Geometric Group Theory: An Introduction

390 Pages·2017·3.257 MB·English

by  Clara Loh

Universitext
Clara Löh
Geometric
Group
Theory
An Introduction
Universitext
Universitext
Serieseditors
SheldonAxler
SanFranciscoStateUniversity
CarlesCasacuberta
UniversitatdeBarcelona
AngusMacIntyre
QueenMary,UniversityofLondon
KennethRibet
UniversityofCalifornia,Berkeley
ClaudeSabbah
Écolepolytechnique,CNRS,UniversitéParis-Saclay,Palaiseau
EndreSüli
UniversityofOxford
WojborA.Woyczyn´ski
CaseWesternReserveUniversity
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Clara Löh
Geometric Group Theory
An Introduction
Clara Löh
Fakultät für Mathematik
Universität Regensburg
Regensburg, Germany

ISSN 0172-5939 ISSN 2191-6675 (electronic)
Universitext
ISBN 978-3-319-72253-5 ISBN 978-3-319-72254-2 (eBook)
http://doi.org/10.1007/978-3-319-72254-2

Library of Congress Control Number: 2017962076
Mathematics Subject Classification (2010): 20F65, 20F67, 20F69, 20F05, 20F10, 20E08, 20E05, 20E06,
20Fxx, 20Exx, 57M07, 53C23, 53C24, 20G15, 05C25

