here is the idea behind the proof: If the Morse Lemma fails, a sequence of “counterexamples” αi to its statement yields a bi-Lipschitz map from an interval to a suitable asymptotic cone Xω of X. Lemma 2.15 then implies that the image of this arc is a geodesic αω in Xω . On the other hand, the sequence of geodesics αi∗ in X connecting the end-points of αi also converges to a geodesic arc in Xω . Since Xω is uniquely geodesic, the resulting geodesic arcs are equal to αω , contradicting the assumption that the distances between quasi-geodesics αi and geodesics αi∗ diverge to inﬁnity. Below is the actual proof. For a quasi-geodesic α : J = [0, a] → X, let α∗ : ∗ J = [0, a∗ ] → X denote the geodesic connecting α(0) to α(a). Deﬁne two numbers: Dα = dist(α, α∗ ) := sup d(α(t), Im(α∗ )), t∈I

Dα∗ = dist(α∗ , α) := sup d(α∗ (t), Im(α)) t∈J ∗

∗

max(Dα , Dα∗ ).

I will prove that Recall that Hausdorﬀ distance between α, α is the quantities Dα are uniformly bounded, since the proof of boundedness of Dα∗ is completely analogous. Suppose that the Morse Lemma fails. Then there exists a sequence fi : Ji → X of (L, A)-quasi-geodesics, so that limi Dαi = ∞. For each i pick a point xi ∈ αi (Ji ) so that d(xi , αi∗ ) is within 1i from Dαi = Di . Now, rescale the metrics on Ji and on X by λi = Di−1 and take ultralimits of the rescaled intervals and the hyperbolic spaces. Then, quasi-isometric (resp. isometric) embeddings αi (resp. αi∗ ) yield bilipschitz (resp. isometric) embeddings α : Jω → Xω = Cone(X),

α∗ : Jω∗ → Cone(X).

By our choice of xi and scaling factors, dist(α, α∗ ) = 1. Since the maps αi were (L, A)-quasi-isometric embeddings, it follows that Jω is ﬁnite iﬀ Jω∗ is ﬁnite. I ﬁrst consider the case when Jω is ﬁnite. Then α, α∗ have common end-points (since the curves αi , αi∗ did). Recall that Xω is a tree. By Lemma 2.15, the images of α, α∗ are the same. This contradicts the fact that dist(α, α∗ ) = 1. Suppose that Jω is inﬁnite, i.e., Jω = R+ . The semi-inﬁnite arcs α(R+ ), α∗ (R+ ) are within unit distance from each other. Let xω = αω (t) be the point represented by the sequence (xi ). Let C = d(α(0), xω ). There exists a geodesic arc β ⊂ Xω of length ≤ 1 connecting points x = α(s), x∗ = α∗ (s∗ ) so that β ∩ Im(α) = x, β∩Im(α∗ ) = x∗ and so that d(x, α(0)) > C. Thus, the simple arc γ = α([0, s])∪β connects the end-points of the geodesic segment γ ∗ = α∗ ([0, s∗ ]). On the other hand, xω ∈ γ \ γ ∗ . This contradicts Lemma 2.15. It remains to prove the Morse Lemma for inﬁnite quasi-geodesics. Such quasigeodesics, say, α : R → X, can be exhausted by ﬁnite quasi-geodesics αi : [−i, i] → X. Applying the Morse Lemma to the quasi-geodesics αi , we get the desired conclusion for α. 1 Maybe

this should be called the 2nd Morse lemma, since the 1st, and more famous, Morse lemma appears in the theory of Morse functions.

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The Morse Lemma also applies to all Gromov-hyperbolic geodesic metric spaces (e.g., Gromov-hyperbolic groups). On the other hand, Morse Lemma fails completely in the case of quasi-geodesics in the Euclidean plane. Exercise 2.19. Let φ : R → R be an L-Lipschitz function. Show that the map f (x) = (x, φ(x)) is a quasi-geodesic in R2 .

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LECTURE 3

Boundary extension and quasi-conformal maps 1. Boundary extension of QI maps of hyperbolic spaces Suppose that X = Hn and f : X → X is a QI map. Then, by the Morse Lemma, f sends geodesic rays uniformly close to geodesic rays: ∀ρ, ∃ρ so that d(f (ρ), ρ ) ≤ D where ρ, ρ are geodesic rays, ρ = (f (ρ))∗ (where the notation ∗ is taken from Theorem 2.18). Let ξ, ξ be the limits of the rays ρ, ρ on the boundary sphere of Hn . Then we set f∞ (ξ) := ξ . Here and in what follows, the limit point of a geodesic ray ρ in Hn is the limit lim ρ(t) ∈ S n−1 = ∂Hn .

t→∞

Exercise 3.1. The point ξ depends only on the point ξ and not on the choice of a ray ρ that limits to ξ. Thus, we obtain the boundary extension f∞ of the quasi-isometry f of Hn to the boundary sphere S n−1 . Exercise 3.2. (f ◦ g)∞ = f∞ ◦ g∞ for all quasi-isometries f, g : X → X. Exercise 3.3. Suppose that d(f, g) < ∞, i.e., there exists C < ∞ so that d(f (x), g(x)) ≤ C for all x ∈ X. Then f∞ = g∞ . In particular, if f¯ is the quasi-inverse of f , then (f¯)∞ is the kp[,,inverse of f∞ . Our next goal is to see that the extensions f∞ are continuous. Actually, they satisfy some further regularity properties which will be critical for the proof of Tukia’s theorem. Let γ be a geodesic ray in Hn and π = πγ : Hn → γ be the orthogonal projection (the nearest-point projection). Then for all x ∈ γ (except for the initial point), Hx := π −1 (x) is an n − 1-dimensional hyperbolic subspace of Hn , which is orthogonal to γ. The projection π extends continuously to a projection π : Hn ∪ S n−1 \ {ξ} → γ, where ξ is the limit point of γ. Clearly, isometries commute with projections π to geodesic rays. The following lemma is a “quasiﬁcation” of the above observation. We leave the proof of the lemma to the reader, since it amounts to nothing but “chasing triangle inequalities.” 147 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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Lemma 3.4. Quasi-isometries quasi-commute with the nearest-point projections. More precisely, let f : Hn → Hn be an (L, A)-quasi-isometry. Let γ be a geodesic ray and γ be a geodesic ray within distance ≤ D(L, A) from the quasigeodesic f (γ). Let π : Hn → γ, π : Hn → γ be nearest-point projections. Then, for some C = C(L, A), we have: d(f π, π f ) ≤ C, i.e., ∀x ∈ Hn , d(f π(x), π f (x)) ≤ C. Let ξ be the limit point of γ. Then, for xi ∈ γ converging to ξ, the boundary spheres Σi of the subspaces Hxi = π −1 (xi ), bound round balls Bi ⊂ S n−1 (containing ξ). These balls form a basis of topology at the point ξ ∈ S n−1 . Since f is a quasi-isometry, the points yi = f (xi ) cannot form a bounded sequence in Hn , hence, lim yi = ξ. Using the above lemma, we see that all the sets f∞ (Bi ) are contained in round balls Bi , whose intersection is the point ξ = f∞ (ξ). Thus, f∞ is continuous and hence, a homeomorphism. We thus obtain f∞

Lemma 3.5. For every quasi-isometry f : Hn → Hn , the boundary extension is a homeomorphism. Corollary 3.6. Hn is QI to Hm if and only if n = m.

2. Quasi-actions The notion of an action of a group on a space is replaced, in the context of quasiisometries, by quasi-action. Recall that an action of a group G on a set X is a homomorphism φ : G → Aut(X), where Aut(X) is the group of bijections X → X. Since quasi-isometries are deﬁned only up to “bounded noise”, the concept of a homomorphism has to be modiﬁed when we use quasi-isometries. Definition 3.7. Let G be a group and X be a metric space. An (L, A)–quasi action of G on X is a map φ : G → QI(X), so that: • φ(g) is an (L, A)-quasi-isometry of X for all g ∈ G. • d(φ(1), idX ) ≤ A. • d(φ(g1 g2 ), φ(g1 )φ(g2 )) ≤ A for all g1 , g2 ∈ G. Thus, Parts 2 and 3 say that φ is “almost” a homomorphism with the error A. In particular, every quasi-action determines a natural homomorphism G → QI(X). Example 3.8. Suppose that G is a group and φ : G → R is a function which determines a quasi-action of G on R by translations (g ∈ G acts on R by translation by φ(g)). Such maps φ are called quasi-morphisms and they appear frequently in GGT. Many interesting groups do not admit nontrivial homomorphisms of R but admit unbounded quasimorphisms. Here is how quasi-actions appear in the context of QI rigidity problems. Suppose that G1 , G2 are groups acting isometrically on metric spaces X1 , X2 and f : X1 → X2 is a quasi-isometry with quasi-inverse f¯. We then deﬁne a conjugate quasi-action φ of G2 on X1 by φ(g) = f¯ ◦ g ◦ f. Exercise 3.9. Show that φ is indeed a quasi-action.

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LECTURE 3. BOUNDARY EXTENSION AND QUASI-CONFORMAL MAPS

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For instance, suppose that X1 = Hn , ψ : G1 X is a geometric action, and suppose that G2 is a group which is QI to G1 (and, hence, by the Milnor-Schwarz Lemma, G2 is QI to X). We then take X2 = G2 (with a word metric). Then the quasi-isometry f : G1 → G2 yields a quasi-action φf,ψ of G2 on Hn . We now apply our extension functor (sending quasi-isometries of Hn to homeomorphisms of the boundary sphere). Then, Exercises 3.1 and 3.2 imply: Corollary 3.10. Every quasi-action φ of a group G on Hn extends (by g → φ(g)∞ ) to an action φ∞ of G on S n−1 by homeomorphisms. Lemma 3.11. The kernel for the action φ∞ is ﬁnite. Proof. The kernel of φ∞ consists of the elements g ∈ G such that d(φ(g), id) < ∞. Since φ(g) is an (L, A)-quasi-isometry of Hn , it follows from Morse Lemma that d(φ(g), id) ≤ C = C(L, A). Thus, such g (as an isometry G → G) moves every point at most by C = C (L, A). However, clearly the set of such elements of G is ﬁnite. Hence, Ker(φ∞ ) is ﬁnite as well. Geometric quasi-actions. The following three deﬁnitions for quasi-actions are direct generalizations of the corresponding deﬁnitions for actions. A quasi-action φ : G X of a group G on a metric space X is called properly discontinuous if for every bounded subset B ⊂ X the set {g ∈ G : φ(g)(B) ∩ B = ∅} is ﬁnite. A quasi-action φ : G X is cobounded if there exists a bounded subset B ⊂ X so that for every x ∈ X there exists g ∈ G so that φ(g)(x) ∈ B (this is an analogue of a cocompact isometric action). Finally, we say that a quasi-action φ : G X is geometric if it is properly discontinuous and cobounded. Exercise 3.12. Suppose that φ2 : G X2 is a quasi-action, f : X1 → X2 is a quasi-isometry and φ1 : G X1 is the conjugate quasi-action. Then φ2 is properly discontinuous (resp. cobounded, resp. geometric) if and only if φ1 is properly discontinuous (resp. cobounded, resp. geometric). 3. Conical limit points of quasi-actions Suppose that φ is a quasi-action of a group G on Hn . A point ξ ∈ S n−1 is called a conical limit point for the quasi-action φ if the following holds: For some (equivalently every) geodesic ray γ ⊂ Hn limiting to ξ, and some (equivalently every) point x ∈ Hn , there exists a constant R < ∞ and a sequence gi ∈ G so that: • limi→∞ φ(gi )(x) = ξ. • d(φ(gi )(x), γ) ≤ R for all i. In other words, the sequence φ(gi )(x) converges to ξ in a closed cone (contained in Hn ) with the tip ξ. Lemma 3.13. Suppose that ψ : G X = Hn is a cobounded quasi-action. Then every point of the boundary sphere S n−1 is a conical limit point for ψ. Proof. Consider the sequence xi ∈ X, xi = γ(i), where γ is a ray in X limiting to a point ξ ∈ S n−1 . Fix a point x0 ∈ X and a ball B = BR (x0 ) so that for every

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MICHAEL KAPOVICH, QUASI-ISOMETRIC RIGIDITY

x ∈ X there exists g ∈ G so that d(x, φ(g)(x0 )) ≤ R. Then, by coboundedness of the quasi-action ψ, there exists a sequence gi ∈ G so that d(xi , φ(gi )(x0 )) ≤ R.

Thus, ξ is a conical limit point.

Corollary 3.14. Suppose that G is a group and f : Hn → G is a quasiisometry. Then every point of S n−1 is a conical limit point for the quasi-action ψ : G Hn , which is induced by conjugating the action G G by f . Proof. The action G G by left multiplication is cobounded, hence, the conjugate quasi-action ψ : G Hn is also cobounded. If φ∞ is a topological action of a group G on S n−1 which is obtained by extension of a quasi-action φ of G on Hn then we will say that conical limit points of the action G S n−1 are the conical limit points for the quasi-action G Hn . 4. Quasiconformality of the boundary extension Can we get a better conclusion than just a homeomorphism for the maps f∞ ? Let f : Hn → Hn be an (L, A)–quasi-isometry. I will work in the upper half-space model of Hn . After composing f with isometries of Hn , we can (and will) assume that: • ξ = 0 ∈ Rn−1 and γ is the vertical geodesic above 0. • 0 = ξ = f∞ (ξ) ∈ Rn−1 . • f∞ (∞) = ∞. In particular, the vertical geodesic γ above ξ maps to a quasi-geodesic within bounded distance from the vertical geodesics γ = γ above ξ = ξ = 0. Consider an annulus A ⊂ Rn−1 given by A = {x : R1 ≤ |x| ≤ R2 } 2 where 0 < R1 ≤ R2 < ∞. We will refer to the ratio R R1 as the eccentricity of A. Then, πγ (A) is an interval of hyperbolic length d = log(R2 /R1 ) in γ. Recall that f quasi-commutes with the orthogonal projection (Lemma 3.4):

d(f ◦ πγ , πγ ◦ f ) ≤ C = C(L, A). Thus, πγ (f (A)) is an interval of the hyperbolic length ≤ c := 2C + Ld + A. Hence, f (A) is contained in the Euclidean annulus A : A = {x : R1 ≤ |x| ≤ R2 },

R2 ≤ ec . R1

We now deﬁne the function

η(r) = ec , c = 2C + L log(r) + A, η(r) = r L e2C+A . Note that η(r), r ≥ 1 is a continuous monotonic function of r so that lim η(r) = 1,

r→1

lim η(r) = ∞.

r→∞

We thus proved,

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Lemma 3.15. The topological annulus f∞ (A) is contained in an annulus A , so that eccentricity of A is ≤ η(r), where r is the eccentricity of A. In particular, round spheres (corresponding to r = 1) map to “quasi–ellipsoids” of eccentricity ≤ e2C+A . This leads to the deﬁnition: Definition 3.16. Let η : [1, ∞) → [1, ∞) be a continuous surjective monotonic function. A homeomorphism f : Rn−1 → Rn−1 is called η-quasi-symmetric1 , if for all x, y, z ∈ Rn we have |x − y| |f (x) − f (y)| ≤η (3.1) |f (x) − f (z)| |x − z| A homeomorphism f is c-weakly quasi-symmetric if |f (x) − f (y)| ≤c (3.2) |f (x) − f (z)| for all x, y, z so that |x − y| = |x − z| > 0. Remark 3.17. It turns out that every weakly quasi-symmetric map is also quasi-symmetric but we will not dwell on this. I will now change my notation and will use n to denote the dimension of the boundary sphere of the hyperbolic n + 1-dimensional space. I will think of S n as the 1-point compactiﬁcation of Rn and will use letters x, y, z, etc., to denote points on Rn . I will also use the notation f for the maps Rn → Rn . We will think of quasi-symmetric maps as homeomorphisms of S n = Rn ∪ ∞, which send ∞ to itself. The following theorem was ﬁrst proven by Tukia in the case of hyperbolic spaces and then extended by Paulin in the case of more general Gromov-hyperbolic spaces. Theorem 3.18 (P. Tukia [35], F. Paulin [25]). Every η-quasi-symmetric homeomorphism f : Rn → Rn extends to an (A(η), A(η))-quasi-isometric map F of hyperbolic space Hn+1 . Proof. Here is the idea of the proof. Since all ideal triangles in Hn+1 are δ-thin, given a triple of distinct points x, y, z ∈ S n we have their center c(x, y, z) ∈ Hn+1 , which is a point within distance ≤ δ from every side of the ideal hyperbolic triangle with the vertices x, y, z. The point c is not uniquely deﬁned, but any two centers are uniformly close to each other. Thus, we can extend the map f to Hn+1 via the formula F (c(x, y, z)) = c(F (x), F (y), F (z)). With this deﬁnition, however, it is far from clear why F is coarsely well-deﬁned. For maps f which ﬁx the point z = ∞ ∈ S n , it is technically more convenient to work instead with the points πα (x), where α is the hyperbolic geodesic connecting y and z. (The points πα (x) and c(x, y, z) will be uniformly close to each other.) This is the approach that we will use below. We deﬁne the extension F as follows. For every p ∈ Hn+1 , let α = αp be the complete vertical geodesic through p. This geodesic limits to points ∞ and x = xp ∈ Rn . Let y ∈ Rn be a point so that πα (y) = p (the point y is non-unique, 1 Quasi-symmetric

maps can be also deﬁned for general metric spaces.

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of course). Let x := f (x), y := f (y), let α ⊂ Hn+1 be the vertical geodesic through x and let p := πα (y ). Lastly, set F (p) := p . I will prove only that F is an (A, A)–coarse Lipschitz, where A = A(η). The quasi-inverse to F will be a map F¯ deﬁned via extension of the map f −1 following the same procedure. I will leave it as an exercise to verify that F¯ is indeed a quasi-inverse to F and to estimate d(F¯ ◦ F, id). Suppose that d(p1 , p2 ) ≤ 1. We would like to bound d(p1 , p2 ) from above. Without loss of generality, we may assume that p1 = en+1 ∈ Hn+1 . It suﬃces to consider two cases: 1. Points p1 , p2 belong to the common vertical geodesic α, x1 = x2 = x and d(p1 , p2 ) ≤ 1. I will assume, for concreteness, that y1 ≤ y2 . Hence, |y2 − x| d(p1 , p2 ) = log ≤ 1. |y1 − x| Since the map f is η-quasi-symmetric, −1 |y2 − x| |y2 − x| |y2 − x | 1 ≤ η ≤η ≤ ≤ η(e). η(e) |y1 − x| |y1 − x | |y1 − x| In particular, d(p1 , p2 ) ≤ C1 = log(η(e)). 2. Suppose that the points p1 , p2 have the same last coordinate, which equals 1 since p1 = en+1 , and t = |p1 − p2 | ≤ e. The points p1 , p2 belong to vertical lines α1 , α2 which limit to points x1 , x2 ∈ Rn . Without loss of generality (by postcomposing f with an isometry of Hn+1 ) we may assume that |x1 − x2 | = 1. Let yi ∈ Rn , yi ∈ Rn be points so that παi (yi ) = pi , παi (yi ) = pi . Then |yi − xi | = |pi − xi | = Ri = 1, |yi We can assume that

R1

xi |

|pi

xi |

i = 1, 2,

Ri

− = − = i = 1, 2. ≤ R2 . Then 1 d(p1 , p2 ) ≤ + log(R2 /R1 ), R1

since we can ﬁrst travel from p1 to the line α2 horizontally (along path of the length 1 R1 ) and then vertically, along α2 (along path of the length log(R2 /R1 )). We then apply the η-quasi-symmetry condition to the triple of points x1 , y1 , x2 and get: t 1 ≤ η ≤ η(e). R1 R1 Setting R3 := |x1 −y2 |, R3 := |x1 −y2 | and applying the η-quasi-symmetry condition to the triple of points x1 , y1 , y2 , we obtain t+1 R3 R3 ≤ η( ) ≤ η ≤ η(e + 1). R1 R1 1 Since R2 ≤ R3 + 1, we get: R2 R3 + 1 ≤ ≤ η(e + 1) + η(e). R1 R1

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Putting it all together, we obtain that in Case 2: d(p1 , p2 ) ≤ η(e) + log (η(e + 1) + η(e)) = C2 . Thus, in general, for p1 , p2 ∈ Hn+1 , d(p1 , p2 ) ≤ 1, we get: d(F (p1 ), F (p2 )) ≤ C1 + C2 = A. Now, for points p, q ∈ Hn+1 , so that d(p, q) ≥ 1, we ﬁnd a chain of points p0 = p, ..., pk+1 = q, where k = d(p, q) and d(pi , pi+1 ) ≤ 1, i = 0, ..., k. Hence, d(F (p), F (q)) ≤ A(k + 1) ≤ Ad(p, q) + A. Hence, the map F is (A, A)–coarse Lipschitz, where A depends only on η. Remark 3.19. One can prove QI rigidity for groups acting geometrically on Hn+1 , n ≥ 2, without using this theorem but the proof would be less clean this way. The drawback of the deﬁnition of quasi-symmetric maps is that we are restricted to maps of Rn rather than S n . In particular, we cannot apply this deﬁnition to Moebius transformations. Definition 3.20. A homeomorphism of S n is called quasi-moebius if it is a composition of a Moebius transformation with a quasi-symmetric map. We thus conclude that every (L, A)-quasi-isometry Hn+1 → Hn+1 extends to a quasi-moebius homeomorphism of the boundary sphere. Unfortunately, this definition of quasi-moebius maps is not particularly useful. One can deﬁne instead quasi-moebius maps by requiring that they quasi-preserve the cross-ratio, but then the deﬁnition becomes quite cumbersome. What we will do instead is to take the limit in the inequality (3.1) as r → 0. Then for every c-weakly quasi-symmetric map f we obtain: (3.3)

∀x,

|f (x) − f (y)| Hf (x) := lim sup sup ≤ c. y,z |f (x) − f (z)| r→0

Here, for each r > 0 the supremum is taken over all points y, z so that r = |x − y| = |x − z|. Definition 3.21. Let U, U be domains in Rn . A homeomorphism f : U → U is called quasiconformal if supx∈U Hf (x) < ∞. A quasiconformal map f is said to have linear dilatation2 H = H(f ), if H(f ) := ess sup Hf (x). x∈U

I will abbreviate quasiconformal to qc. We say that f is 1-quasiconformal if H(f ) = 1. Thus, every H-weakly-quasi-symmetric map f is quasiconformal with H(f ) ≤ H. The advantage of quasiconformality is that every Moebius map f : S n → S n is 1-quasiconformal on S n \ f −1 (∞). In particular, all quasi-moebius maps are qc. Proofs of the converse, which is a much harder theorem (that we will not use), can be found for instance, in [17] and [36]. We can now reformulate Lemma 3.15 as 2 Usually

one uses a diﬀerent quantity, K(f ), to measure the degree of quasiconformality of f , see Appendix 3. However, we will not use K(f ) in these lectures.

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MICHAEL KAPOVICH, QUASI-ISOMETRIC RIGIDITY

Lemma 3.22. Let f : Hn+1 → Hn+1 be an (L, A)-quasi-isometry. Then its boundary extension h = f∞ : S n → S n is quasiconformal with H(h) ≤ c(L, A). Theorem 3.23. Every quasiconformal map f : Rn → Rn is η-quasi-symmetric for some η = η(H(f )). I will assume from now on that n ≥ 2 since for n = 1 the notion of quasiconformality is essentially useless. Example 3.24. 1. Every Moebius transformation of S n is 1-quasiconformal. 2. Every diﬀeomorphism f : S n → S n is quasiconformal. Here is a non-smooth example of a quasiconformal map of R2 . Let (r, θ) be the polar coordinates in R2 and let φ(θ) denote diﬀeomorphisms R+ → R+ and S 1 → S 1 . Then the map f : R2 → R2 , given in polar coordinates by the formula: f (r, θ) = (r, φ(θ)), f (0) = 0, is quasiconformal but is not smooth (unless φ is a rotation). Analytic properties of qc maps. Proofs of the following can be found, for instance, in [17] and [36]. (1) H(f ◦ g) ≤ H(f )H(g), H(f −1 ) = H(f ). These two properties follow directly from the deﬁnition. (2) (J.V¨ ais¨al¨a) Every qc map f is diﬀerentiable a.e. in Rn . Furthermore, its partial derivatives are in Lnloc (Rn ). In particular, they are measurable functions. (3) (J.V¨ ais¨al¨a) The Jacobian Jf of a qc map f is nonzero a.e. in Rn . (4) Suppose that f is a quasiconformal map. For x where Dx f exists and is invertible, we let λ1 ≤ ... ≤ λn denote the singular values of the matrix Dx f . Then λn = Hf (x) λ1 Thus, the image of the unit sphere in the tangent space Tx S n under Dx f is an ellipsoid of eccentricity ≤ H. This is the geometric interpretation of qc maps: They map inﬁnitesimal spheres to inﬁnitesimal ellipsoids of uniformly bounded eccentricity. (5) QC Liouville’s theorem (F. Gehring and Y. Reshetnyak). 1quasiconformal maps are conformal. (Here and in what follows we do not require that conformal maps preserve orientation, only that they preserve angles. Thus, from the viewpoint of complex analysis, we allow holomorphic and antiholomorphic maps of the 2-sphere.) (6) Convergence property for quasiconformal maps (J.V¨ ais¨ al¨ a). Let x, y, z ∈ S n be three distinct points. A sequence of quasiconformal maps (fi ) is said to be “normalized at {x, y, z}” if the limits limi fi (x), limi fi (y), lim fi (z) exist and are all distinct. Then every normalized sequence of quasiconformal maps (fi ) with H(fi ) ≤ H contains a subsequence which converges to an quasiconformal map f with H(f ) ≤ H. (7) Semicontinuity of linear dilatation (P. Tukia; T. Iwaniec and G. Martin). Suppose that (fi ) is a convergent sequence of quasiconformal maps with H(fi ) ≤ H so that the sequence of functions Hfi converges

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155

to a function H in measure: ∀ > 0, lim mes({x : |Hfi (x) − H(x)| > }) = 0. i→∞

(Here mes is the Lebesgue measure on S n .) Then the sequence (fi ) converges to a qc map f so that Hf (x) ≤ H(x) a.e..

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LECTURE 4

Quasiconformal groups and Tukia’s rigidity theorem 1. Quasiconformal groups Recall that we abbreviate quasiconformal to qc. A group G of quasiconformal homeomorphism of S n is called (uniformly) quasiconformal if there exists H < ∞ so that for every g ∈ G, H(g) ≤ H. We will simply say that such a G is a qc group. Example 4.1. 1. Every conformal (Moebius) group is quasiconformal (take H = 1). 2. Suppose that f : S n → S n is quasiconformal with H(f ) ≤ H, and G is a group of conformal transformations of S n . Then the conjugate group Gf := f Gf −1 is uniformly quasiconformal. This follows from the inequality: H(f gf −1 ) ≤ H(f ) · 1 · H = H 2 . 3. Suppose that φ is a quasi-action of a group G on Hn+1 . Then the extension φ∞ deﬁnes an action of G on S n as a qc group. This follows immediately from Lemma 3.22. 4. Conversely, in view of Theorem 3.18, every qc group action G S n extends to a quasi-action G Hn+1 . D. Sullivan [31] proved that for n = 2, every qc group is qc conjugate to a conformal group. This fails for n ≥ 3. For instance, there are qc groups acting on S 3 which are not isomorphic to any subgroup of isometries of H4 , see [32, 16]. Note that Tukia’s examples are solvable and nondiscrete, while Isachenko’s examples are discrete and are virtually isomorphic to free products of surface groups. Our goal is to prove Theorem 4.2 (P. Tukia, [33]). Suppose that G is a (countable) qc group acting on S n , n ≥ 2, so that (almost) every point of S n is a conical limit point of G. Then G is qc conjugate to a group acting conformally on S n . Once we have this theorem, we obtain: Theorem 4.3. Suppose that G = G2 is a group QI to a group G1 acting geometrically on Hn+1 (n ≥ 2). Then G also acts geometrically on Hn+1 . Proof. We already know that a quasi-isometry G1 → G2 yields a quasi-action φ of G on Hn+1 . Every boundary point of Hn+1 is a conical limit point for this quasi-action. We also have a qc extension of the quasi-action φ to a qc group action G S n . Theorem 4.2 yields a qc map h∞ conjugating the group action G S n to a conformal action η : G S n . Every conformal transformation g of S n extends to a unique isometry ext(g) of Hn+1 . Thus, we obtain a homomorphism 157 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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ρ : G → Isom(Hn+1 ), ρ(g) = ext(η(g)); the kernel of ρ has to be ﬁnite since the kernel of the action φ∞ : G S n is ﬁnite. We need to verify that the action ρ of G on Hn+1 is geometric. Let h := ext(h∞ ) be an extension of h∞ to quasi-isometry of Hn+1 . Then ¯ ◦ ρ(g) ◦ h g → h determines a quasi-action ν of G on Hn+1 whose extension to S n is the qc action φ∞ . Thus, there exists C so that for every g ∈ G d(ν(g), φ(g)) ≤ C. It follows that h quasi-conjugates the action ρ and the quasi-action φ. Since the latter was geometric, the former is geometric as well. Remark 4.4. Of course, the action ρ can have nontrivial ﬁnite kernel. ¯ = ρ(G) we obtain: By taking G Corollary 4.5. Let G be a group QI to Hn+1 . Then G contains a ﬁnite ¯ = G/K embeds in Isom(Hn+1 ) as a properly disconnormal subgroup K so that G tinuous cocompact subgroup. Thus, our objective now is to prove Theorem 4.2. 2. Invariant measurable conformal structure for qc groups Let Γ be a group acting conformally on S n = Rn ∪ ∞ and let ds2E be the usual Euclidean metric on Rn . Then conformality of the elements of Γ amounts to saying that for every g ∈ Γ, and every x ∈ Rn (which does not map to ∞ by g) (Dx g)T · Dx g is a scalar matrix (scalar multiple of the identity matrix). Here and in what follows, Dx f is the matrix of partial derivatives of f at x. In other words, the product (Jg,x )− n · (Dx g)T · Dx g 2

is the identity matrix I. Here Jg,x = det(Dx g) is the Jacobian of g at x. This equation describes (in terms of calculus) the fact that the transformation g preserves the conformal structure on S n . More generally, consider Riemannian metrics ds2 on S n (given by symmetric positive-deﬁnite matrices Ax depending smoothly on x ∈ Rn ). A conformal structure on Rn is a metric ds2 on Rn up to multiplication by a conformal factor. It is convenient to use normalized Riemannian metrics ds2 on Rn , where we require that det(Ax ) = 1 for every x. Geometrically speaking, this means that the volume of the unit ball in Tx (Rn ) with respect to the metric ds2 is the same as the volume ωn of the unit Euclidean n-ball. Normalization for a general metric Ax is given by multiplication by det(A)−1/n . We then identify conformal structures on Rn with smooth matrix-valued functions Ax , where Ax is a positive-deﬁnite symmetric matrix with unit determinant. The pull-back g ∗ (ds2 ) of ds2 under a diﬀeomorphism g : S n → S n is given by the symmetric matrices Mx = (Dx g)T Agx Dx g.

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159

If Ax was normalized, then, in order to have normalized pull-back g • (ds2 ) we again rescale: Bx := (Jg,x )− 2n (Dx g)T · Agx · Dx g. 1

How do we use this in the context of qc maps? Since their partial derivatives are measurable functions on Rn , it makes sense to work with measurable Riemannian metrics and measurable conformal structures on Rn . (One immediate beneﬁt is that we do not have to worry about the point ∞.) We then work with measurable matrixvalued functions Ax , otherwise, nothing changes. Given a measurable conformal structure μ, we deﬁne its linear dilatation H(μ) as the essential supremum of the ratios λn (x) H(x) := , λ1 (x) where λ1 (x) ≤ ... ≤ λn (x) are the eigenvalues of Ax . Geometrically speaking, if Ex ⊂ Tx Rn is the unit ball with respect to Ax , then H(x) is the eccentricity of the ellipsoid Ex (with respect to the standard Euclidean metric on Rn ). A measurable conformal structure μ is said to be bounded if H(μ) < ∞. A measurable conformal structure μ on Rn is invariant under a qc group G if g • μ = μ, ∀g ∈ G. In detail: ∀g ∈ G,

(Jg,x )

2 −n

T

(Dx g) · Agx · Dx g = Ax

a.e. in Rn . Theorem 4.6 (D. Sullivan [31], P. Tukia [33]). Every qc group acting on S n , n ≥ 2, admits a bounded invariant measurable conformal structure. Proof. The idea is to start with an arbitrary conformal structure μ0 on Rn (say, the Euclidean structure) and then “average” it over g ∈ G. I will prove this only for countable groups G (which is all we need since we are interested in f.g. groups). Our proof is somewhat diﬀerent from the one given by Sullivan and Tukia. Let Ax be the matrix-valued function deﬁning a normalized Riemannian metric on Rn ; for instance, we can take Ax = I for all x ∈ Rn . Then, since G is countable, for a.e. x ∈ Rn , we have a well-deﬁned matrix-valued function corresponding to g • (μ0 ) on Tx Rn : Ag,x := (Jg,x )− 2n (Dx g)T · Dx g. 1

For such x we let Eg,x denote the unit ball in Tx Rn with respect to g • (μ0 ). From the Euclidean viewpoint, Eg,x is an ellipsoid of volume ωn . This ellipsoid (up to scaling) is the image of the unit ball under the inverse of the derivative Dx g. Since H(g) ≤ H for all g ∈ G, the ellipsoids Eg,x have uniformly bounded eccentricity, i.e., the ratio of the largest to the smallest axis of this ellipsoid is uniformly bounded independently of x and g. Since the volume of Eg,x is ﬁxed, it follows that the diameter of the ellipsoid is uniformly bounded above and below. Let Ux denote the union of the ellipsoids Eg,x . g∈G

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This set has diameter ≤ R for some R independent of x. Note also that Ux is symmetric (about 0). Note that the family of sets {Ux , x ∈ Rn } is invariant under the group G: (Jg,x )−1/n Dx g(Ux ) = Ug(x) , ∀g ∈ G. Lemma 4.7. Given a bounded symmetric subset U of Rn with nonempty interior, there exists a unique ellipsoid E = EU (centered at 0) of smallest volume containing U . The ellipsoid E is called the John–Loewner ellipsoid of U . Existence of such an ellipsoid is clear. Uniqueness is not diﬃcult, but not obvious (see Appendix 2). We then let Ex denote the John-Loewner ellipsoid of Ux . This ellipsoid deﬁnes a measurable function of x to the space of positive-deﬁnite n × n symmetric matrices. In other words, we obtain a measurable Riemannian metric ν on Rn . Uniqueness of the John-Loewner ellipsoid and G-invariance of the sets Ux imply that the action of G preserves νx (up to scaling, of course). One can then get a normalized conformal structure μ by rescaling ν, so that g • μ = μ, ∀g ∈ G. It remains to show that μ is bounded. Indeed, the length of the major semi-axis of Ex does not exceed R while its volume is ≥ V ol(Ux ) ≥ ωn (here we are using the fact that all the matrices Ag,x have unit determinant). Thus, the eccentricity of Ex is uniformly bounded. Hence μ is a bounded measurable conformal structure. 3. Proof of Tukia’s theorem We are now ready to prove Theorem 4.2. As a warm-up, we consider the easiest case, n = 2 (the argument in this case is due to D.Sullivan). In the 2-dimensional case, Theorem 4.2 holds without the conical limit points assumption. Let μ be a bounded measurable conformal structure on S 2 invariant under the group G. The measurable Riemann mapping theorem for S 2 states that every bounded measurable conformal structure μ on S 2 is quasiconformally equivalent to the standard conformal structure μ0 on S 2 , i.e., there exists a quasiconformal map f : S 2 → S 2 which sends μ0 to μ: f • μ0 = μ. ¯ = (Analytically, this theorem amounts to solvability of the Beltrami equation ∂f 2 μ(z)∂f for every measurable Beltrami diﬀerential μ on S .) Since the quasiconformal group G preserves μ on S 2 , it follows that the group Gf = f gf −1 preserves the structure μ0 . Thus, Gf acts as a group of conformal automorphisms of the round sphere, which proves the theorem for n = 2. We now consider the case of arbitrary n ≥ 2. Definition 4.8. A function η : Rn → R is called approximately continuous at a point x ∈ Rn if for every > 0 lim

r→0

mes{y ∈ Br (x) : |η(x) − η(y)| > } = 0. mes Br (x)

Here mes stands for the Lebesgue measure and Br (x) is the r-ball centered at x. In other words, as we “zoom into” the point x, “most” points y ∈ Br (x), have value η(y) close to η(x), i.e., the rescaled functions ηr (x) := η(rx) converge in measure to the constant function.

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LECTURE 4. QUASICONFORMAL GROUPS AND TUKIA’S RIGIDITY THEOREM

161

We will need the following result from real analysis: Lemma 4.9 (See Theorem 3.37 in [1]). For every L∞ function η on Rn , a.e. point x ∈ Rn is an approximate continuity point of η. The functions to which we will apply this lemma are the matrix entries of a (normalized) bounded measurable conformal structure μ(x) on Rn (which we will identify with a matrix-valued function Ax ). Since μ is bounded and normalized, the matrix entries of μ(x) will be in L∞ . We let μ(x) again denote a bounded normalized measurable conformal structure on Rn invariant under G. Since a.e. point in Rn is a conical limit point of G, we will ﬁnd such a point ξ which is also an approximate continuity point for μ(x). Then, without loss of generality, we may assume that the point ξ is the origin in Rn and that μ(0) = μ0 (0) is the standard conformal structure on Rn . We will n+1 . identify Hn+1 with the upper half-space Rn+1 + . Let e = en+1 = (0, ..., 0, 1) ∈ H n+1 . Since 0 Let φ(g)(x) denote the quasi-action of the elements g ∈ G on H is a conical limit point of G, there exists C < ∞ and a sequence gi ∈ G so that limi→∞ φ(gi )(e) = 0 and d(φ(gi )(e), ti e) ≤ c where d is the hyperbolic metric on Hn+1 and ti > 0 is a sequence converging to zero. Let Ti denote the hyperbolic isometry (Euclidean dilation) given by x → ti x, x ∈ Hn+1 . Set g˜i := φ(gi−1 ) ◦ Ti . Then d(φ(˜ gi )(e), e) ≤ Lc + A for all i. Furthermore, each g˜i is an (L, A)-quasi-isometry of Hn+1 for ﬁxed L and A. By applying coarse Arzela-Ascoli theorem, we conclude that the sequence (˜ gi ) coarsely subconverges to a quasi-isometry g˜. Thus, the sequence of quasiconformal gi )∞ subconverges to a quasiconformal map f = (˜ g )∞ . maps fi := (˜ We also have: •

μi := fi• (μ) = (Ti )• (gi )−1 (μ) = (Ti )• μ, since g • (μ) = μ, ∀g ∈ G. Thus, μi (x) = μ(Ti x) = μ(ti x), in other words, the measurable conformal structure μi is obtained by “zooming into” the point 0. Since x is an approximate continuity point for μ, the functions μi (x) converge (in measure) to the constant function μ0 = μ(0). Thus, we have the diagram: fi

μ −→

μi ↓

f

μ −→ μ0 If we knew that the derivatives Dfi subconverge (in measure) to the derivative of Df , then we would conclude that f • μ = μ0 .

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Then f would conjugate the group G (preserving μ) to a group Gf preserving μ0 and, hence, acting conformally on S n . However, derivatives of quasiconformal maps (in general), converge only in the “biting” sense (see [17]), which will not suﬃce for our purposes. Thus, we have to use a less direct argument below. We restrict to a certain round ball B in Rn . Since μ is approximately continuous at 0, for every ∈ (0, 12 ), μi (x) − μ(0) < away from a subset Wi ⊂ B of measure < i , where limi i = 0. Thus, for x ∈ Wi , 1 − < λ1 (x) ≤ ... ≤ λn (x) < 1 + , where λk (x) are the eigenvalues of the matrix Ai,x of the metric μi (x). Thus, √ 1+ √ H(μi , x) < √ ≤ 1 + 4 ≤ 1 + 2. 1− away from subsets Wi . For every g ∈ G, each map γi := fi gfi−1 is conformal with respect to the structure μi and, hence (1 + 2)-quasiconformal away from the set Wi . Since limi mes(Wi ) = 0, we conclude, by the semicontinuity property of qc mappings, that each γ := lim γi is (1 + 2)-quasiconformal. Since this holds for arbitrary > 0 and arbitrary round ball B, we conclude that each γ is conformal (with respect to the standard conformal structure on S n ). Thus, the group Γ = f Gf −1 consists of conformal transformations. 4. QI rigidity for surface groups The proof of Tukia’s theorem mostly fails for groups QI to the hyperbolic plane. The key reason is that quasi-symmetric maps of the circle are diﬀerentiable a.e. but are not absolutely continuous. Thus, their derivative could (and, in the interesting cases will) vanish a.e. on the circle. Nevertheless, the same proof yields: If G is a group QI to the hyperbolic plane, then G acts on S 1 by homeomorphisms with ﬁnite kernel K, so that the action is “discrete and cocompact” in the following sense: Let T denote the set of ordered triples of distinct points on S 1 . Thus, T is an open 3-dimensional manifold; one can compute its fundamental group and see that it is inﬁnite cyclic, furthermore, T is homeomorphic to D2 × S 1 . The action G S 1 , of course, yields an action G T . Then G T is properly discontinuous and cocompact. The only elements of G that can ﬁx a point in T are the elements of K. Thus, Γ = G/K acts freely on T and the quotient T /Γ is a closed 3-dimensional manifold M . It was proven, in a combination of papers by Tukia, Gabai, Casson and Jungreis in 1988—1994, that such a group Γ acts geometrically and faithfully on the hyperbolic plane. Their proof was mostly topological. One can now also derive this result from Perelman’s proof of Thurston’s geometrization conjecture as follows. The inﬁnite cyclic group π1 (T ) will be a normal subgroup of π1 (M ). Then, you look at the list of closed aspherical 3-dimensional manifolds (given by the Geometrization Conjecture) and see that such an M has to be a Seifert manifold, modelled on one of the geometries H2 × R, SL(2, R), N il, E3 , see [27]. In the case of the geometries N il, E3 , one sees that the quotient of π1 by normal inﬁnite cyclic subgroup yields a group Γ which is VI to Z2 . Such a group cannot act on S 1 so

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LECTURE 4. QUASICONFORMAL GROUPS AND TUKIA’S RIGIDITY THEOREM

163

that Γ T is properly discontinuous and cocompact. On the other hand, in the case of the geometries H2 × R, SL(2, R), the quotient by a normal cyclic subgroup will be VI to a group acting geometrically on H2 .

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APPENDIX

1. Hyperbolic space The upper half-space model of hyperbolic n-space Hn is Rn+ = {(x1 , ...xn ) : xn > 0} equipped with the Riemannian metric ds2 =

|dx|2 . x2n

Thus, the length of a smooth path p(t), t ∈ [0, T ] in Hn is given by T |p (t)|e dt. ds = pn (t) p 0 Here |v|e is the Euclidean norm of a vector v and pn (t) denotes the n-th coordinate of the point p(t). The (ideal) boundary sphere of Hn is the sphere S n−1 = Rn−1 ∪ {∞}, where n−1 consists of points in Rn with vanishing last coordinate xn . R Complete geodesics in Hn are Euclidean semicircles orthogonal to Rn−1 as well as vertical straight lines. For instance, if p, q ∈ Hn are points on a common vertical line, then their hyperbolic distance is d(p, q) = | log(pn /qn )|. The group of isometries of Hn is denoted Isom(Hn ). Every isometry of Hn extends uniquely to a Moebius transformation of the boundary sphere S n−1 . The latter are the conformal diﬀeomorphisms of S n−1 in the sense that they preserve (Euclidean) angles. (I do not assume that conformal transformations preserve orientation.) Conversely, every Moebius transformation of S n−1 extends to a unique isometry of Hn . The group M obn−1 of Moebius transformations of S n−1 contains all inversions, all Euclidean isometries of Rn−1 and all dilations. (Compositions of Euclidean isometries and dilations are called similarities.) In fact, a single inversion together with all similarities of Rn−1 generate the full group of Moebius transformations. Furthermore, every similarity of Rn−1 extends to a similarity of Rn+ in the obvious fashion, so that the extension is an isometry of Hn . Similarly, inversions extend to inversions which are also isometries of Hn . Exercise A.1. Show that the group M obn−1 acts transitively on the set of triples of distinct points in S n−1 . The key fact of hyperbolic geometry that we will need is that all triangles in Hn are δ-thin, for δ ≤ 1. Here is an outline of the proof. First, every geodesic 165 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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MICHAEL KAPOVICH, QUASI-ISOMETRIC RIGIDITY

T τ3

τ2 σ3

S

σ2

H

2

p σ1 τ1 R

Figure 1. Hyperbolic triangles S and T . triangle in Hn lies in a 2-dimensional hyperbolic subspace H2 ⊂ Hn , so it suﬃces to consider the case n = 2. Next, consider a geodesic triangle S ⊂ H2 with the sides σ1 , σ2 , σ3 . Then S can be “enlarged” to an ideal hyperbolic triangle T ⊂ H2 , i.e., a triangle all whose vertices belong to the boundary circle S 1 of H2 , see Figure 1. For every point p which belongs to a side σ1 of the triangle S, every geodesic in H2 connecting p to the union of the two opposite sides (τ2 ∪ τ3 ) of T , will have to cross σ2 ∪ σ3 . Thus, S is “thinner” than the triangle T , so it suﬃces to estimate the thinness of T . Since M ob1 acts transitively on triples of distinct points in S 1 , it suﬃces to consider the case where the ideal vertices of the triangle T are the points A1 = ∞, A2 = −1, A3 = 1 in S 1 = R ∪ {∞}. Now, consider points on the side τ1 of T connecting A2 to A3 . Let p denote the top-most point of the Euclidean semicircle τ1 , i.e., p = (0, 1). Then, considering the horizontal Euclidean segment γ connecting p to the point q = (−1, 1) ∈ τ3 , we see that hyperbolic length of γ equals 1 and, hence, d(p, q) ≤ 1. Consider points p ∈ τ1 , so that the ﬁrst coordinate of p is negative. (See Figure 2.) Exercise A.2. The (hyperbolic) length of the horizontal Euclidean segment connecting p to q ∈ τ3 is < 1. The same argument applies to points p with positive ﬁrst coordinate. We thus conclude that for every point in τ1 , the distance to τ2 ∪ τ3 is ≤ 1. Therefore, every hyperbolic triangle is δ-thin for δ ≤ 1. Remark A.3. The optimal thinness constant for hyperbolic triangles is √ arccosh( 2), see e.g. [6, Proposition 6.42]. 2. Least volume ellipsoids Recall that a closed ellipsoid (with nonempty interior) centered at 0 in Rn can be described as E = EA = {x ∈ Rn : ϕA (x) = xT Ax ≤ 1} where A is some positive-deﬁnite symmetric n × n matrix. The volume of such an ellipsoid is given by the formula V ol(EA ) = ωn (det(A))−1/2

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APPENDIX.

8

A1 =

167

T

H

2

p

q q’

p’

R A2 = −1

A3 =1

0

Figure 2. Thinness estimate for the ideal hyperbolic triangle T . where ωn is the volume of the unit ball in Rn . Recall that a subset X ⊂ Rn is centrally-symmetric if X = −X. Theorem A.4 (F. John). For every compact centrally-symmetric subset X ⊂ Rn with nonempty interior, there exists a unique ellipsoid E(X) of least volume containing X. The ellipsoid E(X) is called the John-Loewner ellipsoid of X. Proof. The existence of E(X) is clear by compactness. We need to prove uniqueness. Consider the function f on the space Sn+ of positive deﬁnite symmetric n × n matrices, given by 1 f (A) = − log det(A). 2 Lemma A.5. The function f is strictly convex. Proof. Take A, B ∈ Sn+ and consider the family of matrices Ct = tA+(1−t)B, 0 ≤ t ≤ 1. Strict convexity of f is equivalent to strict convexity of f on such line segments of matrices. Since A and B can be simultaneously diagonalized by a matrix M , we obtain: 1 f (Dt ) = f (M Ct M T ) = − log det(M ) − log det(Ct ) = − log det(M ) + f (Ct ), 2 where Dt is a segment in the space of positive-deﬁnite diagonal matrices. Thus, it suﬃces to prove strict convexity of f on the space of positive-deﬁnite diagonal matrices D = Diag(x1 , ..., xn ). Then, 1 log(xi ) 2 i=1 n

f (D) = −

is strictly convex since log is strictly concave.

