Table of Contents
Preface xi
Acknowledgments xiii
Part 1 Groups and Spaces 1
1 Groups Matt Clay Dan Margalit 3
1.1 Groups 5
1.2 Infinite groups 9
1.3 Homomorphisms and normal subgroups 13
1.4 Group presentations 17
2 …and Spaces Matt Clay Dan Margalit 21
2.1 Graphs 23
2.2 Metric spaces 34
2.3 Geometric group theory: groups and their spaces 40
Part 2 Free Groups 43
3 Groups Acting on Trees Dan Margalit 45
3.1 The Farey tree 46
3.2 Free actions on trees 50
3.3 Non-free actions on trees 56
4 Free Groups and Folding Matt Clay 66
4.1 Topological model for the free group 67
4.2 Subgroups via graphs 70
4.3 Applications of folding 73
5 The Ping-Pong Lemma Johanna Mangahas 85
5.1 Statement, proof, and first examples using ping-pong 85
5.2 Ping-pong with Möbius transformations 90
5.3 Hyperbolic geometry 95
5.4 Final remarks 103
6 Automorphisms of Free Groups Matt Clay 106
6.1 Automorphisms of groups: first examples 106
6.2 Automorphisms of free groups: a first look 108
6.3 Train tracks 110
Part 3 Large scale geometry 123
7 Quasi-isometries Dan Margalit Anne Thomas 125
7.1 Example: the integers 126
7.2 Bi-Lipschitz equivalence of word metrics 127
7.3 Quasi-isometric equivalence of Cayley graphs 130
7.4 Quasi-isometries between groups and spaces 133
7.5 Quasi-isometric rigidity 139
8 Dehn Functions Timothy Riley 146
8.1 Jigsaw puzzles reimagined 147
8.2 A complexity measure for the word problem 149
8.3 Isoperimetry 156
8.4 A large-scale geometric invariant 162
8.5 The Dehn function landscape 163
9 Hyperbolic Groups Moon Duchin 176
9.1 Definition of hyperbolicity 178
9.2 Examples and nonexamples 182
9.3 Surface groups 186
9.4 Geometric properties 193
9.5 Hyperbolic groups have solvable word problem 197
10 Ends of Groups Nic Koban John Meier 203
10.1 An example 203
10.2 The number of ends of a group 206
10.3 Semidirect products 208
10.4 Calculating the number of ends of the braid groups 213
10.5 Moving beyond counting 215
11 Asymptotic Dimension Greg Bell 219
11.1 Dimension 219
11.2 Motivating examples 220
11.3 Large-scale geometry 223
11.4 Topology and dimension 225
11.5 Large-scale dimension 227
11.6 Motivating examples revisited 231
11.7 Three questions 233
11.8 Other examples 234
12 Growth of Groups Eric Freden 237
12.1 Growth series 238
12.2 Cone types 244
12.3 Formal languages and context-free grammars 250
12.4 The DSV method 256
Part 4 Examples 267
13 Coxeter Groups Adam Piggott 269
13.1 Groups generated by reflections 269
13.2 Discrete groups generated by reflections 275
13.3 Relations in finite groups generated by reflections 278
13.4 Coxeter groups 281
14 Right-Angled Artin Groups Robert W. Bell Matt Clay 291
14.1 Right-angled Artin groups as subgroups 293
14.2 Connections with other classes of groups 295
14.3 Subgroups of right-angled Artin groups 298
14.4 The word problem for right-angled Artin groups 301
15 Lamplighter Groups Jennifer Taback 310
15.1 Generators and relators 311
15.2 Computing word length 315
15.3 Dead end elements 318
15.4 Geometry of the Cayley graph 321
15.5 Generalizations 327
16 Thompson's Group Sean Cleary 331
16.1 Analytic definition and basic properties 332
16.2 Combinatorial definition 336
16.3 Presentations 340
16.4 Algebraic structure 345
16.5 Geometric properties 352
17 Mapping Class Groups Tara Brendle Leah Childers Dan Margalit 358
17.1 A brief user's guide to surfaces 359
17.2 Homeomorphisms of surfaces 363
17.3 Mapping class groups 367
17.4 Dehn twists in the mapping class group 370
17.5 Generating the mapping class group by Dehn twists 373
18 Braids Aaron Abrams 384
18.1 Getting started 384
18.2 Some group theory 387
18.3 Some topology: configuration spaces 395
18.4 More topology: punctured disks 400
18.5 Connection: knot theory 404
18.6 Connection: robotics 407
18.7 Connection: hyperplane arrangements 409
18.8 A stylish and practical finale 411
Bibliography 419
Index 437
