Groups, graphs and trees: An introduction to the geometry of infinite groups

John Meier0521895456, 9780521895453, 9780511424427

This outstanding new book presents the modern, geometric approach to group theory, in an accessible and engaging approach to the subject. Topics include group actions, the construction of Cayley graphs, and connections to formal language theory and geometry. Theorems are balanced by specific examples such as Baumslag-Solitar groups, the Lamplighter group and Thompson’s group. Only exposure to undergraduate-level abstract algebra is presumed, and from that base the core techniques and theorems are developed and recent research is explored. Exercises and figures throughout the text encourage the development of geometric intuition. Ideal for advanced undergraduates looking to deepen their understanding of groups, this book will also be of interest to graduate students and researchers as a gentle introduction to geometric group theory.

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Table of contents :
Cover……Page 1
Half-title……Page 3
Title……Page 5
Copyright……Page 6
Contents……Page 9
Preface……Page 11
1.1 Cayley’s Basic Theorem……Page 15
1.2 Graphs……Page 20
1.3 Symmetry Groups of Graphs……Page 24
1.4 Orbits and Stabilizers……Page 29
1.5.1 Generators……Page 31
1.5.2 Cayley’s Better Theorem……Page 33
1.6.1 Dihedral Groups……Page 36
1.6.2 Symmetric Groups……Page 37
1.6.3 The Symmetry Group of a Cube……Page 40
1.6.4 Free Abelian Groups……Page 41
1.7 Symmetries of Cayley Graphs……Page 43
1.8 Fundamental Domains and Generating Sets……Page 44
1.9 Words and Paths……Page 51
Exercises……Page 53
2 Groups Generated by Reflections……Page 58
Exercises……Page 65
3.1.1 Free Groups of Rank n……Page 68
3.1.2 F2 as a Group of Tree Symmetries……Page 69
3.1.3 Free Groups in Nature……Page 73
3.2 F3 is a Subgroup of F2……Page 79
3.3 Free Group Homomorphisms and Group Presentations……Page 81
3.4 Free Groups and Actions on Trees……Page 84
3.5 The Group Z3 * Z4……Page 87
3.6 Free Products of Groups……Page 93
3.7 Free Products of Finite Groups are Virtually Free……Page 97
3.8 A Geometric View of Theorem 3.35……Page 101
3.9 Finite Groups Acting on Trees……Page 103
3.10 Serre’s Property FA and Infinite Groups……Page 104
Exercises……Page 110
4 Baumslag–Solitar Groups……Page 114
Exercises……Page 118
5.1 Normal Forms……Page 119
5.2 Dehn’s Word Problem……Page 123
5.3 The Word Problem and Cayley Graphs……Page 125
5.4 The Cayley Graph of BS(1,2)……Page 129
Exercises……Page 133
6 A Finitely Generated, Infinite Torsion Group……Page 134
Exercises……Page 143
7.1 Regular Languages and Automata……Page 144
7.2 Not All Languages are Regular……Page 150
7.3 Regular Word Problem?……Page 154
7.4 A Return to Normal Forms……Page 155
7.5 Finitely Generated Subgroups of Free Groups……Page 157
Exercises……Page 162
8 The Lamplighter Group……Page 165
Exercises……Page 173
9.1 Gromov’s Corollary, aka the Word Metric……Page 176
9.2 The Growth of Groups, I……Page 182
9.3 Growth and Regular Languages……Page 186
9.4 Cannon Pairs……Page 189
9.5 Cannon’s Almost Convexity……Page 193
Exercises……Page 196
10 Thompson’s Group……Page 201
Exercises……Page 209
11.1 Changing Generators……Page 212
11.2 The Growth of Groups, II……Page 216
11.3 The Growth of Thompson’s Group……Page 219
11.4 The Ends of Groups……Page 222
11.5 The Freudenthal–Hopf Theorem……Page 225
11.6 Two-Ended Groups……Page 226
11.7 Commensurable Groups and Quasi-Isometry……Page 231
Exercises……Page 239
Bibliography……Page 241
Index……Page 244