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MICHAEL KAPOVICH, QUASI-ISOMETRIC RIGIDITY

In particular, whenever V ⊂ Sn+ is a convex subset and f |V is proper, f attains a unique minimum on V . Since log is a strictly increasing function, the same uniqueness assertion holds for the function det−1/2 on Sn+ . Let V = VX denote the set of matrices C ∈ Sn+ so that X ⊂ EC . Since ϕA (x) is linear as a function of A for any ﬁxed x ∈ X, it follows that V convex. Thus, the least volume ellipsoid containing X is unique. 3. Diﬀerent measures of quasiconformality Let M be an n×n invertible matrix with singular values λ1 ≤ ... ≤ λn . Equivalently, these numbers are the square roots of eigenvalues of the matrix M M T . The singular value decomposition yields: M = U Diag(λ1 , ..., λn )V where U, V are orthogonal matrices. We deﬁne the following distortion quantities for the matrix M : • Linear dilatation: λn = M · M −1 , H(M ) := λ1 where A is the operator norm of the n × n matrix A: max

v∈Rn \0

|Av| . |v|

• Inner dilatation: HI (M ) :=

λ1 ....λn | det(M )| = n λ1 M −1 −n

• Outer dilatation HO (M ) :=

λnn M n = λ1 ....λn | det(M )|

• Maximal dilatation K(M ) := max(HI (M ), HO (M )). Exercise A.6. (H(M ))n/2 ≤ K(M ) ≤ (H(M ))n−1 Hint: It suﬃces to consider the case when M = Diag(λ1 , ..., λn ) is a diagonal matrix. As we saw, qc homeomorphisms are the ones which send inﬁnitesimal spheres to inﬁnitesimal ellipsoids of uniformly bounded eccentricity. The usual measure of quasiconformality of a qc map f is its maximal distortion (or maximal dilatation) K(f ), deﬁned as K(f ) := ess sup K(Dx (f )) x

where the essential supremum is taken over all x in the domain of f . Here Dx f is the derivative of f at x (Jacobian matrix). See e.g. J.V¨ ais¨al¨a’s book [36]. A map f is called K-quasiconformal if K(f ) ≤ K. In contrast, the measure of quasiconformality used in these lectures is: H(f ) := ess sup H(Dx f ). x

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APPENDIX.

To relate the two deﬁnitions we observe that 1 ≤ (H(f ))n/2 ≤ K(f ) ≤ (H(f ))n−1 . In particular, K(f ) = 1 if and only if H(f ) = 1.

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169

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Bibliography [1] A. Bruckner, J. Bruckner, B. Thomson, “Real Analysis”, Prentice Hall, 1996. ´ [2] M. Burger, S. Mozes, Lattices in product of trees, Inst. Hautes Etudes Sci. Publ. Math., No. 92 (2000), pp. 151–194. [3] P.-A. Cherix, M. Cowling, P. Jolissaint, P. Julg and A. Valette, “Groups with the Haagerup property. Gromov’s a-T-menability”, Progress in Mathematics, vol. 197. Birkhauser Verlag, Basel, 2001. [4] R. Chow, Groups quasi-isometric to complex hyperbolic space, Transactions of AMS, vol 348 (1996) pp. 1757–1769. [5] A. Casson, D. Jungreis, Convergence groups and Seifert ﬁbered 3-manifolds, Invent. Math., vol. 118 (1994) pp. 441–456. [6] C. Drutu, M. Kapovich, “Geometric Group Theory”, Preprint, 2013. [7] A. Dyubina (Erschler), Instability of the virtual solvability and the property of being virtually torsion-free for quasi-isometric groups, Internat. Math. Res. Notices, vol. 21 (2000) pp. 1097–1101. [8] A. Dyubina (Erschler), I. Polterovich, Explicit constructions of universal R–trees and asymptotic geometry of hyperbolic spaces, Bull. London Math. Soc. vol. 33 (2001), pp. 727–734. [9] A. Eskin, Quasi-isometric rigidity of nonuniform lattices in higher rank symmetric spaces, Journal of AMS, vol. 11 (1998) pp. 321–361. [10] A. Eskin, B. Farb, Quasi-ﬂats and rigidity in higher rank symmetric spaces, Journal of AMS, vol. 10 (1997) pp. 653–692. [11] A. Eskin, D. Fisher, K. Whyte, Quasi-isometries and rigidity of solvable groups, Pure and Applied Mathematics Quarterly, vol. 3 (2007) no. 4, part 1, pp. 927–947. [12] B. Farb, The quasi-isometry classiﬁcation of lattices in semisimple Lie groups, Math. Res. Letters, vol. 4. (1997) pp. 705–717. [13] B. Farb, R. Schwartz, The large-scale geometry of Hilbert modular groups, J. Diﬀerential Geom., vol. 44 (1996), no. 3, 435–478. [14] D. Gabai, Convergence groups are Fuchsian groups, Annals of Math., vol. 136 (1992), pp. 447–510, [15] E. Ghys, P. de la Harpe, Inﬁnite groups as geometric objects (after Gromov), in “Ergodic theory, symbolic dynamics and hyperbolic spaces” (Trieste, 1989), T. Bedford, M. Keane, and C. Series, eds., Oxford Univ. Press, 1991, pp. 299–314. [16] N. A. Isachenko, Uniformly quasiconformal discontinuous groups that are not isomorphic to M¨ obius groups, Dokl. Akad. Nauk SSSR, vol. 313 (1990), no. 5, pp. 1040–1043. [17] T. Iwaniec, G. Martin, “Geometric Function Theory and Non-linear Analysis”, Oxford Univ. Press, 2001. [18] M. Kapovich, B. Leeb, On asymptotic cones and quasi-isometry classes of fundamental groups of nonpositively curved manifolds, Geometric Analysis and Functional Analysis, vol. 5 (1995) no 3, pp. 582–603. [19] M. Kapovich, B. Leeb, Quasi-isometries preserve the geometric decomposition of Haken manifolds, Invent. Math. vol. 128 (1997), no. 2, pp. 393–416. [20] B. Kleiner, B. Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean ´ buildings, Inst. Hautes Etudes Sci. Publ. Math., No. 86 (1997), pp. 115–197. [21] V. Laﬀorgue, A proof of property (RD) for cocompact lattices of SL(3; R) and SL(3; C), J. Lie Theory, vol. 10 (2000), pp. 255–267. [22] B. Leeb, P. Scott, A geometric characteristic splitting in all dimensions, Comment. Math. Helv. vol. 75 (2000), no. 2, pp. 201–215. 171 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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[23] P. Pansu, Metriques de Carnot-Caratheodory et quasiisometries des espaces symmetriques de rang un, Annals of Math., vol. 129 (1989), pp. 1–60. [24] P. Papasoglu, Quasi-isometry invariance of group splittings, Annals of Math., vol. 161 (2005), pp. 759–830. [25] F. Paulin, Un groupe hyperbolique est d´ etermin´ e par son bord, J. London Math. Soc. (2) vol. 54 (1996), no. 1, pp. 50–74. [26] R. Sauer, Homological invariants and quasi-isometry, Geometric and Functional Analysis, vol. 16 (2006) pp. 476–515. [27] P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc. vol. 15 (1983), no. 5, pp. 401–487. ´ [28] R. Schwartz, The quasi-isometry classiﬁcation of rank one lattices, Inst. Hautes Etudes Sci. Publ. Math., No. 82 (1995), pp. 133–168. [29] R. Schwartz, Quasi-isometric rigidity and Diophantine approximation, Acta Math. vol. 177 (1996), no. 1, pp. 75–112. [30] J. Stallings, On torsion free groups with inﬁnitely many ends, Annals of Math., vol. 88 (1968), pp. 312–334. [31] D. Sullivan, On the ergodic theory at inﬁnity of an arbitrary discrete group of hyperbolic motions, in “Riemann Surfaces and Related Topics”: Proceedings of the 1978 Stony Brook Conference, I. Kra and B. Maskit, eds., Ann. Math. Studies 97, Princeton University Press, 1981, pp. 465–496. [32] P. Tukia, A quasiconformal group not isomorphic to a M¨ obius group, Ann. Acad. Sci. Fenn. Ser. A I Math., vol. 6 (1981) pp. 149–160. [33] P. Tukia, On quasiconformal groups, J. d’Anal. Math., vol. 46 (1986), pp. 318–346. [34] P. Tukia, Homeomorphic conjugates of Fuchsian groups, J. Reine Angew. Math., vol. 391 (1988), pp. 1–54. [35] P. Tukia, Convergence groups and Gromov’s metric hyperbolic spaces, New Zealand J. Math. vol. 23 (1994), no. 2, pp. 157–187. [36] J.V¨ ais¨ al¨ a, “Lectures on n-dimensional quasiconformal mappings”, Springer Lecture Notes in Mathematics, vol. 229, 1961.

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https://doi.org/10.1090//pcms/021/06

Geometry of Outer Space Mladen Bestvina

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IAS/Park City Mathematics Series Volume 21, 2012

Geometry of Outer Space Mladen Bestvina Introduction Outer space is a contractible space on which Out(Fn ) acts properly discontinuously. It was introduced by Marc Culler and Karen Vogtmann in [18]. Outer space is analogous to the symmetric space associated to an arithmetic lattice or the Teichm¨ uller space associated to the mapping class group of a surface. It is useful to keep in mind the comparison maps Out(Fn ) → GLn (Z) and M od(S) → Out(π1 (S)). The ﬁrst is obtained by sending an automorphism of Fn to the induced automorphism of the abelianization Zn of Fn . It is always surjective, and for n = 2 it is an isomorphism. The second comparison homomorphism is deﬁned on the (extended) mapping class group of a punctured surface S with χ(S) < 0 and it is always injective. When S is a punctured torus it is an isomorphism. These notes are about the geometry of Outer space. The natural metric on it is not symmetric, but this is imposed by the features of Out(Fn ); e.g. the growth rates of an automorphism and its inverse may be diﬀerent. We give a metric classiﬁcation of automorphisms, following Bers’ proof of Thurston’s classiﬁcation of surface automorphisms. For the most part, the exposition is hands-on, with many exercises involving concrete examples. There are several harder exercises, indicated by asterisks. The last lecture gives a glimpse of the current developments. I would like to thank Yael Algom-Kﬁr, Pritam Ghosh, Brian Mann and Catherine Pfaﬀ for correcting some of the mistakes in an earlier version, and to the organizers of minicourses where I presented some variant of these notes at the following locations: University of Chicago, University of Utah, Goa, Technion, University of Buﬀalo, Yale University, Berlin and Vercors. Special thanks go to Yael Algom-Kﬁr and Catherine Pfaﬀ for carefully reading the manuscript and suggesting several improvements.

c 2014 American Mathematical Society

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LECTURE 1

Outer space and its topology In this lecture we will talk about the topology of Outer space. For more information see the excellent survey [37]. A graph is a cell complex of dimension ≤ 1. The rose Rn is the graph with 1 vertex and n edges. 1.1. Markings A marking of a graph Γ is a homotopy equivalence f : Rn → Γ. This is a convenient way of specifying an identiﬁcation between π1 (Γ) with the free group Fn (thought of as being identiﬁed with π1 (Rn ) once and for all) with a (deliberate) ambiguity of composing with inner automorphisms (no basepoints!). Two marked graphs f : Rn → Γ and f : Rn → Γ are equivalent if there is a homeomorphism φ : Γ → Γ such that φf f (homotopic). In practice one deﬁnes the inverse of a marking, i.e. a homotopy equivalence Γ → Rn . If the edges of Rn are oriented and labeled by a basis a, b, · · · of Fn (thus identifying π1 (Rn ) = Fn ), the inverse marking can be deﬁned by specifying a maximal tree T in Γ, orienting all edges in Γ−T , and labeling them with a (possibly diﬀerent) basis of Fn , expressed as words in a, b, · · · . Such a choice deﬁnes a map Γ → Rn by collapsing T to a point and sending each edge to the edge path speciﬁed by the label. Exercise 1.1. Show that the two marked graphs in Figure 1.1 are equivalent. We follow the convention that capital letters represent inverses of lower case letters. Unlabeled edges form a maximal tree.

b a

aB b

Figure 1.1. Equivalent marked graphs

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1.2. Metric A metric on a ﬁnite graph is an assignment of positive numbers (e), called lengths, to the edges e of Γ. The volume of a ﬁnite metric graph is the sum of the lengths of the edges. A metric on a graph allows one to view the graph as a geodesic metric space, with each edge e having length (e). This point of view lets us assign lengths also to paths in the graph; in particular any closed immersed loop has ﬁnite length (an immersion is a locally injective map, in this case from the circle to the graph). We will consider the triples (Γ, , f ) where Γ is a ﬁnite graph with all vertices of valence ≥ 3, is a metric on Γ with volume 1, and f : Rn → Γ is a marking. Two such triples (Γ, , f ) and (Γ , , f ) are equivalent if there is an isometry (i.e. a length-preserving homeomorphism) φ : Γ → Γ such that φf f . Definition 1.2. Outer space Xn = {(Γ, , f )}/ ∼ is the set of equivalence classes of ﬁnite marked metric graphs with vertices of valence ≥ 3 and of volume 1. We will usually omit equivalence class, and f from the notation, and talk about points Γ ∈ Xn instead of [(Γ, , f )] ∈ Xn . 1.3. Lengths of loops Once π1 (Rn ) is identiﬁed with Fn we can view each nontrivial conjugacy class in Fn as a loop in Rn , up to homotopy. The homotopy class has a unique immersed representative, up to parametrization. If α is a nontrivial conjugacy class and (Γ, , f ) ∈ Xn , deﬁne the length Γ (α) of α in Γ as the length of the immersed loop α|Γ in Γ homotopic to f (α). 1.4. Fn -trees ˜ is a (metric, simplicial) tree, and If Γ is a marked metric graph, the universal cover Γ ˜ The the marking (i.e. the identiﬁcation π1 (Γ) = Fn ) induces an action of Fn on Γ. equivalence relation on marked metric graphs translates to saying that two metric simplicial Fn -trees S, T are equivalent if there is an equivariant isometry S → T . Thus Xn can be alternatively deﬁned as the space of minimal metric simplicial free Fn -trees with covolume 1, up to equivariant isometry. The length of a conjugacy class becomes the translation length in the tree. 1.5. Topology and Action Xn can be naturally decomposed into open simplices. If Γ is a graph and f : Rn → Γ ◦

a marking, the set of possible metrics Σ(Γ) on Γ is an open simplex i = 1} {(1 , 2 , · · · , E ) | i > 0, of dimension E − 1 if E is the number of edges. If T is a forest (i.e. a disjoint union of trees) in Γ and Γ = Γ/T is obtained by collapsing all edges of T to points, ◦

◦

then Σ(Γ ) can be identiﬁed with the open face of Σ(Γ) in which the coordinates of edges in T are 0. Then Γ is said to be obtained from Γ by collapsing a forest, and ◦

Γ is obtained from Γ by blowing up a forest. The union Σ(Γ) of Σ(Γ) with all such

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open faces as T ranges over all forests in Γ is a simplex-with-missing-faces: it can be obtained from the closed simplex Σ∗ (Γ) = {(1 , 2 , · · · , E ) | i ≥ 0, i = 1} by deleting those open faces that assign 0 to a set of edges whose union contains a loop. For example, if Γ is the theta-graph with 2 vertices and 3 edges connecting them, Σ(Γ) is the 2-simplex minus its vertices. Exercise 1.3. The smallest dimension of a Σ(Γ) is n − 1. Exercise 1.4. The largest dimension of a Σ(Γ) is 3n − 4. Top dimensional simplices correspond to 3-valent graphs, codimension 1 simplices to graphs with one valence 4 vertex and all others valence 3, etc. In this way Xn becomes a complex of simplices-with-missing-faces. We deﬁne the simplicial topology on Xn just like on a simplicial complex: a subset U ⊂ Xn is open [closed] if and only if U ∩ Σ(Γ) is open [closed] in Σ(Γ) for every Γ ∈ Xn . One can also put the missing faces in and get a simplicial complex Xn∗ , called the simplicial completion of Xn . This complex is isomorphic to the splitting complex or equivalently to the complex of spheres, see Lecture 4. The fact that Xn∗ is a simplicial complex (e.g. simplices are determined by their vertices) is nontrivial. Exercise 1.5. Show that every Σ(Γ) is contained in only ﬁnitely many Σ(Γ ). Conclude that Xn is locally compact and metrizable. Another way to deﬁne a topology on Xn is via length functions. Let C be the set of all nontrivial conjugacy classes in Fn . The length function is the function L : Xn → (0, ∞)C to the space of functions C → (0, ∞) with the product topology that to Γ assigns (α → Γ (α)). This function is injective (this is the Rigidity of the Length Spectrum, see Exercise 2.8), and if we identify Xn with the image, the subspace topology induces a topology on Xn , the length function topology. This topology is equivalent to the simplicial topology, i.e. L is an embedding, see [22]. See also [17]. 1.6. Thick part and spine For a ﬁxed small > 0 deﬁne the thick part Xn () of Xn as the set of Γ ∈ Xn such that Γ (α) ≥ for every nontrivial conjugacy class α. When > 0 is suﬃciently small the intersection of Xn () with every Σ(Γ) is a nonempty convex set (e.g. 1 ensures that the barycenter of Σ(Γ) is in Xn ()). taking ≤ 3n−3 For each simplex-with-missing-faces Σ(Γ) let S(Γ) be the union of simplices in the barycentric subdivision of the closed simplex Σ∗ (Γ) that are contained in Σ(Γ). Thus S(Γ) is the dual of the missing faces. The spine Kn ⊂ Xn is the union of S(Γ)’s for all Γ ∈ Xn . 1.7. Action of Out(Fn ) There is a natural right action of Out(Fn ) on Xn by precomposing the marking. An element Φ ∈ Out(Fn ) can be thought of as a homotopy equivalence Φ : Rn → Rn and then the action is: [(Γ, , f )] · Φ = [(Γ, , f Φ)] Exercise 1.6. Show that the action is well-deﬁned, i.e. does not depend on the representative in the conjugacy class.

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Figure 1.2. Spine and thick part intersected with a simplex Note that in terms of inverse markings, the action amounts to postcomposing with inverse automorphism. The action is simplicial and it is compatible with the action on conjugacy classes: ΓΦ (α) = Γ (Φ(α)) It is sometimes convenient to write Γ (α) as a pairing Γ, α and then the identity becomes ΓΦ, α = Γ, Φ(α) Exercise 1.7. Show that the point stabilizer Stab(Γ, , f ) is isomorphic to the isometry group Isom(Γ, ) of the underlying graph, with an isometry φ corresponding to the automorphism f −1 φf , where f −1 : Γ → Rn denotes the inverse marking. Exercise 1.8. Show that there are only ﬁnitely many orbits of Σ(Γ)’s. Exercise 1.9. Show that the action leaves the spine and the thick part invariant. Proposition 1.10. The action of Out(Fn ) on Xn is proper. The action on the thick part and on the spine is cocompact. Exercise 1.11. (Combinatorial description of the spine.) Show that the following simplicial complex Pn , the poset of marked graphs is homeomorphic to the spine Kn . The vertices of Pn are marked graphs (Γ, f ) (with Γ having no vertices of valence ≤ 2) modulo equivalence (Γ, f ) ∼ (Γ , f ) if there is a homeomorphism φ : Γ → Γ with φf f . A k-simplex in Pn is induced by a sequence of nontrivial forest collapses Γ0 → Γ1 → · · · → Γk . Exercise 1.12. dim Kn = 2n − 3. Exercise 1.13. There are equivariant deformation retractions from Outer space Xn to the thick part Xn () (for small > 0) and from Xn () to the spine Kn . Reduced Outer space Rn is the subspace of Xn consisting of those graphs that do not have a separating edge. Exercise 1.14. Show that Rn is an equivariant deformation retract of Xn .

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1.8. Rank 2 picture Recall that the comparison map Out(Fn ) → GLn (Z) is an isomorphism when n = 2 and that the (full) mapping class group of a punctured torus is Out(Fn ). Since the symmetric space SL2 (R)/SO2 and Teichm¨ uller space of (T 2 , {p}) is the hyperbolic 2 plane H , it is not surprising that X2 is essentially also (a combinatorial version of) H2 . More precisely, the reduced Outer space in rank 2 is the ﬁlled in Farey graph minus the vertices pictured in Figure 1.3.

Figure 1.3. Reduced Outer space in rank 2. The circle and the vertices of the triangles are not part of the space. Markings of three of the simplices are pictured in Figure 1.4. Observe that there are two ways to blow up a rose R2 to a theta graph and this translates into the fact that reduced Outer space is a surface. To obtain the whole Outer space, we also need to attach simplices corresponding to graphs with separating edges, see Figure 1.5. These simplices have missing vertices and are missing two of the sides. They are attached to the reduced Outer space along the third side. Exercise 1.15. Find an automorphism of F2 that takes the bottom triangle in Figure 1.4 to the upper right triangle. Exercise 1.16. What does the automorphism a → a, b → ab do to the reduced Outer space? It ﬁxes a missing vertex and ... Exercise 1.17. Show that neither Outer space nor reduced Outer space is a manifold when n ≥ 3. Hint: Find a graph in Xn with one vertex of valence 4, the other vertices of valence 3, and all three blowups having no separating edges. 1.9. Contractibility The central fact about Outer space is its contractibility, proved by Marc Culler and Karen Vogtmann. Theorem 1.18 ([18]). Outer space Xn is contractible.

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aB

aB

b

b aB a

b

b a

a

b

b

Figure 1.4. 3 2-simplices with their marked graphs

Figure 1.5. Simplices corresponding to graphs with separating edges

Of course, this means that the thick part and the spine are also contractible. Culler-Vogtmann use combinatorial Morse theory and argue that the spine is contractible. They carefully order the set of roses in Xn : r1 , r2 , · · · and argue that for each i the union of stars of the ﬁrst i roses is contractible. The diﬃcult step is showing that the intersection of the star of the ith rose with the union of the previous stars is contractible.

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An alternative proof, more in the spirit of these notes, was constructed by Skora [33], building on the ideas of Steiner. For each Γ ∈ Xn they construct a (folding) path to Γ from a point in the simplex containing the rose with identity marking and argue that the collection of these paths varies continuously in Γ (this is technically the hard step). These paths then determine a deformation retraction from Xn to a simplex with missing faces. For more on folding paths see Lecture 2. Neither Steiner’s nor Skora’s work was published; for details see [15] or [22]. 1.10. Group theoretic consequences Corollary 1.19. Out(Fn ) is ﬁnitely presented. Proof. Recall that if a group acts freely and cocompactly on a simply connected simplicial complex, then it is ﬁnitely presented. More generally, it is ﬁnitely presented if it acts cocompactly on a simply connected complex with ﬁnitely presented vertex stabilizers and ﬁnitely generated edge stabilizers (see [14]). The action on the spine has ﬁnite stabilizers. Proposition 1.20. Out(Fn ) is virtually torsion-free. In the proof we will use the fact that every ﬁnite subgroup of Out(Fn ) ﬁxes a point of Xn . This is called the (Nielsen) realization theorem, see [39, 16, 29]. For a recent, more intrinsic, proof see [28]. Proof. We claim that the kernel of Out(Fn ) → GLn (Z/3) is torsion-free. Let 1 = Φ ∈ Out(Fn ) have ﬁnite order. By Nielsen realization, Φ is realized as a graph isomorphism φ : Γ → Γ. We may without loss collapse all separating edges of Γ, so every edge is contained in an embedded circle. If Φ is in the kernel, then φ maps any circle to itself preserving orientation. But for each circle C in Γ there is another circle C (because n ≥ 2 without loss) so that C ∩ C is nonempty, connected, and = C. Thus φ is identity. Corollary 1.21. A torsion-free subgroup H of ﬁnite index has a compact classifying space K(H, 1) of dimension 2n − 3, and the virtual cohomological dimension of Out(Fn ) is 2n − 3. To see that vcd(Out(Fn )) ≥ 2n − 3 note that Out(Fn ) contains an abelian subgroup of rank 2n − 3. E.g. for n = 3 we can take the group of automorphisms of the form a → a, b → ap b, c → aq car for p, q, r ∈ Z. Exercise 1.22. Show that this group is indeed isomorphic to Z2n−3 . The next statement does not use contractibility of Xn , only Nielsen realization plus the fact that the action on the spine is cocompact. Corollary 1.23. Out(Fn ) has ﬁnitely many conjugacy classes of ﬁnite subgroups. Exercise∗ 1.24. Find a nontrivial element of ﬁnite order in the kernel of Out(Fn ) → GLn (Z/2). Show that every such element has order 2 and that therefore every ﬁnite subgroup of the kernel is abelian (in fact, a direct sum of Z/2’s). Can you ﬁnd the largest such subgroup? Exercise∗ 1.25. Can you ﬁnd estimates on the size of the largest ﬁnite subgroup of Out(Fn )? For example, the stabilizer of a rose has order 2n n!. Can you ﬁnd a larger ﬁnite group? What about n = 2 and 3? For the answer see [38].

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LECTURE 2

Lipschitz metric, train tracks In this lecture we introduce the Lipschitz metric on Outer space. The deﬁnition, motivated by Thurston’s metric on Teichm¨ uller space [36], dates back to the 1990’s when my former student Tad White proved the key Lemma 2.4. At the time we didn’t have any applications for this metric. It recaptured my own interest when I realized that one can give a classiﬁcation of automorphisms in the style of Bers using this metric. Bers [4] proved the Thurston classiﬁcation theorem for mapping classes using the Teichm¨ uller metric on Teichm¨ uller space. This is the subject of Lecture 3. Francaviglia and Martino were the ﬁrst to study this metric systematically. Much of the material in this section is in their paper [21]. 2.1. Deﬁnitions Let [(Γ, , f )], [(Γ , , f )] ∈ Xn be two points in Outer space. A continuous map φ : Γ → Γ is a diﬀerence of markings map if φf f . We will only consider Lipschitz maps and we denote by σ(φ) the Lipschitz constant of φ. When φ is homotoped rel vertices to a map φ which has constant slope on each edge, then σ(φ ) ≤ σ(φ). We deﬁne the distance: d(Γ, Γ ) = inf log σ(φ) φ

as φ : Γ → Γ ranges over all diﬀerence of markings. Recall the Arzela-Ascoli theorem, which says that any sequence of L-Lipschitz maps between two compact metric spaces has a convergent subsequence. This theorem implies that the inﬁmum above is realized. We will call a diﬀerence of markings φ : Γ → Γ optimal if it has constant slope on each edge and minimizes the Lipschitz constant (which is then the maximal slope). 2.2. Elementary facts Proposition 2.1. • d(Γ1 , Γ3 ) ≤ d(Γ1 , Γ2 ) + d(Γ2 , Γ3 ). • d(Γ, Γ ) ≥ 0 and equality implies Γ = Γ . • d(ΓΦ, Γ Φ) = d(Γ, Γ ). Proof. The ﬁrst claim follows from the general fact that σ(ψφ) ≤ σ(ψ)σ(φ). For the second claim, let φ : Γ → Γ be an optimal map. If d(Γ, Γ ) < 0 then all slopes of φ are < 1. This implies that the volume of the image of φ is < 1, so φ is not surjective. But a homotopy equivalence between ﬁnite graphs without vertices of valence 1 is always surjective. 185 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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If d(Γ, Γ ) = 0 then all slopes of φ must be equal to 1 and the images of diﬀerent edges can intersect only in ﬁnite sets. Thus φ is a quotient map that identiﬁes ﬁnitely many collections of ﬁnitely many points. The only way for such a map to be a homotopy equivalence (or even for Γ and Γ to have the same rank) is for φ to be an isometry, so Γ = Γ . The third claim is an exercise. 2.3. Example To illustrate the deﬁnition, let us compute the distance in the following example, see Figure 2.1. To compute d(A, B) consider the diﬀerence of markings map φ that

B A

B

A

Figure 2.1. A is the rose with edge lengths 12 and B is the theta graph with edge lengths 13 , both in the same 2-simplex. sends the vertex of A to the midpoint of the middle edge of B, the loop on the left homeomorphically to the circle formed by the middle and the left edges, and the loop on the right homeomorphically to the circle formed by the middle and (2) the right edge. The slope of φ on both edges is ( 31 ) = 43 , so d(A, B) ≤ log 43 . We 2

now claim that d(A, B) = log 43 . To see this, observe that each of the two edges in A is a loop of length 12 and any diﬀerence of markings map will map it to a loop homotopic to an immersed loop of length 23 . Thus the length of the image cannot be smaller than 23 , and so the slope of any diﬀerence of markings map on either edge cannot be less than 43 . More generally, we observe: Lemma 2.2. If α is any nontrivial conjugacy class then log

Γ (α) ≤ d(Γ, Γ ) Γ (α)

So for any α we obtain a lower bound on the distance. In our example, the lower bound agrees with the upper bound provided by the explicit diﬀerence of markings map. This determines the distance. We will say that a conjugacy class α is a witness if equality holds in the statement of the Lemma. In a similar way, one can compute that d(B, A) = log 32 by considering the map B → A that collapses the middle edge, and the witness loop formed by the other two edges. Note in particular that d(A, B) = d(B, A). Exercise 2.3. Let A be as above and let C be the graph in the same 1-simplex as A with lengths of edges and 1 − . Show that d(C , A) → ∞ as → 0, but d(A, C ) stays bounded by log 2.

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) Thus the distance function is not even quasi-symmetric, i.e. d(X,Y d(Y,X) can be arbitrarily large. However, a theorem of Handel-Mosher [25] states that the restriction of d to any thick part Xn () is quasi-symmetric. See also [2] for a diﬀerent proof.

2.4. Tension graph, train track structure Here is the crucial fact. It is analogous to Teichm¨ uller’s theorem for Riemann surfaces. It states that witnesses always exist. Lemma 2.4. Suppose d(Γ, Γ ) = log λ. Then there is a conjugacy class α ∈ C such that Γ (α) =λ Γ (α) Note that for any α inequality ≤ holds. So the lemma says that we can deﬁne the distance alternatively as Γ (α) d(Γ, Γ ) = log max α Γ (α) The equality between the min and the max is an instance of the max-ﬂow min-cut principle. The proof of Lemma 2.4 introduces the key idea of train tracks. Proof. Fix a diﬀerence of markings map φ : Γ → Γ with σ(φ) = λ. By Δ = Δφ denote the union of those edges of Γ on which the slope of f is λ. This subgraph of Γ is called the tension graph for φ, and some of its vertices may have valence 1 or 2. Now let v be a vertex of Δ. A direction at v in Δ is a germ of geodesic paths [0, ] → Δ sending 0 to v. Alternatively, it is an oriented edge of Δ with initial vertex at v. Denote the set of these directions by Tv (Δ). Its cardinality is the valence of v in Δ and this set plays the role of the unit tangent space at v. Now φ induces a map (kind of a derivative) φ∗ : Tv (Δ) → Tφ(v) (Γ ) since for small it sends a geodesic γ : [0, ] → Δ to a geodesic φγ : [0, ] → Γ (parametrized with speed λ). Here φ(v) may not be a vertex, in which case Tφ(v) (Γ ) naturally has two directions. Thus we have an equivalence relation on Tv (Δ): d1 ∼ d2 ⇐⇒ φ∗ (d1 ) = φ∗ (d2 ) A train track structure on a graph Δ is simply a collection of equivalence relations on the sets Tv (Δ) for all vertices v. Thus the tension graph is naturally equipped with a train track structure. The deﬁnition is motivated by Thurston’s train tracks on surfaces. It is customary to draw equivalent directions as tangent to each other. The equivalence classes are gates. An immersed path in Δ (thought of as a train route) is legal if whenever it passes through a vertex, the entering and the exiting gates are distinct. Otherwise, a path is illegal. Similarly, a turn (i.e. an unoriented pair of distinct directions) is illegal if the directions are equivalent; otherwise the turn is legal. More informally, legal paths do not make 180◦ turns. Figure 2.2 shows the tension graphs with their train track structures from the examples in Section 2.3. The tension graph of φ : A → B is all of A and the vertex has two gates. For the map B → A the tension graph is a circle formed by two edges and all turns are legal.

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A

B

Figure 2.2. Tension graphs with their train track structures from examples in 2.3. Now we make the following two observations: • if the immersed loop α|Γ representing a conjugacy class α in Γ is contained (α) in Δ and is legal, then ΓΓ(α) = λ, i.e. α is a witness, • if every vertex of Δ has at least two gates, then Δ contains a legal loop; in fact this loop can be chosen to cross every oriented edge at most once. The ﬁrst of these claims is an exercise in deﬁnitions: f has slope λ on each edge of α|Γ and consecutive edges are mapped without backtracking by deﬁnition of legality. For the second claim, keep extending a legal path until the same oriented edge repeats. Of course, in general Δ may have vertices with only one gate. To ﬁnish the proof we will show that φ may be perturbed so that every vertex has at least two gates. Claim: Suppose v is a vertex of Δφ with only one gate. Then φ may be perturbed to φ : Γ → Γ so that σ(φ ) = λ and Δφ Δφ . Repeating this operation will eventually produce a perturbation of φ whose tension graph has at least two gates at every vertex (note that the set of edges where the slope is λ cannot become empty by the assumption that d(Γ, Γ ) = log λ). Proof of Claim. The homotopy φt from φ to φ will be stationary on all vertices except for v, and it will move φ(v) slightly in the direction φ∗ (d), where d ∈ Tv (Δ). All maps φt are linear on edges. Thus the slope is unaﬀected on edges not incident to v, it decreases on edges in Δ incident to v, and it may increase on edges outside Δ incident to v. The perturbation is small so that even the increased slope on such edges is < λ. Thus Δφ ⊂ Δφ but Δφ does not contain v and edges incident to it. Exercise 2.5 ([21]). Show that in any graph with a train track structure with at least two gates at every vertex, there is a legal loop that is either embedded, or it forms a “ﬁgure 8” crossing each edge once, or it forms a “dumbbell”, crossing edges in the two loops once and edges in the connecting arc twice. See Figure 2.3.

Figure 2.3. Possible forms of candidates. Train track structure is suggested by the pictures. We say that an immersed loop in a graph Γ (without any train track structure) is a candidate if it has a form as in Exercise 2.5. Thus, given Γ, Γ there is always a candidate in Γ which is a witness for d(Γ, Γ ). Thus there is a simple algorithm

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to compute distances d(Γ, Γ ) in Outer space. Simply look at the ratio of lengths in Γ and in Γ of all candidate loops in Γ and take the log of the largest such ratio. Exercise 2.6. Let R3 be the rose in X3 with all edges of length 13 and with inverse marking given by a, b, c, and let Γ be another such rose but with inverse marking given by abA, bacB, a. Find all candidates in each that are witnesses for the distance to the other. Exercise 2.7. Consider the automorphism Φ of F4 = a, b, c, d given by a → b → c → d → ADCB (capital letters are inverses of the lowercase letters). (a) Let R be the rose with the identity marking (so the edges correspond to a, b, c, d) and with all lengths 14 . Compute d(R, RΦ). (b) Find the graph Γ in the same simplex as R (i.e. the same marking, but edge lengths can be arbitrary) so that d(Γ, ΓΦ) is minimal. (c) Can you ﬁnd a graph Γ in a small neighborhood of Γ so that d(Γ , Γ Φ) < d(Γ, ΓΦ)? Exercise 2.8. If Γ, Γ are distinct points in Xn show that there are conjugacy classes α, β such that Γ (α) > Γ (α) and Γ (β) < Γ (β). Deduce that the length function L : Xn → (0, ∞)C and the projectivized length function Xn → P(0, ∞)C are injective. This is called the length spectrum rigidity. Exercise 2.9. For a marked graph Γ let KΓ be the ﬁnite set of candidates for Γ and for all marked graphs obtained from Γ by collapsing a forest. Show that lengths of elements of KΓ determine each point of Σ(Γ). But, surprisingly, there is no ﬁnite collection of conjugacy classes whose lengths deﬁne an injection of Xn into RN , see [34]. The following two properties of the Lipschitz metric point out similarities with the ∞ metric. Exercise 2.10. Show that in each simplex straight lines are geodesics (not necessarily parametrized with unit speed). Hint: Let Γ1 , Γ2 , Γ3 be three points along a straight line with Γ2 between the other two. Argue that any witness for Γ1 → Γ3 is also a witness for Γ1 → Γ2 and for Γ2 → Γ3 . Exercise 2.11. Show that geodesics are not unique in general. Speciﬁcally, in rank 2, show that there are geodesics contained in a 2-simplex with endpoints on one edge, but with the geodesic intersecting the interior. Exercise 2.12. Show that the distance function d : Xn × Xn → [0, ∞) is continuous. Hint: It suﬃces to prove continuity on Σ(Γ) × Xn for every simplex Σ(Γ). Now use a variant of Exercise 2.9. 2.5. Folding paths A folding path is determined by an optimal map φ : Γ → Γ such that the tension graph Δφ is all of Γ and every vertex has at least two gates. It is a geodesic path Γt from Γ to Γ and for each t < t it comes with an optimal map Γt → Γt so that the tension graph is all Γt and these maps compose correctly for t < t < t . To deﬁne an initial segment of this path choose > 0 smaller than half the length

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of the shortest edge, and for t ∈ [0, ] deﬁne Γt by identifying segments of length t issuing from any vertex in equivalent directions. Then rescale to make volume equal to 1. For example, for the map φ : A → B considered in Section 2.3, the time t graph before rescaling would have one edge of length 2t and two edges of length 1 − 2t. There are naturally induced maps Γt → B so at t = one can repeat the procedure to continue the path. It is not clear a priori that this deﬁnes a path globally. If φ is simplicial with respect to some subdivisions of Γ and Γ and the lengths of all edges in each subdivision are equal, the procedure amounts to Stallings’ folding, identifying two edges whenever they share a vertex and map to the same edge (but here we do it continuously resulting in a path in Xn ). A very elegant deﬁnition of folding paths is due to Skora [33]. It is most ˜ →Γ ˜ . Consider the conveniently described in terms of the universal cover φ˜ : Γ ˜ graph of φ: ˜ ˜ = {(u, v) ∈ Γ ˜×Γ ˜ | φ(u) = v} Gr(φ) ˜ and deﬁne the vertical t-neighborhood of the graph Gr(φ): ˜ ˜×Γ ˜ | d(φ(u), v) ≤ t} Nt = {(u, v) ∈ Γ ˜×Γ ˜ ˜ . Restrict the horizontal foliation of Γ where d refers to the path metric on Γ ˜ ˜ ˜ by Γ × {v}, v ∈ Γ to Nt and deﬁne Γt as the quotient space where all components ˜ t is a tree and its quotient by the action of Fn is the of leaves are collapsed. Then Γ desired graph Γt (which needs to be rescaled). For t = 0 we have Γt = Γ and for t large Γt = Γ . To get a feel for this deﬁnition, consider the “tent map” φ : [−1, 1] → [0, λ] for λ > 0, which has slope λ on [−1, 0] and slope −λ on [0, 1]. The graph of this map is pictured in Figure 2.4 (with the target thought of as R).

Figure 2.4. Construction of folding paths following Skora. ˜ t comes from projecting to the ﬁrst coordinate and maps Γt → The metric on Γ Γt for t < t from inclusion Nt → Nt . To see that a folding path is always a geodesic take any legal loop in Γ and observe that its image in Γt is legal for Γt → Γt for any t > t and that it is a witness for that map (the slope of the map on each edge is the ratio of lengths of the loop at Γt and Γt ). Example 2.13. Let Γ be the rose in X2 with identity marking, and with (a) = λ−2 and (b) = λ−1 where λ > 0 satisﬁes λ−1 + λ−2 = 1 (see Example 3.6). Let φ : Γ → ΓΦ be the optimal map for Φ given by a → b, b → ab suggested by Φ, so φ

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LECTURE 2. LIPSCHITZ METRIC, TRAIN TRACKS

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has slope λ on both edges and Δ = Γ. The folding path from Γ to ΓΦ amounts to identifying the terminal portion of the edge b around the edge a in the direction of A (note that {A, B} is the only illegal turn). Proposition 2.14 ([21]). d is a geodesic metric. Proof. Choose an optimal map φ : Γ → Γ . If Δφ = Γ (and all vertices have ≥ 2 gates) the folding path is a geodesic from Γ to Γ . If Δφ = Γ start by scaling Δφ up and the edges in the complement up until the tension graph Δ increases. If there are any vertices with one gate, adjust φ. Continue until Δ = Γ and then follow with a folding path. See also [21] and [7] for further discussion. Exercise 2.15. What is the geodesic constructed in this proof in the case Γ = B and Γ = A in the example in Section 2.3? Exercise 2.16. Find geodesics from R to Γ and from Γ to R in Exercise 2.6. Exercise∗∗ 2.17. Can a folding path intersect some Σ(Γ) in a disconnected set?

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LECTURE 3

Classiﬁcation of automorphisms Recall the classiﬁcation of isometries of hyperbolic space. Definition 3.1. Let Φ ∈ Out(Fn ). The displacement function of Φ is the function D = DΦ : Xn → [0, ∞) given by D(Γ) = d(Γ, ΓΦ). We denote τ (Φ) = inf DΦ , the translation length of Φ. Definition 3.2. Φ is • hyperbolic if inf D > 0 and minimum is realized, • elliptic if inf D = 0 and minimum is realized (equivalently, Φ ﬁxes a point of Xn ), • parabolic if the minimum of D is not realized. We now describe the quality of each of these classes, following Bers’ approach to Thurston’s classiﬁcation of mapping classes [4]. 3.1. Elliptic automorphisms Example 3.3. Let Φ : F2 → F2 be given by a → b, b → a. Then Φ is elliptic as it ﬁxes the rose with identity marking and edge lengths 12 . The following is an immediate consequence of the fact that point stabilizers are ﬁnite. Proposition 3.4. Every elliptic automorphism has ﬁnite order. The converse also holds, namely every automorphism of ﬁnite order is elliptic, by Nielsen Realization. Exercise 3.5. Show that the automorphism Φ deﬁned by a → b, b → Ab is elliptic and ﬁnd a ﬁxed point. Hint: First ﬁnd the ﬁxed points of Φ3 . 3.2. Hyperbolic automorphisms Example 3.6. Let Φ be given by a → b, b → ab. Let φ : Γ → Γ be the map suggested by Φ on the rose Γ. Now assign lengths so that φ has the same slope on both edges, say λ. Temporarily assigning 1 to a we see from φ(a) = b that b must have length λ. Then from φ(b) = ab we√ get the equation λ2 = 1 + λ, whose only positive root is the golden ratio λ = 1+2 5 . Now we must rescale to get volume 1, 1 λ = λ−2 and (b) = 1+λ = λ−1 . i.e. we must set (a) = 1+λ Now consider the train track structure on Γ induced by φ. There are 3 gates: {a}, {b} and {A, B}. Observe that φ sends legal paths to legal paths. To prove this observation, one only needs to check: 193 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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• the image of each edge is a legal path, and • legal turns are mapped to legal turns (equivalently, f induces an injective map on the set of gates at every vertex). Both of these are easy to check: ab is a legal path and the map on gates is {a} → {b}, {b} → {a}, {A, B} → {A, B}. In particular, for any m = 1, 2, · · · the map φm : Γ → Γ is optimal with the same train track structure. We now see that d(Γ, ΓΦm ) = m log λ. This implies τ (Φ) = log λ and so Φ is hyperbolic. Exercise 3.7. Prove the assertion in the last sentence. Hint: The triangle inequality. If d(Γ , Γ Φ) < log λ one can get from Γ to ΓΦm via Γ , Γ Φ, · · · , Γ Φm , ΓΦm . For large m this is a contradiction. Exercise 3.8. Suppose φ : Γ → Γ is any map between graphs that sends vertices to vertices and edges to nontrivial immersed paths. Let e1 , · · · , ek be the list of all 1-cells in Γ (ignore orientations). Form the transition matrix M : it’s a k × k matrix whose ij-entry is the number of times f (ej ) crosses ei with either orientation. For example, the transition matrix both for Example 3.6 and for Example 3.5 is M = ( 01 11 ). Assume that some positive power of M has all entries positive. The classical theorem of Perron-Frobenius says that the largest in norm eigenvalue λ of M is > 1, its eigenspace is 1-dimensional and spanned by a vector with all coordinates positive. Show that there is a metric on Γ such that φ has slope λ on every edge. Definition 3.9. Let φ : Γ → Γ be an optimal map. We say that φ is a train track map if Δφ = Γ, every vertex has at least two gates, and φ maps legal paths to legal paths. The slope λ on each edge is the dilatation of φ. The arguments of Example 3.6 prove: Proposition 3.10. If Φ admits a train track representative φ : Γ → Γ then d(Γ, ΓΦ) = τ (Φ). Thus Φ is hyperbolic unless the dilatation is 1, and then Φ is elliptic. More generally, if φ : Γ → Γ is an optimal map representing Φ such that φ(Δφ ) ⊂ Δφ and so that φ : Δφ → Δφ is a train track map, then Φ is hyperbolic. Example 3.11. Let Φ ∈ Out(F3 ) be given by a → b, b → ab, c → ca. Then Φ is hyperbolic with τ (Φ) = log λ with λ the golden ratio just like in Example 3.6. For Γ take the rose with the metric on a, b a scaled down version of the metric in Example 3.6 and let the length of c be close to 1. Then the map Γ → Γ suggested by Φ is a train track map. Theorem 3.12. Every hyperbolic automorphism can be represented by an optimal map φ : Γ → Γ so that Δφ is an invariant subgraph and φ : Δφ → Δφ is a train track map. Note that Δφ may be a proper subgraph of Γ. For a proof see [6]. One can also arrange that φ sends vertices to vertices, but then the train track structure on Δφ has to be modiﬁed by declaring d1 ∼ d2 provided there exists k > 0 such that φk∗ (d1 ) = φk∗ (d2 ). Example 3.13. The following example was obtained by entering a random (surface) automorphism into Peter Brinkmann’s program XTrain available on the

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web at http://math.sci.ccny.cuny.edu/pages?name=XTrain. Capital letters denote inverses of lower case letters. With notation as in Figure 3.1, the map is a → Da, b → CAdaCCAdaCCAda, c → cADacADaccADaccADac, d → ddabccA. The program also tells us that the dilatation λ is a root of the polynomial x4 −10x3 +10x2 −10x+1 and is approximately 9.012144.

b a

d

c

Figure 3.1. A more typical train track map. Train track structure can be computed by looking at the “derivative” map. By a we denote the direction where a begins, and by A where a ends: a → D → a, b → C → C, c → c, d → d, B → A → A. Two directions at a vertex form an illegal turn if they eventually map to the same direction, so A ∼ B and b ∼ C. This is indicated in the diagram. As an exercise, compute the lengths of edges. Example 3.14. Here is another example, this time it does not come from a surface automorphism. The map is given by a → c, b → bcAdEaCb, c → cAeDa, d → BcAeAd, e → BcAe. a d

c

b

e

Figure 3.2. Another train track map. The only nontrivial gate is {d, e}.

When Φ is hyperbolic and Γ achieves the minimum of DΦ , choose a geodesic path from Γ to ΓΦ and take the union of all Φm -translates of the path, m ∈ Z. This is a geodesic line and Φ acts on it by translation by τ (Φ). Such a line is an axis of Φ. Exercise 3.15. Let φ : Γ → Γ be a train track map with dilatation λ. Show that for every conjugacy class α the sequence Γ (Φk (α))/λk , k = 1, 2, · · · is nonincreasing, and it is constant if α|Γ is legal. The limiting values in the exercise are translation lengths of an Fn -action on an R-tree, called the stable tree of Φ.

