Groups, Graphs and Trees: An Introduction to the Geometry of Infinite Groups

by John Meier

• ISBN13: 9780521719773
• ISBN10: 0521719771
• Edition: 1st
• Format: Paperback
• Copyright: 2008-09-15
• Publisher: Cambridge University Press

Summary

Presenting groups in a formal, abstract algebraic manner is both useful and powerful, yet it avoids a fascinating geometric perspective on group theory - which is also useful and powerful, particularly in the study of infinite groups. This 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.

Table of Contents

Preface	p. ix
Cayley's Theorems	p. 1
Cayley's Basic Theorem	p. 1
Graphs	p. 6
Symmetry Groups of Graphs	p. 10
Orbits and Stabilizers	p. 15
Generating Sets and Cayley Graphs	p. 17
More Cayley Graphs	p. 22
Symmetries of Cayley Graphs	p. 29
Fundamental Domains and Generating Sets	p. 30
Words and Paths	p. 37
Groups Generated by Reflections	p. 44
Groups Acting on Trees	p. 54
Free Groups	p. 54
F3 is a Subgroup of F2	p. 65
Free Group Homomorphisms and Group Presentations	p. 67
Free Groups and Actions on Trees	p. 70
The Group Z3 * Z4	p. 73
Free Products of Groups	p. 79
Free Products of Finite Groups are Virtually Free	p. 83
A Geometric View of Theorem 3.35	p. 87
Finite Groups Acting on Trees	p. 89
Serre's Property FA and Infinite Groups	p. 90
Baumslag-Solitar Groups	p. 100
Words and Dehn's Word Problem	p. 105
Normal Forms	p. 105
Dehn's Word Problem	p. 109
The Word Problem and Cayley Graphs	p. 111
The Cayley Graph of BS(1,2)	p. 115
A Finitely Generated, Infinite Torsion Group	p. 120
Regular Languages and Normal Forms	p. 130
Regular Languages and Automata	p. 130
Not All Languages are Regular	p. 136
Regular Word Problem?	p. 140
A Return to Normal Forms	p. 141
Finitely Generated Subgroups of Free Groups	p. 143
The Lamplighter Group	p. 151
The Geometry of Infinite Groups	p. 162
Gromov's Corollary, aka the Word Metric	p. 162
The Growth of Groups, I	p. 168
Growth and Regular Languages	p. 172
Cannon Pairs	p. 175
Cannon's Almost Convexity	p. 179
Thompson's Group	p. 187
The Large-Scale Geometry of Groups	p. 198
Changing Generators	p. 198
The Growth of Groups, II	p. 202
The Growth of Thompson's Group	p. 205
The Ends of Groups	p. 208
The Freudenthal-Hopf Theorem	p. 211
Two-Ended Groups	p. 212
Commensurable Groups and Quasi-Isometry	p. 217
Bibliography	p. 227
Index	p. 230
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