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[×ÊÔ´] How.Groups.Grow

Preface page vii
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
vi Contents
11.3 Groups Defined by Finite Automata 115
11.4 Bartholdi¨CErschler Groups 119
12 Growth and Amenability 121
12.1 Amenability and Intermediate Growth 121
12.2 More 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 HNNExtensions
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
References 187
Index 195
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