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

ISBN-10: 0521719771

ISBN-13: 9780521719773

Edition: 2008

Authors: John Meier
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Description: 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.

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Book details

Copyright year: 2008
Publisher: Cambridge University Press
Publication date: 7/31/2008
Binding: Paperback
Pages: 244
Size: 6.00" wide x 8.75" long x 0.50" tall
Weight: 0.792
Language: English

Preface
Cayley's Theorems
Cayley's Basic Theorem
Graphs
Symmetry Groups of Graphs
Orbits and Stabilizers
Generating Sets and Cayley Graphs
More Cayley Graphs
Symmetries of Cayley Graphs
Fundamental Domains and Generating Sets
Words and Paths
Groups Generated by Reflections
Groups Acting on Trees
Free Groups
F3 is a Subgroup of F2
Free Group Homomorphisms and Group Presentations
Free Groups and Actions on Trees
The Group Z3 * Z4
Free Products of Groups
Free Products of Finite Groups are Virtually Free
A Geometric View of Theorem 3.35
Finite Groups Acting on Trees
Serre's Property FA and Infinite Groups
Baumslag-Solitar Groups
Words and Dehn's Word Problem
Normal Forms
Dehn's Word Problem
The Word Problem and Cayley Graphs
The Cayley Graph of BS(1,2)
A Finitely Generated, Infinite Torsion Group
Regular Languages and Normal Forms
Regular Languages and Automata
Not All Languages are Regular
Regular Word Problem?
A Return to Normal Forms
Finitely Generated Subgroups of Free Groups
The Lamplighter Group
The Geometry of Infinite Groups
Gromov's Corollary, aka the Word Metric
The Growth of Groups, I
Growth and Regular Languages
Cannon Pairs
Cannon's Almost Convexity
Thompson's Group
The Large-Scale Geometry of Groups
Changing Generators
The Growth of Groups, II
The Growth of Thompson's Group
The Ends of Groups
The Freudenthal-Hopf Theorem
Two-Ended Groups
Commensurable Groups and Quasi-Isometry
Bibliography
Index
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