Clara L¨oh Geometric Group Theory An Introduction December 15, 2016 { 15:24 Book project Incomplete draft version! Please send corrections and suggestions to [email protected] Clara L¨oh [email protected] http://www.mathematik.uni-regensburg.de/loeh/ Fakult¨atf¨urMathematik Universit¨atRegensburg 93040 Regensburg Germany 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 31 2.3.2 Free products and free amalgamated products 34 2.E Exercises 38 iv Contents Part II Groups ! Geometry 47 3 Cayley graphs 49 3.1 Review of graph notation 50 3.2 Cayley graphs 53 3.3 Cayley graphs of free groups 56 3.3.1 Free groups and reduced words 57 3.3.2 Free groups ! trees 59 3.3.3 Trees ! free groups 61 3.E Exercises 62 4 Group actions 69 4.1 Review of group actions 70 4.1.1 Free actions 71 4.1.2 Orbits and stabilisers 74 4.1.3 Application: Counting via group actions 77 4.2 Free groups and actions on trees 78 4.2.1 Spanning trees for group actions 79 4.2.2 Completing the proof 80 4.3 Application: Subgroups of free groups are free 83 4.4 The ping-pong lemma 86 4.5 Application: Free subgroups of matrix groups 88 4.5.1 The group SL(2; Z) 88 4.5.2 Regular graphs of large girth 91 4.5.3 The Tits alternative 93 4.E Exercises 96 5 Quasi-isometry 107 5.1 Quasi-isometry types of metric spaces 108 5.2 Quasi-isometry types of groups 114 5.2.1 First examples 117 5.3 The Svarc-Milnorˇ lemma 119 5.3.1 Quasi-geodesics and quasi-geodesic spaces 119 5.3.2 The Svarc-Milnorˇ lemma 121 5.3.3 Applications of the Svarc-Milnorˇ lemma to group theory, geometry, and topology 126 5.4 The dynamic criterion for quasi-isometry 130 5.4.1 Applications of the dynamic criterion 135 5.5 Preview: Quasi-isometry invariants and geometric properties 137 5.5.1 Quasi-isometry invariants 137 5.5.2 Functorial quasi-isometry invariants 138 5.5.3 Geometric properties of groups and rigidity 143 5.E Exercises 145 Contents v Part III Geometry of groups 153 6 Growth types of groups 155 6.1 Growth functions of finitely generated groups 156 6.2 Growth types of groups 159 6.2.1 Growth types 159 6.2.2 Growth types and quasi-isometry 161 6.2.3 Application: Volume growth of manifolds 164 6.3 Groups of polynomial growth 167 6.3.1 Nilpotent groups 168 6.3.2 Growth of nilpotent groups 169 6.3.3 Groups of polynomial growth are virtually nilpotent 170 6.3.4 Application: Virtual nilpotence is a geometric property 172 6.3.5 More on polynomial growth 173 6.3.6 Quasi-isometry rigidity of free Abelian groups 173 6.3.7 Application: Expanding maps of manifolds 174 6.4 Groups of uniform exponential growth 175 6.4.1 Uniform exponential growth 176 6.4.2 Uniform uniform exponential growth 177 6.4.3 The uniform Tits alternative 178 6.4.4 Application: The Lehmer conjecture 179 6.E Exercises 181 7 Hyperbolic groups 187 8 Ends and boundaries 189 9 Amenable groups 191 9.1 Amenability via means 192 9.1.1 First examples of amenable groups 192 9.1.2 Inheritance properties 194 9.2 Further characterisations of amenability 196 9.2.1 Følner sequences 197 9.2.2 Paradoxical decompositions 200 9.2.3 Application: The Banach-Tarski paradox 202 9.2.4 (Co)Homological characterisations of amenability 204 9.3 Quasi-isometry invariance of amenability 205 9.4 Quasi-isometry vs. bilipschitz equivalence 207 9.E Exercises 211 vi Contents Part IV Reference material 217 A Appendix 219 Bibliography 221 Index 231 1 Introduction 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. It is one of the central insights leading to geometric group theory that this process can be reversed to a certain extent: 1. We associate a geometric object with the group in question; this can be an “artificial” abstract construction or a very concrete model space (such as the Euclidean plane or the hyperbolic plane) or action from classical geometric theories. 2. We take geometric invariants and apply these to the geometric objects obtained by the first step. This allows to translate geometric terms such as geodesics, curvature, volumes, etc. into group theory. Usually, in this step, in order to obtain good invariants, one restricts attention to finitely generated groups and takes geometric invariants from large scale geometry (as they blur the difference between different finite generating sets of a given group). 