How Groups Grow

The first book to develop from the basics, with full proofs, the cutting-edge of Group Theory: the growth of groups.

Avinoam Mann (Author)

9781107657502, Cambridge University Press

Paperback / softback, published 15 December 2011

210 pages, 4 b/w illus.
22.4 x 15 x 1.3 cm, 0.32 kg

'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

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.

Preface
1. Introduction
2. Some group theory
3. Groups of linear growth
4. The growth of nilpotent groups
5. The growth of soluble groups
6. Linear groups
7. Asymptotic cones
8. Groups of polynomial growth
9. Infinitely generated groups
10. Intermediate growth: Grigorchuk's first group
11. More groups of intermediate growth
12. Growth and amenability
13. Intermediate growth and residual finiteness
14. Explicit calculations
15. The generating function
16. The growth of free products
17. Conjugacy class growth
Research problems
References.

Subject Areas: Algebra [PBF]