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3.3. Parabolic automorphisms Example 3.16. Let Φ be given by a → a, b → ab. Then τ (Φ) = 0 as can be seen by taking (a) = , (b) = 1 − with → 0. But Φ has inﬁnite order, so it must be parabolic. Definition 3.17. An automorphism Φ ∈ Out(Fn ) is reducible if it can be represented as φ : Γ → Γ so that for some subgraph Γ Γ we have φ(Γ ) ⊆ Γ , and Γ is not a forest (i.e. a disjoint union of trees). Otherwise we say that Φ is irreducible. An automorphism is fully irreducible if every nonzero power is irreducible. Examples 3.3,3.11,3.16 are all reducible (for 3.3 consider the dumbbell graph). Example 3.6 is irreducible and so is Example 3.13, but this is more diﬃcult to prove. Theorem 3.18. Every parabolic automorphism is reducible. Proof. Fix a sequence Γi ∈ Xn such that DΦ (Γi ) → τ (Φ). The key claim is: The sequence Γi leaves every thick part Xn (). The claim implies the theorem by the following argument that goes back to Bers. For large i the graph Γi will contain very small loops. Of course there can be several scales of smallness (e.g. and 2 ) but we are guaranteed to have an arbitrarily large ratio between two consecutive scales. More precisely, for > 0 deﬁne Γi () to be the union of all essential loops (not necessarily immersed) of length < . By construction Φ is represented as φ : Γi → Γi with Lipschitz constant uniformly bounded by some K. Thus f (Γi ()) ⊂ Γi (K), and “large ratio between two consecutive scales” means that Γi (K) deformation retracts to Γi (). Thus φ can be homotoped so that the core subgraph of Γi () is invariant. Formally, one ﬁnds such a large ratio between consecutive scales by considering a long ﬁnite sequence of subgraphs such as Γi (1) ⊃ Γi (1/K) ⊃ Γi (1/K 2 ) ⊃ · · · . They have no contractible components and are nonempty for large i, and in a suﬃciently long chain of nonempty subgraphs two consecutive ones will have the same core (i.e. minimal deformation retract). It remains to prove the claim. The idea is that if the claim fails, we could ﬁnd a graph in Xn where DΦ achieves the minimum. Conceptually, the simplest argument is to translate the claim to a statement about the quotient Xn /Out(Fn ). For clarity, let’s pretend that Xn is a complete Riemannian manifold with Out(Fn ) acting as a deck group by isometries, so that Xn /Out(Fn ) is also a complete Riemannian manifold. The statement that Φ is parabolic amounts to saying that a loop representing Φ in π1 (Xn /Out(Fn )) cannot be homotoped to a loop of length τ (Φ). Projecting a geodesic path from Γi to Γi Φ gives a loop in Xn /Out(Fn ) based at the image [Γi ] of Γi . If all Γi stay in some thick part Xn () then after passing to a subsequence we may assume that [Γi ] → [Γ]. We also note that the projected loops αi stay in some larger compact set, since the distance from a graph with a tiny loop back to the thick part is very large. Thus by Arzela-Ascoli after a further subsequence we have a limiting loop α at [Γ] of length τ (Φ). Now for large i the loops α and αi are homotopic, so they all represent the conjugacy class of Φ. The length of α is τ (Φ), contradiction. There are some technical issues coming from the non-symmetry of the metric on Xn and from the non-freeness of the action. For a less conceptual, but more elementary proof, following Bers, see [6].

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LECTURE 3. CLASSIFICATION OF AUTOMORPHISMS

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If τ (Φ) > 0 and Φ is parabolic, there is an “axis at inﬁnity”. Putting the above discussion together, we obtain the following theorem, originally proved by diﬀerent methods. Theorem 3.19 ([10]). Every irreducible automorphism is represented by a train track map. 3.4. Reducible automorphisms Every automorphism of Fn has a representative φ : Γ → Γ called a relative train track map. It is built from train track maps like an upper triangular block matrix. More precisely, • there is a ﬁltration Γ0 ⊂ Γ1 ⊂ · · · ⊂ Γk = Γ into invariant subgraphs (i.e. φ(Γi ) ⊂ Γi ), and • for every i and every edge e in Γi − Γi−1 the paths φm (e) for m = 1, 2, · · · can backtrack only within Γi−1 . Here we take Γ−1 = ∅. Therefore φ : Γ0 → Γ0 is a train track map. We also assume that the ﬁltration is maximal. The transition matrix is upper triangular, and there are two kinds of strata Γi − Γi−1 : • polynomially growing (PG): φ induces a cyclic permutation of the edges in Γi − Γi−1 , and • exponentially growing (EG): the transition matrix restricted to Γi − Γi−1 grows exponentially and is irreducible (for every entry some power is nonzero in that entry). Example 3.11 has two strata, the lower is (EG) and the upper is (PG). Relative train track maps are useful because cancellation under iterations is controlled. There are various improvements that control cancellation even further; they go under the generic name of “improved relative train tracks”, see [10, 8, 12, 20]. Some of the improvement require ﬁrst passing to a suitable positive power of the automorphism, but this is usually harmless in applications. The following is a partial list of results proved using the improved relative train track technology: • [10] For every automorphism f : Fn → Fn the rank of the ﬁxed subgroup F ix(f ) = {x ∈ Fn | f (x) = x} has rank ≤ n. • [5, 8, 9, 3] Tits Alternative: If H is any subgroup of Out(Fn ) then either H contains F2 or H is virtually abelian. • [19] The abstract commensurator of Out(Fn ) is Out(Fn ). • [12] The mapping torus of any automorphism of Fn satisﬁes quadratic isoperimetric inequality. • [24] If H is any ﬁnitely generated subgroup of Out(Fn ) then either H contains a fully irreducible automorphism, or a ﬁnite index subgroup of H ﬁxes a nontrivial free factor of Fn . 3.5. Growth If Φ ∈ Out(Fn ) and γ is a nontrivial conjugacy class, we can deﬁne the growth rate of γ with respect to Φ: Γ (Φm (γ)) τ (Φ, γ) = lim sup log m m→∞

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for a ﬁxed Γ ∈ Xn . If we decide on some other Γ instead, the ratio of lengths with respect the two is uniformly bounded, so τ (Φ, γ) is independent of the choice of Γ. In fact, as a consequence of the theory, one may replace lim sup by lim in the deﬁnition. If Φ is hyperbolic and we choose φ : Γ → Γ to be as in Theorem 3.12 and if γ is a legal loop in Δφ , then Γ (Φm (γ)) = Γ (γ)λm for m > 0, so τ (Φ, γ) = log λ = τ (Φ). Exercise 3.20. Take the automorphism Φ with a → b, b → ab, c → d, d → cad. √ Show that τ (Φ) = log λ for λ = 1+2 5 and that Γ (Φm (c)) ∼ mλm . Deduce that Φ is parabolic. A (weak) Perron number is a positive real number which is an algebraic integer and it is greater (or equal) than the norm of any of its Galois conjugates. Theorem 3.21. For any Φ ∈ Out(Fn ) there are ﬁnitely many weak Perron numbers λ1 , · · · , λk > 1 so that for any γ τ (Φ, γ) = log λi for some i or τ (Φ, γ) = 0. Moreover, τ (Φ) = max log λi (or 0 if the collection of λi is empty). The numbers λi are the growth rates of the block of the transition matrix corresponding to the strata Γi − Γi−1 . It was shown by Thurston [35] that for every weak Perron number λ > 1 there is an automorphism Φ represented by a train track map φ : Γ → Γ with dilatation λ. For a more detailed information about growth, see [30]. 3.6. Pathologies For those familiar with mapping class groups, the following facts will seem like pathologies. • τ (Φ) may be diﬀerent from τ (Φ−1 ). • Φ may be hyperbolic and Φ−1 parabolic. • If τ (Φ) = 0 then Φ grows polynomially, but not necessarily linearly. • If τ (Φ) = log λ, λ may not be an algebraic unit (but can be any weak Perron number). All of these facts are obstructions to an automorphism being realizable as a homeomorphism of a surface.

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LECTURE 4

Hyperbolic features To what extent is Xn negatively curved? It is not completely clear what we mean by this, since the metric is not symmetric. But even if we could make sense of the question, Xn could not be negatively curved, since it contains “ﬂats” for n ≥ 4. Consider the commuting automorphisms of F4 , Φ: a → b, b → ab, c → c, d → d, Ψ: a → a, b → b, c → d, d → cd. Then there are constants C1 , C2 > 0 so that for the rose R with identity marking and edge lengths 1/4 and for k, l ∈ Z we have C1 (|k| + |l|) ≤ d(R, RΦk Ψl ) ≤ C2 (|k| + |l|) i.e. we have a quasi-isometric embedding of a ﬂat. Exercise 4.1. Prove the inequalities. The ﬁrst negatively curved phenomenon was observed by Yael Algom-Kﬁr, by analogy with Minsky’s theorem [32]. First note that in hyperbolic space Hn every geodesic is strongly contracting: there is a universal constant C so that if B is any metric ball in Hn disjoint from a geodesic line then the image of B under the nearest point projection to has diameter ≤ C. Also note that this property fails in Euclidean space, so we can view it as an indicator of negative curvature. Theorem 4.2 ([1]). Let Φ be a fully irreducible automorphism. Then any orbit {ΓΦk } of Φ is strongly contracting: there is a constant C = C(Φ, Γ) so that if B→ (Δ, R) = {Ω ∈ Xn | d(Δ, Ω) ≤ R} is a ball disjoint from the orbit then the nearest point projection of the ball to the orbit has diameter ≤ C. For Δ ∈ Xn the function k → d(Δ, ΓΦk ) is proper and the minset is the nearest point projection of Δ. To motivate what happens next, let’s look at mapping class groups. If S is a compact surface, deﬁne the curve complex of S to be the simplicial complex C(S) whose vertices are isotopy classes of essential simple closed curves (a curve is essential if it is not homotopic into the boundary or to a point), and a collection of vertices spans a simplex if the corresponding isotopy classes can be represented by pairwise disjoint curves. If S is a hyperbolic surface with totally geodesic boundary, we can work with simple closed geodesics – they are automatically disjoint if they can be isotoped to be disjoint. In the case of the torus, or the torus with one boundary component, the curve complex is a discrete set since non-isotopic essential simple closed curves always intersect. In these cases one modiﬁes the deﬁnition of the curve complex and puts an edge between two vertices if they can be isotoped so that they intersect in one point. The resulting graph is the classical Farey graph, pictured below (cf picture of reduced Outer space in Lecture 1). 199 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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Figure 4.1. Farey graph.

Figure 4.2. Farey graph in upper half plane.

Theorem 4.3 ([31]). The curve complex is hyperbolic. The statement means that the 1-skeleton is δ-hyperbolic for some δ, with respect to the geodesic metric where every edge has length 1. Teichm¨ uller space, like Outer space, is not hyperbolic. It has a coarse map to the curve complex, and this map “kills ﬂats”, so that the curve complex captures hyperbolic aspects of Teichm¨ uller space. Going back to Outer space, there are two analogs of the curve complex.

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LECTURE 4. HYPERBOLIC FEATURES

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4.1. Complex of free factors Fn This complex was deﬁned by Hatcher and Vogtmann in [27]. The deﬁnition is analogous to the Bruhat-Tits building for SLn (Z), which is a the simplicial complex whose vertices are proper vector subspaces of Qn and simplices are chains of subspaces. Recall that a subgroup A < Fn is a free factor if there is a subgroup B < Fn such that Fn = A ∗ B. The vertices of Fn are conjugacy classes of proper (i.e. not 1 nor Fn ) free factors, and a simplex is induced by a chain A0 < A1 < · · · < Ak . Again in rank 2 the complex is a discrete set, so we modify the deﬁnition and put an edge between the conjugacy classes of a and b provided a and b form a basis of F2 . Since the conjugacy class of a rank 1 free factor in F2 corresponds exactly to a simple closed curve in a punctured torus, the graph Fn is also the Farey graph. Theorem 4.4 ([7]). Fn is hyperbolic. 4.2. The complex Sn of free factorizations This complex was introduced by Hatcher in [26] in the form of a complex of spheres. The easiest way to deﬁne it is as the simplicial completion Xn∗ of Xn (see Lecture 1). Thus a vertex of Sn is a 1-edge free splitting of Fn , i.e. a minimal simplicial Fn -tree with trivial edge stabilizers, 1 orbit of edges, and no global ﬁxed points. It is convenient to draw the quotient space of such a tree and label vertices by their stabilizers, as in Bass-Serre theory. Thus a vertex looks like a picture below.

A A

B

Figure 4.3. 1-edge splittings A ∗ B and A∗1 of Fn . Such graphs arise when a marked graph is equipped with a degenerate metric. Similarly, an edge of Sn is a 2-edge splitting, and the endpoints of this edge are 1-edge splittings obtained by collapsing one of two orbits of edges. Theorem 4.5 ([23]). Sn is hyperbolic. 4.3. Coarse projections Recall that when we discuss distance in Fn or Sn we consider points in the 1skeleton only. We will view the distance function on all of Fn or Sn as being deﬁned only “coarsely”, i.e. with a bounded ambiguity. If x, y ∈ Fn we set d(x, y) to be the diameter of the union of the 1-skeletons of the simplices containing x and y respectively, and similarly for Sn . There are coarse Lipschitz projection maps Xn → Sn → Fn . The word “coarse” means that the image of a point is a uniformly bounded set, and “Lipschitz” means that there are constants A, B > 0 so that if two points are at distance ≤ d then the union of their images has diameter ≤ Ad + B. (0) To deﬁne π : Xn → Sn , for Γ ∈ Xn let π(Γ) be the set of vertices of the (0) smallest simplex containing Γ. Similarly, ρ : Sn → Fn (0) is deﬁned to be the set of stabilizers of the vertices of the given splitting, viewed as an Fn -tree.

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Here are some facts. (1)

• Both π : Xn → Sn0 and ρ : Sn → Fn are coarse Lipschitz maps and both are coarsely onto (i.e. there is R > 0 such that every R-ball in the target intersects the image). • Both π and ρ have unbounded point inverses, even when π is restricted to the spine. • A fully irreducible automorphism Φ ∈ Out(Fn ) acts as a hyperbolic isometry on both Fn and Sn . • An automorphism which is not fully irreducible acts with bounded orbits on Fn , but may be hyperbolic on Sn . 4.4. Idea of the proof of hyperbolicity Let Y be a connected graph with edge lengths 1. How does one go about proving that Y is hyperbolic. The deﬁnition requires checking that geodesic triangles are uniformly thin, but in practice it is hard to decide if a given path is a geodesic. The following outline applies to all three complexes mentioned above: the curve complex, the complex of free factors, and the free factorization complex. The strategy is to work with “natural” paths. Suppose one is given a collection of paths P in Y , each joining a pair of vertices, and assume the following: • The collection P is transitive, meaning that every pair of vertices in Y is connected by a path in P, • Each path in P is a reparametrized quasigeodesic, i.e. there are constants L, A so that for every path α : [a, b] → Y in P there is a homeomorphism τ : [a , b ] → [a, b] such that ατ : [a , b ] → Y is an (L, A)-quasi-geodesic. • There is some δ ≥ 0 so that every triangle formed by three paths in P is δ-thin. The following is a variant of hyperbolicity criteria in [31] and [11]. Proposition 4.6. Suppose Y admits a collection of paths P as above. Then Y is hyperbolic. Proof. It suﬃces to prove that any loop in Y of length bounds a disk of area ≤ C log (see [13]). Of course, Y s a graph and has no 2-cells, but one imagines attaching disks to all loops in Y of length bounded by some ﬁxed constant and then the area of a loop is the least number of these attached disks a null-homotopy of the loop crosses, counted with multiplicity. For simplicity, assume that = 3 · 2n for some n > 0 so that we may think of the loop as a polygon with 3 · 2n sides. Such a polygon can be triangulated so that combinatorially it is the n-neighborhood of a ﬁxed triangle in the Farey graph (the 1-neighborhood of a triangle consists of 4 triangles). For each diagonal in this triangulation choose a path in P connecting the same pair of points. Thus we have a map from the Farey-triangulated polygon to Y that takes outer edges to edges in Y and all other edges to paths in P. The area of a thin triangle is bounded by a linear function of its diameter. The diameter of the central triangle and of the triangles adjacent to it is bounded by 2n , but subsequent layers have diameters bounded by 2n−1 , 2n−2 , · · · and each subsequent layer has twice as many triangles as the previous layer. Adding these numbers we get ∼ n · 2n which is about · log .

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LECTURE 4. HYPERBOLIC FEATURES

203

It remains to deﬁne the collection P and check the above properties. For both Fn and Sn the collection of paths is deﬁned by projecting folding paths in Xn using the coarse projection map. The ﬁrst bullet is easy to verify. In practice, the second bullet is veriﬁed by constructing a coarse Lipschitz retraction from Y to the image of a given path in P. Both the second and the third bullets require hard work.

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Bibliography 1. Yael Algom-Kﬁr, Strongly contracting geodesics in outer space, Geom. Topol. 15 (2011), no. 4, 2181–2233. MR2862155 2. Yael Algom-Kﬁr and Mladen Bestvina, Asymmetry of outer space, Geom. Dedicata 156 (2012), 81–92. MR2863547 3. Emina Alibegovi´ c, Translation lengths in Out(Fn ), Geom. Dedicata 92 (2002), 87–93, Dedicated to John Stallings on the occasion of his 65th birthday. MR1934012 (2003m:20045) 4. Lipman Bers, An extremal problem for quasiconformal mappings and a theorem by Thurston, Acta Math. 141 (1978), no. 1-2, 73–98. MR0477161 (57:16704) 5. M. Bestvina, M. Feighn, and M. Handel, Laminations, trees, and irreducible automorphisms of free groups, Geom. Funct. Anal. 7 (1997), no. 2, 215–244. MR1445386 (98c:20045) 6. Mladen Bestvina, A Bers-like proof of the existence of train tracks for free group automorphisms, Fund. Math. 214 (2011), no. 1, 1–12. MR2845630 (2012m:20046) 7. Mladen Bestvina and Mark Feighn, Hyperbolicity of the complex of free factors, arXiv:1107.3308. MR3177291 8. Mladen Bestvina, Mark Feighn, and Michael Handel, The Tits alternative for Out(Fn ). I. Dynamics of exponentially-growing automorphisms, Ann. of Math. (2) 151 (2000), no. 2, 517–623. MR1765705 (2002a:20034) , The Tits alternative for Out(Fn ). II. A Kolchin type theorem, Ann. of Math. (2) 9. 161 (2005), no. 1, 1–59. MR2150382 (2006f:20030) 10. Mladen Bestvina and Michael Handel, Train tracks and automorphisms of free groups, Ann. of Math. (2) 135 (1992), no. 1, 1–51. MR1147956 (92m:20017) 11. Brian H. Bowditch, Intersection numbers and the hyperbolicity of the curve complex, J. Reine Angew. Math. 598 (2006), 105–129. MR2270568 (2009b:57034) 12. Martin R. Bridson and Daniel Groves, The quadratic isoperimetric inequality for mapping tori of free group automorphisms, Mem. Amer. Math. Soc. 203 (2010), no. 955, xii+152. MR2590896 (2011g:20058) 13. Martin R. Bridson and Andr´e Haeﬂiger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR1744486 (2000k:53038) 14. Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, SpringerVerlag, New York, 1994, Corrected reprint of the 1982 original. MR1324339 (96a:20072) 15. Matt Clay, Contractibility of deformation spaces of G-trees, Algebr. Geom. Topol. 5 (2005), 1481–1503 (electronic). MR2186106 (2006i:20026) 16. Marc Culler, Finite groups of outer automorphisms of a free group, Contributions to group theory, Contemp. Math., vol. 33, Amer. Math. Soc., Providence, RI, 1984, pp. 197–207. MR767107 (86g:20027) 17. Marc Culler and John W. Morgan, Group actions on R-trees, Proc. London Math. Soc. (3) 55 (1987), no. 3, 571–604. MR907233 (88f:20055) 18. Marc Culler and Karen Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986), no. 1, 91–119. MR830040 (87f:20048) 19. Benson Farb and Michael Handel, Commensurations of Out(Fn ), Publ. Math. Inst. Hautes ´ Etudes Sci. (2007), no. 105, 1–48. MR2354204 (2008j:20102) 20. Mark Feighn and Michael Handel, Abelian subgroups of Out(Fn ), Geom. Topol. 13 (2009), no. 3, 1657–1727. MR2496054 (2010h:20068) 21. Stefano Francaviglia and Armando Martino, Metric properties of Outer space, Pub. Mat. 55 (2011), 433–473. MR2839451 (2012j:20128)

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22. Vincent Guirardel and Gilbert Levitt, Deformation spaces of trees, Groups Geom. Dyn. 1 (2007), no. 2, 135–181. MR2319455 (2009a:20041) 23. Michael Handel and Lee Mosher, The free splitting complex of a free group I: Hyperbolicity, arXiv:1111.1994. MR3073931 , Subgroup classiﬁcation in Out(Fn ), arXiv:0908.1255. 24. , Parageometric outer automorphisms of free groups, Trans. Amer. Math. Soc. 359 25. (2007), no. 7, 3153–3183 (electronic). MR2299450 (2008c:20045) 26. Allen Hatcher, Homological stability for automorphism groups of free groups, Comment. Math. Helv. 70 (1995), no. 1, 39–62. MR1314940 (95k:20030) 27. Allen Hatcher and Karen Vogtmann, The complex of free factors of a free group, Quart. J. Math. Oxford Ser. (2) 49 (1998), no. 196, 459–468. MR1660045 (99i:20038) 28. Sebastian Hensel, Damian Osajda, and Piotr Przytycki, Realisation and dismantlability, arXiv:1205.0513. 29. D. G. Khramtsov, Finite groups of automorphisms of free groups, Mat. Zametki 38 (1985), no. 3, 386–392, 476. MR811572 (87c:20071) 30. Gilbert Levitt, Counting growth types of automorphisms of free groups, Geom. Funct. Anal. 19 (2009), no. 4, 1119–1146. MR2570318 (2011f:20068) 31. Howard A. Masur and Yair N. Minsky, Geometry of the complex of curves. I. Hyperbolicity, Invent. Math. 138 (1999), no. 1, 103–149. MR1714338 (2000i:57027) 32. Yair N. Minsky, Quasi-projections in Teichm¨ uller space, J. Reine Angew. Math. 473 (1996), 121–136. MR1390685 (97b:32020) 33. Richard K. Skora, Deformations of length functions in groups, preprint, 1989. 34. John Smillie and Karen Vogtmann, Length functions and outer space, Michigan Math. J. 39 (1992), no. 3, 485–493. MR1182503 (93j:20054) 35. William P. Thurston, Entropy in dimension one, preprint 2012. , Minimal stretch maps between hyperbolic surfaces, arxiv:math/9801039. 36. 37. Karen Vogtmann, Automorphisms of free groups and outer space, Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000), vol. 94, 2002, pp. 1–31. MR1950871 (2004b:20060) 38. Shi Cheng Wang and Bruno Zimmermann, The maximum order of ﬁnite groups of outer automorphisms of free groups, Math. Z. 216 (1994), no. 1, 83–87. MR1273467 (95a:20039) ¨ 39. Bruno Zimmermann, Uber Hom¨ oomorphismen n-dimensionaler Henkelk¨ orper und endliche Erweiterungen von Schottky-Gruppen, Comment. Math. Helv. 56 (1981), no. 3, 474–486. MR639363 (83f:57025)

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https://doi.org/10.1090//pcms/021/07

Some Arithmetic Groups that Do Not Act on the Circle Dave Witte Morris

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IAS/Park City Mathematics Series Volume 21, 2012

Some Arithmetic Groups that Do Not Act on the Circle Dave Witte Morris Abstract The group SL(3, Z) cannot act (faithfully) on the circle (by homeomorphisms). We will see that many other arithmetic groups also cannot act on the circle. The discussion will involve several important topics in group theory, such as ordered groups, amenability, bounded generation, and bounded cohomology. Lecture 1 provides an introduction to the subject, and uses the theory of left-orderable groups to prove that SL(3, Z) does not act on the circle. Lecture 2 discusses bounded generation, and proves that groups of the form SL 2, Z[α] do not act on the real line. Lectures 3 and 4 are brief introductions to amenable groups and bounded cohomology, respectively. They also explain how these ideas can be used to prove that actions on the circle have ﬁnite orbits. An appendix provides hints or references for all of the exercises. These notes are slightly expanded from talks given at the Park City Mathematics Institute’s Graduate Summer School in July 2012. The author is grateful to the PCMI staﬀ for their hospitality, the organizers for the invitation to take part in such an excellent conference, and the students for their energetic participation and helpful comments that made the course so rewarding (and improved these notes).

Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta, T1K 3M4, Canada E-mail address: [email protected] c 2014 American Mathematical Society

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LECTURE 1

Left-orderable groups and a proof for SL(3, Z) 1A. Introduction In Geometric Group Theory (and many other ﬁelds of mathematics), one of the main methods for understanding a group is to look at the spaces it can act on. (For example, speakers at this conference have discussed actions of groups on δhyperbolic spaces, CAT(0) cube complexes, Euclidean buildings, and other spaces of geometric interest.) In these lectures, we consider only very simple spaces, namely, the real line R and the circle S 1 . Also, we consider only a single, very interesting class of groups, namely, the arithmetic groups. More precisely, the topic of these lectures is: Main Question 1.1. Let Γ be SL(n, Z), or some other arithmetic group. (1) Does there exist a faithful action of Γ on R? (2) Does there exist a faithful action of Γ on S 1 ? All actions are assumed to be continuous, so the questions ask whether there exists a faithful homomorphism φ : Γ → Homeo(X), where X = R or S 1 . (Recall that a homomorphism is faithful if its kernel is trivial.) A fundamental theorem in the subject tells us that the two seemingly diﬀerent questions in Main Question 1.1 are actually the same for most arithmetic groups (if, as is usual in Geometric Group Theory, we ignore the very minor diﬀerence between a group and its ﬁnite-index subgroups): Theorem 1.2 (Ghys [16], Burger-Monod [7]). Let Γ = SL(n, Z), or some other irreducible arithmetic group, such that no ﬁnite-index subgroup of Γ is isomorphic to a subgroup of SL(2, R). Then: some ﬁnite-index subgroup of Γ has a faithful action on R ⇐⇒ some ﬁnite-index subgroup of Γ has a faithful action on S 1 . Proof. (⇒) Suppose Γ˙ is a ﬁnite-index subgroup of Γ that acts on R. Then ˙Γ also acts on the one-point compactiﬁcation of R, which is homeomorphic to S 1 . (Note that this argument is elementary and very general. It is the opposite direction of the theorem that requires assumptions on Γ, and sometimes requires passage to a ﬁnite-index subgroup.) (⇐) Suppose Γ˙ is a ﬁnite-index subgroup of Γ that acts on S 1 . A major theorem proved independently by Ghys [16] and Burger-Monod [7] tells us that that the action must have a ﬁnite orbit. (We will say a bit about the proof of this ¨ of Γ˙ has theorem in Lectures 3 and 4.) This means that a ﬁnite-index subgroup Γ a ﬁxed point in S 1 . 211 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

212 DAVE WITTE MORRIS, SOME ARITHMETIC GROUPS THAT DO NOT ACT ON S 1

¨ Then {p} is a Γ-invariant ¨ Let p be a point in S 1 that is ﬁxed by Γ. subset, so ¨ acts on S 1 {p}, its complement S 1 {p} is also invariant. This implies that Γ which is homeomorphic to R. Thus, in most cases, it does not matter which of the two versions of Main Question 1.1 we consider. For now, let us look at actions on R. Assumption 1.3. To avoid minor complications, we will assume, henceforth, that all actions are orientation-preserving. This means that an action of Γ on X is a faithful homomorphism φ : Γ → Homeo+ (X), where Homeo+ (X) is the group of orientation-preserving homeomorphisms of X. Since Homeo+ (X) is a subgroup of index 2 in the group of all homeomorphisms, this is just another example of ignoring the diﬀerence between a group and its ﬁnite-index subgroups. Remark 1.4. The expository paper [30] covers the main topics of these lectures in somewhat more depth. See [17] and [33] for introductions to the general theory of group actions on the circle (not just actions of arithmetic groups), and see [31] for an introduction to arithmetic groups. 1B. Examples The following result provides an obstruction to the existence of an action on R. Lemma 1.5. If a group has a nontrivial element of ﬁnite order, then the group does not have a faithful action on R. Proof. It suﬃces to show that every nontrivial element ϕ of Homeo+ (R) has inﬁnite order. Since ϕ is nontrivial, there is some p ∈ R, such that ϕ(p) = p. Assume, without loss of generality, that ϕ(p) > p. The fact that ϕ is an orientation-preserving homeomorphism of R implies that it is a strictly increasing function: x > y =⇒ ϕ(x) > ϕ(y). Therefore (letting x = ϕ(p) and y = p), we have ϕ2 (p) > ϕ(p). In fact, by induction, we have ϕn (p) > ϕn−1 (p) > · · · > ϕ(p) > p, so ϕn (p) > p for every n > 0. This implies ϕn (p) = p, so ϕn is not the identity map. Since n is arbitrary, this means that ϕ has inﬁnite order. Corollary 1.6. If n ≥ 2, then SL(n, Z) does not have a faithful action on R. Proof. It is easy to ﬁnda nontrivial element of ﬁnite order in SL(n, Z). For −1 0 is in SL(2, Z) and has order 2. example, the matrix 0

−1

It is not diﬃcult to show that every arithmetic group has a ﬁnite-index subgroup that has no elements of ﬁnite order [37, Lem. 4.19, p. 232]. This means that Lemma 1.5 does not provide any obstruction at all to the existence of actions of suﬃciently small ﬁnite-index subgroups of Γ. For example:

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LECTURE 1. LEFT-ORDERABLE GROUPS AND A PROOF FOR SL(3, Z)

213

Example 1.7. It is well known that some ﬁnite-index subgroups of SL(2, Z) are free groups. (In fact, every torsion-free subgroup is free [40, Eg. 1.5.3, p. 11, and Prop. 18, p. 36].) Any such subgroup has many faithful actions on R: Exercise 1.8. Show that every ﬁnitely generated free group has a faithful action on R. Here is a much less trivial class of arithmetic groups that act on R: Theorem 1.9 (Agol and Boyer-Rolfsen-Wiest). If Γ is any arithmetic subgroup of SL(2, C), then some ﬁnite-index subgroup of Γ has a faithful action on R. Proof. A very recent and very important theorem of Agol [1] tells us there is a ﬁnite-index subgroup Γ˙ of Γ, such that there is a surjective homomorphism ϕ : Γ˙ Z. Since Z has an obvious nontrivial action on R (by translations), this implies that Γ˙ also acts nontrivially on R (by translations). However, additional eﬀort is required to obtain an action that is faithful. A classic theorem of Burns-Hale [9] provides a cohomological condition that implies the existence of a faithful action: Γ˙ has a faithful action on R if H 1 (Λ; R) is nonzero for every ﬁnitely generated, nontrivial subgroup Λ of Γ˙ ˙ R) is nonzero, which es(see Exercise 1.31(9)). Agol’s theorem tells us H 1 (Γ; ˙ By using 3-manifold tablishes the hypothesis for the special case where Λ = Γ. topology and a fairly simple argument about Euler characteristics, a theorem of Boyer-Rolfsen-Wiest [5, Thms. 3.1 and 3.2] promotes this nonvanishing to obtain the condition for all Λ, and thereby yields a faithful action on R. Examples 1.10. We have seen that some ﬁnite-index subgroups of SL(2, Z) have actions on R. To obtain arithmetic groups that do not act on R (even after passing to a ﬁnite-index subgroup), we need a bigger group. (1) One approach would be to take larger matrices (not just 2 × 2). Later in this lecture, we will see that this works: if n ≥ 3, then no ﬁnite-index subgroup of SL(n, Z) has a faithful action on R. (2) Another possible approach would be to keep the same size of matrix, but enlarge the ring of coeﬃcients: instead of only the ordinary ring of integers Z, consider a ring a algebraic integers O. Lecture 2 outlines a proof that this approach also works: if α is a real √ algebraic integer that is irrational (for example, we could take α = 2), then no ﬁnite-index subgroup of SL 2, Z[α] acts faithfully on R. 1C. The main conjecture In the spirit of Example 1.10, it is conjectured that every “irreducible” arithmetic group that acts on R is contained in a very small Lie group, like SL(2, C): Conjecture 1.11. If Γ is an “irreducible” arithmetic group, then Γ does not have a faithful action on R unless Γ is an arithmetic subgroup of a “very small” Lie group.

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214 DAVE WITTE MORRIS, SOME ARITHMETIC GROUPS THAT DO NOT ACT ON S 1

For the interested reader, the remainder of this section makes the conjecture more precise. However, we will only look at examples of arithmetic groups, not delving deeply into their theory, so, for our purposes, a vague understanding of the conjecture is entirely suﬃcient. Definition 1.12. Saying that Γ is irreducible means that no ﬁnite-index subgroup of Γ is a direct product Γ1 × Γ2 (where Γ1 and Γ2 are inﬁnite). The following simple observation shows that the problem reduces to this case. Exercise 1.13. Show that the direct product Γ1 × Γ2 has a faithful action on R if and only if Γ1 and Γ2 both have faithful actions on R. Technically speaking, instead of saying that the Lie group is “very small,” we should say that it is a simple Lie group whose “real rank” is only 1. In other words, up to ﬁnite index, it belongs to one of the four following families of groups (up to local isomorphism): • SO(1, n) (the isometry group of hyperbolic n-space Hn ), or • SU(1, n) (the isometry group of complex hyperbolic n-space), or • Sp(1, n) (the isometry group of quaternionic hyperbolic n-space), or • F4,1 (the isometry group of the hyperbolic plane over the octonions, also known as the “Cayley plane”). Since SL(2, C) is locally isomorphic to SO(1, 3), this list does include the examples in Theorem 1.9. Remark 1.14. Conjecture 1.11 applies only to actions on R, not actions on S 1 , because some arithmetic groups of large real rank do act on the circle. Namely, if G is a semisimple Lie group that has SL(2, R) as one of its simple factors, then every arithmetic subgroup of G acts on the circle (by linear-fractional transformations). However, it is conjectured that these are the only such arithmetic groups of large real rank [16, p. 200]. 1D. Left-invariant total orders The following exercise translates Conjecture 1.11 into a purely algebraic question about the existence of a certain structure on the group Γ. Definition 1.15. Let Γ be a group. • A total order on a set Ω is a transitive, antisymmetric binary relation ≺ on Ω, such that, for all a, b ∈ Ω, we have either a ≺ b or a b or a = b. • When ≺ is a total order on a group Γ, we can ask that the order structure be compatible with the group multiplication: ≺ is left-invariant if, for all a, b, c ∈ Γ, we have a ≺ b ⇐⇒ ca ≺ cb. See [23] for more about the theory of left-invariant total orders. Exercise 1.16. Let Γ be a countable group. Then Γ has a faithful action on R ⇐⇒ ∃ a left-invariant total order ≺ on Γ.

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LECTURE 1. LEFT-ORDERABLE GROUPS AND A PROOF FOR SL(3, Z)

215

Hint. (⇒) If no nontrivial element of Γ ﬁxes 0, then we may deﬁne a ≺ b ⇐⇒ a(0) < b(0), and this is a left-invariant total order. (Recall that each element of Γ acts on R via in increasing function, so if a(0) < b(0), then c a(0) < c b(0) . If a(0) = b(0), the tie can be broken by choosing some other p ∈ R and comparing a(p) with b(p). (⇐) Note that Γ acts faithfully (by left translation) by automorphisms of the ordered set (Γ, ≺), which is isomorphic (as an ordered set) to a subset of (Q, 1, then Γ contains a ﬁnite-index subgroup of either SL 2, Z[α] (for some α) or a noncocompact arithmetic subgroup of either SL(3, R) or SL(3, C). Combining this with Theorem 2.2 establishes the following observation: Corollary 2.27. Proving the following very special case would establish Conjecture 1.11 under the additional assumption that Γ is not cocompact. Conjecture 2.28. Noncocompact arithmetic subgroups of SL(3, R) and SL(3, C) have no faithful action on R. One possible approach is to use bounded generation: Theorem 2.29 (Lifschitz-Morris [24]). Let Γ be a noncocompact arithmetic subgroup of SL(3, R) or SL(3, C). If some ﬁnite-index subgroup of Γ is boundedly generated by unipotent subgroups, then Γ does not have a faithful action on R. This implies that Conjecture 2.28 is a consequence of the following fundamental conjecture in the theory of arithmetic groups: Conjecture 2.30 (Rapinchuk, 1989). If Γ is a noncocompact, irreducible arithmetic group, and rankR Γ > 1, then Γ contains a ﬁnite-index subgroup that is boundedly generated by unipotent subgroups. In fact, to establish Conjecture 2.28 (and therefore also the entire noncocompact case of Conjecture 1.11), it would suﬃce to prove the special case of Conjecture 2.30 in which Γ is an arithmetic subgroup of either SL(3, R) or SL(3, C).

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LECTURE 3

What is an amenable group? Amenability is a very fundamental notion in group theory — there are literally dozens of diﬀerent deﬁnitions that single out exactly the same class of groups. We will discuss just a few of these many viewpoints, and, for simplicity, we will restrict our attention to discrete groups that are countable, ignoring the important applications of this notion in the theory of topological groups. Much more information can be found in the monographs [35] and [36]. 3A. Ponzi schemes Let us begin with an amusing example that illustrates one of the many deﬁnitions. Example 3.1. Consider the free group F2 = a, b, and let us assume that every element of the group starts with $1. Thus, if f0 (g) denotes the amount of money possessed by element g at time t = 0, then f0 (g) = $1,

for all g ∈ F2 .

Now, everyone will pass their dollar to the person next to them who is closer to the identity. (That is, if g = x1 x2 · · · xn is a reduced word, with xi ∈ {a±1 , b±1 } for each i, then g passes its dollar to g = x1 x2 · · · xn−1 . The identity element has nowhere to pass its dollar, so it keeps the money it started with.) Then, letting f1 denote the amount of money possessed now (at time t = 1), we have f1 (g) = $3

for all g (except that f1 (e) = $5).

Thus, everyone has more than doubled their money. Furthermore, this result was achieved by moving the money only a bounded distance. Such an arrangement is called a Ponzi scheme on the group F2 : Definition 3.2. A Ponzi scheme on a group Γ is a function M : Γ → Γ, such that: (1) M −1 (g) ≥ 2 for all g ∈ Γ (everyone doubles their money if each g passes its dollar to M (g)), and (2) there is a ﬁnite subset S of Γ, such that M (g) ∈ gS for all g ∈ Γ (money moves only a bounded distance). From Example 3.1, we know there is a Ponzi scheme on the free group F2 . However, not all groups have a Ponzi scheme: Exercise 3.3. There does not exist a Ponzi scheme on the abelian group Zn . [Hint: Any group with a Ponzi scheme must have exponential growth, because ft (g) is exponentially large, but the money moves only a linear distance.] 227 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

228 DAVE WITTE MORRIS, SOME ARITHMETIC GROUPS THAT DO NOT ACT ON S 1

More generally, we will see later that no solvable group has a Ponzi scheme (even though solvable groups can have exponential growth). This is because solvable groups are “amenable,” and the nonexistence of a Ponzi scheme can be taken as the deﬁnition of amenability: Theorem 3.4 (Gromov [20, p. 328]). There exists a Ponzi scheme on Γ if and only if Γ is not amenable. This theorem provides a nice description of what it means for a group to not be amenable, but it does not directly provide any positive information about a group that is amenable. Most of the other deﬁnitions we discuss are better for that. 3B. Almost-invariant subsets Instead of using Ponzi schemes, we will adopt the following deﬁnition: Γ is amenable ⇐⇒ Γ has almost-invariant ﬁnite subsets. To see what this means, let us consider an example: Example 3.5. Let Γ = Z2 = a, b, where a = (1, 0) and b = (0, 1). If we let F be a large ball in Γ, then F is very close to being invariant under the left-translation by a and b: #(F ∩ aF ) > (1 − ) #F and #(F ∩ bF ) > (1 − ) #F , where can be as small as we like, if we take F to be suﬃciently large. We say that F is “almost invariant:” Definition 3.6. Let Γ be a group, and ﬁx a ﬁnite subset S of Γ and some > 0. A ﬁnite, nonempty subset F of Γ is almost invariant if #(F ∩ aF ) > (1 − ) #F ,

∀a ∈ S.

Definition 3.7. Γ is amenable if and only if Γ has almost-invariant ﬁnite subsets (for all ﬁnite S and all > 0). Exercises 3.8. Use Deﬁnition 3.7 to show: (1) The free group F2 is not amenable. [Hint: If F is almost invariant, then the ﬁrst letter of most of the words in F must be both a and b.]

(2) If Γ is amenable, S is a ﬁnite subset of Γ, and > 0, then there exists a ﬁnite subset F of Γ, such that #(SF ) < (1 + ) #F , where SF = { sf | s ∈ S, f ∈ F }. (3) Amenability is invariant under quasi-isometry. (This means that you should assume Γ1 is quasi-isometric to Γ2 , and prove that Γ1 is amenable if and only if Γ2 is amenable.) [Hint: Fix c > 1. Show Γ is not amenable iﬀ it has a ﬁnite subset S, such that #(SF ) ≥ c · #F for every ﬁnite subset F of Γ.]

(4) Amenable groups do not have Ponzi schemes. (5) Nonamenable groups have Ponzi schemes. [Hint: Suppose A1 , . . . , An are ﬁnite sets, and a1 , . . . , an ∈ N. Show (by induction on a1 + · · · + an ) that if # i∈I Ai ≥ i∈I ai for every I ⊆ {1, . . . , n}, then there exists Ai ⊆ Ai , such that #Ai = ai and A1 , . . . , An are pairwise disjoint.]

Terminology 3.9. (1) Since the notion of “almost invariant” depends on the choice of S and , many authors say that F is “(S, )-invariant.”

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LECTURE 3. WHAT IS AN AMENABLE GROUP?

229

(2) An almost-invariant set can also be called a “Følner set.” More precisely, a sequence {Fn } of nonempty, ﬁnite subsets of Γ is said to be a Følner sequence if, for every ﬁnite subset S of Γ, every > 0, and every suﬃciently large n, the set Fn is (S, )-invariant. (The set Fn is often called a “Følner set.”) Thus, Deﬁnition 3.7 can be restated as saying that Γ is amenable if and only if it has a Følner sequence. 3C. Average values and invariant measures In many situations, it is diﬃcult to directly employ the almost-invariant sets provided by Deﬁnition 3.7. This section provides some consequences that are often easier to apply. For example, every bounded function on Γ has an average value: Definition 3.10. A mean on ∞ (Γ) is a linear functional A : ∞ (Γ) → C, such that A(ϕ) satisﬁes two axioms that would be expected of the average value of ϕ: • the average value of a constant function is that constant A(c) = c

if c is a constant,

and • the average value of a positive-valued function cannot be negative A(ϕ) ≥ 0 if ϕ ≥ 0. g The mean is left-invariant if A ϕ = A ϕ for all ϕ ∈ ∞ (Γ) and all g ∈ Γ, where ϕg (x) = ϕ(gx). Proposition 3.11. Γ is amenable if and only if there exists a left-invariant mean on ∞ (Γ). Proof (⇒). Choose a sequence {Fn }∞ n=1 of almost-invariant sets with → 0 as n → ∞, and let 1 An (ϕ) = ϕ(x). #Fn x∈Fn

That is, An (ϕ) is the average value of ϕ on the ﬁnite set Fn , so An is obviously a mean on ∞ (Γ). Since the set Fn is almost invariant, the mean An is close to being left-invariant. To obtain perfect left-invariance, we take a limit: A(ϕ) = lim Ank (ϕ), k→∞

where {nk } is a subsequence chosen so that the limit exists. However, if we choose diﬀerent subsequences for diﬀerent functions ϕ, then the limit may not be linear or left-invariant — we need to be consistent in our choice of A(ϕ) for all ϕ. This can be accomplished in various ways: • (logician’s approach) An ultraﬁlter on N tells us which subsequences are “good” and which are “bad.” So the choice of an ultraﬁlter easily leads to a consistent value for A(ϕ). • (analyst’s approach) Deﬁne a linear functional A0 that ◦ takes the value 1 on the constant function 1, and ◦ is 0 on every function of the form ϕg − ϕ. Then the Hahn-Banach Theorem tells us that A0 extends to a linear functional deﬁned on all of ∞ (Γ).

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230 DAVE WITTE MORRIS, SOME ARITHMETIC GROUPS THAT DO NOT ACT ON S 1

• (other viewpoints) Use Zorn’s Lemma, Tychonoﬀ ’s Theorem, or some other version of the Axiom of Choice. The following consequence is very important in the theory of group actions: Corollary 3.12. Suppose • Γ is amenable, and • Γ acts on a compact metric space X (by homeomorphisms). Then there exists a Γ-invariant Borel probability measure μ on X. Proof. Fix a basepoint x0 ∈ X. For any f ∈ C(X), we can deﬁne a function ϕf : Γ → C by restricting f to the Γ-orbit of x0 . More precisely, ϕf (g) = ϕ(gx0 ). Since f is continuous and X is compact, we know that f is bounded. So ϕf is also bounded. Therefore, Proposition 3.11 tells us that it has an average value A(ϕf ), which we call μ(f ). Since A is a mean, it is easy to see that μ is a positive linear functional of ﬁnite norm (in fact, μ = 1). So the Riesz Representation Theorem tells us that μ is a Borel measure on X. Since A is translation-invariant and A(1) = 1, we see that μ is translation-invariant and μ(X) = 1, so μ is a translation-invariant probability measure. Remark 3.13. The converse is true: if every Γ-action on every compact metric space has a Γ-invariant probability measure, then Γ is amenable. So this is another possible choice for the deﬁnition of amenability. Lecture 4 will discuss the “bounded cohomology group” Hbn (Γ; V ), which is deﬁned exactly like the usual group cohomology, except that all cochains are required to be bounded functions. This notion provides another deﬁnition of amenability: Theorem 3.14 (B. E. Johnson [22]). Γ is amenable if and only if Hbn (Γ; V ) = 0 for every Γ-module V that is the dual of a Banach space. Proof of (⇒). Recall that if Γ is a ﬁnite group, and V is a Γ-module (such that multiplication by the scalar |Γ| is invertible), then one can prove H n (Γ; V ) = 0 by averaging: for an n-cocycle α : Γn → V , deﬁne 1 α(g1 , . . . , gn−1 ) = α(g1 , . . . , gn−1 , g). |Γ| g∈Γ

Then α is an (n − 1)-cochain, and δα = ±α. So α is a coboundary, and is therefore trivial in cohomology. Since Γ is amenable, we can do exactly this kind of averaging for any bounded cocycle. See Proposition 4.6 for more details. When Γ is amenable, Proposition 3.11 allows us to take the average value of the characteristic function of any subset of Γ. This leads to von Neumann’s original deﬁnition of amenability [41]: Corollary 3.15. Γ is amenable if and only if there exists a ﬁnitely additive, translation-invariant probability measure that is deﬁned on all of the subsets of Γ. More precisely, if we let 2Γ be the collection of all subsets of Γ, then the conclusion means there is a function μ : 2Γ → [0, 1], such that:

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LECTURE 3. WHAT IS AN AMENABLE GROUP?

231

• μ(X1 ∪ X2 ) = μ(X1 ) + μ(X2 ) if X1 and X2 are disjoint, • μ(Γ) = 1, and • μ(gX) = μ(X) for all g ∈ Γ and X ⊆ Γ. This deﬁnition was motivated by von Neumann’s interest in the famous BanachTarski paradox. The subjects are connected via the following notion: Definition 3.16. A paradoxical decomposition of Γ is a representation ⎛ ⎞ m n ⎝ (disjoint unions), Ai Bj ⎠ Γ= i=1

j=1

such that, for some g1 , . . . , gm , h1 , . . . , hn ∈ Γ, we have m n Γ= gi Ai = h j Bj . i=1

j=1

Exercises 3.17. (1) Show that if Γ is amenable, then Γ does not have a paradoxical decomposition. (2) Find an explicit paradoxical decomposition of a free group. (3) Show that if Γ is not amenable, then Γ has a paradoxical decomposition. [Hint: There exists a Ponzi scheme.]

Remarks 3.18. (1) von Neumann used the German word “messbar” (which can be translated as “measurable”), not the currently accepted term “amenable,” and his condition was not proved to be equivalent to Deﬁnition 3.7 until much later (by Følner [14]). (2) See [42] for much more about the Banach-Tarski paradox, paradoxical decompositions, and the relevance of amenability. We have now seen several proofs that the existence of almost-invariant sets implies some other notion that is equivalent to amenability. Here a proof that goes the other way. Proposition 3.19. If there is an invariant mean A on ∞ (Γ), then Γ has almost-invariant ﬁnite sets. Idea of proof. The dual of 1 (Γ) is ∞ (Γ), so 1 (Γ) is dense in the dual of (Γ), in an appropriate weak topology. Hence, there is a sequence {fn } ⊂ 1 (Γ), such that fn → A. Since A is invariant, we may choose some large n so that fn is close to being invariant. Then, for an appropriate c > 0, the ﬁnite set { x | |f (x)| > c } is almost invariant. ∞

3D. Examples of amenable groups Much of the following exercise can be proved fairly directly by using almostinvariant sets, but it will be much easier to use other characterizations of amenability for some of the parts. Exercises 3.20. Show that all groups of the following types are amenable: (1) ﬁnite groups (2) cyclic groups

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232 DAVE WITTE MORRIS, SOME ARITHMETIC GROUPS THAT DO NOT ACT ON S 1

(3) amenable × amenable [i.e., if Γ1 andΓ2 are amenable, then Γ1 × Γ1 is amenable.] (4) abelian groups (5) amenable by amenable [i.e., if there is a normal subgroup N of Γ, such that N and Γ/N are amenable, then Γ is amenable.]

