How Groups Grow (London Mathematical Society Lecture Note Series, Series Number 395)

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. Cover......Page 1 LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES......Page 2 Title......Page 4 Copyright......Page 5 Contents......Page 6 Preface......Page 8 1 Introduction......Page 12 2.1 Finite Index Subgroups......Page 26 2.2 Growth......Page 29 2.3 Soluble and Polycyclic Groups......Page 36 2.4 Nilpotent Groups......Page 38 2.5 Isoperimetric Inequalities......Page 43 3.1 Linear Growth......Page 47 3.2 Linear Growth Functions......Page 52 4.1 Polynomial Growth of Nilpotent Groups......Page 55 4.2 Groups of Small Degree......Page 61 5.1 Soluble Groups of Polynomial Growth......Page 67 5.2 Uniform Exponential Growth of Soluble Groups......Page 71 6 Linear Groups......Page 74 7 Asymptotic Cones......Page 78 8 Groups of Polynomial Growth......Page 88 9 Infinitely Generated Groups......Page 92 10 Intermediate Growth: Grigorchuk’s First Group......Page 101 11.1 The General Grigorchuk Groups......Page 119 11.2 Groups Acting on Regular Trees......Page 124 11.3 Groups Defined by Finite Automata......Page 126 11.4 Bartholdi–Erschler Groups......Page 130 12.1 Amenability and Intermediate Growth......Page 132 12.2 More Isoperimetric Inequalities......Page 138 13 Intermediate Growth and Residual Finiteness......Page 142 14.1 The Trefoil Group......Page 147 14.2 Wreath Products......Page 150 14.3 Free Products with Amalgamations and HNN-Extensions......Page 152 14.4 Central Products......Page 157 15 The Generating Function......Page 159 16 The Growth of Free Products......Page 169 17 Conjugacy Growth......Page 187 18 Research Problems......Page 196 References......Page 198 Index......Page 206 This Volume Introduces The Subject Of The Growth Of Groups From Scratch, Starting With Basic Definitions And Culminating In The Seminal Results Of Gromov And Grigorchuk And More. It Is Valuable Reading For Researchers From Graduate Students Up Who Want To Be Acquainted With Contemporary Group Theory. 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 -- Avinoam Mann. Includes Bibliographical References (p. [187]-194) And Index. This book introduces the subject of the growth of groups from scratch, starting with basic definitions and culminating in the seminal results of Gromov and Grigorchuk and more. It is valuable reading for researchers from graduate students up who want to be acquainted with contemporary group theory. The first book to develop from the basics, with full proofs, the cutting-edge of Group Theory: the growth of groups