How Groups Grow

By: Avinoam Mann

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Paperback | 15 December 2011

At a Glance

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210 Pages

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22.3 x 14.9 x 1.1

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Growth of groups is an innovative new branch of group theory. This is the first book to introduce the subject from scratch. It begins with basic definitions and culminates in the seminal results of Gromov and Grigorchuk and more. The proof of Gromov's theorem on groups of polynomial growth is given in full, with the theory of asymptotic cones developed on the way. Grigorchuk's first and general groups are described, as well as the proof that they have intermediate growth, with explicit bounds, and their relationship to automorphisms of regular trees and finite automata. Also discussed are generating functions, groups of polynomial growth of low degrees, infinitely generated groups of local polynomial growth, the relation of intermediate growth to amenability and residual finiteness, and conjugacy class growth. This book is valuable reading for researchers, from graduate students onward, working in contemporary group theory.

Industry Reviews

'How Groups Grow is an excellent introduction to growth of groups for everybody interested in this subject. It also touches a variety of adjacent subjects (such as amenability, isoperimetric inequalities, groups generated by automata, etc.) It is written in a very accessible style, with very clear exposition of all main results.' V. Nekrashevych, Bulletin of the American Mathematical Society

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You Can Find This Book In

Non-Fiction Mathematics Algebra Groups & Group Theory

Preface	p. vii
Introduction	p. 1
Some Group Theory	p. 15
Finite Index Subgroups	p. 15
Growth	p. 18
Soluble and Polycyclic Groups	p. 25
Nilpotent Groups	p. 27
Isoperimetric Inequalities	p. 32
Groups of Linear Growth	p. 36
Linear Growth	p. 36
Linear Growth Functions	p. 41
The Growth of Nilpotent Groups	p. 44
Polynomial Growth of Nilpotent Groups	p. 44
Groups of Small Degree	p. 50
The Growth of Soluble Groups	p. 56
Soluble Groups of Polynomial Growth	p. 56
Uniform Exponential Growth of Soluble Groups	p. 60
Linear Groups	p. 63
Asymptotic Cones	p. 67
Groups of Polynomial Growth	p. 77
Infinitely Generated Groups	p. 81
Intermediate Growth: Grigorchuk's First Group	p. 90
More Groups of Intermediate Growth	p. 108
The General Grigorchuk Groups	p. 108
Groups Acting on Regular Trees	p. 113
Groups Defined by Finite Automata	p. 115
Bartholdi-Erschler Groups	p. 119
Growth and Amenability	p. 121
Amenability and Intermediate Growth	p. 121
	p. 127
Intermediate Growth and Residual Finiteness	p. 131
Explicit Calculations	p. 136
The Trefoil Group	p. 136
Wreath Products	p. 139
Free Products with Amalgamations and HNN-Extensions	p. 141
Central Products	p. 146
The Generating Function	p. 148
The Growth of Free Products	p. 158
Conjugacy Growth	p. 176
Research Problems	p. 185
References	p. 187
Index	p. 195
Table of Contents provided by Ingram. All Rights Reserved.

How Groups Grow

At a Glance

Paperback

Industry Reviews

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How to return your order

Defective items

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