Groups, Graphs and Trees: An Introduction to the Geometry of Infinite Groups (London Mathematical Society Student Texts, Series Number 73)

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. 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 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.