Clara L¨oh Geometric Group Theory An Introduction this is a draft version! Clara L¨oh [email protected] http://www.mathematik.uni-regensburg.de/loeh/ Fakult¨atf¨urMathematik Universit¨atRegensburg 93040 Regensburg Germany All drawings by Clara L¨oh this is a draft version! f¨urA ∗ A this is a draft version! this is a draft version! About this book This book is an introduction into geometric group theory. It is certainly not an encyclopedic treatment of geometric group theory, but hopefully it will prepare and encourage the reader to take the next step and learn more ad- vanced aspects of geometric group theory. 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 the applications will need a more extensive background in algebraic topology, Riemannian geometry, and algebra. The exercises are rated in difficulty, from easy* over medium** to hard***. 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 fields and hence require a background in these fields. Moreover, there are exercise sections that develop additional theory in a series of exercises; these exercise sections are marked with +. 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¨atRegensburg: 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). Most of the students had a background in real and complex 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, Micha lMarcinkowski, Andreas Thom, Johannes Witzig, and the anonymous referees for many valuable suggestions and corrections. This work was supported by the GRK 1692 Curvature, Cycles, and Cohomology (Uni- versit¨atRegensburg, funded by the DFG). Regensburg, September 2017 Clara L¨oh this is a draft version! this is a draft version! Contents 1 Introduction 1 Part I Groups 7 2 Generating groups 9 2.1 Review of the category of groups 10 2.1.1 Abstract groups: axioms 10 2.1.2 Concrete groups: automorphism groups 12 2.1.3 Normal subgroups and quotients 16 2.2 Groups via generators and relations 19 2.2.1 Generating sets of groups 19 2.2.2 Free groups 20 2.2.3 Generators and relations 25 2.2.4 Finitely presented groups 29 2.3 New groups out of old 31 2.3.1 Products and extensions 32 2.3.2 Free products and amalgamated free products 34 2.E Exercises 39 this is a draft version! 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 Free groups and reduced words 62 3.3.2 Free groups ! trees 65 3.3.3 Trees ! free groups 66 3.E Exercises 68 4 Group actions 75 4.1 Review of group actions 76 4.1.1 Free actions 77 4.1.2 Orbits and stabilisers 80 4.1.3 Application: Counting via group actions 83 4.1.4 Transitive actions 84 4.2 Free groups and actions on trees 86 4.2.1 Spanning trees for group actions 87 4.2.2 Reconstructing a Cayley tree 88 4.2.3 Application: Subgroups of free groups are free 92 4.3 The ping-pong lemma 95 4.4 Free subgroups of matrix groups 97 4.4.1 Application: The group SL(2; Z) is virtually free 97 4.4.2 Application: Regular graphs of large girth 100 4.4.3 Application: The Tits alternative 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 First examples 125 5.3 Quasi-geodesics and quasi-geodesic spaces 127 5.3.1 (Quasi-)Geodesic spaces 127 5.3.2 Geodesification via geometric realisation of graphs 128 5.4 The Svarc-Milnorˇ lemma 132 5.4.1 Application: (Weak) commensurability 137 5.4.2 Application: Geometric structures on manifolds 139 5.5 The dynamic criterion for quasi-isometry 141 5.5.1 Application: Comparing uniform lattices 146 5.6 Quasi-isometry invariants 148 5.6.1 Quasi-isometry invariants 148 5.6.2 Geometric properties of groups and rigidity 150 5.6.3 Functorial quasi-isometry invariants 151 5.E Exercises 156 this is a draft version! Contents ix Part III Geometry of groups 165 6 Growth types of groups 167 6.1 Growth functions of finitely generated groups 168 6.2 Growth types of groups 170 6.2.1 Growth types 171 6.2.2 Growth types and quasi-isometry 172 6.2.3 Application: Volume growth of manifolds 176 6.3 Groups of polynomial growth 179 6.3.1 Nilpotent groups 180 6.3.2 Growth of nilpotent groups 181 6.3.3 Polynomial growth implies virtual nilpotence 182 6.3.4 Application: Virtual nilpotence is geometric 184 6.3.5 More on polynomial growth 185 6.3.6 Quasi-isometry rigidity of free Abelian groups 186 6.3.7 Application: Expanding maps of manifolds 187 6.4 Groups of uniform exponential growth 188 6.4.1 Uniform exponential growth 188 6.4.2 Uniform uniform exponential growth 190 6.4.3 The uniform Tits alternative 190 6.4.4 Application: The Lehmer conjecture 192 6.E Exercises 194 7 Hyperbolic groups 203 7.1 Classical curvature, intuitively 204 7.1.1 Curvature of plane curves 204 7.1.2 Curvature of surfaces in R3 205 7.2 (Quasi-)Hyperbolic spaces 208 7.2.1 Hyperbolic spaces 208 7.2.2 Quasi-hyperbolic spaces 210 7.2.3 Quasi-geodesics in hyperbolic spaces 213 7.2.4 Hyperbolic graphs 219 7.3 Hyperbolic groups 220 7.4 The word problem in hyperbolic groups 224 7.4.1 Application: \Solving" the word problem 225 7.5 Elements of infinite 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: Products and negative curvature 245 7.6 Non-positively curved groups 246 7.E Exercises 250 this is a draft version! x Contents 8 Ends and boundaries 257 8.1 Geometry at infinity 258 8.2 Ends 259 8.2.1 Ends of geodesic spaces 259 8.2.2 Ends of quasi-geodesic spaces 262 8.2.3 Ends of groups 264 8.3 Gromov boundary 267 8.3.1 Gromov boundary of quasi-geodesic spaces 267 8.3.2 Gromov boundary of hyperbolic spaces 269 8.3.3 Gromov boundary of groups 270 8.3.4 Application: Free subgroups of hyperbolic groups 271 8.4 Application: Mostow rigidity 277 8.E Exercises 280 9 Amenable groups 289 9.1 Amenability via means 290 9.1.1 First examples of amenable groups 290 9.1.2 Inheritance properties 292 9.2 Further characterisations of amenability 295 9.2.1 Følner sequences 295 9.2.2 Paradoxical decompositions 298 9.2.3 Application: The Banach-Tarski paradox 300 9.2.4 (Co)Homological characterisations of amenability 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 Construction and examples 320 A.1.2 Covering theory 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 Construction of the hyperbolic plane 329 A.3.2 Length of curves 330 A.3.3 Symmetry and geodesics 332 A.3.4 Hyperbolic triangles 341 A.3.5 Curvature 346 A.3.6 Other models 347 A.4 An invitation to programming 349 this is a draft version! Contents xi Bibliography 353 Index of notation 367 Index 373 this is a draft version! this is a draft version! 1 Introduction Groups are an abstract concept from algebra, formalising the study of sym- metries of various mathematical objects. What is geometric group theory? Geometric group theory investigates the interaction between algebraic and geometric properties of groups: • Can groups be viewed as geometric objects and how are geometric and algebraic properties of groups related? • More generally: On which geometric objects can a given group act in a reasonable way, and how are geometric properties of these geometric objects/actions related to algebraic properties of the group? How does geometric group theory work? Classically, group-valued invari- ants are associated with geometric objects, such as, e.g., the isometry group or the fundamental group.