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About this book
Thisbookisanintroductiontogeometric group theory.Itiscertainlynotan
encyclopedic treatment of geometric group theory, but hopefully it will pre-
pareandencouragethereadertotakethenextstepandlearnmoreadvanced
aspects of the subject.
The core material of the book should be accessible to third year students,
requiring only a basic acquaintance with group theory, metric spaces, and
point-set topology. I tried to keep the level of the exposition as elementary
as possible, preferring elementary proofs over arguments that require more
machinery in topology or geometry. I refrained from adding complete proofs
for some of the deeper theorems and instead included sketch proofs, high-
lighting the key ideas and the view towards applications. However, many of
theapplicationswillneedamoreextensivebackgroundinalgebraictopology,
Riemannian geometry, and algebra.
Theexercisesareratedindi(cid:14)culty,fromeasy*overmedium**tohard***.
And very hard1* (usually, open problems of some sort). The core exercises
should be accessible to third year students, but some of the exercises aim at
applications in other (cid:12)elds and hence require a background in these (cid:12)elds.
Moreover,thereareexercisesectionsthatdevelopadditionaltheoryinaseries
of exercises; these exercise sections are marked with a +.
This book covers slightly more than a one-semester course. Most of the
material originates from various courses and seminars I taught at the Uni-
versit(cid:127)at Regensburg: the geometric group theory courses (2010 and 2014),
the seminar on amenable groups (2011), the course on linear groups and
heights (2015, together with Walter Gubler), and an elementary course on
geometry(2016).Mostofthestudentshadabackgroundinrealandcomplex
analysis, in linear algebra, algebra, and some basic geometry of manifolds;
some of the students also had experience in algebraic topology and Rieman-
nian geometry. I would like to thank the participants of these courses and
seminars for their interest in the subject and their patience.
I am particularly grateful to Toni Annala, Matthias Blank, Luigi Ca-
puti, Francesca Diana, Alexander Engel, Daniel Fauser, Stefan Friedl, Wal-
ter Gubler, Michal Marcinkowski, Andreas Thom, Johannes Witzig, and the
anonymousrefereesformanyvaluablesuggestionsandcorrections.Thiswork
was supported by the GRK 1692 Curvature, Cycles, and Cohomology (Uni-
versit(cid:127)at Regensburg, funded by the DFG).
Regensburg, September 2017 Clara L(cid:127)oh
v
Contents
1 Introduction 1
Part I Groups 7
2 Generating groups 9
2.1 Review of the category of groups 10
2.1.1 Abstractgroups:axioms 10
2.1.2 Concretegroups:automorphismgroups 12
2.1.3 Normalsubgroupsandquotients 16
2.2 Groups via generators and relations 19
2.2.1 Generatingsetsofgroups 19
2.2.2 Freegroups 20
2.2.3 Generatorsandrelations 25
2.2.4 Finitelypresentedgroups 29
2.3 New groups out of old 31
2.3.1 Productsandextensions 32
2.3.2 Freeproductsandamalgamatedfreeproducts 34
2.E Exercises 39
vii
viii Contents
Part II Groups ! Geometry 51
3 Cayley graphs 53
3.1 Review of graph notation 54
3.2 Cayley graphs 57
3.3 Cayley graphs of free groups 61
3.3.1 Freegroupsandreducedwords 62
3.3.2 Freegroups!trees 65
3.3.3 Trees!freegroups 66
3.E Exercises 68
4 Group actions 75
4.1 Review of group actions 76
4.1.1 Freeactions 77
4.1.2 Orbitsandstabilisers 80
4.1.3 Application:Countingviagroupactions 83
4.1.4 Transitiveactions 84
4.2 Free groups and actions on trees 86
4.2.1 Spanningtreesforgroupactions 87
4.2.2 ReconstructingaCayleytree 88
4.2.3 Application:Subgroupsoffreegroupsarefree 92
4.3 The ping-pong lemma 95
4.4 Free subgroups of matrix groups 97
4.4.1 Application:ThegroupSL(2;Z)isvirtuallyfree 97
4.4.2 Application:Regulargraphsoflargegirth 100
4.4.3 Application:TheTitsalternative 102
4.E Exercises 105
5 Quasi-isometry 115
5.1 Quasi-isometry types of metric spaces 116
5.2 Quasi-isometry types of groups 122
5.2.1 Firstexamples 125
5.3 Quasi-geodesics and quasi-geodesic spaces 127
5.3.1 (Quasi-)Geodesicspaces 127
5.3.2 Geodesi(cid:12)cationviageometricrealisationofgraphs 128
5.4 The S(cid:20)varc{Milnor lemma 132
5.4.1 Application:(Weak)commensurability 137
5.4.2 Application:Geometricstructuresonmanifolds 139
5.5 The dynamic criterion for quasi-isometry 141
5.5.1 Application:Comparinguniformlattices 146
5.6 Quasi-isometry invariants 148
5.6.1 Quasi-isometryinvariants 148
5.6.2 Geometricpropertiesofgroupsandrigidity 150
5.6.3 Functorialquasi-isometryinvariants 151
5.E Exercises 156
Contents ix
Part III Geometry of groups 165
6 Growth types of groups 167
6.1 Growth functions of (cid:12)nitely generated groups 168
6.2 Growth types of groups 170
6.2.1 Growthtypes 171
6.2.2 Growthtypesandquasi-isometry 172
6.2.3 Application:Volumegrowthofmanifolds 176
6.3 Groups of polynomial growth 179
6.3.1 Nilpotentgroups 180
6.3.2 Growthofnilpotentgroups 181
6.3.3 Polynomialgrowthimpliesvirtualnilpotence 182
6.3.4 Application:Virtualnilpotenceisgeometric 184
6.3.5 Moreonpolynomialgrowth 185
6.3.6 Quasi-isometryrigidityoffreeAbeliangroups 186
6.3.7 Application:Expandingmapsofmanifolds 187
6.4 Groups of uniform exponential growth 188
6.4.1 Uniformexponentialgrowth 188
6.4.2 Uniformuniformexponentialgrowth 190
6.4.3 TheuniformTitsalternative 190
6.4.4 Application:TheLehmerconjecture 192
6.E Exercises 194
7 Hyperbolic groups 203
7.1 Classical curvature, intuitively 204
7.1.1 Curvatureofplanecurves 204
7.1.2 CurvatureofsurfacesinR3 205
7.2 (Quasi-)Hyperbolic spaces 208
7.2.1 Hyperbolicspaces 208
7.2.2 Quasi-hyperbolicspaces 210
7.2.3 Quasi-geodesicsinhyperbolicspaces 213
7.2.4 Hyperbolicgraphs 219
7.3 Hyperbolic groups 220
7.4 The word problem in hyperbolic groups 224
7.4.1 Application:\Solving"thewordproblem 225
7.5 Elements of in(cid:12)nite order in hyperbolic groups 229
7.5.1 Existence 229
7.5.2 Centralisers 235
7.5.3 Quasi-convexity 241
7.5.4 Application:Productsandnegativecurvature 245
7.6 Non-positively curved groups 246
7.E Exercises 250
x Contents
8 Ends and boundaries 257
8.1 Geometry at in(cid:12)nity 258
8.2 Ends 259
8.2.1 Endsofgeodesicspaces 259
8.2.2 Endsofquasi-geodesicspaces 262
8.2.3 Endsofgroups 264
8.3 The Gromov boundary 267
8.3.1 TheGromovboundaryofquasi-geodesicspaces 267
8.3.2 TheGromovboundaryofhyperbolicspaces 269
8.3.3 TheGromovboundaryofgroups 270
8.3.4 Application:Freesubgroupsofhyperbolicgroups 271
8.4 Application: Mostow rigidity 277
8.E Exercises 280
9 Amenable groups 289
9.1 Amenability via means 290
9.1.1 Firstexamplesofamenablegroups 290
9.1.2 Inheritanceproperties 292
9.2 Further characterisations of amenability 295
9.2.1 F(cid:28)lnersequences 295
9.2.2 Paradoxicaldecompositions 298
9.2.3 Application:TheBanach{Tarskiparadox 300
9.2.4 (Co)Homologicalcharacterisationsofamenability 302
9.3 Quasi-isometry invariance of amenability 304
9.4 Quasi-isometry vs. bilipschitz equivalence 305
9.E Exercises 309
Part IV Reference material 317
A Appendix 319
A.1 The fundamental group 320
A.1.1 Constructionandexamples 320
A.1.2 Coveringtheory 322
A.2 Group (co)homology 325
A.2.1 Construction 325
A.2.2 Applications 327
A.3 The hyperbolic plane 329
A.3.1 Constructionofthehyperbolicplane 329
A.3.2 Lengthofcurves 330
A.3.3 Symmetryandgeodesics 332
A.3.4 Hyperbolictriangles 341
A.3.5 Curvature 346
A.3.6 Othermodels 347
A.4 An invitation to programming 349