3. We compare the behaviour of such geometric invariants of groups with the algebraic behaviour, and we study what can be gained by this sym- biosis of geometry and algebra. A key example of a geometric object associated with a group is the so- called Cayley graph (with respect to a chosen generating set) together with 2 1. Introduction Z × ZZZ ∗ Z Figure 1.1.: Basic examples of Cayley graphs the corresponding word metric. For instance, from the point of view of large scale geometry, the Cayley graph of Z resembles the geometry of the real line, the Cayley graph of Z × Z resembles the geometry of the Euclidean plane, while the Cayley graph of the free group Z∗Z on two generators has essential features of the geometry of the hyperbolic plane (Figure 1.1; exact definitions of these concepts are introduced in later chapters). More generally, in (large scale) geometric group theoretic terms, the uni- verse of (finitely generated) groups roughly unfolds as depicted in Figure 1.2. The boundaries are inhabited by amenable groups and non-positively curved groups respectively { classes of groups that are (at least partially) accessi- ble. However, studying these boundary classes is only the very beginning of understanding the universe of groups; in general, knowledge about these two classes of groups is far from enough to draw conclusions about groups at the inner regions of the universe: \Hic abundant leones." [22] \A statement that holds for all finitely generated groups has to be either trivial or wrong." [attributed to M. Gromov] Why study geometric group theory? On the one hand, geometric group theory is an interesting theory combining aspects of different fields of math- ematics in a cunning way. On the other hand, geometric group theory has numerous applications to problems in classical fields such as group theory, Riemannian geometry, topology, and number theory. For example, so-called free groups (an a priori purely algebraic notion) can be characterised geometrically via actions on trees; this leads to an elegant proof of the (purely algebraic!) fact that subgroups of free groups are free. Further applications of geometric group theory to algebra and Riemannian geometry include the following: • Recognising that certain matrix groups are free groups; there is a geo- metric criterion, the so-called ping-pong-lemma, that allows to deduce freeness of a group by looking at a suitable action (not necessarily on a tree). 3 elementary amenable solvable polycyclic nilpotent amenableAbelian groups 1 Z ?! finite groups non-positivelyfree curved groups groups hyperbolic groups CAT(0)-groups Figure 1.2.: The universe of groups (simplified version of Bridson's universe of groups [22]) • Recognising that certain groups are finitely generated; this can be done geometrically by exhibiting a good action on a suitable space. • Establishing decidability of the word problem for large classes of groups; for example, Dehn used geometric ideas in his algorithm solving the word problem in certain geometric classes of groups. • Recognising that certain groups are virtually nilpotent; Gromov found a characterisation of finitely generated virtually nilpotent groups in terms of geometric data, more precisely, in terms of the growth function. • Proving non-existence of Riemannian metrics satisfying certain cur- vature conditions on certain smooth manifolds; this is achieved by 4 1. Introduction translating these curvature conditions into group theory and looking at groups associated with the given smooth manifold (e.g., the funda- mental group). Moreover, a similar technique also yields (non-)splitting results for certain non-positively curved spaces. • Rigidity results for certain classes of matrix groups and Riemannian manifolds; here, the key is the study of an appropriate geometry at infinity of the groups involved. • Group-theoretic reformulation of the Lehmer conjecture; by the work of Breuillard et al., the Lehmer conjecture in algebraic number theory is equivalent to a problem about growth of certain matrix groups.