How Groups Grow
How Groups Grow
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GBPGrowth 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.
- The first book to introduce this exciting area of group theory with full proofs
- Presents the most important results of the subject in a reader friendly manner, requiring the minimum of prerequisites
- Groups of polynomial growth of low degrees are completely classified, up to commensurability
- Contains results not previously published in book form
Reviews & endorsements
'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
Product details
February 2012Adobe eBook Reader
9781139211994
0 pages
0kg
4 b/w illus.
This ISBN is for an eBook version which is distributed on our behalf by a third party.
Table of Contents
- 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.
