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Author Mann, Avinoam, 1937- author
Title How groups grow / Avinoam Mann
Imprint Cambridge : Cambridge University Press, 2012
book jacket
Descript 1 online resource (ix, 199 pages) : digital, PDF file(s)
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unmediated n rdamedia
volume nc rdacarrier
text file PDF rda
Series London Mathematical Society lecture note series ; 395
London Mathematical Society lecture note series ; 395
Note Title from publisher's bibliographic system (viewed on 05 Oct 2015)
1 Introduction 1 -- 2 Some Group Theory 15 -- 2.1 Finite Index Subgroups 15 -- 2.2 Growth 18 -- 2.3 Soluble and Polycyclic Groups 25 -- 2.4 Nilpotent Groups 27 -- 2.5 Isoperimetric Inequalities 32 -- 3 Groups of Linear Growth 36 -- 3.1 Linear Growth 36 -- 3.2 Linear Growth Functions 41 -- 4 The Growth of Nilpotent Groups 44 -- 4.1 Polynomial Growth of Nilpotent Groups 44 -- 4.2 Groups of Small Degree 50 -- 5 The Growth of Soluble Groups 56 -- 5.1 Soluble Groups of Polynomial Growth 56 -- 5.2 Uniform Exponential Growth of Soluble Groups 60 -- 6 Linear Groups 63 -- 7 Asymptotic Cones 67 -- 8 Groups of Polynomial Growth 77 -- 9 Infinitely Generated Groups 81 -- 10 Intermediate Growth: Grigorchuk's First Group 90 -- 11 More Groups of Intermediate Growth 108 -- 11.1 The General Grigorchuk Groups 108 -- 11.2 Groups Acting on Regular Trees 113 -- 11.3 Groups Defined by Finite Automata 115 -- 11.4 Bartholdi-Erschler Groups 119 -- 12 Growth and Amenability 121 -- 12.1 Amenability and Intermediate Growth 121 -- 12.2tMore Isoperimetric Inequalities 127 -- 13 Intermediate Growth and Residual Finiteness 131 -- 14 Explicit Calculations 136 -- 14.1 The Trefoil Group 136 -- 14.2 Wreath Products 139 -- 14.3 Free Products with Amalgamations and HNN-Extensions 141 -- 14.4 Central Products 146 -- 15 The Generating Function 148 -- 16 The Growth of Free Products 158 -- 17 Conjugacy Growth 176 -- 18 Research Problems 185
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
Subject Group theory
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