Lectures on Profinite Topics in Group Theory

by Klopsch, Benjamin; Nikolov, Nikolay; Voll, Christopher; Segal, Dan

• ISBN13: 9781107005297
• ISBN10: 1107005299
• Format: Hardcover
• Copyright: 2011-03-21
• Publisher: Cambridge Univ Pr

Summary

In this book, three authors introduce readers to strong approximation methods, analytic pro-p groups and zeta functions of groups. Each chapter illustrates connections between infinite group theory, number theory and Lie theory. The first explains how methods from linear algebraic groups can be utilised to study the finite images of linear groups. The second introduces the theory of compact p-adic analytic groups. The final chapter provides an overview of zeta functions associated to groups and rings. Derived from an LMS/EPSRC Short Course for graduate students, this book provides a concise introduction to a very active research area and assumes less prior knowledge than existing monographs or original research articles. Accessible to beginning graduate students in group theory, it will also appeal to researchers interested in infinite group theory and its interface with Lie theory and number theory.

Table of Contents

Preface	p. ix
Editor's introduction	p. 1
An introduction to compact p-adic Lie groups	p. 7
Introduction	p. 7
From finite p-groups to compact p-adic Lie groups	p. 10
Nilpotent groups	p. 10
Finite p-groups	p. 11
Lie rings	p. 12
Applying Lie methods to groups	p. 13
Absolute values	p. 15
p-adic numbers	p. 16
p-adic integers	p. 17
Preview: p-adic analytic pro-p groups	p. 18
Basic notions and facts from point-set topology	p. 19
First series of exercises	p. 21
Powerful groups, profinite groups and pro-p groups	p. 25
Powerful finite p-groups	p. 25
Profinite groups as Galois groups	p. 28
Profinite groups as inverse limits	p. 29
Profinite groups as profinite completions	p. 30
Profinite groups as topological groups	p. 31
Pro-p groups	p. 32
Powerful pro-p groups	p. 33
Pro-p groups of finite rank - summary of characterisations	p. 34
Second series of exercises	p. 35
Uniformly powerful pro-p groups and Zp-Lie lattices	p. 39
Uniformly powerful pro-p groups	p. 39
Associated additive structure	p. 40
Associated Lie structure	p. 41
The Hausdorff formula	p. 42
Applying the Hausdorff formula	p. 43
The group GLd(Zp), just-infinite pro-p groups and the Lie correspondence for saturable pro-p groups	p. 44
The group GLd(Zp) - an example	p. 44
Just-infinite pro-p groups	p. 46
Potent filtrations and saturable pro-p groups	p. 47
Lie correspondence	p. 48
Third series of exercises	p. 49
Representations of compact p-adic Lie groups	p. 53
Representation growth and Kirillov's orbit method	p. 53
The orbit method for saturable pro-p groups	p. 54
An application of the orbit method	p. 56
References for Chapter I	p. 57
Strong approximation methods	p. 63
Introduction	p. 63
Algebraic groups	p. 64
The Zariski topology on Kn	p. 64
Linear algebraic groups as closed subgroups of GLn (K)	p. 66
Semisimple algebraic groups: the classification of simply connected algebraic groups over K	p. 73
Reductive groups	p. 76
Chevalley groups	p. 77
Arithmetic groups and the congruence topology	p. 77
Rings of algebraic integers in number fields	p. 78
The congruence topology on GLn(k) and GLn(O)	p. 78
Arithmetic groups	p. 80
The strong approximation theorem	p. 82
An aside: Serre's conjecture	p. 84
Lubotzky's alternative	p. 85
Applications of Lubotzky's alternative	p. 87
The finite simple groups of Lie type	p. 87
Refinements	p. 87
Normal subgroups of linear groups	p. 89
Representations, sieves and expanders	p. 89
The Nori-Weisfeiler theorem	p. 90
Unipotently generated subgroups of algebraic groups over finite fields	p. 92
Exercises	p. 93
References for Chapter II	p. 95
A newcomer's guide to zeta functions of groups and rings	p. 99
Introduction	p. 99
Zeta functions of groups	p. 99
Zeta functions of rings	p. 101
Linearisation	p. 103
Organisation of the chapter	p. 104
Local and global zeta functions of groups and rings	p. 105
Rationality and variation with the prime	p. 106
Flag varieties and Coxeter groups	p. 108
Counting with p-adic integrals	p. 110
Linear homogeneous diophantine equations	p. 114
Local functional equations	p. 116
A class of examples: 3-dimensional p-adic anti-symmetric algebras	p. 125
Global zeta functions of groups and rings	p. 126
Variations on a theme	p. 127
Normal subgroups and ideals	p. 127
Representations	p. 129
Further variations	p. 137
Open problems and conjectures	p. 138
Subring and subgroup zeta functions	p. 138
Representation zeta functions	p. 139
Exercises	p. 140
References for Chapter III	p. 141
Index	p. 145
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