(6) (7) (8) (9)

solvable groups subgroups of amenable groups quotients of amenable groups [i.e., if every ﬁnitely generated subgroup of Γ is locally amenable groups amenable, then Γ is amenable.]

[i.e., if A is a collection of amenable groups that is totally ordered under inclusion, then A is amenable.] (11) groups of subexponential growth [i.e., if there is a ﬁnite generating set S of Γ, such that limn→∞ (#S n )/en = 0 for every > 0, then Γ is amenable.]

(10) direct limits of amenable groups

Remarks 3.21. (1) From Exercise 3.20, we see that any group obtained from ﬁnite groups and abelian groups by repeatedly taking extensions, subgroups, quotients, and direct limits must be amenable. These “obvious” examples of amenable groups are said to be elementary amenable. (2) The so-called “Grigorchuk group” is an example of a group with subexponential growth that is not elementary amenable [18]. (3) A group is said to be subexponentially amenable if it can be constructed from groups of subexponential growth by repeated application of extensions, subgroups, quotients, and direct limits. The “Basilica group” is an example of an amenable group that is not subexponentially amenable [4]. Thus, the following obvious inclusions are proper: {ﬁnite} elementary subexponentially amenable . amenable amenable {abelian} {solvable} Remark 3.22. By combining Exercise 3.8(1) with Exercise 3.20(7), we see that if Γ has a nonabelian free subgroup, then Γ is not amenable. The converse is often called the “von Neumann Conjecture,” but it was shown to be false in 1980 when Ol’shanskii proved that the “Tarski monster” is not amenable. This is a group in which every element has ﬁnite order, so it certainly does not contain free subgroups [34]. N. Monod [27] has recently constructed counterexamples that are much less complicated. Warning 3.23. In the theory of topological groups, it is not true that every subgroup of an amenable group is amenable — only the closed subgroups need to be amenable. In particular, many amenable topological groups contain nonabelian free subgroups (but such subgroups cannot be closed). 3E. Applications to actions on the circle We are discussing amenability in these lectures because it plays a key role in the proofs of Ghys [16] and Burger-Monod [7] that large arithmetic groups must always have a ﬁnite orbit when they act on the circle (cf. Theorem 1.2). Here is a much simpler example of the connection between amenability and ﬁnite orbits: Proposition 3.24. Suppose • Γ is amenable, and

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LECTURE 3. WHAT IS AN AMENABLE GROUP?

233

• Γ acts on S 1 (by orientation-preserving homeomorphisms). Then either (1) the abelianization of Γ is inﬁnite, or (2) the action has a ﬁnite orbit. Proof. From Corollary 3.12, we know there is a Γ-invariant probability measure μ on S 1 . Case 1. Assume μ has an atom. This means there exists some point assumption p ∈ S 1 that has positive measure: μ {p} > 0. Since μ is Γ-invariant, every point in the orbit of p must have the same measure. However, since μ is a probability measure, we know that the sum of the measures of these points is ﬁnite. Therefore the orbit of p must be ﬁnite (since the sum of inﬁnitely many copies of the same positive number is inﬁnite). Case 2. Assume μ has no atoms. Assume, for simplicity, that the support of μ is all of S 1 . (That is, no nonempty open interval has measure 0.) Then the assumption of this case implies that, after a continuous change of coordinates, the measure μ is simply the Lebesgue measure on S 1 . Since Γ preserves this measure (and is orientation-preserving), this implies that Γ acts on the circle by rotations. Since the group of rotations is abelian, we conclude that the abelianization of Γ is inﬁnite. (Or else the image of Γ in the rotation group is ﬁnite, which means that every orbit is ﬁnite.) Remark 3.25. If we assume that Γ is inﬁnite and ﬁnitely generated, then the conclusion of the proposition can be strengthened: it can be shown that the abelianization of Γ is inﬁnite [28], so there is no need for alternative (2). Large arithmetic groups always have ﬁnite abelianization, so it might seem that the theorem of Ghys and Burger-Monod could be obtained directly from Proposition 3.24. Unfortunately, that is not possible, because arithmetic groups are not amenable (since they contain free subgroups). Instead, Ghys’s proof is based on the following more sophisticated observations: Proposition 3.26. Suppose • Γ is amenable, • Γ acts by (continuous) linear maps on a locally convex vector space V , and • C is a nonempty, compact, convex, Γ-invariant subset of V . Then Γ has a ﬁxed point in C. Proof. Γ acts on the compact set C by homeomorphisms, so Corollary 3.12 provides a Γ-invariant probability measure μ on C. Let p be the center of mass of μ. Then p is ﬁxed by Γ, since μ is Γ-invariant. Also, since C is convex, we know p ∈ C. Remark 3.27. Proposition 3.26 has a converse: if Γ has a ﬁxed point in every nonempty, compact, convex, Γ-invariant set, then Γ is amenable. So this ﬁxed-point property provides yet another possible deﬁnition of amenability. Corollary 3.28 (Furstenberg). Suppose • Γ is an arithmetic subgroup of SL(3, R) = G, • Γ acts on S 1 (by homeomorphisms),

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234 DAVE WITTE MORRIS, SOME ARITHMETIC GROUPS THAT DO NOT ACT ON S 1

• P =

⎡ ∗ ⎣

∗ ∗

⎤ ∗ ∗⎦ ∗

⊂ G, and

• Prob(S 1 ) = {probability measures on S 1 }, with the natural weak∗ topology. Then there exists a Γ-equivariant measurable function ψ : G/P → Prob(S 1 ). Proof. Let C=

measurable Γ-equivariant ψ : G → Prob(S 1 )

(where functions that diﬀer only on a set of measure 0 are identiﬁed). It is easy to see that C is convex. Then, since the Banach-Alaoglu Theorem tells us that weak∗ -closed, convex, bounded sets are compact, we see that C is compact in an appropriate weak topology. Also, P acts continuously on C, via ψ p (g) = ψ(gp). We know that solvable groups are amenable (see Exercise 3.20(6)). Although we have only been considering discrete groups, the same is true for topological groups in general. So P is amenable (because it is solvable). Therefore (a generalization of) Proposition 3.26 tells us that P has a ﬁxed point. This means there is a Γequivariant map ψ : G → Prob(S1 ), such that ψ(gp) = ψ(g) (a.e.). Ignoring a minor issue about sets of measure 0, this implies that ψ factors through to a well-deﬁned Γ-equivariant function ψ : G/P → Prob(S1 ). We omit the proof of the main step in Ghys’s argument: Theorem 3.29 (Ghys [16]). The function ψ provided by Corollary 3.28 is constant (a.e.). From this, it is easy to complete the proof: Corollary 3.30 (Ghys [16]). If Γ is any arithmetic subgroup of SL(3, R), then every action of Γ on the circle has a ﬁnite orbit. Proof. From Theorem 3.29, we know there is a constant function ψ : G/P → Prob(S 1 ) that is Γ-equivariant (a.e.). • Since ψ is constant, its range is a single point μ (a.e.). • Since ψ is Γ-equivariant, its range is a Γ-invariant set. So μ is Γ-invariant. Since μ ∈ Prob(S 1 ), then the proof of Proposition 3.24 shows that either (1) the abelianization of Γ is inﬁnite, or (2) the action has a ﬁnite orbit. Since the abelianization of every arithmetic subgroup of SL(3, R) is ﬁnite, we conclude that there is a ﬁnite orbit, as desired. Remark 3.31. See [17] for a nice exposition of Ghys’s proof for the special case of lattices in SL(n, R). (A slightly modiﬁed proof of the general case that reduces the amount of case-by-case analysis is in [44].) A quite diﬀerent (and very interesting) version of the proof is in [3].

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LECTURE 4

Introduction to bounded cohomology M. Burger and N. Monod [7, 8] developed a sophisticated machinery to calculate bounded cohomology groups, and used it to prove that actions of arithmetic groups on the circle have ﬁnite orbits. (See [25] for an exposition.) We will discuss only some elementary aspects of bounded cohomology, and describe how it is related to actions on the circle, without explaining the fundamental contributions of Burger-Monod. See [26] for a more comprehensive introduction to bounded cohomology and its applications. (Almost all of the information in this lecture can be found there.) The widespread interest in this subject was inspired by a paper of Gromov [19]. 4A. Deﬁnition Recall 4.1. For a discrete group Γ, the cohomology group H n (Γ; R) is deﬁned as follows. • Any function c : Γn → R is an n-cochain, and the set of these cochains is denoted C n (Γ). • A certain coboundary operator δn : C n (Γ) → C n+1 (Γ) is deﬁned. Here are the deﬁnitions for the smallest values of n: for c ∈ R,

δ0 c (g1 ) = 0 δ1 c (g1 , g2 ) = c(g1 g2 ) − c(g1 ) − c(g2 )

for c : Γ → R.

• Then H n (Γ; R) =

Z n (Γ) ker δn n-cocycles = n . = Image δn−1 n-coboundaries B (Γ)

(Note that, for simplicity, we take the coeﬃcients to be R, not a general Γ-module.) Definition 4.2. The bounded cohomology group Hbn (Γ; R) is deﬁned in exactly the same way as H n (Γ; R), except that all cochains are required to be bounded functions. Example 4.3. Hb0 (Γ; R) and Hb1 (Γ; R) are very easy to compute: • It is easy to check that H 0 (Γ; R) = { Γ-invariants in R } = R = { the set of constants }. • The same calculation shows that Hb0 (Γ) is the set of bounded constants. Then, since it is obvious that every constant is a bounded function, we have Hb0 (Γ; R) = R. • It is easy to check that H 1 (Γ; R) = { homomorphisms Γ → R }. 235 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

236 DAVE WITTE MORRIS, SOME ARITHMETIC GROUPS THAT DO NOT ACT ON S 1

• The same calculation shows that Hb1 (Γ; R) is the set of bounded homomorphisms into R. Since a homomorphism into R can never be bounded (unless it is trivial), this means Hb1 (Γ; R) = {0}. Thus, Hb0 (Γ; R) and Hb1 (Γ; R) give no information at all about Γ. (One of them is always R, and the other is always {0}.) So Hbn (Γ; R) is only interesting when n ≥ 2. These groups are not easy to calculate: Example 4.4. For the free group F2 , we have: ∞-dimensional n = 2, 3 n Hb (F2 ; R) = open problem n > 3. Open Problem 4.5. Find some countable group Γ, such that you can calculate Hbn (Γ; R) for all n > 0 (and Hbn (Γ; R) = 0 for some n > 0). Bounded cohomology is easy to calculate for amenable groups: Proposition 4.6 (B. E. Johnson [22]). If Γ is amenable, then Hbn (Γ; R) = 0 for all n > 0. Proof. From Proposition 3.11, we know there is a left-invariant mean A : ∞ (Γ; R) → R. Any element of Hbn (Γ; R) is represented by a bounded function c : Γn → R, such that δn c = 0. To simplify the notation, let us assume n = 2. For each g ∈ Γ, we can deﬁne a bounded function cg : Γ → R by cg (x) = c(g, x). Then, by deﬁning c(g) = A(cg ) ∈ R, we have c : Γ → R. Now, for g1 , g2 , x ∈ Γ, we have 0 = δ2 c (g1 , g2 , x) = c(g1 , g2 ) − c(g1 , g2 x) + c(g1 g2 , x) − c(g2 , x). Applying A to both sides (considered as functions of x), and recalling that A is left-invariant, we obtain 0 = c(g1 , g2 ) − c(g1 ) + c(g1 g2 ) − c(g2 ), so c = −δ1 c ∈ Image(δ1 ). Therefore [c] = 0 in Hb2 (Γ; R).

Remarks 4.7. (1) The bounded cohomology Hbn (X; R) of a topological space X is deﬁned by stipulating that a cochain in C n (X) is bounded if it is a bounded function on the space of singular n-simplices. (2) (Brooks [6], Gromov [19]) Hbn (X; R) = Hbn π1 (X); R . (3) Forgetting that the cochains are bounded yields a comparison homomorphism Hbn (Γ; R) → H n (Γ; R). It is very interesting to ﬁnd situations in which this map is an isomorphism. (4) (Thurston) If M is a closed manifold of negative curvature, then the comparison map Hbn (M ; R) → H n (M ; R) is surjective for n ≥ 2. However, it can fail to be injective.

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LECTURE 4. INTRODUCTION TO BOUNDED COHOMOLOGY

237

4B. Application to actions on the circle Definition 4.8. If ρ : Γ → Homeo+ (S 1 ) is a homomorphism, then, for each g ∈ Γ, covering-space theory tells us that ρ(g) can be lifted to a homeomorphism g of the universal cover, which is R. However, the lift depends on the choice of a basepoint, so it is not unique — lifts can diﬀer by an element of π1 (S 1 ) = Z. Speciﬁcally, if g! is another lift of ρ(g), then ∃n ∈ Z, ∀t ∈ R, g!(t) = g(t) + n. Therefore, for any g, h ∈ Γ, there exists c(g, h) ∈ Z, such that " + c(g, h), ∀t ∈ R, g h(t) = gh(t) " are two lifts of gh. because g h and gh It is easy to verify that c is a 2-cocycle: c(h, k) − c(gh, k) + c(g, hk) − c(g, h) = 0 for g, h, k ∈ Γ, and that choosing a diﬀerent lift g only changes c by a coboundary. Therefore, c determines a well-deﬁned cohomology class α ∈ H 2 (Γ; Z), which is called the Euler class of ρ. Exercise 4.9. Show that the Euler class of ρ is trivial if and only if ρ lifts to a homomorphism ρ : Γ → Homeo+ (R). Remark 4.10. The Euler class can also be deﬁned more naturally, by noting that if we let H be the set consisting of all possible lifts of all elements of Homeo+ (S 1 ), then we have a short exact sequence {e} → Z → H → Homeo+ (S 1 ) → {e} with Z in the center of H. Any such central extension is determined by a well deﬁned cohomology class α0 ∈ H 2 Homeo+ (S 1 ); Z , and the Euler class is obtained using the homomorphism ρ to pull this class back to Γ. Exercise 4.11. Choose a basepoint in R (say, 0), and assume the lift g is chosen with 0 ≤ g(0) < 1 for all g ∈ Γ. Show c is bounded. Definition 4.12. Although we only deﬁned bounded cohomology with real coeﬃcients, the same deﬁnition can be applied with Z in place of R. Therefore, if we choose c as in Exercise 4.11, then it represents a bounded cohomology class [c] ∈ Hb2 (Γ; Z), which is called the bounded Euler class of the action. Remark 4.13. It can be shown that the bounded Euler class is a well-deﬁned invariant of the action (independent of the choice of basepoint, etc.). Proposition 4.14 (Ghys [15]). The bounded Euler class is trivial if and only if Γ has a ﬁxed point in S 1 . Proof. (⇐) We may assume the ﬁxed point is the basepoint 0 ∈ S 1 . Then we may choose g with g(0) = 0. So c(g, h) = 0 for all g, h. (⇒) We have c(g, h) = ϕ(gh) − ϕ(g) − ϕ(h) for some bounded ϕ : Γ → Z. Letting g!(t) = g(t) + ϕ(g), we have # so Γ ! is a lift of Γ to Homeo+ (R), and • g! ! h = gh, • |! g (0)| ≤ |g(0)| + |ϕ(g)| ≤ 1 + ϕ∞ .

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238 DAVE WITTE MORRIS, SOME ARITHMETIC GROUPS THAT DO NOT ACT ON S 1

! Hence, the Γ-orbit of 0 is a bounded set, so it has a supremum in R. This supremum ! so its image in S 1 is a ﬁxed point of Γ. is a ﬁxed point of Γ, Corollary 4.15. If Hb2 (Γ; Z) = 0, then every orientation-preserving action of Γ on S 1 has a ﬁxed point. The following result is easier to apply, because it uses real coeﬃcients for the cohomology, instead of integers: Corollary 4.16. If • Hb2 (Γ; R) = 0, • H 1 (Γ; R) = 0, and • Γ is ﬁnitely generated, then every orientation-preserving action of Γ on S 1 has a ﬁnite orbit. Proof. The short exact sequence 0→Z→R→T→0 yields a long exact sequence of bounded cohomology: Hb1 (Γ; T) → Hb2 (Γ; Z) → Hb2 (Γ; R). By assumption, the group at the right end is 0, so the map on the left is surjective. Therefore, the bounded Euler class is the coboundary of some (bounded) 1-cocycle α : Γ → T. I.e., α is a homomorphism to T. Since H 1 (Γ; R) = 0 (and Γ is ﬁnitely generated), we know α is trivial on some ﬁnite-index subgroup Γ of Γ. Then the bounded Euler class δ1 α is trivial on Γ , so Proposition 4.14 tells us that Γ has a ﬁxed point p. Since Γ has ﬁnite index, we see that the Γ-orbit of p is ﬁnite. Theorem 4.17 (Ghys [16], Burger-Monod [7]). If Γ is any arithmetic subgroup of SL(n, R), with n ≥ 3, then every action of Γ on S 1 has a ﬁnite orbit. Outline of Burger-Monod proof. Burger and Monod showed (in a much more general setting) that the comparison map Hb2 (Γ; R) → H 2 (Γ; R) is injective. Since it is known that H 2 (Γ; R) = 0 (if n is suﬃciently large), we conclude that Hb2 (Γ; R) = 0. The other hypotheses of Corollary 4.16 are well known to be true. 4C. Computing Hb2 (Γ; R) To calculate Hb2 (Γ; R), we would like to understand the kernel of the comparison map Hb2 (Γ; R) → H 2 (Γ; R). For this, we introduce some notation: Definition 4.18. • A function α : Γ → R is a: ◦ quasimorphism if α(gh) − α(g) − α(h) is bounded (as a function of (g, h) ∈ Γ × Γ); ◦ near homomorphism if it is within a bounded distance of a homomorphism. • We use Quasi(Γ, R) and Near(Γ, R) to denote the space of quasimorphisms and the space of near homomorphisms, respectively. Note that Near(Γ, R) ⊂ Quasi(Γ, R).

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LECTURE 4. INTRODUCTION TO BOUNDED COHOMOLOGY

239

Proposition 4.19. The kernel of the comparison map Hb2 (Γ; R) → H 2 (Γ; R) is

Quasi(Γ, R) . Near(Γ, R)

Proof. Let c be a bounded 2-cocycle, such that c is trivial in H 2 (Γ; R). Then c = δ1 α, for some α : Γ → R. Thus, for all g, h ∈ Γ, we have |α(gh) − α(g) − α(h)| = |δ1 α (g, h)| = |c(g, h)| ≤ c∞ is bounded. So α is a quasimorphism. This establishes that Quasi(Γ, R) maps onto the kernel of the comparison map, via α → δ1 α. Now suppose α ∈ Quasi(Γ, R), such that δ1 α is trivial in Hb2 (Γ; R). The triviality of δ1 α means there is a bounded function c : Γ → R, such that δ1 α = δ1 c. Then δ1 (α − c) = 0, so α − c is a homomorphism. Since c is bounded, this means that α is within a bounded distance of a homomorphism; i.e., α ∈ Near(Γ, R). Example 4.20 (Brooks [6]). We can construct many quasimorphisms on the free group F2 : • As a warm-up, recall that there is an obvious homomorphism ϕa , deﬁned by letting ϕa (x) be the (signed) number of occurrences of a in the reduced representation of x. For example, ϕa (a2 ba3 b2 ab−3 a−7 b2 ) = 2 + 3 + 1 − 7 = −1. There is an analogous homomorphism ϕb , and every homomorphism F2 → R is a linear combination of these two. • Similarly, for any nontrivial reduced word w, we can let ϕw (x) be the (signed) number of disjoint occurrences of w in the reduced representation of x. For example, ϕab (a2 ba3 b2 ab−3 a−7 b2 ) = 1 + 1 − 1 = 1. This is a quasimorphism. Exercise 4.21. Verify that ϕw is a quasimorphism, for any reduced word w. With these quasimorphisms in hand, it is now easy to prove a fact that was mentioned in Example 4.4: Exercise 4.22. Show that Hb2 (F2 ; R) is inﬁnite-dimensional. [Hint: Verify that

ϕak (k ≥ 2) is not within a bounded distance of the linear span of {ϕb , ϕa , ϕak+1 , ϕak+2 , ϕak+3 , . . .},

by ﬁnding a word x, such that ϕak (x) is large, but the others vanish on x.]

Exercises 4.23. (1) Show that if Γ is boundedly generated (by cyclic subgroups), then the kernel of the comparison map Hb2 (Γ; R) → H 2 (Γ; R) is ﬁnite-dimensional. [Hint: Every quasimorphism Z → R is a near homomorphism.]

(2) Show the free group F2 is not boundedly generated (by cyclic subgroups). (3) Show that every quasimorphism is bounded on the set of commutators {x−1 y −1 xy}. (4) Show that if Γ is amenable group, and the abelianization of Γ is ﬁnite, then Γ does not have unbounded quasimorphisms. [Hint: Amenable groups do not have bounded cohomology.]

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240 DAVE WITTE MORRIS, SOME ARITHMETIC GROUPS THAT DO NOT ACT ON S 1

(5) Show that SL(3, Z) has no unbounded quasimorphisms. [Hint: Use Remark 2.21.]

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APPENDIX

Hints for exercises Exercise 1.8. Every ﬁnitely generated free group is a subgroup of the free group F2 = a, b, so we need only consider this one example. Choose any faithful action of F2 on the circle. (For example, use the Ping-Pong Lemma [33, Lem. 2.3.9], which provides a suﬃcient condition for two homeomorphisms to generate a free group, or note that F2 is a subgroup of PSL(2, R), which acts faithfully on the circle by linear-fractional transformations.) Since F2 is free, we can lift this to an action on the line (simply by choosing any lift of the two generators a and b). Since it projects down to a faithful action on the circle, this action on the line must also be faithful. Exercise 1.13. Since any open interval is homeomorphic to R, we may let Γ1 act on the open interval (0, 1) (ﬁxing all points in the complement), and let Γ2 act on the open interval (1.2) (ﬁxing all points in the complement). These actions commute, so they deﬁne an action of Γ1 × Γ2 . Exercise 1.16. Details are in [17, Thm. 6.8]. Exercise 1.18. By left-invariance, we have ab = a · b a · e = a e

and

e = a−1 · a a−1 · e = a−1 .

Exercise 1.21(1). Straightforward matrix multiplication veriﬁes that z = [x, y] and that z commutes with both x and y. Exercise 1.21(2). Since z = [x, y], we have xy = yxz. By induction on k (and using the fact that z commutes with x), then xk y = yxk z k . By induction on (and using the fact that z commutes with y), then xk y = y xk (z k ) for k, ∈ Z+ . Exercise 1.21(3). To apply Exercise 1.31(3), note that H has a chain of normal subgroups {e} z z, x H, and each quotient is isomorphic to Z (hence, has an obvious left-invariant order). $ % Exercise 1.26(1). Either calculate that k − 1 , k + 1 = k , and that k com mutes with both k − 1 and k + 1 , or observe that some permutation matrix conjugates the ordered triple k − 1 , k , k + 1 to (x, z, y). Exercise 1.26(2). Details are in [43, §3]. Exercise 1.31(1). The restriction of a left-invariant total order is a left-invariant total order. Exercise 1.31(2). We may assume Γ is ﬁnitely generated (see Exercise 1.31(7)), so Γ∼ = Z × · · · × Z. Now use Exercise 1.13 or Exercise 1.31(3). 241 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

242 DAVE WITTE MORRIS, SOME ARITHMETIC GROUPS THAT DO NOT ACT ON S 1

Exercise 1.31(3). Let ≺∗ and ≺∗ be left-invariant total orders on N and Γ/N , respectively. Then we deﬁne g≺h

⇐⇒

gN ≺∗ hN

or

gN = hN and h−1 g ≺∗ e.

The left-invariance of ≺∗ and ≺∗ implies the left-invariance of ≺. Exercise 1.31(4). Recall that the ascending central series {e} = Z0 Z1 · · · Zc = Γ is deﬁned inductively by Zi /Zi−1 = Z Γ/Zi−1 ). Fix i ≥ 2 and let Γ = Γ/Zi−2 . For any nontrivial z ∈ Zi , there exists g ∈ Γ, such that [g, z] is a nontrivial element of Zi−1 , which is torsion-free (by induction). Since [g, z n ] = [g, z]n , this implies that Zi /Zi−1 is torsion-free. It is also abelian, so we can apply Exercise 1.31(2) and Exercise 1.31(3). Exercise 1.31(5). This is not at all obvious, but speciﬁc examples are given on page 52 of [23]. Here is the general philosophy. Assume G is a nontrivial, ﬁnitely generated, left-orderable group. If G is solvable (or, more generally, if G is “amenable”), then it can be shown that the abelianization of G is inﬁnite [28]. So any torsion-free solvable group with ﬁnite abelianization provides an example. Exercise 1.31(6). (⇒) Choose i so that gii e. Then every element of the semigroup is e. (⇐) The condition implies it is possible to choose a semigroup P in Γ, such that e ∈ / P and, for every nonidentity element g of Γ, either g ∈ P or g −1 ∈ P . Deﬁne x ≺ y ⇐⇒ x−1 y ∈ P . Details can be found in [23, Thm. 3.1.1, p. 45]. Exercise 1.31(7). Use Exercise 1.31(6). Exercise 1.31(8). Let g1 , . . . , gn be nontrivial elements of Γ. For each i, there is a left-orderable group Hi , and a homomorphism ϕi : Γ → Hi , such that ϕi (gi ) = e. For the resulting homomorphism ϕ into H1 × · · · × Hn , we have ϕ(gi ) = e for all i. Now apply Exercise 1.31(6). Exercise 1.31(9). Given nontrivial elements g1 , . . . , gn of Γ, the assumption provides a nontrivial homomorphism ρ : g1 , . . . , gn → R. Assume ρ is trivial on g1 , . . . , gk , and nontrivial on the rest. By induction on n, we can choose 1 , . . . , k ∈ {±1}, such that the semigroup generated by {g1 , . . . , gk } does not contain e. For i > k, choose i so that ρ(gii ) > 0. Then the semigroup generated by g1 , . . . , gn does not contain e, so Exercise 1.31(6) applies. (Details can be found on page 50 of [23].) Exercise 2.11(1). If Γ = H1 · · · Hn , then Γ/N = H1 · · · Hn , where Hi is the image of Hi in Γ/N . Exercise 2.11(2). Call the subgroup N . Then N is a normal subgroup, so Exercise 2.11(1) tells us that Γ/N is boundedly generated by cyclic groups. However, every element of Γ/N has ﬁnite order, so all of these cyclic groups are ﬁnite.

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APPENDIX. HINTS FOR EXERCISES

243

Exercise 2.11(3). (⇐) For a normal subgroup N of Γ, it is easy to see that if N and Γ/N are boundedly generated, then Γ is boundedly generated. Also note that ﬁnite groups are (obviously) boundedly generated. (⇒) To present the main idea with a minimum of notation, let us assume ˙ and let Γ = HK is the product of just two cyclic groups. Let K˙ = K ∩ Γ, {k1 , . . . , kn } be a set of coset representatives for K˙ in K. There exists a ﬁniteindex subgroup H˙ of H, such that the conjugate H˙ kj is contained in Γ˙ for every j. ˙ we may Let {h1 , . . . , hm } be a set of coset representatives for H˙ in H. Given g ∈ Γ, choose h ∈ h and k ∈ K, such that ˙ j k) ˙ = (hi kj )h˙ kj k. ˙ g = hk = (hi h)(k Therefore, if we let 1 , . . . , r be a list of the elements in Γ˙ ∩ {hi kj }, then ˙ Γ˙ = 1 · · · r H˙ k1 · · · H˙ kn K. Exercise 2.11(4). Let Γ be a free subgroup of ﬁnite index in SL(2, Z). Then Exercise 4.23(2) tells us that Γ is not boundedly generated, so Exercise 2.11(3) implies that SL(2, Z) also is not boundedly generated by cyclic groups. Since U and V are cyclic, this completes the proof. Exercise 2.11(5). The argument is somewhat similar to Exercise 2.11(3)(⇒). Let Γ˙ be the subgroup that is under consideration, and let us assume, for simplicity, ˙ and let that Γ = HK is the product of just two cyclic groups. Let K˙ = K ∩ Γ, ˙ {k1 , . . . , kn } be a set of coset representatives for K in K. The key point is to observe that, by deﬁnition, Γ˙ contains a ﬁnite-index subgroup of each H kj , so we may choose a ﬁnite-index subgroup H˙ of H, such that H˙ kj is contained in Γ˙ for every j. Let {h1 , . . . , hm } be a set of coset representatives for H˙ in H. For g ∈ Γ, we have ˙ j k) ˙ = (hi kj )h˙ kj k˙ ∈ (hi kj )Γ. ˙ g = hk = (hi h)(k Therefore, {hi kj } contains a set of coset representatives, so the index of Γ˙ is at most mn. Exercise 2.25. The Triangle Inequality implies that every orbit of every cyclic group is bounded. Now, for any x ∈ X, any R ∈ R+ , and any cyclic subgroup Hi of Γ, this implies there exists ri ∈ R+ , such that Hi x is contained in the ball Bri (x). By the Triangle Inequality, we have Hi ·BR (x) ⊆ BR+ri (x). By induction, if Γ = H1 · · · Hn , then Γx ⊆ Br1 +···+rn (x). Exercise 3.3. Suppose M is a Ponzi scheme on Γ, and M (g) ∈ gS for all g. Let k be the maximum word length of an element of S. Then, since M is (at least) 2-to-1, we know that the ball of radius r + k has at least twice as many elements as the ball of radius r. So Γ has exponential growth. Exercise 3.8(1). Assume, without loss of generality, that (at least) 3/4 of the elements of F do not start with a−1 . Then 3/4 of aF starts with a and 3/4 of baF starts with b. If F is almost invariant, this implies that almost half of F starts with both a and b. Exercise 3.8(2). Let n = #S, and choose F so that #(F ∩ aF ) > 1 − (/n) #F for all a ∈ S. Then #(aF F ) < /n for all a ∈ S, so #(SF ) − #F < n · (/n) = .

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244 DAVE WITTE MORRIS, SOME ARITHMETIC GROUPS THAT DO NOT ACT ON S 1

Exercise 3.8(3). It will suﬃce to prove the hint, since it gives a condition that is invariant under quasi-isometry. Suppose Γ is not amenable. Then there exist S and , such that #(SF ) ≥ (1+)#F , for every ﬁnite subset F of Γ. Choosing n large enough that (1+)n > c, then we have #(S n F ) ≥ c · #F . The other direction is immediate from Exercise 3.8(2). Exercise 3.8(4). Suppose M is a Ponzi scheme on Γ, and M (g) ∈ gS for all g. From Exercise 3.8(2) (and replacing F with F −1 to convert left-translations into right-translations), we know there is a ﬁnite set F , such that #(F S −1 ) < 2 · #F . This is impossible, since M is (at least) 2-to-1 and M −1 (F ) ⊆ F S −1 . Exercise 3.8(5). In the special case where ai = 1 for all i, the hint is known as Hall’s Marriage Theorem, and can be found in many combinatorics textbooks. The general case is proved similarly. Since there are only ﬁnitely many possibilities for each set Ai , a standard diagonalization argument shows that the result is also valid for an inﬁnite sequence A1 , A2 , . . . of ﬁnite sets and an inﬁnite sequence {ai } ⊆ N. From the hint to Exercise 3.8(3), there exists S ⊂ Γ, such that #(F S −1 ) ≥ 2 · #F for every ﬁnite F ⊂ Γ. For y ∈ Γ, let Ay = yS −1 and ay = 2. Then there exists Ay ⊆ Ay , such that #Ay = 2 and the sets {Ay }y∈Γ are pairwise disjoint. Deﬁne M (g) = y ∈ gS for all g ∈ Ay . Exercise 3.17(1). Let a=

m

μ(Ai )

and

i=1

b=

n

μ(Bj ),

j=1

where μ is a ﬁnitely additive, translation-invariant probability measure on Γ. Then, since A1 , . . . , Am , B1 , . . . , Bn are pairwise disjoint, we have a + b = μ(Γ) = 1. &m On the other hand, since Γ = i=1 gi Ai , we have m m m gi Ai ≤ μ(gi Ai ) = μ(Ai ) = a, 1 = μ(Γ) = μ i=1

i=1

i=1

and, similarly, b = 1. This is a contradiction. Exercise 3.17(2). Let A1 , A2 , B1 , and B2 be the reduced words that start with a, a−1 , b, or b−1 , respectively. (Also add e to one of these sets.) Then F2 = a−1 A1 ∪ aA2 = b−1 B1 ∪ bB2 . Exercise 3.17(3). Let M be a Ponzi scheme, and choose S such that M (g) ∈ Sg. Let A contain a single element of M −1 (x), for every x ∈ Γ, and let B be the complement of A. Then, for s ∈ S, let As = { g ∈ A | M (g) = sg } and Bs = { g ∈ B | M & (g) = sg }.& By construction, these sets are pairwise disjoint, and we have Γ = s∈S sAs = s∈S sBs . Exercise 3.20(1). This is obvious from almost any characterization of amenability. For example, letting F = Γ yields a nonempty, ﬁnite set that is invariant, not merely almost-invariant. Exercise 3.20(2). By Exercise 3.20(1), it suﬃces to consider the inﬁnite cyclic group Z. A long interval {0, 1, 2, . . . , n} is almost-invariant.

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APPENDIX. HINTS FOR EXERCISES

245

Exercise 3.20(3). The Cartesian product of two almost-invariant sets is almostinvariant. Exercise 3.20(4). We may assume Γ is ﬁnitely generated (by Exercise 3.20(9)), so it is a direct product of ﬁnitely many cyclic groups. Exercise 3.20(5). This is diﬃcult to do with almost-invariant sets, because multiplication by an element of G/N will act by conjugation on N , which may cause distortion. It is perhaps easiest to apply Proposition 3.26. Since N is amenable, it has ﬁxed points in C. The set C N of such ﬁxed points is a closed, convex subset. Also, it is Γ-invariant (because N is normal). So Γ acts on C N . Since N is trivial on this set, the action factors through to Γ/N , which must have a ﬁxed point. This is a ﬁxed point for Γ. Exercise 3.20(6). By deﬁnition, a solvable group is obtained by repeated extensions of abelian groups, so this follows from repeated application of Exercise 3.20(5). Exercise 3.20(7). This is another case that is diﬃcult to do with almost-invariant sets. Instead, note that if there is a Ponzi scheme on some subgroup of Γ, then it could be reproduced on all of the cosets, to obtain a Ponzi scheme on all of Γ. This establishes the contrapositive. Exercise 3.20(8). This is immediate from Corollary 3.12 (or Proposition 3.26), because any action of Γ/N is also an action of Γ. It also follows easily from Corollary 3.15, since any subset of Γ/N pulls back to a subset of Γ. Exercise 3.20(9). Given S and , let H be the subgroup generated by S, so H is ﬁnitely generated. If H is amenable, then it contains an almost-invariant set, which is also an (S, )-invariant set in G. Exercise & 3.20(10). This is immediate from Exercise 3.20(9), because any ﬁnite subset of A must be contained in one of the sets in A. Exercise 3.20(11). See the hint to Exercise 3.8(3). Exercise 4.9. (⇐) By assumption, we may choose the lifts in such a way that " for all g and h. So c = 0. g h = gh (⇒) We have c(g, h) = ϕ(gh) − ϕ(g) − ϕ(h) for some ϕ : Γ → Z. Then, letting ρ(g)(t) = g(t) + ϕ(g), we have ρ(g) ρ(h) = ρ(gh), so ρ is a homomorphism. " Exercise 4.11. We have c(g, h) = g h(0) − gh(0). Note that " • 0 ≤ gh(0) < 1, and • 0 ≤ g(0) ≤ g h(0) < g(1) = g(0) + 1 < 1 + 1 = 2, so both terms on the right-hand side are bounded. Exercise 4.21. In fact ϕw (xy) never diﬀers by more than 1 from ϕw (x) + ϕw (y). There is a diﬀerence only if some occurrence of w (or w−1 ) overlaps the boundary between x and y, and there cannot be two such occurrences that are disjoint. Exercise 4.22. Let x = (ak bab−1 )n (a−(k−1) b2 a−1 b−1 a−1 b−1 )n .

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246 DAVE WITTE MORRIS, SOME ARITHMETIC GROUPS THAT DO NOT ACT ON S 1

Exercise 4.23(1). Write Γ = H1 · · · Hr . Any quasimorphism on Γ is determined, up to bounded error, by its restriction to the cyclic subgroups H1 , . . . , Hr . Also, it is not diﬃcult to show that every quasimorphism Z → R is a near homomorphism. (Or this can be deduced from Proposition 4.6 and Proposition 4.19.) So the restriction of ϕ to each Hi is a near homomorphism. Since the homomorphisms from Z to R form a one-dimensional space, we conclude that the dimension of Quasi(Γ; R)/∞ (Γ; R) is at most r. Exercise 4.23(2). Compare Exercise 4.22 with Exercise 4.23(1). Exercise 4.23(3). We have

ϕ(x−1 y −1 xy) = ϕ(x−1 y −1 ) + ϕ(xy ± C = ϕ(x−1 ) + ϕ(y −1 ) + ϕ(x) + ϕ(y ± 3C = ϕ(x−1 x) + ϕ(y −1 y) ± 5C = 2ϕ(e) ± 5C,

so |ϕ(x−1 y −1 xy)| ≤ 2|ϕ(e)| + 5C. Exercise 4.23(4). Let ϕ : Γ → R be a quasimorphism. From Proposition 4.6 and Proposition 4.19, we see that ϕ is within bounded distance of a homomorphism. However, since the abelianization of Γ is ﬁnite, there are no nontrivial homomorphisms Γ → R. Therefore ϕ is bounded. Exercise 4.23(5). Every elementary matrix is a commutator (recall that [xk , y] = z k ), so Exercise 4.23(3) implies that ϕ is bounded on the set of elementary matrices. Since every element of SL(3, Z) is the product of a bounded number of these elementary matrices (see Remark 2.21), a simple estimate shows that ϕ is bounded.

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Bibliography [1] I. Agol: The virtual Haken conjecture (preprint). http://arxiv.org/abs/1204.2810 MR3104553 [2] J. C. Ault: Right-ordered locally nilpotent groups, J. London Math. Soc. (2) 4 (1972) 662– 666. http://dx.doi.org/10.1112/jlms/s2-4.4.662 MR0294201 (45:3274) [3] U. Bader, A. Furman, A. Shaker: Superrigidity, Weyl groups, and actions on the circle (preprint). http://arxiv.org/abs/math/0605276 [4] L. Bartholdi and B. Vir´ ag: Amenability via random walks, Duke Math. J. 130 (2005), no. 1, 39–56. http://dx.doi.org/10.1215/S0012-7094-05-13012-5 MR2176547 (2006h:43001) [5] S. Boyer, D. Rolfsen, and B. Wiest: Orderable 3-manifold groups, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 1, 243–288. http://dx.doi.org/10.5802/aif.2098 MR2141698 (2006a:57001) [6] R. Brooks: Some remarks on bounded cohomology, in: Riemann Surfaces and Related Topics (Stony Brook, N.Y., 1978). Princeton Univ. Press, Princeton, N.J., 1981, pp. 53–63. ISBN: 0-691-08264-2 MR624804 (83a:57038) [7] M. Burger and N. Monod: Bounded cohomology of lattices in higher rank Lie groups, J. Eur. Math. Soc. 1 (1999) 199–235. http://dx.doi.org/10.1007/s100970050007 MR1694584 (2000g:57058a) [8] M. Burger and N. Monod: Continuous bounded cohomology and applications to rigidity theory, Geom. Funct. Anal. 12 (2002), no. 2, 219–280. http://dx.doi.org/10.1007/ s00039-002-8245-9 MR1911660 (2003d:53065a) [9] R. G. Burns and V. W. D. Hale: A note on group rings of certain torsion-free groups, Canad. Math. Bull. 15 (1972), 441–445. http://dx.doi.org/10.4153/CMB-1972-080-3 MR0310046 (46:9149) [10] D. Carter and G. Keller: Bounded elementary generation of SLn (O), Amer. J. Math. 105 (1983), 673–687. http://www.jstor.org/stable/2374319 MR704220 (85f:11083) [11] D. Carter and G. Keller: Elementary expressions for unimodular matrices, Comm. Algebra 12 (1984), 379–389. http://dx.doi.org/10.1080/00927878408823008 MR737253 (86a:11023) [12] D. Carter, G. Keller, and E. Paige: Bounded expressions in SL(n, A) (unpublished). [13] V. Chernousov, L. Lifschitz, and D. W. Morris: Almost-minimal nonuniform lattices of higher rank, Michigan Math. J. 56, no. 2, (2008), 453–478. http://arxiv.org/abs/0705.4330 MR2492403 (2009m:22016) [14] E. Følner: On groups with full Banach mean value, Math. Scand. 3 (1955), 243–254. http:// www.mscand.dk/issue.php?year=1955&volume=3 MR0079220 (18:51f) ´ Ghys: Groupes d’hom´ [15] E. eomorphismes du cercle et cohomologie born´ ee, in: The Lefschetz Centennial Conference, Part III (Mexico City, 1984). Contemp. Math., vol. 58, Part III. Amer. Math. Soc., Providence, 1987, pp. 81–106. ISBN: 0-8218-5064-4 MR893858 (88m:58024) ´ Ghys: Actions de r´ [16] E. eseaux sur le cercle, Invent. Math. 137 (1999) 199–231. http://dx.doi. org/10.1007/s002220050329 MR1703323 (2000j:22014) ´ Ghys: Groups acting on the circle, Enseign. Math. 47 (2001) 329–407. http://dx.doi. [17] E. org/10.5169/seals-65441 MR1876932 (2003a:37032) [18] R. Grigorchuk and I. Pak: Groups of intermediate growth: an introduction, Enseign. Math. (2) 54 (2008), no. 3–4, 251–272. http://dx.doi.org/10.5169/seals-109938 MR2478087 (2009k:20101) [19] M. Gromov: Volume and bounded cohomology. Publ. Math. IHES 56 (1982) 5–99. http:// archive.numdam.org/article/PMIHES_1982__56__5_0.pdf MR686042 (84h:53053)

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248 DAVE WITTE MORRIS, SOME ARITHMETIC GROUPS THAT DO NOT ACT ON S 1

[20] M. Gromov: Metric structures for Riemannian and non-Riemannian spaces. Birkh¨ auser, Boston, 1999. ISBN: 0-8176-3898-9 MR1699320 (2000d:53065) [21] C. Hooley: On Artin’s conjecture, J. Reine Angew. Math. 225 (1967) 209–220. http://dx. doi.org/10.1515/crll.1967.225.209 MR0207630 (34:7445) [22] B. E. Johnson: Cohomology in Banach Algebras. Memoirs Amer. Math. Soc., no. 127. American Mathematical Society, Providence, R.I., 1972. ISBN: 0-8218-1827-1 MR0374934 (51:11130) [23] V. M. Kopytov and N. Ya. Medvedev: Right-Ordered Groups. Plenum, New York, 1996. ISBN: 0-306-11060-1 MR1393199 (97h:06024a) [24] L. Lifschitz and D. W. Morris: Bounded generation and lattices that cannot act on the line, Pure Appl. Math. Q. 4 (2008), no. 1, part 2, 99–126. http://arxiv.org/abs/math/0604612 MR2405997 (2009b:22011) [25] N. Monod: Continuous Bounded Cohomology of Locally Compact Groups. Springer, Berlin, 2001. ISBN: 3-540-42054-1 MR1840942 (2002h:46121) [26] N. Monod: An invitation to bounded cohomology, in: Proc. Internat. Congress Math., Madrid, Spain, 2006, vol. 2, pp. 1183–1211. http://www.mathunion.org/ICM/ICM2006.2/ Main/icm2006.2.1183.1212.ocr.pdf MR2275641 (2008e:22011) [27] N. Monod: Groups of piecewise projective homeomorphisms (preprint). http://arxiv.org/ abs/1209.5229 MR3047655 [28] D. W. Morris: Amenable groups that act on the line, Algebr. Geom. Topol. 6 (2006), 2509– 2518. http://dx.doi.org/10.2140/agt.2006.6.2509 MR2286034 (2008c:20078) [29] D. W. Morris: Bounded generation of SL(n, A) (after D. Carter, G. Keller and E. Paige), New York J. Math. 13 (2007) 383–421. http://nyjm.albany.edu/j/2007/13-17.html MR2357719 (2008j:20145) [30] D. W. Morris: Can lattices in SL(n, R) act on the circle?, in Geometry, Rigidity, and Group Actions, U of Chicago Press, Chicago, 2011. http://arxiv.org/abs/0811.0051 MR2807831 (2012c:37048) [31] D. Morris: Introduction to Arithmetic Groups (preprint). http://arxiv.org/abs/math/ 0106063 ´ [32] A. Navas: Actions de groupes de Kazhdan sur le cercle, Ann. Sci. Ecole Norm. Sup. (4) 35 (2002), no. 5, 749–758. http://www.numdam.org/item?id=ASENS_2002_4_35_5_749_0 MR1951442 (2003j:58013) [33] A. Navas: Groups of Circle Diﬀeomorphisms, U of Chicago Press, Chicago, IL, 2011. ISBN: 978-0-226-56951-2, http://arxiv.org/abs/math/0607481 MR2809110 [34] A. Yu. Ol’shanskii and M. Sapir: Non-amenable ﬁnitely presented torsion-by-cyclic groups, Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 63–71. http://dx.doi.org/10.1090/ S1079-6762-01-00095-6 MR1852901 (2002f:20041) [35] A. L. T. Paterson: Amenability. American Mathematical Society, Providence, RI, 1988. ISBN: 0-8218-1529-6 MR961261 (90e:43001) [36] J.–P. Pier: Amenable Locally Compact Groups. Wiley, New York, 1984. ISBN: 0-471-89390-0 MR767264 (86a:43001) [37] V. Platonov and A. Rapinchuk: Algebraic Groups and Number Theory, Academic Press, New York, 1994. ISBN: 0-12-558180-7. MR1278263 (95b:11039) [38] A. H. Rhemtulla: Right-ordered groups, Canad. J. Math. 24 (1972) 891–895. http://dx.doi. org/10.4153/CJM-1972-088-x MR0311538 (47:100) [39] J.–P. Serre: A Course in Arithmetic. Springer, New York-Heidelberg, 1973. ISBN: 0-38790040-3 MR0344216 (49:8956) [40] J.–P. Serre: Trees. Springer, New York, 1980. ISBN: 3-540-10103-9 MR607504 (82c:20083) [41] J. von Neumann: Zur allgemeinen Theorie des Masses, Fund. Math. 13, no. 1 (1929) 73–116. pldml.icm.edu.pl/mathbwn/element/bwmeta1.element.bwnjournal-article-fmv13i1p6bwm [42] S. Wagon: The Banach-Tarski Paradox. Cambridge U. Press, Cambridge, 1993. ISBN: 0-52145704-1 MR803509 (87e:04007) [43] D. Witte: Arithmetic groups of higher Q-rank cannot act on 1-manifolds, Proc. Amer. Math. Soc. 122 (1994) 333–340. http://www.jstor.org/stable/2161021 MR1198459 (95a:22014) [44] D. Witte and R. J. Zimmer: Actions of semisimple Lie groups on circle bundles, Geom. Dedicata 87 (2001) 91–121. http://dx.doi.org/10.1023/A:1012068331112 MR1866844 (2002j:57068)

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https://doi.org/10.1090//pcms/021/08

Lectures on Lattices and Locally Symmetric Spaces Tsachik Gelander

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IAS/Park City Mathematics Series Volume 21, 2012

Lectures on Lattices and Locally Symmetric Spaces Tsachik Gelander Introduction The aim of this short lecture series is to expose the students to the beautiful theory of lattices by, on one hand, demonstrating various basic ideas that appear in this theory and, on the other hand, formulating some of the celebrated results which, in particular, shows some connections to other ﬁelds of mathematics. The time restriction forces us to avoid many important parts of the theory, and the route we have chosen is naturally biased by the individual taste of the speaker.

