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

John Meier

doi:10.1017/CBO9781139167505

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.

'Meier has the rare ability to make complex concepts accessible to novices while simultaneously challenging the experienced. … With well-chosen illustrations and examples, Meier succeeds brilliantly in this unique approach.'

Source: SciTech Book News

'… totally accessible to undergraduate students and would be a good textbook for an advanced undergraduate course in group theory, or a graduate course in geometric group theory.'

Source: Mathematical Review

'… an excellent introduction to geometric group theory. … Carefully chosen examples are an essential part of the exposition and they really help to understand general constructions.'

Source: EMS Newsletter

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Groups, Graphs and Trees

An Introduction to the Geometry of Infinite Groups

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Contents

Frontmatter
pp i-vi

Contents
pp vii-viii

Preface
pp ix-xii

1 - Cayley's Theorems
pp 1-43

2 - Groups Generated by Reflections
pp 44-53

3 - Groups Acting on Trees
pp 54-99

4 - Baumslag–Solitar Groups
pp 100-104

5 - Words and Dehn's Word Problem
pp 105-119

6 - A Finitely Generated, Infinite Torsion Group
pp 120-129

7 - Regular Languages and Normal Forms
pp 130-150

8 - The Lamplighter Group
pp 151-161

9 - The Geometry of Infinite Groups
pp 162-186

10 - Thompson's Group
pp 187-197

11 - The Large-Scale Geometry of Groups
pp 198-226

Bibliography
pp 227-229

Index
pp 230-231

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Groups, Graphs and Trees

An Introduction to the Geometry of Infinite Groups

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