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How Groups Grow


Part of London Mathematical Society Lecture Note Series

  • Date Published: December 2011
  • availability: Available
  • format: Paperback
  • isbn: 9781107657502

£ 49.99

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About the Authors
  • 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.

    • 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
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    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

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    Product details

    • Date Published: December 2011
    • format: Paperback
    • isbn: 9781107657502
    • length: 210 pages
    • dimensions: 224 x 150 x 13 mm
    • weight: 0.32kg
    • contains: 4 b/w illus.
    • availability: Available
  • Table of Contents

    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

  • Author

    Avinoam Mann, Hebrew University of Jerusalem
    Avinoam Mann is a Professor Emeritus at his Alma Mater, the Einstein Institute of Mathematics in the Hebrew University of Jerusalem. He has published over a hundred papers on group theory and co-authored the influential book Analytic Pro-p Groups.

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