Einstein Institute of Mathematics, The Hebrew University of Jerusalem Jerusalem, 91904, Israel c 2014 American Mathematical Society

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LECTURE 1

A brief overview on the theory of lattices The purpose of the ﬁrst lecture is to introduce the student to the theory of lattices. While some parts of this lecture will be done consistently with full detail (some of which are given as exercises), other parts should be considered as a story aiming to give a broader view on the theory. 1. Few deﬁnitions and examples Let G be a locally compact group equipped with a left Haar measure μ, i.e. a Borel regular measure which is ﬁnite on compact sets, positive on open sets and invariant under left multiplications (i.e. μ(gA) = μ(A) for every g ∈ G and measurable A ⊂ G) — by Haar’s theorem such μ exists and is unique up to normalization. The group G is called unimodular if μ is also right invariant, or equivalently if it is symmetric in the sense that μ(A) = μ(A−1 ) for every measurable set A. Note that G is compact iﬀ μ(G) < ∞. For example: • Discrete groups, abelian groups, compact groups and Perfect groups (i.e. groups that are equal to their commutator subgroup) are unimodular. • The group of aﬃne transformations of the real line is not unimodular. A closed subgroup H ≤ G is said to be co-ﬁnite if the quotient space G/H admits a non-trivial ﬁnite G invariant Borel regular measure. Like in Haar’s theorem, it can be shown that a G invariant measure on G/H, if exists, is unique up to scaling (see [39, Lemma 1.4]). A lattice in G is a co-ﬁnite discrete subgroup. A discrete subgroup Γ ≤ G is a lattice iﬀ it admits a ﬁnite measure fundamental domain, i.e. a measurable set Ω of ﬁnite measure which forms a set of right coset representatives for Γ in G — we will always normalize the measures so that vol(G/Γ) = μ(Ω). Let us denote Γ ≤L G to express that Γ is a lattice in G. Exercise 1. Show that every two fundamental domains have the same measure. Exercise 2. Deduce from Exercise 1 that if G admits a lattice then it is unimodular. We shall say that a closed subgroup H ≤ G is uniform if it is cocompact, i.e. if G/H is compact. Exercise 3. A uniform discrete subgroup Γ ≤ G is a lattice. Note that if G = SL2 (R) and H is the Borel subgroup of upper triangular matrices, then H is uniform but not co-ﬁnite. Examples: (1) If G is compact every closed subgroup is coﬁnite. The lattices are the ﬁnite subgroups. 253 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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TSACHIK GELANDER, LATTICES

(2) If G is abelian, a closed subgroup H ≤ G is coﬁnite iﬀ it is uniform. (3) Let G be the Heisenberg group of 3×3 upper triangular unipotent matrices over R and let Γ = G(Z) = G ∩ SL3 (Z) be the integral points. Then Γ is a uniform lattice in G. (4) Let T be a k regular tree equipped with a k coloring of the edges such that neighboring edges have diﬀerent colors. Let G = Aut(T ) be the group of all automorphisms of T considered with the open-compact topology and let Γ be the group of those automorphisms that preserve the coloring. Then Γ is a uniform lattice in G. (To study more about lattices in Aut(T ), see [4].) (5) SLn (Z) is a non-uniform lattice in SLn (R) (see [40] or [39, Ch. X]). (6) Let Σg be a closed surface of genus g ≥ 2. Equip Σg with a hyperbolic structure and ﬁx a base point and a unit tangent vector. The action of the fundamental group π1 (Σg ) via Deck transformations on the universal ˜ g yields an embedding of π1 (Σg ) in PSL2 (R) ∼ cover H2 = Σ = Isom(H2 )◦ and the image is a uniform lattice. 2. Lattices resemble their ambient group in many ways. Here are few illustrations of this phenomenon: Let G be a locally compact group and Γ ≤L G a lattice. (1) G is amenable iﬀ Γ is amenable. (2) G has property (T) iﬀ Γ has property (T). (3) Margulis’ normal subgroup theorem [30, Ch. VIII]: If G is a simple Lie group of real rank ≥ 2 (e.g. SLn (R) for n ≥ 3) then Γ is just inﬁnite, i.e. has no inﬁnite proper quotients, or in other words, every normal subgroup of Γ is of ﬁnite index. (4) Borel density theorem: If G is a non-compact simple real algebraic group then Γ is Zariski dense in G (a good reference for this result is [22]). Items (1) and (2) can be deduced directly from the deﬁnition of co-ﬁniteness and we recommend them as exercises, but they can be found in many places (for an excellent reference for property (T ), see [6]). Note that both amenability and property (T ) can be expressed as ﬁxed point properties. 3. Some basic properties of lattices In this section some basic results about lattices are proved. Students who wish to accomplish a more comprehensive background are highly encouraged to read [39, Ch 1] as well as [41]. Let G be a locally compact second countable group. Lemma 1.1 (Compactness criterion). Suppose Γ ≤L G, let π : G → G/Γ be the quotient map and let gn ∈ G be a sequence. Then π(gn ) → ∞ (i.e. eventually leaves every compact set) iﬀ there is a sequence γn ∈ Γ \ {1} such that gn γn gn−1 → 1. Before proving this lemma, let us try to give an intuitive explanation. Thinking of G/Γ as a generalisation of a manifold, the elements γ g = gγg −1 (with γ = 1) correspond to the generalisation of (homotopically) nontrivial loops. Thus the lemma “says” that a ﬁnite volume manifold is compact iﬀ it admits arbitrarily short nontrivial loops.

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255

Proof. Suppose that π(gn ) does not go to inﬁnity. Since G is locally compact and, by deﬁnition of the quotient topology, π is an open map, there is a bounded sequence {hn } ⊂ G with π(hn ) = π(gn ), ∀n. A subsequence hnk converges to some g0 . Let W be an identity neighborhood which intersects Γg0 (= g0 Γg0−1 ) trivially, and let V be a symmetric identity neighborhood satisfying V 3 ⊂ W . For suﬃciently hn gn large k we have g0 h−1 nk ∈ V which implies that Γ k = Γ k intersects V trivially. Conversely, suppose that π(gn ) → ∞. Let W be a an arbitrary identity neighborhood in G and let V be a relatively compact symmetric identity neighborhood satisfying V 2 ⊂ W . Let K be a compact subset of G such that vol(π(K)) > vol(G/Γ) − μ(V ). Since π(gn ) → ∞, there is n0 such that n ≥ n0 implies that π(V gn )∩π(K) = ∅. The volumes inequality above then implies that vol(π(V gn )) < μ(V ) and we conclude that V gn does not inject to the quotient, i.e. that V gn ∩ V gn γ = ∅ for some γ ∈ Γ \ {1}, hence γ gn ∈ V 2 ⊂ W . Since G is second countable, one deduces that there are γn ∈ Γ such that γngn → 1. Example 1.2. Consider SL2 (Z) ≤L SL2 (R), let n 0 1 0 gn = = and γ n 0 n−1 1 1 then gn γn gn−1 → 1, and hence π(gn ) → ∞. For general G, let us say that h ∈ G is unipotent if the closure of its conjugacy class contains the identity. Let us say that a sequence {hn } ⊂ G is asymptotically (or approximated) unipotent, if there are gn ∈ G such that hgnn → 1. Corollary 1.3. A lattice Γ ≤L G admits non-trivial approximated unipotents in G iﬀ it is non-uniform. It is worth mentioning that a celebrated theorem of Kazhdan and Margulis [29] states that if G is a real algebraic semisimple group, then every non-uniform lattice in G admits non-trivial unipotents. Exercise 4. Let G be a totally disconnected locally compact group and Γ ≤L G. Show that if Γ is non-uniform then Γ admits torsion, i.e. non-trivial elements of ﬁnite order. Moreover, show that in that case Γ admits element of arbitrarily large ﬁnite order. Hint: Make use of V. Dantzig’s theorem, namely that every totally disconnected locally compact group admits an open compact subgroup. We shall now explain some basic results established in the beautiful paper A. Selberg [41]. Lemma 1.4 (Recurrence). Let Γ ≤L G, let g ∈ G and let Ω ⊂ G be an open set. Then Ω−1 g n Ω ∩ Γ = ∅ inﬁnitely often. This is immediate from Poincare recurrence theorem, but let us sketch an argument: Proof. Since π(Ω) has positive measure while vol(G/Γ) is ﬁnite, we can ﬁnd k, m ∈ N with arbitrarily large gap, so that g k · π(Ω) and g m · π(Ω) are not disjoint. This means that π(g m−k Ω) ∩ π(Ω) = ∅ which is equivalent to Ω−1 g n Ω ∩ Γ = ∅ with n = m − k.

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256

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Exercise 5 (A weak version of Borel’s density theorem). Let Γ ≤L SLn (R). Deduce from the last lemma that • Γ admits regular elements, and • Span(Γ) = Mn (R). (The less motivated student, may ﬁnd the details in [41].) Proposition 1.5. Let Γ ≤UL G (a uniform lattice) and γ ∈ Γ. Let CG (γ) be the centralizer of γ in G. Then Γ ∩ CG (γ) is a uniform lattice in CG (γ). Proof. There are two ways to prove this, one by constructing a compact fundamental domain for Γ∩CG (γ) in CG (γ) and one by showing that the projection of CG (γ) to G/Γ is closed. Let us describe the ﬁrst approach. Let Ω be a relatively compact fundamental domain for Γ in G. Let δ1 , . . . , δm ∈ Γ be chosen such that δi γδi−1 , i = 1, . . . , m exhaust the ﬁnite set Ω−1 γΩ ∩ γ Γ . We claim that Ω := ∪m i=1 Ωδi ∩ CG (γ) is a fundamental domain for Γ ∩ CG (γ) in CG (γ). Indeed, given h ∈ CG (γ) we can express h as ωδ with ω ∈ Ω and δ ∈ Γ, so δ = ω −1 h and we may ﬁnd 1 ≤ i ≤ m so that γ δ = w−1 γw = γ δi (i.e. δi−1 δ ∈ CG (γ)). Thus h = (ωδi )(δi−1 δ) ∈ Ω · CΓ (γ). Exercise 6. Show (with the aid of Exercise 5) that if Γ ≤UL SLn (R) then Γ admits a diagonalizable subgroup isomorphic to Zn−1 . Here is a geometric interpretation of the last exercise: Let X = G/K be the symmetric space of SLn (R) (see Lecture 3), and let M = Γ\X be a compact Xmanifold (or orbifold). Then M admits a ﬂat totally geodesic imbedded (n − 1)torus. In fact any simple closed geodesic is contained in such a torus. This fact generalizes without diﬃculties to arbitrary Riemannian symmetric space X where n − 1 is replaced by rank(X) = rankR (G). However, the analogous statement for general CAT(0) (or even Hadamard) spaces is the wide open, well known, Flat Closing Problem. Exercise 7. Find an element γ in SL3 (Z) whose centraliser in SL3 (Z) is cyclic. Deduce that the analog of Proposition 1.5 cannot hold for general non-uniform lattices. In spite of that, the analog of Exercise 6 does hold for non-uniform lattices as well, by a theorem of Prasad and Raghunathan [38]. 4. A theorem of Mostow about lattices in solvable groups The discussion in this section is taken from [2]. If G is abelian and H ≤ G a co-ﬁnite subgroup then G/H is a group with ﬁnite Haar measure, hence compact. A similar result holds for general nilpotent groups: Proposition 1.6. Let G be a nilpotent locally compact group and H ≤ G a closed subgroup. Then H has ﬁnite covolume if and only if H is cocompact. First prove:

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LECTURE 1. A BRIEF OVERVIEW ON THE THEORY OF LATTICES

257

Exercise 8. Every nilpotent locally compact group is unimodular. We will also make use of the following: Exercise 9. Let G be a locally compact group and H ≤ F ≤ G closed subgroups. If H is co-ﬁnite then so is F . Furthermore if F normalizes H than F/H is compact.1 Proof of Proposition 1.6. Let us prove the “only if” direction. Let H ≤ G be a co-ﬁnite subgroup, let Z = Z(G) be the center of G and let F = Z · H. Arguing by induction on the nilpotency degree, we infer that F/Z is uniform in G/Z and, hence, that F is uniform in G. It is thus suﬃcient to prove that H is uniform in F . The latter fact is clear since H is normal in F by deﬁnition. Let us note that the “if” part of 1.6 holds in the much greater generality of amenable groups. Indeed, if G is amenable and H ≤ G is a closed uniform subgroup, the compact G-space G/H (as any compact G-space) admits a G invariant probability measure. The “only if” direction however, is more involved for nonnilpotent groups. For solvable Lie groups Mostow proved the following classical result: Theorem 1.7 ([34]). Let G be a connected solvable Lie group. Then every co-ﬁnite subgroup of G is uniform. Let us give an elementary proof to Mostow’s theorem. Exercise 10. (a) Show that if G is a connected Lie group and Γ ≤L G is a ﬁnitely generated (or more generally compactly generated) abelian co-ﬁnite subgroup, then Γ is uniform. (One can dig out an argument for this fact from the proof of Theorem 3.1 in [39], but I would recommend trying to establish a direct argument. In fact, this is true also for general locally compact G.) (b) Show that a connected solvable Lie group is Noetherian in the sense that every closed subgroup is compactly generated. (Hint: deduce the general case from the abelian case using induction on the solvability degree.) Proof of Theorem 1.7. We shall prove the result for every compactly generated solvable Lie group (note that a connected Lie group is compactly generated). Let G be a compactly generated solvable Lie group and H ≤ G a co-ﬁnite subgroup. Up to replacing G and H by ﬁnite index subgroups we may assume that the commutator G is nilpotent. (Indeed, by the Ado–Iwasawa theorem G admits an almost faithful complex linear representation ρ : G → GLd (C), and after replacing G by a ﬁnite index subgroup, the Zariski closure of the image is connected, in which case, by Lie’s theorem the commutator is nilpotent.) We may argue by induction on the nilpotency degree of G , where the base case when G is trivial follows from Proposition 1.6. Let Z be the center of G . By induction (G/Z)/(HZ/Z) is compact, hence, we are left to show that H is uniform in E := HZ, which is again compactly generated by Exercise 10 (b). Clearly F = H ∩ E is normal in E. Dividing by F we are left to prove that H/F is cocompact in E/F . Since H/F is abelian, the result follows from Exercise 10 (a). 1 A more general, but slightly harder, statement is: For any locally compact groups H ≤ F ≤ G, H is co-ﬁnite in G iﬀ it is co-ﬁnite in F and F is co-ﬁnite in G, see [39, Lemma 1.6].

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The analog of Mostow’s theorem holds whenever G is a linear group over a local ﬁeld, and when G is compactly generated with nilpotent commutator (see [2]). However, contrary to a conjecture of Benoist and Quint there are solvable groups which admits non-uniform lattices: ∞ n < ∞. Example 1.8 (See [2]). Let pn , n ∈ N be primes such that n=1 pnp−1 Let G be the compact by discrete metabelian group G=(

∞

F∗pn ) (

n=1

∞

Fpn )

n=1

and let Γ be the set of sequences (an , an − 1) where an ∈ Fpn and an = 1 for all but ﬁnitely manny n’s. It can be shown that Γ is a non-uniform lattice in G. Exercise 11 (Completing details in the Example 1.8). (1) Show that in the aﬃne group FF∗ over a ﬁeld F, the set {(a, a−1) : a ∈ F} forms a subgroup. Deduce that Γ is a subgroup of G. (2) Show that Γ is discrete in G. (3) For m ∈ N let Gm := (

∞

F∗pn ) (

n=1

m

Fpn ) and Γm = Γ ∩ Gm .

n=1

Show that [Gm : Gm−1 ] = pm and [Γm : Γm−1 ] = pm − 1. (4) Deduce ∞ that if we normalise that Haar measure on G so that its compact subgroup n=1 F∗pn has measure 1 then vol(Gm /Γm ) =

m pn vol(Gm ) = . |Γm | p −1 n=1 n

(5) Making use of the data that G is a direct limit of the open compact subgroups Gn , deduce that: • Γ is nonuniform in G — indeed, neither of the GN contains a fundamental domain for Γ in G, since the sequence of co-volumes in (4) does not stabilises, pn • vol(G/Γ) = ∞ n=1 pn −1 < ∞ and hence Γ is a lattice. 5. Existence of lattices The very existence of (uniform and non-uniform) lattices is an interesting question. As we have seen in Exercise 2 groups which are not unimodular cannot admit lattices. There are also examples of nilpotent Lie groups which admit no lattices (see [39, Ch. 2]). The discussion above shows that certain solvable groups admit only uniform lattices. In [3] it is shown that there are locally compact simple (amenable as well as non-amenable) groups which admit no lattices at all. In the remaining lectures we will mostly restrict ourselves to the case where G is a semisimple Lie group. By a classical theorem of Borel [9] every connected semisimple Lie group admits plenty of uniform and non-uniform lattices.

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LECTURE 1. A BRIEF OVERVIEW ON THE THEORY OF LATTICES

259

6. Arithmeticity One of the highlights of the theory of lattices is the connection with arithmetic groups. This is illustrated in the following celebrated theorems2 : Theorem 1.9 (Borel–Harish-Chandra, see [35]). Let G be an algebraic group deﬁned over Q which has no Q-characters. Then G(Z) ≤L G(R). Furthermore, G(Z) ≤UL G(R) iﬀ G has no Q-co-characters. Definition 1.10. Let G be a Lie group. We shall say that a subgroup Γ ≤ G is arithmetic if there is a Q algebraic group H and a surjective homomorphism with compact kernel f : H(R) G such that f (H(Z)) contains Γ as a subgroup of ﬁnite index. If G has no non-trivial real characters (homomorphisms to R∗ ), it follows that so does H(R) and hence by Theorem 1.9 that H(Z) is a lattice in H(R). Since f has compact kernel, the image f (H(Z)) is still discrete, and hence a lattice in G = f (H(R)). √ 2 √ 2 Example 1.11. Let f (x, y, z) = x2 + √y − 2z , and consider the Q[ 2]-group G = SO(f ) and the subgroup Γ = G(Z[ 2]). Let H = RQ[√2]/Q G be the algebraic group obtained by restriction of scalars (see [35, 2.1.2]) and let H = H(R). Then H ∼ = SO(2, 1) × SO(3) and Γ is isomorphic to H(Z). Since SO(3) is compact, Γ projects faithfully to an arithmetic lattice in SO(2, 1) and it can be shown that it has no unipotent. Now SO(2, 1) acts properly by isometries on the hyperbolic plan H2 and hence Γ\H2 is a compact hyperbolic orbifold. This yields an interesting information about the algebraic structure of Γ. For instance, it is known that every such orbifold is ﬁnitely covered by a Riemann surface, and hence Γ admits a ﬁnite index subgroup which is a surface group. Moreover, every surface group admits a presentation with 2g generators and one relator, where g is the genus and can be computed from the area of the surface (the covolume of the corresponding lattice, with respect to an appropriate normalisation), using the Gauss–Bonnet theorem. Amazingly, in some cases the converse of the Borel–Harish-Chandra theorem is also true. Recall that the rank of a Lie group is the minimal dimension of a centralizer of an element. The rank of SLn (R) is n − 1 and the rank of SO(n, 1) is one. Theorem 1.12 (Margulis arithmeticity theorem [30]). If G is a simple Lie group of rank ≥ 2 then every lattice is arithmetic. In the fourth lecture we will discuss some rigidity theorems and describe how Margulis’ superrigidity theorem implies the arithmeticity theorem. Superrigidity and arithmeticity hold also for lattices in the rank one groups Sp(n, 1) and F4−20 as proved in [18] and [27]. On the other hand it is clear that SL2 (R) admits non-arithmetic lattices. Indeed, Teichmuller theory produces continuously many pairwise non-isometric hyperbolic structures on a surface of genus g ≥ 2, yielding continuously many non-conjugate lattices, while it can be shown that only countably of them can be arithmetic. A beautiful construction of Gromov and 2I

recommend the book by D. Witte–Morris [50] for an introduction to the theory of arithmetic groups

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Piatetsky-Shapiro [26] produces ﬁnite volume non-arithmetic real hyperbolic manifolds in every dimension n ≥ 3, by gluing two pieces of arithmetic ones, proving that SO(n, 1), n ≥ 2 admits non-arithmetic lattices.

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LECTURE 2

On the Jordan–Zassenhaus–Kazhdan–Margulis theorem 1. Zassenhaus neighborhood Given two subsets of a group A, B ⊂ G we denote by {[A, B]} := {[a, b] : a ∈ A, b ∈ B} the set of commutators [a, b] = aba−1 b−1 . We deﬁne recursively A(n) := {[A, A(n−1) ]} where A(0) := A. By the Ado–Iwasawa theorem every Lie group is locally isomorphic to a linear Lie group. By explicit computation using sub-multiplicativity of matrix norms, one proves: Lemma 2.1. Every Lie group G admits an open identity neighborhood U such that U (n) → 1 in the sense that it is eventually included in every identity neighborhood. As pointed out, it is enough to explain this for G = GLd (R). Write a = 1 + X, b = 1 + Y with X, Y ∈ Md (R) and suppose X ≤ and Y ≤ δ ≤ . By continuity of the inverse map, for suﬃciently small we have a−1 , b−1 ≤ 2. Thus by sub-multiplicity of the norm: aba−1 b−1 − 1 = (ab − ba)a−1 b−1 = (XY − Y X)a−1 b−1 ≤ 2X · Y · a−1 · b−1 ≤ 8δ, hence for < 18 and U = U := {a ∈ GLd (R) : a − 1 < } we see that Ω(n) tends to 1 at an exponential speed. Exercise 12. Let Δ be a group generated by a set S ⊂ Δ. If S (N ) = {1} for some n then Δ is nilpotent of class ≤ N . Corollary 2.2. If Δ ≤ G is a discrete subgroup then Δ ∩ U is nilpotent. Proof. Since Δ is discrete, there is an identity neighborhood V which intersects Γ trivially. By Lemma 2.1 S := Δ ∩ U satisﬁes S (n) → 1, hence for some N , S (N ) ⊂ V ∩ Δ = {1} and the result follows from the previous exercise. Furthermore, taking Ω = U with suﬃciently small we can even guarantee that every discrete group with generators in Ω is contained in a connected nilpotent group: 261 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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Theorem 2.3 (Zassenhaus (1938), Kazhdan–Margulis (1968)). Let G be a Lie group. There is an open identity neighborhood Ω ⊂ G such that every discrete subgroup Δ ≤ G which is generated by Δ ∩ Ω is contained in a connected nilpotent Lie subgroup of N ≤ G. Moreover Δ ≤UL N . The idea is that near the identity the logarithm is well deﬁned and two elements commute iﬀ their logarithms commute. For a complete proof see [39, Theorem 8.16] or [43, Section 4.1]. A set Ω as in the theorem above is called a Zassenhaus neighborhood. 2. Jordan’s theorem Since connected compact nilpotent groups are abelian, we deduce the following classical result: Theorem 2.4 (Jordan 1878). For a compact Lie group K there is a constant m ∈ N such that every ﬁnite subgroup Δ ≤ K admits an abelian subgroup of index ≤ m. Proof. Let Ω be a Zassenhaus neighborhood in K, let U be a symmetric identity neighborhood satisfying U 2 ⊂ Ω, and set m := μ(K) μ(U) . Given a ﬁnite subgroup F ≤ K set A = F ∩ Ω. By the remark preceding the theorem A is abelian. Now if f1 , . . . , fm+1 are m+1 elements in F then for some 1 ≤ i = j ≤ m+1 we have fi U ∩ fj U = ∅ implying that fi−1 fj ∈ F ∩ Ω ⊂ A. Thus [F : A] ≤ m. Note that since any connected Lie group G admits a unique maximal compact subgroup K up to conjugation, one can state Jordan’s theorem for non-compact connected Lie groups as well. (Originally, it was stated for G = GLn (C).) 3. Approximations by ﬁnite transitive spaces Let us make a short detour before continuing the discussion about discrete groups. Suppose that K is a metric group. An -quasi morphism f : F → K from an abstract group F is a map satisfying d(f (ab), f (a)f (b)) ≤ , ∀a, b ∈ F . We shall say that K is quasi ﬁnite if for every there a ﬁnite group F and an -quasi morphism into K with an -dense image (i.e. ∀k ∈ K, ∃a ∈ F with d(f (a), k) ≤ ). Relying on Jordan’s theorem, Turing showed [44]: Theorem 2.5 (Turing 1938). A compact connected Lie group is quasi ﬁnite iﬀ it is abelian (i.e. a torus). Recall that a metric space is said to be transitive if its isometry group acts transitively. With the aid of Turing’s theorem one can classify the metric spaces which can be approximated by ﬁnite transitive ones: Theorem 2.6 ([25]). A metric space is a limit of ﬁnite transitive spaces (in the Gromov–Hausdorﬀ topology) iﬀ it admits a transitive compact group of isometries whose identity connected component is abelian. The lines of the proof are as follows. Given a metric space X, one shows that there is a δ0 > 0 and a function : (0, δ0 ) → R>0 whose limit at 0 is 0, such that for any ﬁnite metric space F with dGH (X, F) < δ ≤ δ0 there is a natural (δ) quasi morphism from the ﬁnite group Isom(F) to Isom(X). The result is then proved relying on structure theorems for compact groups and on 2.5 (see [25] for details).

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LECTURE 2. ON THE JORDAN–ZASSENHAUS–KAZHDAN–MARGULIS THEOREM

263

It follows from Theorem 2.5 and the Peter–Weyl theorem that if X is approximable by ﬁnite transitive spaces then its connected components are inverse limits of tori, hence the only manifolds that can be approximated are tori. In particular we obtain the following result which answers a question of I. Benjamini and can be interpreted as the non-existence of a perfect soccer ball: Corollary 2.7. S 2 cannot be approximated by ﬁnite homogeneous spaces. Theorem 2.6 has also some graph theoretic applications. For instance one can deduce that any sequence of distance-transitive graphs with normalized diameter and bounded geometry converges, in the Gromov–Hausdorﬀ sense, to a circle (see [25, Corollary 1.6]). 4. Margulis’ lemma Coming back from this short detour, let us present another classical result: Theorem 2.8. (The Margulis lemma, [43, Section 4.1]) Let G be a Lie group acting properly by isometries on a Riemannian manifold X. Given x ∈ X there are = (x) > 0 and m = m(x) ∈ N such that if Δ ≤ G is a discrete subgroup which is generated by the set ΣΔ,x, := {γ ∈ Δ : d(γ · x, x) ≤ } then Δ admits a subgroup of index ≤ m which is contained in a (closed) connected nilpotent Lie group. Furthermore, if G acts transitively on X then and m are independent of x. Proof. The properness of the action implies that the set C = {g ∈ G : d(g · x, x) ≤ 1} is compact. Let V ⊂ G be a relatively compact open symmetric set such that V 2 is a Zassenhaus neighborhood. Setting vol(C · V ) ] and = 1/m m=[ vol(V ) one can prove the theorem arguing as in the proof of 2.4. An extra complication arises from the fact that Δ and F = Δ ∩ V 2 are inﬁnite, but this can be taken care of by observing that whether a connected graph has more than m vertices or not, can be seen by looking at a ball of radius m in the graph. Thus, assuming in contrary that the Schreier graph of Δ/F has more than m vertices, we could ﬁnd m + 1 elements γ1 , . . . , γm+1 in the m-ball (ΣΔ,x, )m which belong to mutually diﬀerent cosets of F . However, by the choice of we have that (ΣΔ,x, )m ⊂ C, hence for some 1 ≤ i = j ≤ m + 1 we have γi V ∩ γj V = ∅, i.e. γi−1 γj ∈ V 2 ∩ Δ ⊂ F , a contradiction. A diﬀerential–geometric proof of the Margulis lemma, which provides more information, can be found in [5]. 5. Crystallographic manifolds In the special case of X = Rn since homotheties commute with isometries it follows that = ∞ — i.e. that any ﬁnitely generated1 discrete group of isometries of Rn is virtually (i.e. admits a ﬁnite index subgroup which is) contained in a connected 1 Since

any discrete subgroup of Isom(Rn ) is f.g. this assumption is in fact redundant.

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nilpotent group. Indeed, given any > 0 and a ﬁnite set Σ generating a discrete subgroup of Isom(Rn ), one can rescale the metric (or alternatively, apply a homothety) so that the displacement of Σ at an arbitrary point x ∈ Rn becomes less than . Moreover, it is easy to verify that the connected nilpotent subgroups of the group G = Isom(Rn ) ∼ = On (R) Rn are abelian. Given an isometry γ of Rn one n can decompose R , considered as an aﬃne space, to min(γ)⊕min(γ)⊥ where min(γ) is the aﬃne subspace on which γ acts by a translation and min(γ)⊥ is an arbitrarily located orthogonal complement. Clearly for γ non-elliptic (i.e. which does not ﬁx a point) min(γ) has positive dimension, and it is not hard to show that if Λ is a set of commuting non-elliptic isometries then ∩γ∈Λ min(γ) has positive dimension and is Λ-invariant. Exercise 13. Complete the details above as follows: (1) Show that every connected nilpotent subgroup of On (R) Rn is abelian. (Hint: use the fact that a compact connected nilpotent group is abelian.) (2) Show that an isometry of Rn whose linear part has no nonzero invariant vector must have a ﬁxed point. Deduce the existence of the above decomposition min(γ) ⊕ min(γ)⊥ by ﬁrst decomposing Rn as a linear space according to the liner part of γ. (3) Show that the min-sets of arbitrarily many commuting isometries intersect nontrivially. Thus, we deduce: Theorem 2.9 (Bieberbach (1911) — Hilbert’s 18’th problem). Let Γ be a torsion free group acting properly discontinuously by isometries on Rn . Then Γ admits a ﬁnite index subgroup isomorphic to Zk (k ≤ n) which acts by translations on some k dimensional invariant subspace, and k = n iﬀ Γ is uniform. In particular, every crystallographic manifold is ﬁnitely covered by a torus. A well known result, commonly attributed to Selberg states that every ﬁnitely generated linear group is virtually torsion free (c.f. [39, Corollary 6.13]). Thus Theorem 2.9 holds without the assumption that the discrete group Γ ≤ Isom(Rn ) is torsion free.

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LECTURE 3

On the geometry of locally symmetric spaces and some ﬁniteness theorems 1. Hyperbolic spaces Consider the hyperbolic space Hn and its group of isometries G = Isom(Hn ). Recall that G◦ ∼ = PSO(n, 1) is a rank one simple Lie group. For g ∈ G denote by dg (x) := d(g · x, x) the displacement function of g at x ∈ Hn . Let |g| = inf dg and min(g) = {x :∈ Hn : dg (x) = |g|}. Note that dg is a convex function which is smooth outside min(g). The isometries of Hn split to 3 types (c.f. [43, Section 2.5]): • elliptic — those that admit ﬁxed points in Hn . • hyperbolic — isometries for which dg attains a positive minimum. In that case min(g) is a g-invariant geodesic, called the axis of g. • parabolic — isometries for which inf dg = 0 but have no ﬁxed points in Hn . The ﬁrst two types are called semisimple. One way to prove that every isometry is of one of these forms is to consider n the visual compactiﬁcation H = Hn ∪ ∂Hn , where ∂Hn can be deﬁned as the set of geodesic rays up to bounded distance (the student is refereed to P.E. Caprace’s course — given in parallel — for a detailed description of this compactiﬁcation). n n The action of G on H extends to a continuous action on H and H is homeomorphic n to a closed ball in R . By Brouwer’s ﬁxed point theorem every g ∈ G admits a n ﬁxed point in H . Exercise 14. Suppose n ≥ 2 and let g ∈ G. • If g has 3 ﬁxed points on ∂Hn then g ﬁxes point wise the hyperbolic plane in Hn determined by these 3 points, and in particular g is elliptic. • If g is non-elliptic and has exactly 2 ﬁxed points at ∂Hn then g is hyperbolic and its axis is the geodesic connecting these ﬁxed points. • If g has exactly one ﬁxed point at ∂Hn then g is parabolic. By considering the upper half space model for Hn it is easy to see that a parabolic isometry preserves the horospheres around its ﬁxed point at inﬁnity, and that each such horosphere, considered with its intrinsic metric, is isometric to Rn−1 . Exercise 15. Suppose that g, h ∈ G commute, then • if g is hyperbolic, then h is semisimple, • if g and h are both hyperbolic then they share a common axis, 265 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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• if g and h are parabolics, they have the same ﬁxed point at ∂Hn . Exercise 16. A discrete subgroup Δ ≤ G admits a common ﬁxed point in Hn if and only if it is ﬁnite. It follows that a discrete group Γ ≤ G acts freely on Hn if and only if it is torsion free. 2. The thick–thin decomposition Let Γ ≤ G be a torsion free discrete subgroup. We denote by M = Γ\Hn the associated complete hyperbolic manifold. Note that Γ is a lattice iﬀ M has ﬁnite volume. We denote by InjRad(x) the injectivity radius at x. Let (Hn ) be the Margulis’ constant of Hn (see Theorem 2.8 in the previous lecture) and set = 1 n 10 (H ) (it is helpful for some arguments to work with a constant which is strictly smaller than (Hn )). Let M 0 such that for every ﬁnite subset F of Γ |∂S F | ε|F |. Exercise: Show that amenability is preserved under quasi-isometry. Exercise: Show that the non-abelian free groups Fk are non-amenable. It follows from the last exercise that if a countable group contains a free subgroup, it is non-amenable. The converse is not true. In fact there are ﬁnitely generated torsion groups (i.e. every element is of ﬁnite order) which are non-amenable. Adyan and Novikov showed that if n is a large ( 665) odd integer, the Burnside groups B(n, k) := a1 , . . . , ak |γ n = 1 ∀γ are inﬁnite, and in fact, as Adyan later proved [1], they are non-amenable. C. Invariant means Amenable groups were introduced by John von Neumann in 1929 ([96]). His deﬁnition was in terms of invariant means (amenable = admits a mean). Definition 1.3. An invariant mean on a countable group Γ is a ﬁnitely additive probability measure m deﬁned on the set of all subsets of Γ, which is invariant under the group action by left translations, i.e. m(γA) = m(A) for all γ ∈ Γ and A ⊂ Γ. It is easily checked that an invariant mean is the same thing as a continuous linear functional m : ∞ (Γ) → R such that • f 0 ⇒ m(f ) 0, • ∀γ ∈ Γ, γ∗ m = m, • m(1) = 1. where γ∗ m is the push forward of m by the left translation by γ, and 1 is the constant function equal to 1 on Γ. It is worth observing that if Γ has an invariant mean, then it has a bi-invariant mean, that is a mean m as above which is also invariant under right translations: just take the mean of f (y −1 x) with respect to y, then with respect to x. Folner [52] showed the following: Proposition 1.4 (Folner criterion). A group Γ is amenable (in the sense of Deﬁnition 1.1 above) if and only if it admits an invariant mean. The proof of the existence of the invariant mean from the Folner sequence follows by taking a weak- limit in ∞ (Γ) of the “approximately invariant” probability measures |F1n | 1Fn . For the converse, one needs to approximate m in the weak topology by functions in 1 , then take appropriate level sets of these functions. For the 1 We

allow multiple edges between two distinct points, but no loop at a given vertex.

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LECTURE 1. AMENABILITY AND RANDOM WALKS

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details of this proof and that of the next proposition, we refer the reader to the appendix of the book by Bekka, de la Harpe and Valette [8]. We will also give an alternate argument for the converse in the exercises at the end of this lecture using Tarski’s theorem on paradoxical decompositions. There is also a related characterization of amenability in terms of actions on compact metric spaces: Proposition 1.5. A group Γ is amenable if and only if every action of Γ by homeomorphisms on a compact (metric) space X preserves a Borel probability measure. Sketch of proof. If one averages any probability measure on X by Folner sets and takes a weak limit, one obtains an invariant probability measure. Conversely Γ acts on the space of means on Γ. This is a convex compact space (for the weak-* topology) and if Γ preserves a probability measure on it, it must ﬁx its barycenter (which will be an invariant mean). D. Random walks on groups, the spectral radius and Kesten’s criterion In his 1959 Cornell thesis [78], Kesten studied random walks on Cayley graphs of ﬁnitely generated groups and he established yet another characterization of amenability relating it to the rate of decay of the probability of return to the identity, and to the spectrum of the Markov operator associated to the random walk. Before we state Kesten’s theorem, let us ﬁrst give some background on random walks on groups. This will be useful later on in Lectures 3 and 4 when we discuss the Bourgain-Gamburd method. Suppose Γ is ﬁnitely generated and μ is a ﬁnitely supported symmetric (i.e. ∀γ ∈ Γ, μ(γ) = μ(γ −1 )) probability measure on Γ whose support generates Γ. We can associate to μ an operator Pμ on 2 (Γ), the Markov operator, by setting for f ∈ 2 (Γ) Pμ f (x) = f (γ −1 x)μ(γ). γ∈Γ

Clearly Pμ is self-adjoint (because μ is assumed symmetric) and moreover Pμ ◦ Pν = Pμ∗ν , for any two probability measures μ and ν on Γ, where μ ∗ ν denotes the convolution of the two measures, that is the new probability measure deﬁned by μ(xγ −1 )ν(γ). μ ∗ ν(x) := γ∈Γ

The convolution is the image of the product measure μ ⊗ ν under the product map Γ × Γ → Γ, (x, y) → xy and is the probability distribution of the product random variable XY , if X is a random variable taking values in Γ with distribution μ and Y is a random variable with distribution ν independent of X. The probability measure μ induces a random walk on Γ, i.e. a stochastic process (Sn )n1 deﬁned as Sn = X 1 · . . . · X n , where the Xi ’s are independent random variables with the same probability distribution μ on Γ. The process (Sn )n1 is a Markov chain and px→y := μ(x−1 y) are

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332

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the transition probabilities. This means that the probability that Sn+1 = y given that Sn = x is px→y , independently of n 1. When μ is the probability measure 1 μ = μS := δs , |S| s∈S

where δs is the Dirac mass at s ∈ S and S is a ﬁnite symmetric generating set for Γ, we say that μ and its associated process (Sn )n1 is the simple random walk on (Γ, S). It corresponds to the nearest neighbor random walk on the Cayley graph G(Γ, S), where we jump at each stage from one vertex to a neighboring vertex with equal probability. Kesten was the ﬁrst to understand that studying the probability that the random walk returns to the identity at time n could be useful to classify inﬁnite groups2 . This quantity is P roba(Sn = 1) = μn (1), where we have denoted the n-th convolution product of μ with itself by μn := μ ∗ . . . ∗ μ. We will denote the identity element in Γ sometimes by 1 sometimes by e. Proposition 1.6. Here are some basic properties of μn . • μ2n (1) is non-increasing, • μ2n (x) μ2n (1) for all x ∈ Γ. Note that μ2n+1 (1) can be zero sometimes (e.g. the simple random walk on the free group), but μ2n (1) is always positive. The Markov operator Pμ is clearly a contraction in 2 (and in fact in all p , p 1), namely ||Pμ || 1. A basic tool in the theory of random walks on groups is the spectral theorem for self-adjoint operators applied to Pμ . This will yield Kesten’s theorem and more. Proof of Proposition 1.6. Let δx be the Dirac mass at x. Observe that Pμn = Pμn and that μn (x) = Pμn δe (x) = Pμn δe , δx (in 2 (Γ) scalar product). Denoting Pμ by P for simplicity it follows that μ2(n+1) (1) = P n δe , P n+2 δe ||P n δe || · ||P 2 P n δe || ||P n δe ||2 = μ2n (1). and that μ2n (x) = P 2n δe , δx ||P n δe || · ||P n δx || = μ2n (1), where the last equality follows from the fact that P n δx (y) = P n δe (yx−1 ).

Proposition-Definition 1.7 (Spectral radius of the random walk). The spectral radius ρ(μ) of the Markov operator Pμ acting on 2 (Γ) is called the spectral radius of the random walk. Note that since Pμ is self-adjoint, its spectral radius coincides with its operator norm ||Pμ ||, and with max{|t|, t ∈ spec(Pμ )}. Let us apply the spectral theorem for self-adjoint operators to Pμ . This gives a resolution of identity E(dt) (measure taking values into self-adjoint projections), 2 His secret goal was to use his criterion to establish that the Burnside groups are inﬁnite by showing that they are non-amenable, see the comments at end of [78].

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LECTURE 1. AMENABILITY AND RANDOM WALKS

333

and a probability measure η(dt) := E(dt)δe , δe on the interval [−1, 1] such that for all n 1, tn η(dt). (1) P n δe , δe = [−1,1]

Definition 1.8 (Spectral measure). The spectral measure of the random walk is the measure η associated to the Markov operator Pμ by the spectral theorem as above. Exercise: Show that E(dt)δx , δx = η(dt) for all x ∈ Γ and that the other spectral measures E(dt)f, g, with f, g ∈ 2 (Γ) are all absolutely continuous w.r.t η. We can now state: Theorem 1.9 (Kesten). Let Γ be a ﬁnitely generated group and μ a symmetric probability measure with ﬁnite support generating Γ. 1 • ∀n 1, μ2n (1) ρ(μ)2n and limn→+∞ (μ2n (1)) 2n = ρ(μ), • (Kesten’s criterion) ρ(μ) = 1 if and only if Γ is amenable. Proof of the first item. The existence of the limit and the upper bound follows from the subadditive lemma (i.e. if a sequence an ∈ R satisﬁes an+m an + am , for all n, m ∈ N, then ann converges to inf n1 ann ). Indeed μ2(n+m) (1) μ2n (1)μ2m (1) (the chance to come back at 1 at time 2n + 2m is at least the chance to come back at time 2n and to come back again at time 2n + 2m). Take logs. In order to identify the limit as the spectral radius, we apply the spectral 1 1 theorem (see equation (1) above) to Pμ , so that μ2n (1) 2n = P 2n δe , δe 2n takes the form 1 t2n η(dt) 2n . [−1,1]

However when n → +∞, this tends to max{|t|, t ∈ spec(Pμ )} = ρ(μ).

Below we sketch a proof of Kesten’s criterion via an analytic characterization of amenability in terms of Sobolev inequalities. The following proposition subsumes Kesten’s criterion. Proposition 1.10. Let Γ be a group generated by a ﬁnite symmetric set S and let μ be a symmetric probability measure whose support generates Γ. The following are equivalent: (1) Γ is non-amenable, (2) there is C = C(S) > 0 such that ||f ||2 C||∇f ||2 for every f ∈ 2 (Γ), (3) there is ε = ε(S) > 0 such that maxs∈S ||s · f − f ||2 ε||f ||2 for all f ∈ 2 (Γ), (4) ρ(μ) < 1. Here ∇f is the function on the set of edges of the Cayley graph of Γ associated with S given by ∇f (e) = |f (e+ ) − f (e− )|, where e+ and e− are the end-points of the edge e. Proof. Note that condition (3) does not depend on the generating set (only the constant ε may change). For the equivalence between (3) and (4) observe further that a ﬁnite collection of unit vectors in a Hilbert space average to a vector of norm

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strictly less than 1 if and only if the angle between at least two of them is bounded away from zero (and the bounds depend only on the number of vectors). The equivalence between (2) and (3) is clear because ||∇f ||2 is comparable (up to multiplicative constants depending on the size of S only) to maxs∈S ||s · f − f ||2 . Condition (2) easily implies (1), because the linear isoperimetric inequality |∂S F | ε|F | is immediately derived from (2) by taking f = 1F the indicator function of F . The only less obvious implication is (1) ⇒ (2) as we need to go from sets to arbitrary functions. The idea to do this is to express f as a sum of indicator functions of sublevel sets. Namely, for t 0, let At = {γ ∈ Γ; f (γ) > t}. Then for x∈Γ +∞ +∞ f (x) = 1t 0 and in fact much more is showed that ncd/2 d/2 2n true (namely n μ (e) converges to a non-zero constant and there are gaussian estimates for μ2n (x) depending on d(e, x), see the work of Alexopoulos [2] and Hebisch-Saloﬀ-Coste [67]). Note that this implies in particular that every group of exponential growth 1 must have a decay of the probability of return at least in exp(n− 3 ). This rate is achieved by polycyclic groups (that is solvable discrete subgroups of GLd (C)) as was shown by Alexopoulos [3]. The theorem is a special case of a more general result proved by Varopoulos which says that if u(t) is the solution to the ODE u = 0, u + ψ(u)2 where ψ(u) := inf{n, VS (n) > 1/u} for u ∈ (0, 1), then μ2n (e) u(n) Varopoulos’s proof is a reﬁnement of Kesten’s argument used in the proof of Proposition 1.10 above, in which the Sobolev inequality is weakened so as to make the constant depend on the size of the support of f (Nash inequality). We refer the reader to the survey [104] and the book [126] for the details of this argument. The possible growth behaviors of ﬁnitely generated groups are still quite mysterious. For example I think it is an open question to determine whether every real number 0 can arise as the exponential growth rate lim n1 log |S n | of a ﬁnitely generated group. A consequence of the uniform Tits alternative (see Theorem 2.15 below) is that the exponential growth rate of non-virtually solvable linear groups (linear = subgroup of GLd over some ﬁeld) is bounded away from 0 by a positive constant depending only on d and not on the ﬁeld. The situation for groups of intermediate growth is also very interesting. Grigorchuk ([56, 57]) proved in the early 1980’s that there exist ﬁnitely generated groups whose growth function is not exponential, yet not polynomial either. We refer the reader to the nice recent exposition [58] for the description of Grigorchuk’s examples. Very recently Bartholdi and Erschler [6] showed that for every α ∈ [0.77, 1] there exists a ﬁnitely generated group Γ = S and constants c1 , c2 > 0 such that

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for all large n 1,

exp(c1 nα ) VS (n) exp(c2 nα ) Their construction builds on Grigorchuk’s own constructions using certain permutational wreath products. Grigorchuk conjectures [59] the following Gap Conjecture: if given any ﬁnitely generated group either Γ is virtually nilpotent (⇔ of polynomial √ growth by Gromov’s theorem [60]) or there is c > 0 such that VS (n) exp(c n) for all large n. It is known when Γ is residually nilpotent (i.e. ∀γ ∈ Γ \ {1} there is a nilpotent quotient in which the image of γ is non-trivial), and recently Grigorchuk reduced the conjecture to two cases: residually ﬁnite groups and simple groups. It is again an open problem to determine the possible rates of decay of μ2n (e) for an arbitrary group. For example can all rates of the form exp(−nα ) for α ∈ [ 15 , 1] d , d ∈ N, be achieved ? (Pittet and Saloﬀ-Coste showed that the values α = d+2 d are achieved by wreath products Z F for a ﬁnite group F ). See [103] and [50, Theorem 2] for more on this topic. F. Exercise: Paradoxical decompositions, Ponzi schemes and Tarski numbers We conclude this lecture by proving, in the form of an exercise, yet another characterization of amenability, which is due to Tarski [121] following an argument from [39]. Theorem 1.13 (Tarski). A group is non-amenable if and only if it is paradoxical. Let us deﬁne “paradoxical”. Let Γ be a group acting on a set X. This Γ-action is said to be N -paradoxical if one can partition X into n + m N disjoint pieces X = A 1 ∪ . . . ∪ A n ∪ B1 ∪ . . . ∪ Bm

in such a way that there are elements a1 , . . . , an ∈ Γ and b1 , . . . , bm ∈ Γ such that we get new partitions of X into disjoint pieces n m ai Ai and bj Bj X= i=1

j=1

We say that Γ is paradoxical if it is N -paradoxical for some ﬁnite N ∈ N for the action of Γ on itself by left translations. 1) Prove that the non-abelian free group F2 and in fact any group Γ containing the free group F2 is 4-paradoxical. 2) Suppose that Γ is a 4-paradoxical group and Γ = A1 ∪ A2 ∪ B1 ∪ B2 is a paradoxical decomposition as deﬁned above. Show that Γ plays ping-pong on itself, −1 where the ping-pong players are a := a−1 1 a2 and b := b1 b2 . Deduce that Γ contains a non-abelian free subgroup F2 . 3) Deﬁne the Tarski number T (Γ) of a group Γ to be the smallest integer N if it exists such that Γ is N -paradoxical. By the above T (Γ) = 4 if and only if Γ contains F2 . Show that if Γ is amenable, then T (Γ) = +∞. 4) Suppose that Γ is ﬁnitely generated with symmetric generating set S and is endowed with the corresponding word metric d (i.e. d(x, y) := inf{n ∈ N, x−1 y ∈ S n }). Given k ∈ N, let Gk be the bi-partite graph obtained by taking two copies Γ1 and Γ2 of Γ as the left and right vertices respectively and by placing an edge

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LECTURE 1. AMENABILITY AND RANDOM WALKS

337

between γ ∈ Γ1 and γ ∈ Γ2 if and only if d(γ, γ ) k in the word metric of Γ. Verify that there is some ﬁnite k ∈ N such that Gk admits a (2, 1) perfect matching3 if and only if there exists a surjective 2-to-1 mapping φ : Γ → Γ with the property that supγ∈Γ d(γ, φ(γ)) < +∞ (we call such φ a “Ponzi scheme”). 5) Show that the property of 4) takes place if and only if Γ is paradoxical. 6) Prove the following version of Hall’s marriage lemma for inﬁnite bi-partite graphs. Let k be a positive integer (we will need the result for k = 2 only). Suppose B is a bi-partite graph whose set of left vertices is countable inﬁnite as is the set of right vertices. Suppose that for every ﬁnite subset of left vertices L, the number of right vertices connected to some vertex in L has size at least k|L|, while for every ﬁnite subset R of right vertices, the number of left vertices connected to some vertex in R has size at least |R|. Show that B admits a (k, 1) perfect matching. [Hint: ﬁrst treat the case k = 1, then reduce to this case.] 7) Using 6) prove that if Γ is a non-amenable ﬁnitely generated group, then there is k 1 such that Gk has a (2, 1) perfect matching. 8) Conclude the proof of Tarski’s theorem for arbitrary (not necessarily ﬁnitely generated) groups. Remark. There are ﬁnitely generated groups with ﬁnite Tarski number > 4. For example the large Burnside groups with odd exponent. See [39] for some more examples.

3 By deﬁnition this is a subset of edges of G such that the induced bi-partite graph has the k property that every vertex on the left hand side is connected to exactly two vertices on the right hand side, while every vertex on the right hand side is connected to exactly one vertex on the left hand side.

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LECTURE 2

The Tits alternative and Kazhdan’s property (T ) A. The Tits alternative. Linear groups over a ﬁeld K, namely subgroups of GLd (K), form a very interesting large class of groups. While there are few general tools to study arbitrary ﬁnitely generated groups (often one has to resort to combinatorics and analysis as we did in Lecture 1 for example), the situation is very diﬀerent for linear groups as a wide range of techniques (including algebraic number theory and algebraic geometry) becomes available. Hence proving that a group of geometric origin is linear can have a big pay oﬀ (e.g. braid groups, or the more recent example of small simpliﬁcation groups, cf. Sageev’s lectures in this volume). Jacques Tits determined in 1972 which linear groups are amenable as a consequence of his famous alternative: Theorem 2.1 (Tits alternative [122]). Let Γ be a ﬁnitely generated linear group (over some ﬁeld K). Then • either Γ is virtually solvable (i.e. has a solvable ﬁnite index subgroup), • or Γ contains a non-abelian free subgroup F2 . Remark. Virtually solvable subgroups of GLd (K) have a subgroup of ﬁnite index which can be triangularized over the algebraic closure (Lie-Kolchin theorem). In particular, Corollary 2.2. A ﬁnitely generated linear group is amenable if and only if it is virtually solvable. Indeed free subgroups are non-amenable and subgroups of amenable groups are amenable. The proof of the Tits alternative uses a technique called “ping-pong” used to ﬁnd generators of a non-abelian free subgroup in a given group. The basic idea is to exhibit a certain geometric action of the group Γ on a space X and two elements a, b ∈ Γ, the “ping-pong players” whose action on X have the following particular behavior: Lemma 2.3 (Ping-pong lemma). Suppose a group Γ acts on a set X and there are two elements a, b ∈ Γ and 4 disjoint (non-empty) subsets A+ , A− , B + , and B − of X such that • a maps Y \ A− into A+ , • a−1 maps Y \ A+ into A− , • b maps Y \ B − into B + , and • b−1 maps Y \ B + , into B − . where Y := A+ ∪A− ∪B + ∪B − . Then a and b are free generators of a free subgroup a, b F2 in Γ. 339 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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Proof. The subset A+ is called the attracting set for a and A− the repelling set, and similarly for the other letters. Pick a reduced word w in a and b and their inverses. Say it starts with a. Pick a point p not in A+ and not in the repelling set of the last letter of w (note that there is still room to choose such a p) Then the above ping-pong rules show that w · p belongs to A+ hence is not equal to p. In particular w acts non trivially on X and hence is non trivial in Γ. Remark. There are other variants of the ping-pong lemma (e.g. it is enough that there are disjoint non-empty subsets A and B such that any (positive or negative) power of a sends B inside A and any power of b sends A inside B (e.g. take A := A+ ∪ A− and B := B + ∪ B − above). But the above is the most commonly used in practice. On Tits’s proof. Tits’s proof uses algebraic number theory and representation theory of linear algebraic groups to construct a local ﬁeld K, namely R, C or a ﬁnite extension of Qp or Fp ((t)), and an irreducible linear representation of Γ in GLm (K) whose image is unbounded. If Γ is not virtually solvable, one can take m 2. Then he shows that one can change the representation (passing to an exterior power) and exhibit an element γ of Γ which is semisimple (i.e. diagonalizable in a ﬁeld extension) and has the property that both γ and γ −1 have a unique eigenvalue (counting multiplicity) of maximal modulus (such elements are called proximal elements). Then one considers the action of Γ on the projective space of the representation X := P(K m ) and observes that the powers γ n , n ∈ Z, have the following contracting behavior on X. If we decompose K m = Kv + ⊕ Hγ into the direct sum of the eigenline Kv + of maximal modulus of γ and the complementary γinvariant subspace Hγ , we see that the positive powers γ n , n 1 push any compact subset of P(K m ) which is disjoint from P(Hγ ) inside a small neighborhood around the point P(Kv + ), if n is large enough. Using the irreducibility of the action, one then ﬁnds a conjugate cγc−1 of γ such that a := γ n and b = cγ n c−1 exhibit the desired “ping-pong” behavior for all large enough n and thus generate a free subgroup. For details, see the original article [122] or e.g. [65] and [16]. It turns out that one can give a shorter proof of the corollary, which by-passes the proof of the existence of a free subgroup. This was observed by Shalom [115] and the argument, which unlike the proof of the Tits alternative does not require the theory of algebraic groups, is as follows. Sketch of a direct proof of Corollary 2.2. If Γ is virtually solvable, then it is amenable (see Lecture 1). So we focus on the converse. Let us ﬁrst assume that Γ is an unbounded subgroup of GLn (k), for some local ﬁeld k, which acts strongly irreducibly on kn (i.e. it does not preserve any ﬁnite union of proper linear subspaces). If Γ is amenable, then it must preserve a probability measure on P(kn ) (by Proposition 1.5). However recall: Lemma 2.4 (Furstenberg’s Lemma). Suppose μ is a probability measure on the projective space P(kn ). Then the stabilizer of μ in PGLn (k) is compact unless μ is degenerate in the sense that it is supported on a ﬁnite number of proper (projective) linear subspaces. For the proof of this lemma, see Zimmer’s book [127], Furstenberg’s beautifully written original note [53], or just try to prove it yourself. Clearly the stabilizer of a degenerate measure preserves a ﬁnite union of proper subspaces. This contradicts our assumption of strong irreducibility.

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LECTURE 2. THE TITS ALTERNATIVE AND KAZHDAN’S PROPERTY (T )

341

To complete the proof, it remains to see that if Γ is not virtually solvable, then we can always reduce to the case above. This was proved by Tits at the start of his proof of Theorem 2.1. It follows from two claims. Claim 1. A non virtually solvable subgroup Γ of GLd (K) has a ﬁnite index subgroup which has a linear representation in a vector space of dimension at least 2 which is absolutely strongly irreducible (i.e. it preserves no ﬁnite union of proper vector subspaces deﬁned over any ﬁeld extension of K). Claim 2. If a ﬁnitely generated subgroup Γ of GLd (K) acts absolutely strongly irreducibly on K d , d 2, and K is a ﬁnitely generated ﬁeld, then K embeds in a local ﬁeld k in such a way that Γ is unbounded in GLd (k). We present a sketch of the proof of the two claims in the form of exercises. The interested reader can also consult the original paper by Tits. Exercise. Prove Claim 1. The proof of Claim 2 requires some basic algebra and number theory and proceeds as follows. Exercise. Prove that if a subgroup of GLd (K) acts irreducibly (K=algebraic closure) and all of its elements have only 1 in their spectrum (i.e. are unipotents), then d = 1 (hint: use Burnside’s theorem that the only K-subalgebra of Md (K) d acting irreducibly on K is all of Md (K).) d

Exercise.(Burnside) Suppose Γ is a subgroup of GLd (K) acting irreducibly on K such that every element of Γ has order dividing a ﬁxed N ∈ N. Then Γ is ﬁnite (hint: use again the same Burnside’s theorem to ﬁnd that every m ∈ Md (K) can d2 be expressed as m = 1 tr(mγi )ξi for some γi ∈ Γ, ξi ∈ Md (K), see e.g. [44]). Exercise. Show that a ﬁnitely generated ﬁeld K contains only ﬁnitely many roots of unity and that if x ∈ K is not a root of unity, then there is a local ﬁeld k with absolute value | · | such that K embeds in k and |x| = 1 (hint: this is based on Kronecker’s theorem that if a polynomial in Z[X] has all its roots within the unit disc, then all its roots are roots of unity; see [122, Lemma 4.1] for a full proof). Exercise. Use the last three exercises to prove Claim 2. B. Kazhdan’s property (T ) Let us go back to general (countable) groups and introduce another spectral property of groups, namely Kazhdan’s property (T ). Our goal here is to give a very brief introduction to the notion ﬁrst introduced by Kazhdan in [77]. Many excellent references exist on property (T ) starting with the 1989 Ast´erisque monograph by de la Harpe and Valette [66], the recent book by Bekka, de la Harpe and Valette [8] for the classical theory; see also Shalom 2006 ICM talk [117] for more recent developments. Let π be a unitary representation of Γ on a Hilbert space Hπ . We say that π admits (a sequence of) almost invariant vectors if there is a sequence of unit vectors vn ∈ Hπ (||vn || = 1) such that ||π(γ)vn − vn || converges to 0 as n tends to +∞ for every γ ∈ Γ.

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Definition 2.5 (Kazhdan’s property (T )). A group Γ is said to have Kazhdan’s property (T ) if every unitary representation π admitting a sequence of almost invariant vectors admits a non-zero Γ-invariant vector. Groups with property (T ) are sometimes also called Kazhdan groups. A few simple remarks are in order following this deﬁnition: • The deﬁnition resembles that of non-amenability, except that we are now considering all unitary representations of Γ and not just the left regular representation 2 (Γ) (given by λ(γ)f (x) := f (γ −1 x)). Indeed Proposition 1.10(3) above shows that a group is amenable if and only if the regular representation on 2 (Γ) admits a sequence of almost invariant vectors. • Property (T ) is inherited by quotient groups of Γ (obvious from the deﬁnition). • Finite groups have property (T ) (simply average an almost invariant unit vector over the group). • If Γ has property (T ) and is amenable, then Γ is ﬁnite (indeed 2 (Γ) has a non-zero invariant vector iﬀ the constant function 1 is in 2 (Γ) and this is iﬀ Γ is ﬁnite). A ﬁrst important consequence1 of property (T ) is the following: Proposition 2.6. Every countable group with property (T ) is ﬁnitely generated. Proof. Let Sn be an increasing family of ﬁnite subsets of Γ such that Γ = n Sn . Let Γn := Sn be the subgroup generated by Sn . We wish to show that Γn = Γ for all large enough n. Consider the left action of Γ on the coset space Γ/Γn and the unitary representation πn it induces on 2 functions on that coset space, 2 (Γ/Γn ). Let π = ⊕n πn be the Hilbert direct sum of the 2 (Γ/Γn )’s with the natural action of Γ on each factor. We claim that this unitary representation of Γ admits a sequence of almost invariant vectors. Indeed let vn be the Dirac mass at [Γn ] in the coset space Γ/Γn . We view vn as a (unit) vector in π. Clearly for every given γ ∈ Γ, if n is large enough γ belongs to Γn and hence preserves vn . Hence ||π(γ)vn − vn || is equal to 0 for all large enough n and the (vn )n form a family of almost invariant vectors. By Property (T ), there is a non-zero invariant vector ξ := n ξn . The Γ-invariance of ξ is equivalent to the Γ-invariance of all ξn ∈ 2 (Γ/Γn ) simultaneously. However observe that if ξn = 0, then Γ/Γn must be ﬁnite (otherwise a non-zero constant function cannot be in 2 ). Since there must be some n such that ξn = 0, we conclude that some Γn has ﬁnite index in Γ. But Γn itself is ﬁnitely generated. It follows that Γ is ﬁnitely generated. So let Γ have property (T ), and let S be a ﬁnite generating set for Γ. Then from the very deﬁnition we observe that there must be some ε = ε(S) > 0 such that for every unitary representation π of Γ without non-zero Γ-invariant vectors, one has: max ||π(s)v − v|| ε||v||, s∈S

for every vector v ∈ Hπ . 1 This

was partly the motivation for the introduction of property (T ) by Kazhdan in 1967 (at age 21). He used it to prove that non-uniform lattices in (higher rank) semisimple Lie groups are ﬁnitely generated. Nowadays new proofs exist of this fact, which are purely geometric and give good bounds on the size of the generating sets, see Gelander’s lecture notes in this volume.

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LECTURE 2. THE TITS ALTERNATIVE AND KAZHDAN’S PROPERTY (T )

343

And conversely it is clear that if there is a ﬁnite subset S in Γ with the above property, then every unitary representation of Γ with almost invariant vectors has an invariant vector. Hence this is equivalent to Property (T ). Definition 2.7 (Kazhdan constant). The (optimal) number ε(S) > 0 above is called a Kazhdan constant for the ﬁnite set S. Another important property of Kazhdan groups is that they have ﬁnite abelianization: Proposition 2.8. Suppose Γ is a countable group with property (T ). Then Γ/[Γ, Γ] is ﬁnite. Proof. Indeed, Γ/[Γ, Γ] is abelian hence amenable. It also has property (T ), being a quotient of a group with property (T ). Hence it is ﬁnite (see itemized remark above). This implies in particular that the non-abelian free groups do not have property (T ) although they are non-amenable. In fact Property (T ) is a rather strong spectral property a group might have. I tend to think of it as a rather rare and special property a group might have (although in some models of random groups, almost every group has property (T )). Exercise. Show that if Γ has a ﬁnite index subgroup with property (T ), then it has property (T ). And conversely, if Γ has property (T ), then every ﬁnite index subgroup also has property (T ) (hint: induce the representation from the ﬁnite index subgroup to Γ). In fact establishing Property (T ) for any particular group is never a simple task. In his seminal paper in which he introduced Property (T ) Kazhdan proved that Property (T ) for simple Lie groups of rank2 at least 2. Then he deduced (as in the above exercise) that Property (T ) is inherited by all discrete subgroups of ﬁnite co-volume in the Lie group G (i.e. lattices). Theorem 2.9 (Kazhdan 1967, [77]). A lattice in a simple real Lie group of real rank at least 2 has property (T ). There are several proofs of Kazhdan’s result for Lie groups (see e.g. Zimmer’s book [127] and Bekka-delaHarpe-Valette [8] for two slightly diﬀerent proofs). Those proofs rely on establishing a “relative property (T )” for the pair (SL2 (R) R2 , R2 ). This relative property (T ) means that every unitary representation of the larger group with almost invariant vectors admits a non-zero vector which is invariant under the smaller group. One proof of this relative property makes use of Furstenberg’s lemma above (Lemma 2.4). The proof extends to simple groups deﬁned over a local ﬁeld with rank at least 2 (over this local ﬁeld). A very diﬀerent proof of Kazhdan’s theorem was recently given by V. Laﬀorgue using Schur’s products, see [82]. The Lie group SL2 (R) admits a lattice isomorphic to a free group (e.g. the fundamental group of an non-compact hyperbolic surface of ﬁnite co-volume). Hence SL2 (R) does not have property (T ). A similar argument can be made for SL2 (C). 2 In fact he proved it for rank at least 3 by reducing the proof to SL (R) since every simple real Lie 3 group of rank at least 3 contains a copy of SL3 (R), but it was quickly realized by others (treating the case of Sp4 (R)) that the argument extends to groups of rank 2 as well.

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In 1969 Kostant [79] gave a precise description of the spherical irreducible unitary representations of an arbitrary simple real Lie group of rank one. From it he was able to prove that the rank one groups Sp(n, 1) and F4−20 have property (T ), while the other rank one groups SU (n, 1) and SO(n, 1) (including SL2 (R) and SL2 (C)) do not have property (T ). The discrete group SLn (Z) is a lattice in SLn (R) and hence has property (T ) by Kazhdan’s theorem. Nowadays (following Burger [35] and Shalom [116]) there are more direct proofs that SLn (Z) has property (T ) using bounded generation. Recently property (T ) was established for SLn (R), n 3, where R is an arbitrary ﬁnitely generated commutative ring with unit and for ELn (R), where R is an arbitrary ﬁnitely generated associative ring with unit. Theorem 2.10 (Ershov and Jaikin-Zapirain [51]). Let R be a (not necessarily commutative) ﬁnitely generated ring with unit and ELn (R) be the subgroup of n × n matrices generated by the elementary matrix subgroups Idn + REij . If n 3, then ELn (R) has property (T ). In particular, if Zx1 , . . . , xk denotes the free associative algebra on k generators, ELn (Zx1 , . . . , xk ) has property (T ) for all k 0 and n 3. As an other special case, the so-called universal lattices ELn (Z[x1 , ..., xk ]) = SLn (Z[x1 , ..., xk ]), where Z[x1 , ..., xk ] is the ring of polynomials on k (commutative) indeterminates has property (T ) when n 3. This remarkable result extends earlier works of Kassabov, Nikolov, Shalom and Burger on various special cases (see [74, 75, 116, 35] and references therein) and is based on a new method for proving property (T ) originating in the work of Dymara and Januszkiewicz [47] (see Kassabov’s beautiful paper [72] on this subject). The Kazhdan constant in the above theorem behaves 1 for large n and k. asymptotically as √n+k An important tool in some of these proofs (e.g. see Shalom ICM talk [117]) is the following characterization of property (T ) in terms of aﬃne actions of Hilbert spaces. Theorem 2.11 (Delorme-Guichardet). A group Γ has property (T ) if and only if every action of Γ by aﬃne isometries on a Hilbert space must have a global ﬁxed point. See [66] or [8] for a proof. Kazhdan groups enjoy many other ﬁxed point properties (e.g. Serre showed that they cannot act on trees without a global ﬁxed point) and related rigidity properties (see e.g. the lectures by Dave Morris in this summer school). Although the above class of examples of groups with property (T ) all come from the world of linear groups, Kazhdan groups also arise geometrically, for example as hyperbolic groups through Gromov’s random groups. The following holds: Theorem 2.12. In the density model of random groups, if the density is < 12 , then the random group is inﬁnite and hyperbolic with overwhelming probability. If the density is > 13 , then the random group has property (T ) with overwhelming probability. 1 It is unknown whether 13 is the right threshold for property (T ). Below 12 random groups have small cancellation C (1/6) and Ollivier and Wise proved that below 16 they act freely and co-compactly on a CAT (0) cube complex and are Haagerup, hence they do not have property (T ).

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LECTURE 2. THE TITS ALTERNATIVE AND KAZHDAN’S PROPERTY (T )

345

For a proof of the above see Zuk [128], Ollivier [100], Gromov [63], Ghys’ Bourbaki talk [54] and the recent [80]. In fact Zuk proved a similar result for a slightly diﬀerent model of random groups (the so-called triangular model) and Ollivier [100] sketches a reduction of the above to Zuk’s theorem from [128]. Full details have now been provided by Kotowski and Kotowski [80]. The proof of this result is based on the following celebrated geometric criterion for property (T ). Let Γ be a group generated by a ﬁnite symmetric set S (with e ∈ / S). Let L(S) be the ﬁnite graph whose vertices are the elements of S and an edge is drawn between two vertices s1 and s2 iﬀ s−1 1 s2 belongs to S. Suppose that L(S) is connected (this is automatic if S is replaced say by S ∪ S 2 \ {e}). Theorem 2.13 (local criterion for property (T )). Let Γ be a group generated by a ﬁnite symmetric set S (with e ∈ / S) such that the ﬁrst non-zero eigenvalue of the Laplacian on the ﬁnite graph L(S) is > 12 . Then Γ has property (T ). For a short proof, see Gromov’s random walks in random groups paper [63] and the end of Ghys’ Bourbaki talk [54]. The criterion is due to Zuk, BallmannZwiatkowski, and originated in the work of Garland, see the above references for more historical comments. For certain groups of geometric origin, such as Out(Fn ) and the mapping class groups, determining whether they have property (T ) or not is very hard. For example it is not known whether Out(Fn ) has property (T ) for n 4 (even open for Aut(Fn ), not true for n = 2, 3 though). For the mapping class group also it is problematic; see the work of Andersen. C. Uniformity issues in the Tits alternative, non-amenability and Kazhdan’s property (T ) A well-known question of Gromov from [62] is whether the various invariants associated with an inﬁnite group (such as the rate of exponential growth, the isoperimetric constant of a non-amenable group, the Kazhdan constant of a Kazhdan group, etc) can be made uniform over the generating set. For example we say: Definition 2.14 (uniformity). Consider the family of all ﬁnite symmetric generating sets S of a given ﬁnitely generated group. Γ. We say that Γ • has uniform exponential growth if ∃ε > 0 such that lim n1 log |S n | ε, for all S, • is uniformly non-amenable if ∃ε > 0 such that |∂S A| ε|A| for S, • has uniform property (T ) if ∃ε > 0 such that maxS ||π(s)v − v|| ε||v|| for all S and all unitary representations of Γ with no non-zero invariant vector. • satisﬁes the uniform Tits alternative if ∃N ∈ N > 0 such that S N contains generators of a non-abelian free subgroup F2 . Note that there are some logical implications between these properties. For example if Γ satisﬁes the uniform Tits alternative, or if Γ (is inﬁnite and) has uniform property (T ), then Γ is uniformly non-amenable (exercise). Similarly if Γ is uniformly non-amenable, then Γ has uniform exponential growth. Uniform exponential growth holds for linear groups of exponential growth (Eskin-Mozes-Oh [49], see also [17] in positive characteristic), for solvable groups of

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EMMANUEL BREUILLARD, LECTURES ON EXPANDER GRAPHS

exponential growth (Osin), but fails for general groups: John Wilson [125] gave an example of a non-amenable group (even containing F2 ) whose exponential growth is not uniform. In fact Bartholdi and Erschler recently proved in [7] that every countable group embeds in a ﬁnitely generated group of non-uniform exponential growth. They also show that if G is any ﬁnitely generated group with exponential growth, then the permutational wreath product G012 X G, where G012 is the ﬁrst Grigorchuk group (of sub-exponential growth) permuting the orbit X of the right-most branch of the binary rooted tree, has non-uniform exponential growth. The uniform Tits alternative is known to hold for non-elementary Gromov hyperbolic groups (Koubi [81]) and for non-virtually solvable linear groups by work of Breuillard-Gelander [17]. In this case the uniformity is even stronger as one has: Theorem 2.15 (Uniform Tits alternative [24]). Given d ∈ N, there is N = N (d) ∈ N such that for any ﬁeld K and any ﬁnite symmetric set S ⊂ GLd (K) one has S N contains two generators of a non-abelian free subgroup F2 unless S is virtually solvable. The uniformity in the ﬁeld in the above theorem requires some non-trivial number theory (see [23]). This result implies that the rate of exponential growth is bounded below by a positive constant depending only on d (= the number of rows of the matrix) and not on the ﬁeld. So the uniform exponential growth is also uniform in the ﬁeld. However this is known to hold only for non-virtually solvable groups. The solvable case remains an open problem, already for K = C: is the rate of exponential growth uniform over all virtually solvable subgroups of GLd (C) ? In fact even the case of solvable subgroups of GL2 (C) is open. One can show however that if this is indeed the case, then this would imply the Lehmer conjecture from number theory [25]. Besides, the analogous uniform Tits alternative for free semi-groups does not hold ([25, Thm. 1.7]). Although it is a result about inﬁnite linear groups, the above uniform Tits alternative has applications to ﬁnite groups as well. It turns out that the uniformity in the ﬁeld allows one to transfer information from the inﬁnite world to the ﬁnite world (we will see more of that in the remainder of this course). For example the following can be derived from Theorem 2.15 Corollary 2.16. There is N = N (d) ∈ N and ε = ε(d) > 0 such that if S is a generating subset of SLd (Fp ) (p arbitrary prime number), then S N contains two elements a, b which generate SLd (Fp ) and have no relation of length (log p)ε . In other words the Cayley graph G(SLd (Fp ), {a±1 , b±1 }) has girth at least (log p)ε . It is an open question (connected to whether all Cayley graphs of SLd (Fp ) are uniformly expanders) whether one can replace (log p)ε with C log p, for some C > 0, in the above result. Uniform property (T ) is even more mysterious. Examples were constructed by Osin and Sonkin [101] (every inﬁnite hyperbolic group with property (T ) has a quotient with uniform property (T )). Osin showed on the other hand that hyperbolic groups do not have uniform property (T ) and Gelander and Zuk showed that any countable group which maps densely in a connected Lie group, and this includes all co-compact lattices in semi-simple real Lie groups, does not have uniform property (T )). But it is an open problem to determine whether SLn (Z) has uniform property (T ) for n 3. See the article by Lubotzky and Weiss [90] for further discussion.

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LECTURE 3

Property (τ ) and expanders There are many excellent existing texts for the material in this lecture, starting with Lubotzky’s monograph [86] and recent AMS survey paper [87]. For expander graphs and their use in theoretical computer science, we refer the reader to the survey by Hoory, Linial and Wigderson [69]. Computer scientists also have numerous lecture notes on expander graphs available on the web (e.g. Linial and Wigderson). We give here only a brief introduction. A. Expander graphs We start with a deﬁnition. Definition 3.1 (Expander graph). Let ε > 0. A ﬁnite connected k-regular graph G is said to be an ε-expander if for every subset A of vertices in G, with |A| 12 |G|, one has the following isoperimetric inequality: |∂A| ε|A|, where ∂A denotes the set of edges of G which connect a point in A to a point in its complement Ac . The optimal ε as above is sometimes called the discrete Cheeger constant of the graph: (2)

h(G) =

inf

A⊂G,|A| 12 |G|

|∂A| , |A|

Just as in Lecture 1, when we discussed the various equivalent deﬁnitions of amenability, it is not a surprise that this deﬁnition turns out to have a spectral interpretation. Given a k-regular graph G, one can consider the Markov operator (also called averaging operator, or sometimes Hecke operator in reference to the Hecke graph of an integer lattice) on functions on vertices on G deﬁned as follows: 1 f (y), (3) P f (x) = k x∼y where we wrote x ∼ y to say that y is a neighbor of x in the graph. This operator is easily seen to be self-adjoint on the ﬁnite dimensional Euclidean space 2 (G). Moreover it is a contraction, namely ||P f ||2 ||f ||2 and hence its spectrum is real and contained in [−1, 1]. We can write the eigenvalues of P in decreasing order as μ0 = 1 μ1 . . . μ|G| . The top eigenvalue μ0 must be 1, because the constant function 1 is clearly an eigenfunction of P , with eigenvalue 1. On the other hand, since G is connected 1 is the only eigenfunction (up to scalars) with eigenvalue 1. This is immediate by the maximum principle (if P f = f and f 347 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

348

EMMANUEL BREUILLARD, LECTURES ON EXPANDER GRAPHS

achieves its maximum at x, then f must take the same value f (x) at each neighbor of x, and this value spreads to the entire graph). Hence the second eigenvalue μ1 is strictly less than 1. Instead of P , we may equally well consider Δ := Id − P , which is then a nonnegative self-adjoint operator. This operator is called the combinatorial Laplacian in analogy with the Laplace-Beltrami operator on Riemannian manifolds. 1 f (y). Δf (x) := f (x) − k x∼y Its eigenvalues are traditionally denoted by λ0 = 0 < λ1 . . . λ|G| and: λi (G) = 1 − μi (G). As promised, here is the connection between the spectral gap an the edge expansion. Proposition 3.2 (Discrete Cheeger-Buser inequality). Given a connected kregular graph, we have: 1 1 λ1 (G) h(G) 2λ1 (G) 2 k The proof of this proposition is basically an exercise and follows a similar line of argument as the proof we gave in Lecture 1 of the Kesten criterion relating the Folner condition and the spectral radius of the averaging operator (Proposition 1.10). See Lubotzky’s book [86] for a detailed derivation and the references therein for the original papers (e.g. [46]). We note in passing that, since P is self-adjoint, the following holds: ||P ||20 = max |μi | i =0

where

20

is the space of functions on G with zero average, and P f, f μ1 = sup{ ; f (x) = 0} ||f ||22 x∈G

and hence (4)

λ1 = inf{

||∇f ||22 Δf, f 1 ; f (x) = 0} = inf{ ; f (x) = 0}. 2 ||f ||2 k ||f ||22 x∈G

x∈G

Expander graphs are very important to theoretical computer science (e.g. in the construction of good error correcting codes, see [69]). Typically one wants to have a graph of (small) bounded degree (i.e. k is bounded) but whose number of vertices is very large. For this it is convenient to use the following deﬁnition: Definition 3.3 (family of expanders). Let k 3. A family (Gn )n of k-regular graphs is said to be a family of expanders if the number of vertices |Gn | tends to +∞ and if there is ε > 0 independent of n such that for all n λ1 (Gn ) ε. Although almost every random k-regular graph is an expander (Pinsker 1972), the ﬁrst explicit construction of an inﬁnite family of expander graphs was given using Kazhdan’s property (T ) and is due to Margulis [95] (see below Proposition 3.9).

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LECTURE 3. PROPERTY (τ ) AND EXPANDERS

349

Clearly an ε-expander graph of size N has diameter at most O( 1ε log |G|). But more is true. A very important feature of expander graphs is the fact that the (lazy) simple random walk on such a graph equidistributes as fast as could possibly be towards the uniform probability distribution. This is made precise in the following proposition: Proposition 3.4 (Random walk characterization of expanders). Suppose G is a k-regular graph and let Q = αId+(1−α)P be the lazy averaging operator on 2 (G) (here α ∈ (0, 1) and P is the original averaging operator deﬁned in (3)). Assume that λ1 (G) ε, then there is C = C(ε, k, α) > 0 such that if n C log |G| then 1 1 max |Qn δx , δy − | 10 . x,y∈G |G| |G| Conversely for every C > 0 there is ε = ε(C, k, α) > 0 such that if the k-regular graph G satisﬁes 1 1 | max |Qn δx , δx − 10 , x∈G |G| |G| for some n C log |G|, then G is an ε-expander. Here Qn δx , δy has a probabilistic interpretation: it is equal to the transition probability from x to y at time n, namely the probability that the α-lazy simple random walk (i.e. the walk that either stays put with probability α or jumps to a nearest neighbor with equal probability (1 − α)/k) starting at x visits y at time n. Note that if G is a Cayley graph, then Qn δx , δy depends only on yx−1 , and in particular Qn δx , δx = Qn δe , δe . Remark 3.5. It is important in this proposition to consider the lazy walk, that is the operator Q with α > 0, rather than just the simple walk with operator P , because of possible issues with negative eigenvalues of P close to −1 (see exercise below). The spectrum of Q is given by α + (1 − α)Spec(P ). In particular the eigenvalues of Q are all −1 + 2α, so (5)

1 − (1 + α)λ1 (G) ||Q||20 (G) max{1 − 2α, 1 − (1 + α)λ1 (G)}.

1 Proof of Proposition 3.4. The function fx := δx − |G| 1 has zero mean on G, hence √ 1 |Qn δx , δy − | = |Qn fx , δy | ||Q||n ||fx ||||δy || 2||Q||n . |G|

Now this is at most 1/|G|10 as some as n Cε log |G| for some Cε > 0. Conversely observe that trace(Qn ) = x∈G Qn δx , δx , and hence summing the estimates for Qn δx , δx , we obtain 1 , |trace(Qn ) − 1| |G|9 But on the other hand trace(Qn ) = 1 + μn1 + . . . + μn|G| , where the μi ’s are the eigenvalues of Q, hence 1 max |μi |n μn1 + . . . + μn|G| , i =0 |G|9 thus recalling that |G|1/ log |G| = e, we obtain the desired upper bound on ||Q||20 (G) = maxi =0 |μi |, hence the lower bound on λ1 (G) via (5).

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350

EMMANUEL BREUILLARD, LECTURES ON EXPANDER GRAPHS

This fast equidistribution property is usually considered as a feature of expander graphs, a consequence of the spectral gap. We will see in the last lecture, when explaining the Bourgain-Gamburd method, that the proposition can also be used in the reverse direction, that is to establish the spectral gap. Exercise. Show that the eigenvalue −1 appears in the spectrum of the averaging operator P (deﬁned in (3)) if and only if the graph G is bi-partite, that is |G| = 2|A| for some subset of vertices A ⊂ G and every edge has one end point in A and the other in the complement G \ A. If G is a Cayley graph with generating set S, then G is bi-partite iﬀ G has an index two subgroup H such that S ∩ H = ∅. In fact for non bi-partite Cayley graphs, there is no need to consider the lazy walk: non bi-partite expander Cayley graphs are characterized by the fast equidistribution of their simple random walk (not just the lazy one). Indeed: Exercise 3.6 (There are no almost bi-partite expander Cayley graphs, [21]). If G is a Cayley graph with generating set S of size k, which is not bi-partite, then the smallest eigenvalue μ|G| of the averaging operator P satisﬁes μ|G| −1 + ck λ1 (G)2 , for some constant c = ck depending on k only. So, unless G has an index 2 subgroup, a lower bound on λ1 implies an upper bound on the norm of P . In particular Proposition 3.4 holds with Q = P (i.e. α = 0) when G is a Cayley graph which is not bi-partite. Here is a hint for the exercise: use Proposition 3.2 to show the existence of a subset A of size roughly |G|/2 such that |ss AΔA| = o(|G|), ∀s, s ∈ S, then use expansion to show that each right translate Ag is very close to either A or sA). See the last appendix in [21] for a proof. Next we describe another spectral estimate, which is special to Cayley graphs. The Cheeger constant (see (2)) of a k-regular graph G is obviously at least 2k/|G|. By the Cheeger inequality (right hand side of Proposition 3.2) this gives a lower bound for the ﬁrst eigenvalue of the Laplacian in 1/|G|2 up to constants. It turns out that when the graph G is a Cayley graph, one can improve this bound and replace the size of the graph |G| by the diameter D(G). Indeed we have (see e.g. Diaconis and Saloﬀ-Coste [45, Cor. 1]): Proposition 3.7 (spectral gap from diameter). Suppose G is a Cayley graph of a ﬁnite group G associated to a ﬁnite symmetric generating set of size k. Then 2 , λ1 (G) k · D(G)2 where D(G) is the diameter of the graph G. Proof. Write D = D(G). Any y ∈ G can be written y = s1 · . . . · sD , there each si is either 1 or one of the k generators. For any function f on G apply Cauchy-Schwarz and get |f (x) − f (xy)|2 D

D

|f (xwi ) − f (xwi−1 )|2

i=1

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LECTURE 3. PROPERTY (τ ) AND EXPANDERS

351

where w0 = 1 and wi = s1 · . . . · si . Summing over x we get

|f (x) − f (xy)|2 D

x∈G

Now assuming that

x

D

|f (xsi ) − f (x)|2 D2 ||∇f ||22 .

i=1 x∈G

f (x) = 0 and summing over y, we obtain 2|G| · ||f ||22 D2 |G| · ||∇f ||22

The desired inequality follows from the variational characterization of λ1 (G) (i.e. (4)). The above lower bound on λ1 is not enough to prove that G is an expander, but it is already useful in some applications (e.g. in [48]). For Riemannian manifolds there is a rich theory relating the λ1 to the volume or the diameter of the manifold in presence of curvature bounds. See for instance Cheng’s paper [42] for an analogue of the above proposition and Chavel’s books [40, 41] for a comprehensive introduction. Many of these results have graph theoretic analogues. For more about random walks on ﬁnite graphs and groups and the speed of equidistribution, the cut-oﬀ phenomenon, etc, see the survey by Saloﬀ-Coste [110]. Exercise. Show that the above proposition fails for general ﬁnite k-regular graphs (hint: connect two large bulbs by an edge). For Cayley graphs the following simple reformulation of the expander property is very useful (see below Proposition 3.11 and Theorem 4.2). Given a Cayley graph G(G, S) of a ﬁnite group G, let α(G) be the inﬁmum of all values α > 0 such that the following holds. For every unitary representation (ρ, V ) of G and every vector v ∈ V one has max ||ρ(g)v − v|| α max ||ρ(s)v − v||. g∈G

s∈S

Proposition 3.8 (Representation theoretic reformulation). If G = G(G, S) is a k-regular Cayley graph of a ﬁnite group G, then we have λ1 (G) α(G)−1 2kλ1 (G) 2 Exercise. Prove the above proposition (hint: every linear representation of G decomposes into irreducible components, each of which appears in 2 (G)). In particular a family of k-regular Cayley graphs Gn is a family of expanders, if and only if the values α(Gn ) are uniformly bounded. B. Property (τ ) Margulis [95] was the ﬁrst to construct an explicit family of k-regular expander graphs. For this he used property (T ) through the following observation: Proposition 3.9 ((T ) implies (τ )). Suppose Γ is a group with Kazhdan’s property (T ) and S is a symmetric set of generators of Γ of size k = |S|. Let Γn Γ be a family of ﬁnite index subgroups such that the index [Γ : Γn ] tends to +∞ with n. Then the family of k-regular Schreier graphs G(Γ/Γn , S) forms a family of expanders.

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352

EMMANUEL BREUILLARD, LECTURES ON EXPANDER GRAPHS

Recall that the Schreier graph G(Γ/Γ0 , S) of a coset space Γ/Γ0 associated to a ﬁnite symmetric generating set S of Γ is the graph whose vertices are the left cosets of Γ0 in Γ and one connects gΓ0 to hΓ0 if there is s ∈ S such that gΓ0 = shΓ0 . Proof. The group Γ acts on the ﬁnite dimensional Euclidean space 20 (Γ/Γn ) of functions with zero average on the ﬁnite set Γ/Γn . Denote the resulting unitary representation of Γ by πn . Property (T ) for Γ gives us the existence of a Kazhdan constant ε = ε(S) > 0 such that maxs∈S ||π(s)v − v|| ε||v|| for every unitary representation π of Γ without invariant vectors. In particular, this applies to the πn since they have no non-zero Γ-invariant vector. This implies that the graphs Gn := S(Γ/Γn , S) are ε-expanders, because if A ⊂ Gn has size at most half 2 2 of the graph, then v := 1A − |A| |G| 1 is a vector in 0 (Γ/Γn ) and ||πn (s)v − v|| = 2

||πn (s)1A − 1A ||2 = |sAΔA|, while ||v||2 = 2|A|(1 − |∂A| ε2 |A|.

|A| |Gn | )

|A|. In particular

So we see that Cayley graphs (or more generally Schreier graphs) of ﬁnite quotients of ﬁnitely generated groups can yield families of expanders. This is the case for the family of Cayley graphs of SL3 (Z/mZ) associated to the reduction mod m of a ﬁxed generating set S in SL3 (Z). To characterize this property, Lubotzky introduced the following terminology: Definition 3.10 (Property (τ )). A ﬁnitely generated group Γ with ﬁnite symmetric generating set S is said to have property (τ ) with respect to a family of ﬁnite index normal subgroups (Γn )n if the family of Cayley graphs G(Γ/Γn , Sn ), where Sn = SΓn /Γn is the projection of S to Γ/Γn , is a family of expanders. If the family (Γn )n runs over all ﬁnite index normal subgroups of Γ, then we say that Γ has property (τ ). Proposition 3.9 above shows that every group with property (T ) has property (τ ). The converse is not true and property (τ ) is in general a weaker property which holds more often. For example Lubotzky and Zimmer [91] showed that an irreducible lattice Γ in a semisimple real Lie group G without compact factors has property (τ ) as soon as one of the simple factors of the ambient semisimple Lie group has property (T ). Note however that for Γ to have property (T ) it is necessary that all factors of G have property (T ). Exercise. Show that if Γ is amenable and has property (τ ), then Γ has only ﬁnitely many ﬁnite index subgroups. Exercise. Recall that the regular representation of a ﬁnite group G contains an isomorphic copy of each irreducible representation of G (see e.g. [114]). Let H G a subgroup and S a symmetric generating set for G. Show that λ1 (G(G/H, S)) λ1 (G(G, S)), where G(G/H, S) is the associated Schreier graph. Deduce that if Γ has property (τ ), then the family of all Schreier graphs G(Γ/Γn , S), where Γn ranges over all ﬁnite index (not necessarily normal) subgroups of Γ, is a family of expanders. Property (τ ) is stable under quotients (obviously). In particular groups with property (τ ) have ﬁnite abelianization, just as Kazhdan’s groups. As property (T ), property (τ ) is also stable and under passing to and from a ﬁnite index subgroup:

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LECTURE 3. PROPERTY (τ ) AND EXPANDERS

353

Proposition 3.11. Suppose Γ Γ is a subgroup of ﬁnite index in Γ. Then Γ has property (τ ) if and only if Γ has property (τ ).

sketch. Let S a ﬁnite symmetric generating set for Γ and S ⊂ S 2[Γ:Γ ] be a ﬁnite generating set for Γ , which is obtained as usual from the ReidemeisterSchreier rewriting process, so that if {γi }i is a set of representatives of the cosets of Γ in Γ contained in S [Γ:Γ ] , then for every i and s ∈ S there is j and s ∈ S such that sγi = γj s (see [93, sec 2.3]). Suppose Γ has (τ ). Then thanks to Proposition 3.8, there is an upper bound on α(Gn ) for all Cayley graphs associated to ﬁnite quotients of Γ . It is easy to check that this upper bound lifts to an upper bound on α(Gn ) for the Cayley graphs of all ﬁnite quotients of Γ, therefore Γ has (τ ). In the converse direction, suppose Γ has (τ ) and let (ρ, V ) be a linear representation of a ﬁnite quotient of Γ . Then the induced representation (ρ , W ) to Γ is a linear representation of a ﬁnite quotient of Γ. Its ambient space W is the set of functions on Γ to V such that f (xγ ) = ρ(γ )f (x) for all x ∈ Γ, γ ∈ Γ, and Γ acts by left translations. Now suppose that v ∈ V is almost ﬁxed by the elements s ∈ S , then the function f ∈ W deﬁned by f (γi ) = v for each i is almost ﬁxed by every element s ∈ S, hence almost ﬁxed by all of Γ, by property (τ ) and Proposition 3.8. It follows that v is almost ﬁxed by all γ ∈ Γ and we are done. For more details see [90, Prop. 3.9.]. Arithmetic lattices in semisimple algebraic groups deﬁned over Q admit property (τ ) with respect to the family of all congruence subgroups. Namely: Theorem 3.12 (Selberg, Burger-Sarnak, Clozel). Let G ⊂ GLd is a semisimple algebraic Q-group, Γ = G(Z) := G(Q) ∩ GLd (Z) and Γm = Γ ∩ ker(GLd (Z) → GLd (Z/mZ)), then Γ has property (τ ) with respect to the Γm ’s. This property is also called the Selberg property because in the case of G = SL2 it is a consequence (as we will see below) of a celebrated theorem of Selberg 3 theorem, which asserts that the non-zero eigenvalues of the Laplace[113], the 16 Beltrami laplacian on the hyperbolic surfaces of ﬁnite co-volume H2 / ker(SL2 (Z) → SL2 (Z/mZ)) are bounded below by a positive constant independent of m (in fact 3 16 ). The general case was established by Burger-Sarnak [36] and Clozel [43]. This connects property (τ ) for lattices with another interesting feature of some lattices, namely the congruence subgroup property. This property of an arithmetic lattice asks that every ﬁnite index subgroup contains a congruence subgroup (i.e. a subgroup of the form G(Z) ∩ ker(GLd (Z) → GLd (Z/mZ)). Exercise. Show that if G(Z) has both the Selberg property and the congruence subgroup property, then it has property (τ ) with respect to all of its ﬁnite index subgroups (hint: see the second exercise after Deﬁnition 3.10). An interesting open problem in this direction is to determine whether or not lattices in SO(n, 1) can have property (τ ) or not. Lubotzky and Sarnak conjecture that they do not, and this would also follow from Thurston’s conjecture that such lattices have a subgroup of ﬁnite index with inﬁnite abelianization (now proved in dimension 3 !). 3 theorem and property (τ ) is provided by the The link between Selberg’s 16 following general fact, which relates the combinatorial spectral gap of a Cayley (or

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354

EMMANUEL BREUILLARD, LECTURES ON EXPANDER GRAPHS

Schreier) graph of ﬁnite quotients of the fundamental group of a manifold with the spectral gap for the analytic Laplace-Beltrami operator on the Riemannian manifold. Recall that given a connected Riemannian manifold M the Laplace-Beltrami operator is a non-negative self-adjoint operator on L2 functions (L2 with respect to the Riemannian volume measure). If M is compact, the spectrum of this operator is discrete λ0 (M ) = 0 < λ1 (M ) . . . (e.g. see [9]). The fundamental group Γ = π1 (M ) acts freely and co-compactly on the uni by isometries (for the lifted Riemannian metric on M

). Given a versal cover M

base point x0 ∈ M , the set

; d(x, x0 ) < d(x, γ · x0 ) ∀γ ∈ Γ \ {1}} (6) FM = {x ∈ M

. Moreover the group is a (Dirichlet) fundamental domain for the action of Γ on M Γ is generated by the ﬁnite symmetric set S := {γ ∈ Γ; γFM ∩ FM = ∅}. We can now state: Theorem 3.13 (Brooks [28], Burger [33]). Let M be a compact Riemannian manifold with fundamental group Γ = π1 (M ). Let S be the ﬁnite symmetric generating set of Γ obtained from a Dirichlet fundamental domain FM as above. Then there are constants c1 , c2 > 0 depending on M only such that for every ﬁnite cover M0 of M c1 λ1 (M0 ) λ1 (G(Γ/Γ0 , S)) c2 λ1 (M0 ), where Γ0 is the fundamental group of M0 and G(Γ/Γ0 , S)) the Schreier graph of the ﬁnite coset space Γ/Γ0 associated to the generating set S. We deduce immediately: Corollary 3.14. Suppose (Mn )n is a sequence of ﬁnite covers of M . Then there is a uniform lower bound on λ1 (Mn ) if and only if Γ := π1 (M ) has property (τ ) with respect to the sequence of ﬁnite index subgroups Γn := π1 (Mn ). The proof consists in observing that the Schreier graph can be drawn on the manifold M0 as a dual graph to the decomposition of M0 into translates of the fundamental domain FM . The geometry of this Schreier graph closely resembles that of the cover M up to a bounded disturbance depending on M0 only. For the proof of this Brooks-Burger transfer principle, we refer the reader to the Appendix, where we give a complete treatment and further discussion. The result also extends to non-compact hyperbolic manifolds of ﬁnite co-covolume (see [10, Section 2] and [48, Appendix]). In a similar spirit, with similar proof, Brooks showed: Proposition 3.15 (Brooks [27]). If M is a compact Riemannian manifold and M0 a normal cover of M with Galois group Γ. Then λ0 (M0 ) = 0 if and only if Γ is amenable. For more on property (τ ) we refer the reader to the forthcoming book by Lubotzky and Zuk [92].

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LECTURE 4

Approximate groups and the Bourgain-Gamburd method A. Which ﬁnite groups can be turned into expanders? In [90], Lubotzky and Weiss asked the question of whether the property of being an expander is a group property. Namely given a sequence of ﬁnite groups (Gn )n generated by a ﬁxed number of elements k 1, is it true that if one can ﬁnd a sequence of k-regular Cayley graphs of the Gn ’s which is an expander family, then the family of all Cayley graphs of the Gn ’s on k generators is an expander family? In other words is being an expander independent of the choice of the generating set? It turned out that the answer to this question is no in general. An example was produced in [4] using the so-called zig-zag product construction. Somewhat later Martin Kassabov, in a remarkable breakthrough [73], managed to turn the family of symmetric groups Sn into a family of expanders with generating sets of bounded size. On the other hand Sn is not a family of expanders when generated by the transposition (12) and the long cycle (12 . . . n) because the diameter of the associated Cayley graph is at least n2 log |Sn | (see [86, Prop. 8.1.6.]), while as we already observed the diameter of an expander graph G is always at most O(log |G|). See also [74] for other examples involving SLn (Fp ) for ﬁxed p. However there are classes of groups for which an answer is known or at least is expected. To begin with, the following observation of Lubotzky and Weiss [90] shows that solvable groups of ﬁxed derived length cannot be turned into expanders: Proposition 4.1 (Finite solvable groups are never expanders). Fix , k ∈ N. Suppose Gn is a family of k-regular Cayley graphs of ﬁnite solvable groups Gn with derived length . Then {Gn } is not a family of expanders. Proof. The free solvable group Γ := Fk /D (Fk ) on k generators is a ﬁnitely generated solvable group. Hence amenable. By Kesten’s criterion (Theorem 1.9) the probability of return μ2m (e) to the identity of the simple (or lazy simple) random walk on Γ decays subexponentially in m. Each Gn is a homomorphic image of Γ in such a way that the Cayley graph Gn is a quotient of that of Γ . Hence the probability of return of the simple (or lazy simple) random walk on Gn is always at least μ2m (e). However if the Gn form an expander family, then Proposition 3.4 ensures that there is c, ρ < 1 such that μ2m (e) < ρm if m c log |Gn |. A contradiction if |Gn | → +∞. In fact further arguments (see [90, Theorem 3.6]) show that one needs at least log() (|G|) generators to turn a ﬁnite -solvable group G into an expander graph. For = 1, i.e. for abelian groups, we thus need log |G| generators to make a Cayley graph with ﬁrst eigenvalue of the Laplacian bounded away from 0. It is interesting 355 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

356

EMMANUEL BREUILLARD, LECTURES ON EXPANDER GRAPHS

to observe that Alon and Roichman [5] showed that for an arbitrary ﬁnite group G, O(log |G|) generators are always suﬃcient to produce to expanding Cayley graph, in fact a random set of O(log |G|) elements in G generates an expanding Cayley graph with high probability as |G| → +∞. See [83] for a simple proof of this fact using the Ahlswede-Winter large deviation bounds for sums of independent non-negative self-adjoint operators on a Hilbert space. If solvable groups cannot be made into expanders, what about simple groups ? As we saw above, the answer depends on the generating set. But remarkably the following holds: Theorem 4.2 (Simple groups as expanders). There is k 2 and ε > 0 such that every ﬁnite simple group G has a k-regular Cayley graph which is an ε-expander. The proof of this result is due to Kassabov, Lubotzky and Nikolov [76] who proved it except for the family of Suzuki groups {Suz(22n+1 )}n , which was later settled in [20]. Of course the proof is based on the classiﬁcation of ﬁnite simple groups. The case of the sub-family PSL2 (Fq ) was proved by Lubotzky [88], building on the PSL2 (Fp ) case (i.e. Theorem 3.12, which boils down in this case to Selberg’s 3 16 theorem via the Brooks-Burger transfer principle), and using further deep facts from the theory of automorphic forms, but as we will see shortly (in Theorem 4.11 below) there are now new methods to settle this case and indeed all ﬁnite simple groups of Lie type and bounded rank together. The case of PSLn (Fq ) with n going to inﬁnity and q arbitrary is due to Kassabov [74], and can now be seen as a consequence of Theorem 2.10 proving property (T ) for the non-commutative universal lattices EL3 (Zx1 , . . . , xk ). Recall that for a ring R, ELd (R) denotes the subgroup of GLd (R) generated by the elementary matrices Id + REij . Kassabov’s beautiful idea is to take advantage of the following straightforward observation: EL3 (M atn×n (Fq )) EL3n (Fq ) and argue that M atn×n (Fq ) can be generated as an associative ring by 2 elements, and hence is a quotient of the free associative algebra Zx1 , x2 . The universal lattice EL3 (Zx1 , x2 ) is ﬁnitely generated by the elementary matrices Id ± Eij and Id ± xm Eij . That makes 36 generators. Now Theorem 2.10 says that this group has property (T ) and we thus conclude (as in Proposition 3.9) that the quotients EL3n (Fq ) SL3n (Fq ) are uniformly expanders, for all n 1 and all prime powers q, with respect to the corresponding projected generating set (still with 36 generators). In order to go from PSLn (Fq ) with n divisible by 3 to PSLn (Fq ) for all n and ﬁnally to G(Fq ) for every group of Lie type G, one uses bounded generation. Nikolov [97] shows that every G(Fq ) can be written as a product of a bounded number of groups isomorphic to PSL3n (Fq ) (up to the center). Now Proposition 3.8, which is a representation theoretic reformulation of the expander property easily implies the following: if a group G can be written as G = H1 · . . . · Hn , with n bounded (we say that G is boundedly generated by the Hi ’s), and each Hi has a generating set Σi of bounded size with respect to which it is an ε-expanding Cayley graph, then G too has a generating set of bounded size (the union of the Σi ’s) with respect to which it is ε -expanding for some ε depending only on ε and n. This settles the remaining cases for Theorem 4.2 for simple groups of Lie type.

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LECTURE 4. APPROXIMATE GROUPS AND THE BOURGAIN-GAMBURD METHOD 357

Finally the case of alternating groups An was settled by Kassabov in a tour-deforce paper [73] which blends some of the above ideas, in particular by embedding large powers of SLk (F2 ) inside An and using the above expansion result for SLk (F2 ), together with some ideas of Roichman [107, 108] involving character bounds for certain representations of An . See [76] for a sketch. B. The Bourgain-Gamburd method Up until the Bourgain-Gamburd breakthrough [11] in 2005, the only known ways to turn SLd (Fp ) into an expander graph (i.e. to ﬁnd a generating set of small size whose associated Cayley graph has a good spectral gap) was either through property (T ) (as in the Margulis construction, i.e. Proposition 3.9) when d 3 or through the Selberg property when d = 2 via the Brooks-Burger transfer principle between combinatorial expansion of the Cayley graphs and the spectral gap for the Laplace-Beltrami Laplacian on towers of covers of hyperbolic manifolds (see Proposition 3.13 and the appendix) . This poor state of aﬀairs was particularly well illustrated by the embarrassingly open question of Lubotzky, the Lubotzky 1-2-3 problem, which asked whether the subgroups Γi := Si SL2 (Z) for i = 1, 2 and 3 given by 1 ±i 1 0 , } Si = { 0 1 ±i 1 have property (τ ) with respect to the family of congruence subgroups Γi ∩ker(SL2 (Z) → SL2 (Z/pZ)) as p varies among the primes. The (positive!) answer for i = 1 and 2 3 theorem, because both Γ1 and Γ2 are subgroups follows as before from Selberg’s 16 of ﬁnite index in SL(2, Z) (even Γ1 = SL2 (Z)). However Γ3 has inﬁnite index in SL2 (Z) (its limit set on the projective line P(R2 ) is a Cantor set) and therefore none of these methods apply. Bourgain and Gamburd changed the perspective by coming up with a more head-on attack on the problem showing fast equidistribution of the simple random walk directly by more analytic and combinatorial means. This idea originates in the work of Sarnak and Xue [112]. As we saw in Proposition 3.4 this is enough to yield a spectral gap. One of these combinatorial ingredients was the notion of an approximate group (deﬁned below) which was subsequently studied for its own sake and lead in return to many more applications about property (τ ) and expanders as we are about to describe. Let us now state the Bourgain-Gamburd theorem: Theorem 4.3 (Bourgain-Gamburd [11]). Given k 1 and τ > 0 there is ε = ε(k, τ ) > 0 such that every Cayley graph G(SL2 (Z/pZ), S) of SL2 (Z/pZ) with symmetric generating set S of size 2k and girth at least τ log p is an ε-expander. We recall that the girth of a graph is the length of the shortest loop in the graph. Conjecturally all Cayley graphs of SL2 (Z/pZ) are ε-expanders for a uniform ε, and this was later established for almost all primes in Breuillard-Gamburd [15] using the Uniform Tits alternative. But the Bourgain-Gamburd theorem is the ﬁrst instance of a result on expanders where a purely geometric property, such as large girth, is shown to imply a spectral gap. The Bourgain-Gamburd result answers positively the Lubotzky 1-2-3 problem:

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358

EMMANUEL BREUILLARD, LECTURES ON EXPANDER GRAPHS

Corollary 4.4. Every non-virtually solvable subgroup Γ in SL2 (Z) has property (τ ) with respect to the congruence subgroups Γp := Γ∩ker(SL2 (Z) → SL2 (Z/pZ)) as p varies among the primes. Proof. Let S be a symmetric generating set for Γ. By the Tits alternative (or using the fact that SL2 (Z) is virtually free), there is N = N (Γ) > 0 such that S N contains two generators of a free group a, b. Now in order to prove the spectral gap for the action of S on 2 (Γ/Γp ) it is enough to prove a spectral gap for the action of a and b. Indeed suppose there is f ∈ 20 (Γ/Γp ) such that maxs∈S ||s · f − f || ε||f ||. Then writing a and b as words in S of length at most N , we conclude that ||a · f − f || N ε||f || and ||b · f − f || N ε||f ||. Since N depends only on Γ and not on p we have reduced the problem to proving spectral gap for a, b and we can thus assume that Γ = a, b is a 2-generated free subgroup of SL2 (Z). Then it is easy to verify that the logarithmic girth condition holds for this new Γ. Indeed the size of the matrices w(a, b), where w is a word of length n do not exceed max{||a±1 ||, ||b±1 ||}n , hence w(a, b) is not killed modulo p if p is larger that max{||a±1 ||, ||b±1 ||}n , that is if n is smaller that τ log p for some τ = τ (a, b) > 0. We can then apply the theorem and we are done. Before we go further, let us recall the following: Theorem 4.5 (Strong Approximation Theorem, Nori [99], Matthews-Vasserstein-Weisfeiler [98, 124]). Let Γ be a Zariski-dense subgroup of SLd (Z). Then its projection modulo p via the map SLd (Z) → SLd (Z/pZ) is surjective for all but ﬁnitely many primes p. This is a deep result. The proof of Matthews-Vasserstein-Weisfeiler is based on ﬁnite group theory and uses the classiﬁcation of ﬁnite simple groups. Nori’s approach is diﬀerent, via algebraic geometry. There are also alternate proofs by Hrushovski-Pillay [70] via model theory and by Larsen-Pink [85]. Those proofs avoid the classiﬁcation of ﬁnite simple groups and have a broader scope. However in the special case of SL2 (Z) this result is just an exercise (once one observes that the only large subgroups of SL2 (Z/pZ) are dihedral, diagonal, or upper triangular). It will be important for us, because it says that Γ/Γp = SL2 (Z/pZ) as soon as p is large enough, and we will use several key features of SL2 (Z/pZ) in the proof of Theorem 4.3. We are now ready for a sketch of the Bourgain-Gamburd theorem. 1 Let ν = |S| s∈S δs be the symmetric probability measure supported on the generating set S. Our ﬁrst task will be to make explicit the connection between the decay of the probability of return to the identity and the spectral gap, pretty much as we did in Lecture 3. We may write: 1 2n Pν δx , δx ν 2n (e) = Pν2n δe , δe = |Gp | x∈Gp

where we have used the fact that the Cayley graph is homogeneous (i.e. vertex transitive) and hence the probability of return to the e starting from the e is the same as the one of returning to x starting from x, whatever x ∈ Gp may be, so Pν2n δe , δe = Pν2n δx , δx . A key ingredient here is that we will make use of an important property of ﬁnite simple groups of Lie type (such as SL2 (Z/pZ)) which is that they have no non-trivial

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LECTURE 4. APPROXIMATE GROUPS AND THE BOURGAIN-GAMBURD METHOD 359

ﬁnite dimensional complex representation of small dimension. This goes back to Frobenius for SL2 (Z/pZ) and is due to Landazuri and Seitz [84] for arbitrary ﬁnite simple groups of Lie type. For SL2 (Z/pZ) this says the following: Lemma 4.6 (Quasi-randomness). The dimension of a non-trivial irreducible (complex) representation of SL2 (Z/pZ) is at least p−1 2 . Proof. Let V be a non-trivial irreducible SL2 (Z/pZ) complex linear representation, and view it as a representation of the upper-triangular unipotent subgroup U . The subgroup U and its conjugates generate SL2 (Z/pZ), so conjugating if necessary, we may assume that the U action on V is non-trivial. Then V splits into U -invariant isotopic components Vχ , each corresponding to a character χ ∈ U ∗ . These components are permuted by the normalizer N (U ). However the conjugation action of N (U ) on U is isomorphic to the action of the subgroup of squares {x2 ; x ∈ (Z/pZ) \ {0}} by multiplication on U (Z/pZ, +). Besides {0} it has just two orbits of size p−1 2 . Hence all Vχ for each χ in one of the non-trivial orbits must occur in V . It follows that dim V p−1 2 . A ﬁnite group is called “quasi-random” if it has no non trivial irreducible character of small dimension (how small depends on the quality of the desired quasirandomness). The term quasirandomness is derived from a paper of Gowers [55] in which he describes some combinatorial consequences of this property. Its use in the current context goes back to the seminal paper of Sarnak and Xue [112]. A consequence of this fact is the following high multiplicity trick : the eigenvalues of Pν on 20 (Γ/Γp ) all appear with multiplicity at least p−1 2 . This is indeed true, because when p is large enough, then Γ/Γp = Gp := SL2 (Z/pZ), by the above strong approximation theorem. And the regular representation 2 (SL2 (Z/pZ)) can be decomposed into irreducible (complex) linear representations, each of which appears with a multiplicity equal to its dimension1 . The operator Pν preserves each one of these invariant subspaces, and hence its non-trivial eigenvalues appear with 1 p−1 3 a multiplicity at least equal to p−1 2 by Lemma 4.6 above. Since 2 |Gp | , we get 1 3 1 2n 2n 2n 2n 2n |Gp | (μ + μ1 + . . . + μ|Gp |−1 ) μ1 ν (e) = |Gp | 0 |Gp | where the μi ’s are the eigenvalues of Pν , μ0 = 1 and Gp = SL2 (Z/pZ), and means larger than up to a positive multiplicative constant. Hence 2 2n 3 μ2n 1 ν(e) |G| , So if we knew that 1 ν 2n (e) |Gp |1−β for some small β < 13 and for n of size say at most C log |Gp | for some constant C > 0, we would deduce the following spectral gap: μ1 e−

1/3−β C

0, where C > 0 and β < 13 are constants independent of p. Note that we have not used the girth assumption yet. We will do so now (and will use it one more time towards the end of the argument). This tells us that the Cayley graph looks like a tree (a 2k-regular homogeneous tree) on any ball of radius < τ log p (note that the Cayley graph is vertex transitive, so it looks the same when viewed from any point). In particular the random walk behaves exactly like a random walk on a free group on k-generators at least for times n < τ log p. However, we saw in Lecture 1, that ν 2n (e) ρ(ν)2n for every n, where ρ(ν) is the spectral radius of the random walk. For the simple random walk on a free group Fk , the value of the spectral radius is well-known. It is √ 2k − 1 −Ck < 1, := ρ=e k as was computed by Kesten, see [78]. Hence for n τ log p τ3 log |Gp | we have: (8)

ν 2n (e)

1 |Gp |α

where α = α(τ ) = Ck τ /3 > 0. However α(τ ) will typically be small, and our task is now to bridge the gap between (8), which holds at time n τ log p and (7), which we want to hold before C log p for some constant C independent of p. Hence we need ν 2n (e) to keep decaying at a certain controlled rate for the time period τ log p n C log p. This decay will be slower than the exponential rate taking place at the beginning thanks to the girth condition, but still signiﬁcant. And this is where approximate groups come into the game. C. Approximate groups Approximate groups were introduced around 2005 by T. Tao, who was motivated both by their appearance in the Bourgain-Gamburd theorem and because they form a natural generalization to the non-commutative setting of the objects studied in additive combinatorics such as ﬁnite sets of integers with bounded doubling (i.e. sets A ⊂ Z such that |A + A| K|A| for some ﬁxed parameter K 1). Definition 4.7 (Approximate subgroup). Let G be a group and K 1 a parameter. A ﬁnite subset A ⊂ G is called a K-approximate subgroup of G if the following holds: • A−1 = A, 1 ∈ A, • there is X ⊂ G with X = X −1 , |X| K, such that AA ⊂ XA. Here K should be thought as being much smaller than |A|. In practice it will be important to keep track of the dependence in K. If K = 1, then A is the same thing as a ﬁnite subgroup. Another typical example of an approximate group is

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LECTURE 4. APPROXIMATE GROUPS AND THE BOURGAIN-GAMBURD METHOD 361

an interval [−N, N ] ∈ Z, or any homomorphic image of it. More generally any homomorphic image of a word ball in the free nilpotent group of rank r and step s is a C(r, s)-approximate group (a nilprogression). A natural question regarding approximate groups is to classify them and Tao coined this the “non-commutative inverse Freiman problem” (in honor of G. Freiman who classiﬁed approximate subgroups of Z back in the 60’s, see [120]). Recently Breuillard-Green-Tao proved such a classiﬁcation theorem [19] for arbitrary approximate groups showing that they are essentially built as extensions of a ﬁnite subgroup by a nilprogression. However this very general classiﬁcation theorem does not come with good enough bounds to prove expansion. For linear groups and groups of Lie type such as SL2 (Z/pZ) a much stronger classiﬁcation theorem can be derived: Theorem 4.8 (Pyber-Szabo [105], Breuillard-Green-Tao [18]). Suppose G is a simple algebraic group of dimension d deﬁned over a ﬁnite ﬁeld Fq (such as SLn (Fq )). Let A be a K-approximate subgroup of G(Fq ). Then • either A is contained in a proper subgroup of G(Fq ), • or |A| K C , • or |A| |G(Fq )|/K C . where C = C(d) > 0 is a constant independent of q. This result can be interpreted by saying that there are no non-trivial approximate subgroups of simple algebraic groups (disregarding the case when A is contained in a proper subgroup). Theorem 4.8 was ﬁrst proved by H. Helfgott [68] for SL2 (Fp ), p prime, by combinatorial means (using the Bourgain-Katz-Tao sum-product theorem [12]). The general case was later established independently by Pyber-Szabo and BreuillardGreen-Tao using tools from algebraic geometry and the structure theory of simple algebraic groups. For a sketch of the argument, we refer the reader to the survey papers [26] and [106]. Let us now go back to the proof of the Bourgain-Gamburd theorem. The connection with approximate groups appears in the following lemma: Lemma 4.9 (2 -ﬂattening lemma). Suppose μ is a probability measure on a group G and K 1 is such that 1 ||μ ∗ μ||2 ||μ||2 . K Then there is a K C -approximate subgroup A of G such that • μ(A) K1C • |A| K C ||μ||−2 2 , where C and the implied constants are absolute constants. For the proof of this lemma, see the original paper of Bourgain-Gamburd [11] or [123, Lemma 15]. It is based on a remarkable graph theoretic lemma, the BalogSzemeredi-Gowers lemma, which allows one to show the existence of an approximate group whenever we have a set which is an approximate group only in a weak statistical sense. Namely if A ⊂ G is such that the probability that ab belongs to A for a random choice (with uniform distribution) of a and b in A is larger than say 1 C K , then A has large intersection with some K -approximate group of comparable size.

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EMMANUEL BREUILLARD, LECTURES ON EXPANDER GRAPHS

The above lemma combined with Theorem 4.8 implies the desired controlled decay of ν 2n (e) in the range τ log p n C log p, namely (recall that ν 2n (e) = ||ν n ||22 ): Corollary 4.10. There is a constant ε > 0 such that ||ν n ∗ ν n ||2 ||ν n ||1+ε 2 for all n τ log p and as long as ||ν n ||22

1 1

|Gp |1− 10

say.

Indeed, if the lower bound failed to hold at some stage, then by the 2 ﬂattening lemma, there would then exist a pε -approximate subgroup A of Gp of 1 1 size < |Gp |1− 10 such that ν n (A) pCε . By the classiﬁcation theorem, Theorem 4.8, A must be a contained in a proper subgroup of SL2 (Z/pZ). But those all have a solvable subgroup of bounded index. In fact proper subgroups of SL2 (Z/pZ) are completely known (see e.g. [11, Theorem 4.1.1] and the references therein) and besides a handful of bounded subgroups, they are contained either in the normalizer of the diagonal subgroup, or in a Borel subgroup (upper triangular ma1 trices). Hence there is 2-step solvable subgroup A of Gp such that ν n (A) pCε for some n between p and C log p. But ν n (A) is essentially non-increasing, τ log n m −1 n−m that is ν (A) = )ν (xA) max ν n−m (xA) and so ν 2(n−m) (A) x ν (x 1 n−m 2 n 2 (xA) (ν (A)) p2Cε for all m. In particular there is n0 = n − m < ν τ 1 log p for which ν n0 (A) pCε . However at time n0 , we are before the girth 10 bound and the random walk is still in the tree. But in a free group the only 2-step solvable subgroups are cyclic subgroups, so subsets of elements whose second commutator vanish must in fact commute and thus be contained in a cyclic subgroup: they occupy a very tiny part of the free group ball of radius n0 . This contradicts 1 . See [11, Lemma 3] for more details. the lower bound pCε The proof is now complete as we have now a device, namely Corollary 4.10, to go from (8) to (7) by applying this upper bound iteratively a bounded number of times. This ends the proof of Theorem 4.3 and we are done. D. Random generators and the uniformity conjecture The Bourgain-Gamburd method has been used and reﬁned by many authors in the past few years. For example, it is powerful enough to prove that a random Cayley graph of SL2 (Z/pZ) is expanding. This is already contained in the original paper by Bourgain and Gamburd [11]. Recently the method has been pushed to yield the following: Theorem 4.11 (Random Cayley graphs, Breuillard-Green-Guralnick-Tao [21]). Given k 2 and d 1, there is ε, γ > 0, such that the probability that k elements chosen at random in G(Fq ) generate G(Fq ) and turn it into an ε-expander is at least 1 − O( |G(F1q )|γ ). Here G is any simple algebraic group of dimension at most d over Fq . In particular, this yields another proof of Lubotzky’s result in [88], which produced an expanding generating set of ﬁxed size in PSL2 (Fq ). The proof of Theorem 4.11 follows the Bourgain-Gamburd method outlined above via Theorem 4.8. Besides the classiﬁcation of approximate groups, the main new diﬃculty compare to the SL2 case is to prove that the random walk does not concentrate too much on

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LECTURE 4. APPROXIMATE GROUPS AND THE BOURGAIN-GAMBURD METHOD 363

proper subgroups. Proving the non-concentration estimate (i.e. the ﬁnal stage of the Bourgain-Gamburd method) requires showing that certain word varieties are non trivial in G and this is performed by establishing the existence of strongly dense free subgroups of G(Fp ), namely free subgroups all of whose non-abelian subgroups are Zariski-dense, see [22] Theorem 4.11 can be seen as an approximation towards the following conjecture: Conjecture 4.12 (Uniformity conjecture). Given k 2 and d 1, there is ε > 0, such that, for every prime power q and every simple algebraic group of dimension at most d over Fq , all k-regular Cayley graphs of G(Fq ) are ε-expander graphs. In other words the Lubotzky-Weiss independence problem mentioned at the beginning of this lecture is expected to have an aﬃrmative answer in the case of bounded rank ﬁnite simple groups. And indeed all examples so far of sequences of ﬁnite simple groups with both expanding and non-expanding generating sets have unbounded rank. In fact following a method described in [90, Cor. 4.4] it was recently shown [118] that for all ε > 0, every ﬁnite simple group of Lie type with large enough rank (depending on ε) has a generating set with at most 10 elements with respect to which it is not an ε-expander. The only progress to date towards the above is in [15], where it is shown how the uniform Tits alternative (Theorem 2.15) can be used to prove that the family {SL2 (Fp )}p∈P for some inﬁnite family P (indeed of density one) of prime numbers p, is uniformly expanding. E. Super-strong approximation The Bourgain-Gamburd method has also been very successful in establishing the Selberg property (i.e. property (τ ) with respect to congruence subgroups) for new examples of ﬁnitely generated linear groups. In particular all thin groups, that is discrete Zariski-dense subgroups Γ of semisimple Lie groups G which are not lattices, are expected to have the Selberg property. One speaks of super-strong approximation, in reference to Theorem 4.5, because not only are the congruence quotients of Γ generating those of G, but their associated Cayley graphs are expanders. This is still conjectural in full generality, but here is a representative example of what is known: Theorem 4.13 (Super-strong approximation, Bourgain-Varju [13]). If Γ SLd (Z) is a Zariski-dense subgroup, then it has property (τ ) with respect to the family of congruence subgroups Γ ∩ ker(SLd (Z) → SLd (Z/nZ)), where n is an arbitrary integer. This theorem can be viewed as a vast generalization of Selberg’s theorem, and indeed it gives a diﬀerent proof (via the Brooks-Burger dictionary mentioned in Lecture 3) of the uniform spectral gap for the ﬁrst eigenvalue of the Laplacian on the congruence covers of the modular surface H2 / SL2 (Z) (although not such a good 3 of course). Despite its resemblance with Corollary 4.4, the proof of this bound as 16 theorem is much more involved, in particular the passage from n prime to arbitrary n requires much more work. See already Varju’s thesis [123] for the special case of square free n. In a similar spirit, one has the following extension for perfect groups:

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EMMANUEL BREUILLARD, LECTURES ON EXPANDER GRAPHS

Theorem 4.14 (Salehi-Golseﬁdy, Varju [109]). A ﬁnitely generated subgroup of GLd (Q) has property (τ ) with respect to congruence quotients modulo square free integers if and only if the connected component of its Zariski closure is perfect. This result has had several interesting applications to sieving in orbits (e.g. [10]) and other counting problems in groups (e.g. [89]). We refer the reader to the survey paper [14] and to the articles in the recent MSRI proceedings volume devoted to super-strong approximation, starting by Sarnak’s overview [111] for more information on these recent developments.

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APPENDIX

The Brooks-Burger transfer The goal of this appendix is to give a derivation of the Brooks-Burger transfer principle, which relates the ﬁrst eigenvalue of the Laplacian on a cover of a compact manifold M with the ﬁrst eigenvalue of the combinatorial Laplacian on the Cayley-Schreier graph associated to this cover via a ﬁxed choice of generators of the fundamental group of M . A similar statement holds for the Cheeger constant and we will brieﬂy sketch the proof of that as well. Let M be a compact connected Riemannian manifold. Let Γ be the fundamental group of M deﬁned with respect to some base point. It acts by isometries on the

. We denote by F the Dirichlet fundamental domain based at universal cover M

, namely p0 ∈ M for the Γ action on M

; d(x, p0 ) < d(x, γ · p0 ), for all γ ∈ Γ}. F := {x ∈ M If p = γ · p0 is in the Γ-orbit of p0 , then we denote by F(p) = γ · F the corresponding Dirichlet fundamental domain based at p. Associated to this fundamental domain is the ﬁnite symmetric (i.e. S = S −1 ) set S of those elements s ∈ Γ such that the distance between F(s · p0 ) and F(p0 )

is at most equal to the diameter of M . It is a generating set for Γ. in M We denote by G(Γ, S) the associated Cayley graph. Given a subgroup Γ Γ, we let G(Γ/Γ , S) be the corresponding Cayley-Schreier graph (a quotient of G(Γ, S)). Theorem 1. There are constants c1 , . . . , c4 > 0 depending on M only, such that for every ﬁnite degree cover M of M , with fundamental group Γ Γ, we have (9)

c1 λ1 (M ) λ1 (G(Γ/Γ , S)) c2 λ1 (M ),

and similarly for the Cheeger constant: (10)

c3 h(M ) h(G(Γ/Γ , S)) c4 h(M ).

(I). Discussion The Cheeger constant result is due to Brooks, while the Laplace eigenvalue statement is essentially due to Burger. Although all of the ideas to prove (10) are present in [27, 29] the statement ﬁrst appears in [30]. As for (9) it is part of Marc Burger’s EPFL thesis [32], where a proof of the lower bound for λ1 (M ) can be found (see also [34]). Burger also proved the upper bound for λ1 (M ) in (9) in the special case when M is a rank-one locally symmetric space using the harmonic analysis of spherical functions (see [33]). In the generality of Theorem 1 the inequalities (9) were ﬁrst stated in [31] and the argument given below for the lower bound follows a suggestion from [31]. We hope that this appendix will help record these arguments in one place and give the right amount of detail. 365 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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EMMANUEL BREUILLARD, LECTURES ON EXPANDER GRAPHS

Note that statements (9) and (10) are not immediately equivalent: indeed the Cheeger-Buser inequalities (see 3.2) relating h and λ1 are not strong enough: it allows to upper bound λ1 by (a constant times) h, but the lower bound is in terms of h2 , and could therefore be much smaller than h. What the CheegerBuser inequalities allow you to deduce from statement (10) is the fact that λ1 (Mn ) tends to zero if and only if λ1 (G(Γ/Γn , S)) tends to zero for any sequence of ﬁnite covers Mn with fundamental groups Γn whose index in Γ grow to inﬁnity. This is the version stated by Lubotzky and Zimmer in [91] and in Lubotzky’s book [86, Theorem 4.3.2]. From a philosophical point of view, Theorem 1 is not very surprising, because it is clear that the large scale geometry of a ﬁnite cover M of high degree of M is basically governed by that of the associated Cayley-Schreier graph, which serves the purpose of a skeleton for the covering space M . What is a bit less clear is that we have this nice control of the quantities by ﬁxed multiplicative constants. The constants ci ’s are eﬀective and we will give bounds on them, which can be explicitly computed in terms of the geometry and spectrum of the fundamental domain F only. Note in particular that if one rescales the metric on the base manifold M by a factor T , then λ1 (M ) is changed into by a factor T12 . This aﬀects the constants ci ’s accordingly, while the Cayley-Schreier graph G(Γ \Γ) remains unchanged. We may thus assume without loss of generality that the diameter of M is equal to 1 say. See the discussion after the proof. We also note that the choice of the generating set in the above statement is somewhat arbitrary and it remains valid (albeit with diﬀerent constants ci ’s) for any other ﬁnite symmetric generating set. See the discussion after the proof. (II). Notation We denote by || · ||2 the L2 norm on M , M or Γ/Γ alike. The Rayleigh quotient ||∇f ||2 of a function f on M is the quantity ||f ||22 . Recall that the ﬁrst eigenvalue of the 2 Laplace operator on M admits the following variational characterization: ||∇f ||22 λ1 (M ) := inf{ , f = 0} ||f ||22 M Moreover there exists a non zero eigenfunction of the Laplace operator on M for the eigenvalue λ1 (M ) and its Rayleigh quotient realizes the above inﬁmum. The same holds for functions on the Cayley-Schreier graph G(Γ/Γ , S), namely (see Lecture 3) ||∇F ||22 λ1 (G(Γ/Γ , S)) := inf{ , f (x) = 0} ||F ||22 x∈Γ/Γ

Here the nabla sign ∇ turns a function F on Γ/Γ into a function on the edges of the Cayley-Schreier graph, namely ∇F (e) = |F (p) − F (q)|, if e is an edge with end points p and q.

. For p = γ · p0 , γ ∈ Γ, a Let vol be the Riemannian measure on M and M

point in the orbit of the base point p0 ∈ M , let F(p) = γ · F(p0 ) be the Dirichlet fundamental domain based at p. The F(p) are all isometric to each other. They

with piecewise are connected (even star-shaped around p) open submanifolds of M smooth boundary of measure zero. They form a tessellation of the universal cover

. M

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APPENDIX. THE BROOKS-BURGER TRANSFER

367

We denote by V the volume of M and by NV the valency of the Cayley-Schreier graph, namely the cardinality of the generating set S. (III). Proof of the lower bound for λ1 (M ) We use a variant of Burger’s argument given in [34]. For a Lipschitz function f on M with zero average, we need to build a function F on Γ \Γ, with zero average and with comparable Rayleigh quotient. This is done by setting: 1 f. F (p) = vol(F) F (p) Obviously p F (p) = 0. We need to upper bound ||∇F ||2 in terms of ||∇f ||2 . For this we introduce the ﬁrst Neumann eigenvalue μ1 (F) of the fundamental domain

. It has the following variational characterization (see [40, Chapter 1]) F ⊂M

|∇u|2 (11) μ1 (F) := inf{ F 2 ; u = 0}, u F F where u is an arbitrary Lipschitz function on F (in particular it does not have to descend to a continuous function on the base manifold M ). The basic idea of Burger’s argument is that if λ1 (M ) were very small, in particular much smaller than μ1 (F), then any eigenfunction f of the Laplace operator on M with eigenvalue λ1 (M ) would be almost constant on each one of the fundamental domains F(p), making F and f very close. Note that these are all isometric to F. We now formalize this idea and pass to the details. Given a pair of adjacent vertices p ∼ q in the Cayley of Γ, the union of the

two associated fundamental domains F(p) ∪ F(q) inside the universal cover M is a bounded connected subset with piecewise smooth boundary. Hence its ﬁrst Neumann eigenvalue μ1 (F(p) ∪ F(q)) is positive. Set μ := min{μ1 (F(p)), μ1 (F(p) ∪ F(q))} > 0 q∼p

, the quantity μ Since Γ acts transitively on the fundamentals domains F(p) in M just deﬁned is independent of p. We now choose an eigenfunction f for the Laplace operator on M with eigenvalue λ1 (M ). From the variational characterization of the ﬁrst Neumann eigenvalue we have the following Poincar´e inequality: 1 2 (f − F (p)) |∇f |2 μ F (p) F (p) and

F (p)∪F (q)

(f −

F (p) + F (q) 2 1 ) 2 μ

F (p)∪F (q)

|∇f |2

(q) 2 where F(p) is now considered inside M . Hence, writing ( F (p)−F ) 2((f − 2 F (p)+F (q) 2 2 F (p)) +(f − ) ) on F(p) and similarly on F(q), we get (recall V = vol(M )) 2 2 F (p) − F (q) 2 ) |∇f |2 V( 2 μ F (p)∪F (q)

hence taking squares and summing over neighbors 16NV 2 |F (p) − F (q)| |∇f |2 , (12) V · μ M p∼q

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EMMANUEL BREUILLARD, LECTURES ON EXPANDER GRAPHS

where NV is the valency of the graph. On the other hand we may decompose f orthogonally on each F(p) as 2 f − F (p) f 2 = V · F (p)2 + F (p)

F (p)

hence, summing over p (13)

V ||F ||22 − ||f ||22 1 ||∇f ||22 = λ1 (M ) ||f ||22 . μ μ

In particular F is not identically zero if λ1 (M ) < μ. Combining (13) and (12), we get ||∇F ||22 16NV λ1 (M ) . 2 ||F ||2 μ 1 − λ1 (M ) μ

Therefore if λ1 (M ) we obtain the desired bound λ1 (G(Γ/Γ , S)) c2 λ1 (M ) 32NV with c2 = μ . On the other hand, if λ1 (M ) μ2 , then, since at any case λ1 (G(Γ/Γ , S)) 2NV , we obviously have the desired bound in that case too. μ 2,

(IV). Proof of the upper bound for λ1 (M ) Starting with any function f on the vertex set Γ/Γ with zero average, we need to build a function F on M with zero average and comparable Rayleigh quotient. Given ε > 0, let CM,ε (p) be the set of points x ∈ F(p) such that the distance between x and the complement (F(p))c is at least ε. As ε → 0, the measure vol(CM,ε (p)) tends to V := vol(F(p)). Let F = F(p0 ) for some base point p0 . Without loss of generality we may normalize the Riemannian metric on M so that the diameter of M is equal to 1. Let f be a Γ -invariant function on Γ with zero average on Γ \Γ. We now deﬁne

by setting its value on each a Γ -invariant Lipschitz continuous function Fε on M CM,ε (p) to be f (p) and by ﬁlling in using a weighted average as follows: 1 (14) Fε (x) = f (p)dp (x), p dp (x) p ( 1 −d(x,C

(p)))+

M,ε when x lies outside CM,ε (p) (we denoted y + := where dp (x) = 2 d(x,CM,ε (p)) max{0, y}). The function F above is deﬁned outside the union of all CM,ε (p)’s

equal to but it clearly extends by continuity to a continuous function on all of M f (p) inside each CM,ε (p). Observe ﬁnally that in the sum deﬁning Fε at x, only a bounded number of terms are non-zero, namely the number of tiles intersecting the ball of radius 12 at x, and this number is bounded independently of x. It is clear that Fε is Γ -invariant. We now choose ε > 0. The crucial point is that ε has to be chosen independently of the cover M . We do so by picking ε ∈ (0, 14 ) small enough so that

(15)

vol(CM,ε ) − NV vol(F \ CM,ε )

1 vol(F) 2

where NV (the valency of our Cayley-Schreier graph) is the number of fundamental domains F(p) containing a point at distance at most 1 from F. This is possible since vol(F \ CM,ε ) tends to zero as ε → 0.

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APPENDIX. THE BROOKS-BURGER TRANSFER

369

1 Lemma 2. Setting Gε = Fε − vol(M F the associated zero mean function ) M ε on M , we have 1 ||Gε ||2L2 (M ) ||f ||2 vol(F), (16) 2 4 11 1 |f (p) − f (q)| |∇Gε (x)| (17) 1 ε p,q d(x,F (p)) 2 d(x,F (q)) 2 Before proving the above lemma, let us explain ﬁrst how to conclude the proof of the left hand side inequality in part (2) of Theorem 1. Let N NV be the

. In maximal number of fundamental domains intersecting a ball of radius 12 in M the second displayed equation above, for any given x, the sum is restricted to at most N 2 couples (p, q). Moreover these couples are neighbors (p ∼ q) in the CayleySchreier graph, because F(q) has a point at distance at most 1 from F(p). Hence by Cauchy-Schwarz: 16 |∇Gε (x)|2 2 N 2 |f (p) − f (q)|2 1d(x,F (p)) 12 1d(x,F (q)) 12 ε p∼q Integrating we obtain: ||∇Gε ||2L2 (M )

16 2 8N 2 NV N |f (p) − f (q)|2 NV vol(F) = vol(F)||∇f ||22 2 2 ε ε p∼q

From these two estimates, it readily follows that ||∇Gε ||2L2 (M ) ||Gε ||2L2 (M ) where c1 :=

ε2 16NV N 2 .

1 ||∇f ||L2 (Γ \Γ) c1 ||f ||2L2 (Γ \Γ) 2

We thus obtain the desired lower bound c1 λ1 (M ) λ1 (G(Γ/Γ , S)).

It only remains to prove the lemma. Proof of Lemma 2. Consider the ﬁrst estimate. Since f has zero average on Γ \Γ we have Fε = Fε M

and thus |

M

M \∪p CM,ε (p)

Fε |

M \∪p CM,ε (p)

max{f (p), d(x, F(p))

1 } 2

which becomes by Cauchy-Schwarz 2 vol(M \ ∪p CM,ε (p)) 1 F f (p)2 ε vol(M ) M vol(M ) M (18) d(x,F (p)) 12 vol(F \ CM,ε )||f ||22 NV , where NV is the number of fundamental domains F(p) at distance at most 1 from F. On the other hand clearly ||Fε ||2 f (p)2 vol(CM,ε (p)) = vol(CM,ε )||f ||22 p

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EMMANUEL BREUILLARD, LECTURES ON EXPANDER GRAPHS

So combining the last inequality with (18) we obtain: 2 1 1 2 2 ||Gε ||2 = ||Fε || − Fε vol(F)||f ||22 vol(M ) M 2 as desired, as soon as 1 vol(F). 2 This yields the ﬁrst estimate in Lemma 2 and we now turn to the second estimate. We compute ∇Gε = ∇Fε as 1 ( f (p)∇d )( d ) − ( f (p)d )( ∇dp ) ∇Fε = p p p ( p dp )2 p p p p 1 = (f (p) − f (q))(∇dp )dq ( p dp )2 p,q vol(CM,ε ) − NV vol(F \ CM,ε )

From the deﬁnition of dp (x) it is a simple matter to verify the following bound from which the desired estimate in Lemma 2 follows directly: dq ∇dp 4 (19) 2 1d(x,F (p)) 12 1d(x,F (q)) 12 . ε dm m

Indeed ﬁrst note that |∇dp |

1d(x,CM,ε (p)) 12

2d(x, CM,ε (p))2 While if d(x, CM,ε (p)) ε, then d(x, CM,ε (q)) ε for every q = p, and thus dq (x) 1 1 . On the other hand m dm (x)d(x, CM,ε (p)) dp (x)d(x, CM,ε (p)) 2 − ε and 2ε putting this together yields (19), when d(x, CM,ε (p)) ε. If on the other hand d(x, CM,ε (p)) ε, then |∇dp | 2ε12 . But every x ∈ M \ ∪m CM,ε (m) belongs to at least one F(m), and hence 1 as ε < 14 . So we always have m dm 4ε . It follows that dq ∇dp 1 dq 4ε 2 2 2ε m dm m dm

has dm (x)

1 2 −ε

ε

1 4ε ,

2 . ε

So we do get (19) in all cases and this ends the proof the lemma and of part (2) of Theorem 1. (V). Dependence of the constants on the geometry of M . The constants ci ’s, i = 1, . . . , 4 in Theorem 1 depend only on the geometry of the fundamental domain F(p). Recall that for c1 and c2 , we had found: c1 =

ε2 32NV , , c2 = 16NV3 μ

Here NV is the number of fundamental domains at distance at most 1 from the fundamental domain F(p0 ) associated to a base point p0 . We denoted by μ a positive lower bound for the non-zero Neumann eigenvalues of F(p) and F(p)∪F(q) as deﬁned by (11). The constant ε = ε(M ) is deﬁned by (15). Due to Gromov’s compactness theorem (see e.g. [60] and [102]) the set of Riemannian metrics on a compact manifold M with bounded diameter, bounded curvature and a lower bound on the injectivity radius, is pre-compact. As a consequence there is a uniform bound C = C(n, D, κ, r) > 0 such that if M is a

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APPENDIX. THE BROOKS-BURGER TRANSFER

371

compact n-dimensional Riemannian manifold with diameter at most D, sectional curvature |KM | κ and injectivity radius at least r > 0, then the constants ci ’s from Theorem 1 lie in [ C1 , C]. Using the standard comparison theorems in Riemannian geometry, it is easy to get an explicit control of NV in terms of the parameters (n, D, κ, r). For μ one can use Cheeger’s inequality (cf. [37]) and the estimate of the Cheeger constant for Dirichlet domains obtained in [38, Lemma 5.1.]. Controlling ε explicitly seems a bit more challenging however (although note that one is allowed to regularize the boundary of F without altering the associated graph.) (VI). A sketch of the proof of the Cheeger constant inequalities. For the bound c3 h(M ) h(G(Γ/Γ , S)), take a set A of at most half of the vertices of the graph G(Γ/Γ , S) which almost realizes the combinatorial Cheeger constant, i.e. |∂A0 |/|A0 | 2h(G(Γ/Γ , S)) say. Then consider the hypersurface deﬁned as the boundary of the union of the Dirichlet fundamental domains F(p) with p ∈ A0 . The area of the hypersurface is clearly bounded by |∂A0 |voln−1 (∂F), while the volume n−1 (∂F ) enclosed is |A0 |vol(F). This yield the desired inequality with c3 = vol2vol(F ) . The proof of the opposite inequality is more delicate. The diﬃculty (dubbed “the problem of hairs” in [38]) is that we have little information on the hypersurfaces that may realize or almost realize the Cheeger constant: in particular there is no guarantee that the hypersurface does not intersect every single fundamental domain F(p). There are two possible strategies to overcome this diﬃculty. The ﬁrst is the one chosen by Brooks [29], p100–102. It consists in looking for a minimizing hypersurface for the Cheeger constant. Typically no smooth minimizer exists, but one can use a non-trivial result from geometric measure theory according to which there is a minimizing integral current T with some strong regularity properties and constant mean curvature, such that the Cheeger constant is realized for T . The fact that the (Ricci or sectional) curvature of the covers M are uniformly bounded implies that the mean curvature of the current is uniformly bounded. In turn this implies that the intersection of T with any ﬁxed small ball has controlled area, and thus the area of T is controlled by the number of domains F(p) intersecting it, and we are done. The second strategy avoids the use of currents and geometric measure theory and instead uses the standard comparison theorems in Riemannian geometry. The idea is due to Buser [38, Sec. 4] who used it for a slightly diﬀerent purpose. We consider an almost minimizing smooth hypersurface X = ∂A = ∂B separating M into two disjoint connected pieces A and B, with voln−1 (X) 2h(M ) min{vol(A), vol(B)} say. And we modify it by setting := {x ∈ M ; vol(A ∩ B(x, r)) > 1 vol(B(x, r))} A := {x ∈ M ; vol(A ∩ B(x, r)) < 12vol(B(x, r))}, B 2 where r > 0 is a number deﬁned a posteriori. The two sets are again disjoint, and = {x ∈ M ; vol(A ∩ B(x, r)) = 1 vol(B(x, r))}. Now pick a their boundary is X 2 and write maximal r-separated set {xi }i in X vol(X ∩ B(xi , r)) h(B(xi , r)) min{vol(A ∩ B(xi , r), vol(B ∩ B(xi , r))} h(B(xi , r)) vol(B(xi , r)), 2

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where h(B(xi , r)) is the Cheeger constant of this ball, which can be bounded below using the bounded curvature assumption ([38, Lemma 5.1]). Summing over i, one gets 2r ) r h(M ) min{vol(A), vol(B)}. vol(X To conclude, let C0 (resp. A0 , B0 ) be the subset of fundamental domains F(p) 2r non trivially (resp. are contained in A \X 2r , B \X 2r ,). Then which intersect X |C| (|A0 |, |B0 |) are controlled by vol(X2r ) (resp. vol(A), vol(B)). The three sets are disjoint, in particular |∂A0 |, |∂B0 | NV |C0 | and the result follows. For more details, see [38, Sec. 4]. (VII). Additional remarks Remark 3 (On the precise deﬁnition of the Schreier graph). There is a certain amount of indeterminacy in the very deﬁnition of a Schreier graph of a ﬁnite quotient Γ/Γ . We chose to deﬁne it as the quotient of the Cayley graph associated to S with respect to the action of Γ . This produces a graph which may have some loops and double edges (note that already the Cayley graph may have double edges). However we may just as well consider the graph obtained from this one by removing all loops and keeping only one edge in case of multiple edges. The resulting graph is then a graph in the common sense of the word. The inequalities of Theorem 1 remain valid (albeit with slightly diﬀerent constants) for this new graph. In the case ofthe Laplace eigenvalues inequalities, this is because the Dirichlet forms D(f, g) = p∼q |f (q) − f (p)|2 of the two graphs are comparable up to multiplicative constants. And hence C1 λ1 (g) λ1 (g ) Cλ1 (g) for some C > 0, where g and g are the old and new graph. Similarly one has C1 h(g) h(g ) Ch(g) for the Cheeger constant. Remark 4 (Change of generating set). In the statement of Theorem 1, we could have taken any other ﬁxed generating set S for the Cayley graph of Γ at the expense of modifying the constants ci ’s. If S is another generating set, then there is an in N also S ⊂ S N . This implies that the Dirichlet forms teger N such that S ⊂ S and 2 D(f, g) = p∼q |f (q)−f (p)| associated to two the Cayley-Schreier graphs G(Γ \Γ) relative to S and S are comparable up to multiplicative constants independent of Γ , and hence so are their C1 λ1 (G(Γ \Γ, S) λ1 (G(Γ \Γ, S ) Cλ1 (G(Γ \Γ, S), where C = C(S, S ) > 0 is independent of Γ . It is worth pointing out however that one must keep the same generating for all ﬁnite index subgroups Γ , that is the generating set cannot be allowed to vary with the ﬁnite index subgroup Γ without violating the uniformity of the constants ci ’s. See the discussion of uniform property (T ) and (τ ) in Lecture 3. Remark 5 (Added in proof). The Burger-Brooks transfer extends to all eigenvalues of the Laplace operator and holds more generally for all discretizations X of the manifold M . In [94], isoperimetric inequalities from Chavel’s book [41] are used to extend the above arguments and to show that under Ricci and injectivity radius lower bounds the eigenvalues λk (M ) and λk (X) are comparable up to multiplicative constants independent of k 1. In particular, Theorem 1 and (9) holds also for λk .

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https://doi.org/10.1090//pcms/021/11

Cube Complexes, Subgroups of Mapping Class Groups, and Nilpotent Genus Martin R. Bridson

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IAS/Park City Mathematics Series Volume 21, 2012

Cube Complexes, Subgroups of Mapping Class Groups, and Nilpotent Genus Martin R. Bridson 1. Introduction These notes are based on my lecture at PCMI in July 2012. They are structured around two sets of results, one concerning groups of automorphisms of surfaces and the other concerning the nilpotent genus of groups. The ﬁrst set of results exempliﬁes the theme that even the nicest of groups can harbour a diverse array of complicated ﬁnitely presented subgroups: we shall see that the ﬁnitely presented subgroups of the mapping class groups of surfaces of ﬁnite type can be much wilder than had been previously recognised. The second set of results ﬁts into the quest to understand which properties of a ﬁnitely generated group can be detected by examining the group’s ﬁnite and nilpotent quotients and which cannot. These two topics appear to have little in common and neither has any obvious connection to the study of non-positively curved cube complexes; they are chosen for exactly these reasons. My purpose is to describe the resolution of various longstanding problems in a way that emphasizes the broad applicability of a certain template for constructing interesting examples of ﬁnitely presented groups. This template, described in Section 8, can be applied in many other contexts. It reﬁnes a construction that I articulated in [17], with improvements based on recent advances in the understanding of right-angled Artin groups (RAAGs) and nonpositively curved cube complexes (particularly the virtually special cube complexes of Haglund and Wise). The new results concerning subgroups of mapping class groups are from [16] while the new results concerning nilpotent genera of groups are from [29] (which is part of a wider project with Alan Reid from the University of Texas). Two of the results that we shall discuss are the following. Mathematical Institute, Andrew Wiles Building, Oxford OX2 6GG, U.K. E-mail address: [email protected] The author was a Clay Senior Scholar in Residence at PCMI in 2012. He thanks the Clay Mathematics Institute for this honour. He also thanks the Royal Society for the Wolfson Research Merit Award that supports his research. Published with the consent of the Clay Mathematics Institute. Key words and phrases. Mapping class groups, RAAGs, cube complexes, nilpotent genus, decision problems 2010 Mathematics Subject Classiﬁcation. Primary 20F65, 57M60, 20E26; Secondary 20F36, 20F67, 20F10

c 2014 American Mathematical Society

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382 MARTIN R. BRIDSON, CUBES, MAPPING CLASS GROUPS AND NILPOTENT GENUS

Theorem 1.1 ([16]). If the genus of a surface S is suﬃciently large, then the isomorphism problem for the ﬁnitely presented subgroups of the mapping class group Mod(S) is unsolvable. Theorem 1.2 ([29]). There exist pairs of ﬁnitely generated, residually torsionfree nilpotent groups N → Γ so that N has the same ﬁnite and nilpotent quotients as Γ but Γ is ﬁnitely presented while H2 (N, Q) is inﬁnite dimensional. Several of the results that we shall discuss concern decision problems for groups. I shall assume that the reader is familiar with the basic vocabulary associated to such problems, and recommend [56] as a pleasant introduction to the subject. 2. Subgroups of mapping class groups Throughout, S will denote a compact, connected, orientable surface, which is allowed to have non-empty boundary. Most of the results that we will discuss remain valid for compact surfaces with ﬁnitely many punctures. The mapping class group of S will be denoted1 Mod(S); this is the group of isotopy classes of orientationpreserving homeomorphisms S → S, where maps and isotopies are required to ﬁx the boundary point-wise. I shall assume that the reader is familiar with the basic ideas and vocabulary concerning mapping class groups, as described in [42] and [50] for example. 2.1. The ﬁrst subgroups Cyclic subgroups: Nielsen-Thurston theory describes the individual elements of Mod(S): an element φ ∈ Mod(S) is reducible if it is the class of a homeomorphism that leaves invariant a non-empty collection of homotopically-essential circles on S, none of which is homotopic to a boundary component; Thurston proved that the irreducible elements ψ of inﬁnite order are pseudo-Anosov, which implies in particular that if c is a loop that is not homotopic into ∂S then, in any ﬁxed metric, the length of the shortest loop in the homotopy class ψ n [c] grows exponentially with |n|. Nielsen proved that the elements of ﬁnite order in Mod(S) are precisely those mapping classes which contain a diﬀeomorphism f such that f d = idS for some d. Finite subgroups: Kerckhoﬀ’s resolution of the Nielsen Realisation Problem [51] shows that every ﬁnite subgroup G < Mod(S) arises as a group of isometries of a metric of constant curvature on S; equivalently, G has a ﬁxed point in the natural action of Mod(S) on the Teichm¨ uller space T (S). The action of Mod(S) on T (S) is proper and there is an equivariant retraction onto a spine where the action of Mod(S) is cocompact [59]. It follows that there are only ﬁnitely many conjugacy classes of ﬁnite subgroups in Mod(S); see [20]. This does not remain true if one replaces Mod(S) by a ﬁnitely presented subgroup (Theorem 2.4). Abelian subgroups: The Dehn twists in disjoint curves on S have inﬁnite order and commute. On a closed surface of genus g the maximum number of disjoint, non-homotopic, essential simple closed curves that one can ﬁt is 3g − 3; if S has b boundary components then one can add a further b curves parallel to the boundary components. Thus we obtain free abelian groups of rank 3g − 3 + b generated by Dehn twists. Birman-Lubotzky-McCarthy [14] proved that every abelian subgroup 1 reﬂecting

the more historic name, Teichm¨ uller modular group

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2. SUBGROUPS OF MAPPING CLASS GROUPS

383

of Mod(S) is ﬁnitely generated and has rank no greater than 3g − 3 + b (see also [50]). Free subgroups: These abound in Mod(S), indeed Ivanov2 [50] and McCarthy [54] proved that mapping class groups satisfy a Tits Alternative: if a subgroup G < Mod(S) is not virtually abelian, then it contains a free subgroup. More explicitly, if φ, ψ ∈ Mod(S) are pseudo-Anosov, then for some n > 0, the subgroup φn , ψ n is free (and cyclic only if φ and ψ share an axis in Teichm¨ uller space). A more involved argument due to Dahmani, Guirardel and Osin [36] shows that if ψ ∈ Mod(S) is pseudo-Anosov, then for suitable m > 0, the normal closure ψ m is a free group freely generated by the conjugates of ψ m . Direct Products: If S and S are disjoint subsurfaces of S, then by extending diﬀeomorphisms to be the identity on the complement we see that Mod(S ) × Mod(S ) is a subgroup of Mod(S). 2.2. Wreath products The wreath product A B of groups is the semidirect product B ⊕b∈B Ab , where the Ab are isomorphic copies of A permuted by left translation. Proposition 2.1. If S is a compact surface with non-empty boundary and G is a ﬁnite group, then there is a closed surface Sg and a monomorphism Mod(S)G → Mod(Sg ). To prove this, one ﬁrst reduces to the case where S has one boundary component. Then one takes a closed surface on which G acts eﬀectively, deletes a family of open discs centred at the points of a free G-orbit, and attaches a copy of S to each of the resulting boundary circles, extending homeomorphisms of each copy of S by the identity on the complement. (See [18] for details.) For a group H with ﬁnite-index K H, there is a standard embedding H → K (H/K). Corollary 2.2. If a group H has a subgroup of ﬁnite index that embeds in the mapping class group of a compact surface with boundary, then H embeds in the mapping class group of a closed surface. Remark 2.3. There is considerable ﬂexibility in the above construction, but one cannot hope to embed H in Mod(S) for all surfaces of suﬃciently high genus. Indeed there are constraints even for ﬁnite groups [53]. 2.3. Non-subgroups Reducibility and its Consequences. The Reduction Theory of Ivanov [50] allows one to prove all manner of results concerning the subgroups of mapping class groups by induction on the complexity of the surface. To explain this, we ﬁrst extend the deﬁnition of reducibility to subgroups: G < Mod(S) is reducible if it leaves invariant a non-empty collection of homotopically-essential circles on S, none of which is homotopic to a boundary component. Ivanov [50] proves that if G < Mod(S) is inﬁnite and irreducible then it contains a pseudo-Anosov element. There are subgroups of ﬁnite index P < Mod(S) in the mapping class that consist entirely of pure automorphisms: [φ] is pure if, on the complement of a tubular neighbourhood of a set of disjoint curves that φ ﬁxes, the restriction of 2 Most

of the results of Ivanov that I quote were proved in his earlier papers, but [50] provides an excellent, coherent account of his work.

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384 MARTIN R. BRIDSON, CUBES, MAPPING CLASS GROUPS AND NILPOTENT GENUS

φ to each component is either trivial or pseudo-Anosov. The kernel of the action of Mod(S) on H1 (S, Z3 ) is pure, for example [50, p.4]. With this fact in hand, it follows from Ivanov’s theorem that after passing to a subgroup of ﬁnite index, any G < Mod(S) ﬁts into a short exact sequence where the kernel (which is central) is a free virtually abelian group generated by multi-twists in the reducing curves for G and the quotient is a direct product of subgroups of mapping class groups of smaller surfaces, each containing a pseudo-Anosov on that surface. From there, it is not diﬃcult to prove, for example, that every amenable subgroup is abelian (cf. [50], [14]). And using [36] one sees that Mod(S) does not contain inﬁnite images of groups that are Z-averse in the sense of [30], from which it follows that all homomorophisms to Mod(S) from irreducible lattices in higherrank semisimple Lie groups have ﬁnite image (cf. [43]). Other obstructions to embedding come from global properties of Mod(S) that are inherited by subgroups; for example, if H < Mod(S) then H must be residuallyﬁnite [44], and all cyclic subgroups of H must be quasi-isometrically embedded [41]. 2.4. Subgroups that are not ﬁnitely presented We are concerned here almost entirely with subgroups that are ﬁnitely presented, or at least ﬁnitely generated. But it would be remiss of me not to mention that Mod(S) has a host of natural subgroups where these ﬁniteness conditions may fail. Foremost among these is the Torelli group, which is the kernel of the action of Mod(S) on H1 (S, Z). This is ﬁnitely generated if the genus of S is at least 3 but it is not known if it is ﬁnitely presented. The kernel of the action Mod(S) on π1 (S) modulo any later term of the lower central series is not even ﬁnitely generated. Connecting to the second part of this talk, we note that the Torelli group is residually torsion-free nilpotent [5]. 2.5. On the diﬃculty of identifying ﬁnitely presented subgroups For the most part, the heart of geometric and combinatorial group theory lies with the study of ﬁnitely presented groups, but this restriction brings with it real challenges. In any context, it is easy to ﬁnd ﬁnitely generated subgroups of a given group G: one can simply take a ﬁnite subset S ⊂ G and consider S. But typically there will be no algorithm to decide which ﬁnite subsets of G generate ﬁnitely presentable subgroups (this is the case in a direct product of free groups already). Moreover, even if one is given information that guarantees that S has a ﬁnite presentation, there is no general procedure that will produce such a presentation, even when the ambient group is as benign as G = GL(n, Z); see [30]. 2.6. Direct products of free groups If H1 , H2 < Mod(S) are supported on disjoint subsurfaces of S, then they commute. One can embed g disjoint one-holed tori in a surface of genus g, and the mapping class group of a one-holed torus is a central extension of SL(2, Z), which contains non-abelian free groups. Thus, if S has genus g (and any number of boundary components), then Mod(S) contains the direct product D of g non-abelian free groups. In D, what (ﬁnitely generated or ﬁnitely presented) subgroups might we ﬁnd? The answer to this question leads us in the main direction of this lecture. But ﬁrst we pause to record a consequence of Corollary 2.2.

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3. FIBRE PRODUCTS AND SUBDIRECT PRODUCTS OF FREE GROUPS

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Theorem 2.4. There exist closed surfaces S and ﬁnitely presented subgroups H < Mod(S) such that H has inﬁnitely many conjugacy classes of elements of ﬁnite order. Proof: The direct product Γ of n copies of SL(2, Z) contains as a subgroup of ﬁnite index the direct product D of n non-abelian free groups. D embeds in the mapping class group of a surface of genus n with 1 boundary component, so Γ embeds in Mod(S) for some closed surface S. It is proved in [20] that Γ contains ﬁnitely presented subgroups with inﬁnitely many conjugacy classes of elements of order 4. (See also [15].) 3. Fibre products and subdirect products of free groups We shall develop our discussion of these subgroups around two constructions. Construction 3.1. Fibre Products. Let Q = A | R be a ﬁnitely presented group. Let F be the free group on A and let p : F → Q be the surjection implicit in the notation. The kernel of p is ﬁnitely generated if and only if Q is ﬁnite, but regardless of what Q is, the ﬁbre product P = {(u, v) | p(u) = p(v)} < F × F will be generated by the ﬁnite set ΣQ = {(a, a), (r, 1); a ∈ A, r ∈ R}. This observation provides complicated ﬁnitely generated subgroups of F × F : Mihailova [57] and Miller [55] were the ﬁrst to see if Q has an unsolvable word problem, then P has an unsolvable conjugacy problem (cf. Proposition 7.6) and there is no algorithm to decide which words in the generators of F × F deﬁne elements of P (Proposition 7.5); and since P ∼ = F × F if and only if Q = 1, there can be no algorithm to decide isomorphism among the ﬁnitely generated subgroups of F × F , because there is no algorithm that can determine which ﬁnitely presented groups (with generating sets of a ﬁxed cardinality) are trivial. There isn’t even an algorithm that, given a ﬁnite subset Σ ⊂ F × F , can calculate the ﬁrst homology of Σ (see [27]). There are uncountably many 2-generator groups, and via ﬁbre products one can deduce from this that there are uncountably many non-isomorphic subgroups P < F × F (see [10]). Hence: Proposition 3.2. The mapping class group of any surface of genus at least 2 contains uncountably many non-isomorphic subgroups. These ﬁbre products, though, do not give us complicated ﬁnitely presented subgroups of mapping class groups, because P is ﬁnitely presentable if and only if Q is ﬁnite [45]. Indeed, Baumslag and Roseblade [10] proved that F × F has no ﬁnitely presented subgroups other than the obvious ones: if G < F × F is ﬁnitely presented, then G is either free or else it has a subgroup of ﬁnite index that is a product of two free groups (its intersections with the direct factors). 3.1. Finitely presented examples A celebrated construction of Stallings [65] and Bieri [12] shows that there are interesting ﬁnitely presented subgroups in the direct product of three (or more) free groups.

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386 MARTIN R. BRIDSON, CUBES, MAPPING CLASS GROUPS AND NILPOTENT GENUS

Construction 3.3. The Stallings-Bieri Groups. Let h : F × · · · × F → Z be a homomorphism that restricts to an epimorphism on each of the n factors. Stallings [65] (in the case n = 3) and Bieri [12] proved that the kernel SBn has a classifying space with a ﬁnite (n − 1)-skeleton, but Hn (SBn , Z) is not ﬁnitely generated. In the light of the discussion in Construction 3.1, one might anticipate that these examples are the tip of an iceberg of pathology akin to the wildness that we saw among the ﬁnitely generated subgroups of F × F . But Bridson and Miller [28] (cf. [25]) proved, roughly speaking, that variations on the construction of Stallings and Bieri account for all the ﬁnitely presented subdirect products of free groups. Recall that a subgroup of a direct product H < G1 × · · · × Gd is termed a subdirect product if its projection to each of the factors is onto, and H is said to be full if all of the intersections H ∩ Gi are non-trivial. Theorem 3.4 ([28]). If H < F1 × · · · × Fn = D is a full subdirect product of ﬁnitely generated free groups, then there is a subgroup of ﬁnite index D0 < D such that H contains the (n − 1)st term of the lower central series of D0 . 4. A new level of complication We now come to the cluster of new results concerning subgroups of mapping class groups. These theorems show that the ﬁnitely presented subgroups of mapping class groups can be vastly more complicated than those studied hitherto. The proofs of these theorems will be outlined in later sections, where they are used to illustrate a general technique for constructing wild subgroups in varied contexts. Readers unfamiliar with decision problems may wish to consult [56]. For an account of the history of the problems settled by the following theorems, see [40]. Theorem 4.1 ([16]). If the genus of S is suﬃciently large, then the isomorphism problem for the ﬁnitely presented subgroups of Mod(S) is unsolvable. In more detail, there is a recursive sequence Δi (i ∈ N) of ﬁnite subsets of Mod(S), together with ﬁnite presentations Δi | Θi of the subgroups they generate, such that there is no algorithm that can determine whether or not Δi | Θi ∼ = Δ0 | Θ0 . Theorem 4.2 ([16]). If the genus of S is suﬃciently large, then there is a ﬁnitely presented subgroup of Mod(S) with unsolvable conjugacy problem. Theorem 4.3 ([16]). If the genus of S is suﬃciently large, then there are ﬁnitely presented subgroups of Mod(S) for which the membership problem is unsolvable. The Dehn function of a ﬁnitely presented group Γ = A | R estimates the complexity of the word problem by counting the number of times one has to apply the deﬁning relations in order to prove that a word w in the generators represents the identity in the group: Area(w) is deﬁned to be the least integer N for which there is an equality N θi ri±1 θi−1 w= i=1

in the free group F (A), with ri ∈ R, and the Dehn function of A | R is δ(n) := max{Area(w) | w =Γ 1, |w| ≤ n},

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5. THE NILPOTENT GENUS OF A GROUP

387

where |w| denotes word-length. Mapping class groups have quadratic Dehn functions [60], as do the ﬁnitely presented subgroups described in the previous sections. Theorem 4.4 ([16]). If the genus of S is suﬃciently large, then there are ﬁnitely presented subgroups of Mod(S) whose Dehn functions are exponential. One might hope to prove some of these theorems by focussing on subgroups of direct products of free groups, but restrictions that follow from Theorem 3.4 dash this hope. The following theorem was proved in [28] and extended to all ﬁnitely presented subgroups of residually-free groups in [26]. Theorem 4.5 ([28]). The conjugacy problem is solvable for every ﬁnitely presented subgroup of a direct product of free (or surface) groups, and there is a uniform solution to the membership problem for all such subgroups. In Section 6 we consider how we might enlarge the class of direct products of free groups so as to obtain wilder subgroups (via ﬁbre product constructions) while retaining enough geometry to provide embeddings into mapping class groups. But ﬁrst we turn to a diﬀerent topic. 5. The nilpotent genus of a group This section is based on joint work with Alan Reid from the University of Texas [29]. If each ﬁnite subset of a group Γ injects into some nilpotent (or ﬁnite) quotient of Γ, then one expects to be able to detect many properties of Γ from the totality of its nilpotent (or ﬁnite) quotients. Which properties can be detected and which cannot? Forms of this question have stimulated a lot of research into discrete and proﬁnite groups over the last forty years, and there has been a particular resurgence of interest recently, marked by several notable breakthroughs. Here we focus on the nilpotent quotients. Recall that a group Γ is said to be residually nilpotent (resp. residually torsionfree nilpotent) if for each non-trivial γ ∈ Γ there exists a nilpotent (resp. torsion-free nilpotent) group Q and a homomorphism φ : Γ → Q with φ(γ) = 1. Thus Γ is residually nilpotent if and only if Γn = 1, where Γn is the n-th term of the lower central series of Γ, deﬁned inductively by setting Γ1 := Γ and Γn+1 := [x, y] : x ∈ Γn , y ∈ Γ. We say that two residually nilpotent groups Γ and Λ have the same nilpotent genus if they have the same nilpotent quotients; this is equivalent to requiring that Γ/Γc ∼ = Λ/Λc for all c ≥ 1. Examples 5.1. Examples of ﬁnitely generated residually torsion-free nilpotent groups include free groups Fn (hence residually free groups such as surface groups and limit groups), right-angled Artin groups (RAAGs) [38], the Torelli subgroup of the mapping class group [5], and IAn < Out(Fn ), the kernel of the natural map Out(Fn ) → GL(n, Z) (see [3], [5], [30]), and the corresponding subgroup in the outer automorphism group of any RAAG [66]. A group is termed parafree if it is residually nilpotent and has the same genus as a free group. The existence of families of parafree groups that are not free gives a ﬁrst inkling of the diversity that can exist within a ﬁxed nilpotent genus. One such family was discovered by Gilbert Baumslag (see [6]): Gij = a, b, c | a = [ci , a].[cj , b].

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388 MARTIN R. BRIDSON, CUBES, MAPPING CLASS GROUPS AND NILPOTENT GENUS

In [7] Baumslag surveyed the state of the art concerning groups of the same nilpotent genus and compiled a list of open problems that are of particular importance in the ﬁeld. Here we shall concentrate on three problems whose resolution will serve to emphasize how diﬀerent groups within a given genus can be. (Other problems on Baumslag’s list challenge the reader to establish commonalities across a genus; cf. [29] Theorem C.) Problems 5.2. Do there exist pairs of groups of the same nilpotent genus such that • one is ﬁnitely presented and the other is not; or • both are ﬁnitely presented, one has a solvable conjugacy problem but the other does not; or • one has ﬁnitely generated second homology H2 (−, Z) and the other does not? These questions are settled by the following compilation of results from [29]. Theorem 5.3. (1) There exist pairs of ﬁnitely presented, residually torsion-free nilpotent groups P → Γ of the same nilpotent genus such that Γ has a solvable conjugacy problem and P does not. (2) There exist pairs of ﬁnitely generated, residually torsion-free nilpotent groups N → Γ of the same nilpotent genus such that Γ is ﬁnitely presented while H2 (N, Q) is inﬁnite dimensional (so in particular N is not ﬁnitely presented). Remark 5.4. The pairs of groups that are constructed in [29] to prove this theorem have the additional property that the inclusion map induces an isomorphism of proﬁnite and pro-nilpotent completions. 5.1. Criteria for pro-nilpotent equivalence The following theorem of John Stallings [64] provides a useful criterion for establishing that groups have the same nilpotent genus: If a homomorphism of groups u : N → Γ induces an isomorphism on H1 (−, Z) and an epimorphism on H2 (−, Z), then uc : N/Nc → Γ/Γc is an isomorphism for all c ≥ 1. Given a short exact sequence of groups 1 → N → G → Q → 1, the LyndonHochschild-Serre (LHS) spectral sequence (which is explained on page 171 of [31]) calculates the homology of G in terms of the homology N and Q. The terms on the 2 = Hp (Q, Hq (N, Z)), where the action of E 2 page of the spectral sequence are Epq Q on H∗ (N, Z) is induced by the action of G on N by conjugation. The following proposition is proved in [29] by using this spectral sequence to see that N → Γ satisﬁes the hypotheses of Stallings’ theorem. u

Proposition 5.5. Let 1 → N → Γ → Q → 1 be a short exact sequence of groups and let uc : N/Nc → Γ/Γc be the homomorphism induced by u : N → Γ. Suppose that N is ﬁnitely generated, that Q has no non-trivial ﬁnite quotients, and that H2 (Q, Z) = 0. Then uc is an isomorphism for all c ≥ 1. In particular, if Γ is residually nilpotent then N and Γ have the same nilpotent genus. Corollary 5.6. Under the hypotheses of Proposition 5.5, the inclusion P → Γ × Γ of the ﬁbre product induces an isomorphism P/Pc → Γ/Γc × Γ/Γc for every nil × Γ nil . c ∈ N and hence an isomorphism Pnil → Γ

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6. CUBES, RAAGS AND CAT(0) i

389

j

Proof. We have inclusions N × N → P → Γ × Γ and the proposition implies that i and j ◦ i induce isomorphisms modulo any term of the lower central series, and therefore j does as well. 5.2. Relation with proﬁnite genus Let Γ be a ﬁnitely generated group. If one orders the normal subgroups of ﬁnite index N < Γ by reverse inclusion, then the quotients Γ/N form an inverse system whose limit = lim Γ/N Γ ←− nil is the proﬁnite completion of Γ. Similarly, the pro-nilpotent completion, denoted Γ is the inverse limit of the nilpotent quotients of Γ. Every homomorphism of discrete → Γ and a homomorphism groups u : H → G induces a homomorphism u ˆ : H u nil : Hnil → Γnil . If H, G are residually nilpotent, they lie in the same nilpotent genus if and only nil . If G is nilpotent then G = G nil ∼ nil . if H =G There are ﬁnitely generated nilpotent groups H ∼ G that have the same ﬁnite = = G but H nil ∼ nil . The situation is quite diﬀerent if the quotients; thus H = G isomorphism of proﬁnite completions is induced by a homomorphism of the discrete groups. Proposition 5.7 ([29]). Let u : P → Γ be a pair of ﬁnitely generated, residually ﬁnite groups, and for each c ≥ 1, let uc : P/Pc → Γ/Γc be the induced is an isomorphism, then uc is an isomorphism for homomorphism. If u : P → Γ nil . all c ≥ 1, hence Pnil ∼ =Γ With this observation in hand, we see that (modulo variations in the ﬁniteness assumptions) Proposition 5.5 is a weak form of the following proposition, which originates in the work of Platonov and Tavgen [61] where the ﬁrst pairs of ﬁnitely generated groups P → Γ satisfying the hypotheses of Proposition 5.7 were constructed. (Such pairs of groups are now known as Grothendieck pairs.) This proposition also played an important role in the Bridson-Grunewald construction of Grothendieck pairs where both P and Γ are ﬁnitely presented [23]. Proposition 5.8. Let 1 → N → Γ → Q → 1 be a short exact sequence of groups with Γ ﬁnitely generated and let P be the associated ﬁbre product. Suppose that Q = 1 is ﬁnitely presented, has no proper subgroups of ﬁnite index, and H2 (Q, Z) = 0. Then (1) P → Γ × Γ induces an isomorphism P → Γ × Γ; → Γ. (2) if N is ﬁnitely generated then N → Γ induces an isomorphism N 6. Cubes, RAAGs and CAT(0) In this section we shall see right-angled Artin groups (RAAGs) emerge as a generalisation of direct products of free groups. RAAGs have a similar cubical geometry to F × · · · × F and are residually torsion-free nilpotent; moreover it is easy to get them to act on surfaces. But, crucially for us, they harbour a much greater array of ﬁnitely presented subgroups. We take up the theme of Construction 3.3, retaining the notation. The original proofs of Stallings and Bieri are essentially algebraic. Bestvina and Brady [11] discovered a geometric proof that motivated their Morse theory for

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390 MARTIN R. BRIDSON, CUBES, MAPPING CLASS GROUPS AND NILPOTENT GENUS

cubical complexes. If we regard F as the fundamental group of a compact simplicial graph Y , then D = F × · · · × F is the fundamental group of X = Y × · · · × Y , which has a natural cubical structure. This cube complex is non-positively curved in the sense of Alexandrov, i.e. locally CAT(0). (The standard reference for CAT(0) spaces is [24], but the lectures of Caprace and Sageev in this volume cover more than enough for our needs.) The vertex set of the universal cover is D and the homomorphism h : D → Z ˜ → R. Bestvina can be extended linearly across cells to give a Morse function X and Brady [11] determine the ﬁniteness properties of the kernel of h by examining the way in which the sublevel sets of this Morse function change as one passes through critical points (vertices). They extended this analysis to the larger class of cubical complexes deﬁned below, and in this way settled long-standing questions concerning the relationship between diﬀerent ﬁniteness properties of groups. A right angled Artin group (RAAG) is a group given by a presentation of the form A = v1 , . . . , vn | [vi , vj ] = 1 ∀(i, j) ∈ E. Thus A is encoded by a graph with vertex set {v1 , . . . , vn } and edge set E ⊂ V × V . The prototype F2 ×· · ·×F2 is the RAAG associated to the 1-skeleton of the simplicial join S0 ∗· · ·∗S0 . The Salvetti complex is the classifying space for A obtained by gluing standard tori (cubes with opposite faces identiﬁed) along coordinate faces according to the commuting relations in the presentation; it has non-positive curvature. I have portrayed right angled Artin groups (RAAGs) as a natural generalisation of direct products of free groups, but in many ways this fails to do them justice. They have gained prominence in recent years as an extremely important class of groups whose simple description belies their rich structure. From the point of view of this article, their three most important features are the richness of their subgroup structure, the ease with which they can be made to act on a great range of objects, and their residual properties. 6.1. RAAGs everywhere Whenever one has n automorphisms αi of an object X, some of which commute, say [αi , αj ] = 1 if (i, j) ∈ E, then one has an action of the RAAG associated to the n-vertex graph with edge-set E. Roughly speaking, this action will be faithful if the αi that do not commute are unrelated. One such setting is that of surface automorphisms: if two simple closed curves on a surface are disjoint, then the Dehn twists in those curves commute, but if one has a set of curves no pair of which can be homotoped oﬀ each other, then suitable powers of the twists in those curves freely generate a free group. (Signiﬁcantly sharper results of this sort are proved in [52] and [33].) It follows that any RAAG can be embedded in the mapping class group of any surface S of suﬃciently high genus: it suﬃces that the dual of the graph deﬁning A can be embedded in S. (This is explained by Crisp and Wiest in [35].) The surface can have boundary and punctures. With more care, one can arrange for the embedding of the RAAG to lie in the Torelli subgroup T (S) < Mod(S) (cf. Koberda [52]). Proposition 6.1. Every RAAG embeds in the mapping class group of any surface of suﬃciently high genus.

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6. CUBES, RAAGS AND CAT(0)

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6.2. Some properties of RAAGs With the preceding proposition in hand, we see that the theorems stated in Section 4 will follow if we can construct RAAGs with subgroups of the desired kind. Similarly, we shall solve Baumslag’s problems (5.2) by constructing suitable RAAGs and exploiting the following theorem proved by Droms in his thesis [38] (cf. [39] and [67]). Theorem 6.2 ([38]). RAAGs are residually torsion-free nilpotent. Further important properties of RAAGs include the fact that they are linear [49] even over Z [37]; they are conjugacy separable [58]; they are RFRS in the sense of Agol [2]; and their quasi-convex subgroups are virtual retracts (and so are closed in the proﬁnite topology) [46]. 6.3. Special cube complexes What has really brought RAAGs to the fore in recent years is the richness of their subgroup structure. We saw hints of this in the work of Bestvina and Brady, but the spectacular extent of this richness truly emerged from the theory of special cube complexes initiated by Haglund and Wise [47] and advanced in many subsequent papers, particularly by Wise and his co-authors. (The lectures of Sageev have much more on this.) Definition 6.3. A non-positively curved cube complex X is special if it admits a locally isometric embedding into the Salvetti complex K(A, 1) of a RAAG A. Remark 6.4. A locally isometric map between compact non-positively curved spaces induces an injective map on fundamental groups ([24], p.201), so the fundamental groups of special cube complexes are subgroups of RAAGs. The deﬁnition of special is not a very practical one, but Haglund and Wise [47] prove that it is equivalent to a short list of conditions on the behaviour of hyperplanes in the given cube complex (see Sageev’s lectures). Theorem 6.5 ([47]). A non-positively curved cube complex is special if and only if its hyperplanes are 2-sided, do not self-cross, do not self-osculate, and do not inter-osculate. This remarkable insight makes it possible to verify specialness (of X or some ﬁnite cover of it) in many instances. Thus we have a putative machine for constructing interesting subgroups of RAAGs (hence residually torsion-free nilpotent groups that are subgroups of mapping class groups): • Cubulate groups, i.e. ﬁnd methods for exhibiting large classes of groups as fundamental groups of compact non-positively curved cube complexes. (This is the central theme of Sageev’s lectures.) • Use the Haglund-Wise criterion to prove that these cube complexes X are special, or at least that ﬁnite-sheeted covers of them are special (i.e. X is virtually special). This programme, widely promoted by Dani Wise, has proved extremely successful. Two results are of particular importance for our purposes, one for each of the steps articulated above. To state the ﬁrst, we need the vocabulary of small cancellation groups.

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392 MARTIN R. BRIDSON, CUBES, MAPPING CLASS GROUPS AND NILPOTENT GENUS

The symmetrisation R∗ of a set of words R over an alphabet X consists of all cyclic permutations of w and w−1 with w ∈ R. The set R and the presentation X | R are said to satisfy the C (λ) small cancellation condition if pieces (i.e. recurring subwords) are bounded in length: if there exist distinct w, w ∈ R∗ such that w ≡ uv and w ≡ uv , then |u| < λ|w|. If R is ﬁnite and λ ≤ 1/6, then Γ∼ = X | R is hyperbolic (in the sense of Gromov) and torsion-free3 . Theorem 6.6 ([68]). If a group G has a ﬁnite presentation that is C (1/6), then G acts properly and cocompactly on a CAT(0) cube complex. Many groups are now proved to be virtually special. The crowning achievment, following much work of Wise [69] and others, is Agol’s theorem. Theorem 6.7 ([1]). Let X be a compact non-positively curved cube complex. If π1 X is hyperbolic, then X is virtually special. This theorem has many consequences – the most important to date being Agol’s resolution of the virtual ﬁbering conjecture for hyperbolic 3-manifolds (cf. Section 11)– but the two that concern us here are the following. Corollary 6.8. If a hyperbolic group H is the fundamental group of a compact non-positively curved cube complex, then H embeds in the mapping class group of inﬁnitely many (closed) surfaces, and some subgroup of ﬁnite-index H0 < H is residually torsion-free nilpotent. Proof. The ﬁrst assertion follows from Proposition 6.1 and Corollary 2.2, the second from Theorem 6.2. 6.4. The mapping class genus of virtually special groups Corollary 6.8 assures us that the following quantity, which one might call the mapping class genus, is a well-deﬁned invariant of hyperbolic groups that can be cubulated. It seems diﬃcult to compute but may well provide a rich ﬁeld of exploration. The case of Kleinian groups is already intriguing. Here, Sg denotes the closed orientable surface of genus g. mcg(Γ) := min{g | ∃ Γ → Mod(Sg )}. 7. Rips, ﬁbre products and 1-2-3 We take up the theme of Construction 3.1. The lack of ﬁnite presentability in the ﬁbre products that we constructed P < F × F can be traced to the fact that a non-trivial normal subgroup of inﬁnite index in a free group cannot be ﬁnitely generated. The key to getting around this problem is to express groups as quotients of hyperbolic groups instead of free groups, with a gain in the ﬁniteness properties of the kernel. This idea is due to E. Rips [62]. Theorem 7.1 ([62]). There is an algorithm that, given a ﬁnite group-presentation Q, will construct a short exact sequence q

1→N →H→Q→1 where H is C (1/6) small-cancellation, Q is the group with presentation Q, and N is a 2-generator group. 3 In saying this, I’m skipping over a technicality about how to treat words in R that are proper powers.

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7. RIPS, FIBRE PRODUCTS AND 1-2-3

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Remark 7.2. The discussion in the previous section shows that H is virtually special. In fact, one can arrange this more directly: the Rips construction is very ﬂexible and variations by diﬀerent authors have imposed extra conditions on H; before Agol’s work, Haglund and Wise [47] used this ﬂexibility to arrange for H to be virtually special (at the cost of adding more generators to N ). In the spirit of Construction 3.1, we focus on the ﬁbre product P = {(h, h ) ∈ H × H | q(h) = q(h )}. We want P to be ﬁnitely presented, but in general it will not be (cf. Lemma 7.7). However, if Q is of type F3 (i.e., has a classifying space K(Q, 1) with ﬁnite 3-skeleton), then the 1-2-3 Theorem of [8] assures us that P will be ﬁnitely presented. q

Theorem 7.3 ([8]). Let 1 → N → Γ → Q → 1 be a short exact sequence of groups. If N is ﬁnitely generated, Γ is ﬁnitely presented, and Q is of type F3 , then the associated ﬁbre product P < Γ × Γ is ﬁnitely presented. One may wonder if N can also be made ﬁnitely presented, but it cannot. Lemma 7.4. If Q is inﬁnite, then the subgroup N < Γ in the Rips construction is not ﬁnitely presented. Proof. Being a small cancellation group, Γ has cohomological dimension 2. Bieri [13] proved that a ﬁnitely presented normal subgroup of inﬁnite index in a group of cohomological dimension 2 must be free. But in the Rips construction N is visibly not free: it is a 2-generator group that has non-trivial relations, and it is not cyclic because non-elementary hyperbolic groups do not have cyclic normal subgroups. 7.1. Finite and nilpotent quotients Proposition 5.5 and its corollary show that if Q has no ﬁnite quotients and H2 (Q, Z) = u 0 then the pair of groups N → Γ produced by the Rips construction is such that u induces an isomorphism modulo each term of the lower central series, and so does the inclusion of the ﬁbre product P → Γ × Γ. Likewise, Proposition 5.8 tells us that these inclusions induce isomorphisms of proﬁnite completions. 7.2. Rips translates foibles from Q to N and P The translation of properties from Q to N and P has to be analysed according to context. The ones that interest us here concern ﬁnite and nilpotent quotients (as described above), decision problems and ﬁniteness conditions. If Q has an unsolvable word problem, this manifests itself in unsolvable decision problems of a diﬀerent type for N and P . p

Proposition 7.5 ([8]). Let 1 → N → Γ → Q → 1 be a short exact sequence of groups, with Γ ﬁnitely generated, and let P < Γ × Γ be the associated ﬁbre product. If the word problem in Q is unsolvable, then the membership problem for P < Γ × Γ is unsolvable. Proof. We ﬁx a ﬁnite generating set X for Γ and work with the generators X = {(x, 1), (1, x) | x ∈ X} for Γ × Γ. Given a word w = x1 . . . xn in the free group on X, we consider the word (x1 , 1) . . . (xn , 1) in the free group on X . This word deﬁnes an element of P if and only if p(w) = 1 in Q, and we are assuming that there is no algorithm that can determine which words in the symbols p(x) equal the identity in Q.

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394 MARTIN R. BRIDSON, CUBES, MAPPING CLASS GROUPS AND NILPOTENT GENUS p

Proposition 7.6 ([8]). Let 1 → N → Γ → Q → 1 be a short exact sequence of groups, with Γ torsion-free and hyperbolic, and let P < Γ × Γ be the associated ﬁbre product. If the word problem in Q is unsolvable, then the conjugacy problem is unsolvable in N and in P . Sketch of proof: Fix ﬁnite generating sets B for Γ and A for N . Fix a ∈ N {1}. For each b ∈ B and = ±1, let ub, be a word in the free group on A so that b ab− = ub, in G. Given an arbitrary word w in the letters B, one can use the relations b ab− = ub, to convert waw−1 into a word w in the letters A. Now ask if w is conjugate to a in N . The answer is “yes” if w ∈ N , and consideration of centralizers shows that it is “no” if w ∈ N . Thus w is conjugate to a in N if and only if p(w) = 1 in Q, and we are assuming that there is no algorithm that can decide if this is the case. The argument for P is similar but more involved; see [8, Section 3]. The following lemma is proved using the LHS spectral sequence; see [29, Section 6]. Lemma 7.7. Let 1 → N → G → Q → 1 be a short exact sequence of ﬁnitely generated groups. If H3 (G, Q) is ﬁnite dimensional but H3 (Q, Q) is inﬁnite dimensional, then H2 (N, Q) is inﬁnite dimensional. 8. Examples template The following template can be employed in any context where one is interested in demonstrating diverse or extreme behaviour among the ﬁnitely presented subgroups of groups in a class of groups C with the property that every RAAG embeds in some Γ ∈ C. • Feed designer groups Q into the Rips construction to obtain 1 → N → H → Q → 1. • Pass to a subgroup of ﬁnite index in H to obtain q

1 → N0 → H0 → Q0 → 1, N0 ﬁnitely generated, H0 special (subgroup of a RAAG) and Q0 < Q ﬁnite-index. • Pass to the ﬁbre product P = {(h, h ) ∈ H0 × H0 | q(h) = q(h )} and note that P is a subgroup of a RAAG. • Note that by the 1-2-3 Theorem, P is ﬁnitely presented4 if Q is of type F3 . • Embed the RAAG containing P in Γ ∈ C. This is a rather general and loosely stated template, but it does have remarkably wide applicability. We’ll use it to resolve the problems for mapping class groups and nilpotent genus that are the focus of our story. But to make use of the template one has to resolve the following diﬃculties: 4 If

there is an algorithm to construct a ﬁnite 3-skeleton for a K(Q, 1), then one can construct a ﬁnite presentation for P in an algorithmic manner, but this requires further argument [26].

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9. PROOFS FROM THE TEMPLATE

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• If you are interested in constructing ﬁnitely presented groups P < H0 × H0 with some property P then you must ﬁrst identify a related property P such that if Q has P then P < H0 × H0 has P (cf. subsection 7.2). • You have to prove that there exist groups of type F3 with property P. • You have to ensure that P is inherited by subgroups of ﬁnite index Q0 < Q; this might be inherent to P but you might have to control the ﬁnite-index subgroups of Q, perhaps arranging that there are none other than Q itself; cf. [21] and [23]. 8.1. Crafting designer groups The construction of input groups Q for the template is very particular to the situation at hand and typically requires ad hoc innovation. In order to prove Theorems 4.2, 4.3 and 5.3 we need groups with the following properties: Examples 8.1. (1) There are inﬁnite groups of type F3 that have no non-trivial ﬁnite quotients. The ﬁrst such examples were constructed by Graham Higman [48] and a general method for constructing such groups is described in [21] and [23]. (2) There are groups of type F3 that have an unsolvable word problem. Examples of this sort were constructed by Collins and Miller [34]. (3) There exist inﬁnite groups Q of type F3 that, simultaneously, have no nontrivial ﬁnite quotients, have an unsolvable word problem, and H2 (Q, Z) = 0; see [22, Theorem 3.1]. (4) There are ﬁnitely presented groups Δ with no non-trivial ﬁnite quotients, ˜ Q) = ∞. Such groups are so that Hi (Δ, Z) = 0 for i = 1, 2 but dim H3 (Δ, constructed in [29]. 9. Proofs from the template Proposition 6.1 and Corollary 2.2 tell us that Theorems 4.3 and 4.2 will follow if we can prove the same results for subgroups of RAAGs or virtually special groups. Likewise, since RAAGs are residually torsion-free nilpotent, in order to prove Theorem 5.3 it is enough to exhibit RAAGs with the stated properties. In each case, we shall use the Examples Template to construct suitable RAAGs. I shall state the key points and I encourage the reader to check the details. 9.1. Proof of Theorems 4.3 and 4.2 Let property P be the insolubility of the word problem. Apply the template with Q as in Example 8.1(2) and appeal to Propositions 7.5 and 7.6. 9.2. Proof of Theorem 5.3(1) Let property P be the insolubility of the word problem. Apply the template with Q as in Example 8.1(3) and appeal to Corollary 5.6 and Proposition 7.6. 9.3. Proof of Theorem 5.3(2) Apply the template with Q as in Example 8.1(4) and appeal to Proposition 5.5 and Lemma 7.7.

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396 MARTIN R. BRIDSON, CUBES, MAPPING CLASS GROUPS AND NILPOTENT GENUS

10. The isomorphism problem for subgroups of RAAGs and Mod(S) To prove Theorem 4.1 we use a criterion due to Bridson and Miller [27]. Theorem 10.1 ([27]). Let 1 → N → Γ → L → 1 be an exact sequence of groups. Suppose that (1) Γ is torsion-free and hyperbolic, (2) N is inﬁnite and ﬁnitely generated, and (3) L is a non-abelian free group. If F is a non-abelian free group, then the isomorphism problem for ﬁnitely presented subgroups of Γ × Γ × F is unsolvable. In more detail, there is a recursive sequence Δi (i ∈ N) of ﬁnite subsets of Γ×Γ×F , together with ﬁnite presentations Δi | Θi of the subgroups they generate, such that there is no algorithm that can determine whether or not Δi | Θi ∼ = Δ0 | Θ0 . The subgroup Gi presented by Δi | Θi is obtained by ﬁxing a splitting Γ = N L and deﬁning Gi to be the subgroup generated by N × N and {(φi (x), x) | x ∈ F } where φi : F → L × L is a homomorphism whose image is a subdirect product. A key feature of the construction in [27] is that the ﬁniteness properties of centralisers in Gi are intimately connected to the question of whether φi is onto. We saw in Section 3 that there is no algorithm that can determine if a ﬁnite subset of a direct product of free groups generates the product, and this provides a seed of undecidability that propogates through the construction. Corollary 10.2. There exist RAAGs in which the isomorphism problem for ﬁnitely presented subgroups is unsolvable. Proof. We apply the template of Section 8 with Q a non-abelian free group and deﬁne Γ = H0 . If H0 is a subgroup of the RAAG A, then Γ × Γ × F will be a subgroup of the RAAG A × A × F . In the light of Proposition 6.1, Theorem 4.1 follows from this corollary. Remark 10.3. Baumslag and Miller [9] proved that the isomorphism problem is unsolvable in the class of ﬁnitely presented residually torsion-free nilpotent groups. Corollary 10.2 provides an alternative proof. 11. Dehn functions One can prove Theorem 4.4 without the Rips construction or the 1-2-3 Theorem: one can deduce it directly from the fact that the fundamental groups of closed hyperbolic 3-manifolds are virtually special, applying Proposition 6.1 to embed the RAAG A of the following proposition into mapping class groups. Proposition 11.1. There exist right-angled Artin groups A and ﬁnitely presented subgroups P < A such that P has an exponential Dehn function. Proof. Let M be a closed, orientable, hyperbolic 3-manifold that ﬁbres over the circle. Then π1 M = Σ Z where Σ is the fundamental group of a closed surface of genus at least 2, and Γ = π1 M × π1 M contains P := (Σ × Σ) Z, the inverse image of the diagonal in Γ/(Σ×Σ) = Z×Z. The Dehn function of P is exponential; see [19] Theorem 2.5. The growth of a Dehn function is preserved on passage to subgroups of ﬁnite index, and by [1] there is a subgroup of ﬁnite index in Γ that embeds in a right-angled Artin group.

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BIBLIOGRAPHY

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398 MARTIN R. BRIDSON, CUBES, MAPPING CLASS GROUPS AND NILPOTENT GENUS

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Published Titles in This Series 21 Mladen Bestvina, Michah Sageev, and Karen Vogtmann, Editors, Geometric Group Theory, 2014 20 Benson Farb, Richard Hain, and Eduard Looijenga, Editors, Moduli Spaces of Riemann Surfaces, 2013 19 Hongkai Zhao, Editor, Mathematics in Image Processing, 2013 18 Cristian Popescu, Karl Rubin, and Alice Silverberg, Editors, Arithmetic of L-functions, 2011 17 Jeﬀery McNeal and Mircea Mustat ¸˘ a, Editors, Analytic and Algebraic Geometry, 2010 16 Scott Sheﬃeld and Thomas Spencer, Editors, Statistical Mechanics, 2009 15 Tomasz S. Mrowka and Peter S. Ozsv´ ath, Editors, Low Dimensional Topology, 2009 14 Mark A. Lewis, Mark A. J. Chaplain, James P. Keener, and Philip K. Maini, Editors, Mathematical Biology, 2009 13 Ezra Miller, Victor Reiner, and Bernd Sturmfels, Editors, Geometric Combinatorics, 2007 12 Peter Sarnak and Freydoon Shahidi, Editors, Automorphic Forms and Applications, 2007 11 Daniel S. Freed, David R. Morrison, and Isadore Singer, Editors, Quantum Field Theory, Supersymmetry, and Enumerative Geometry, 2006 10 Steven Rudich and Avi Wigderson, Editors, Computational Complexity Theory, 2004 9 Brian Conrad and Karl Rubin, Editors, Arithmetic Algebraic Geometry, 2001 8 Jeﬀrey Adams and David Vogan, Editors, Representation Theory of Lie Groups, 2000 7 Yakov Eliashberg and Lisa Traynor, Editors, Symplectic Geometry and Topology, 1999 6 Elton P. Hsu and S. R. S. Varadhan, Editors, Probability Theory and Applications, 1999 5 Luis Caﬀarelli and Weinan E, Editors, Hyperbolic Equations and Frequency Interactions, 1999 4 Robert Friedman and John W. Morgan, Editors, Gauge Theory and the Topology of Four-Manifolds, 1998 3 J´ anos Koll´ ar, Editor, Complex Algebraic Geometry, 1997 2 Robert Hardt and Michael Wolf, Editors, Nonlinear partial diﬀerential equations in diﬀerential geometry, 1996 1 Daniel S. Freed and Karen K. Uhlenbeck, Editors, Geometry and Quantum Field Theory, 1995

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Geometric group theory refers to the study of discrete groups using tools from topology, geometry, dynamics and analysis. The field is evolving very rapidly and the present volume provides an introduction to and overview of various topics which have played critical roles in this evolution. The book contains lecture notes from courses given at the Park City Math Institute on Geometric Group Theory. The institute consists of a set of intensive short courses offered by leaders in the field, designed to introduce students to exciting, current research in mathematics. These lectures do not duplicate standard courses available elsewhere. The courses begin at an introductory level suitable for graduate students and lead up to currently active topics of research. The articles in this volume include introductions to CAT(0) cube complexes and groups, to modern small cancellation theory, to isometry groups of general CAT(0) spaces, and a discussion of nilpotent genus in the context of mapping class groups and CAT(0) groups. One course surveys quasi-isometric rigidity, others contain an exploration of the geometry of Outer space, of actions of arithmetic groups, lectures on lattices and locally symmetric spaces, on marked length spectra and on expander graphs, Property tau and approximate groups. This book is a valuable resource for graduate students and researchers interested in geometric group theory.

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