Lectures on Profinite Topics in Group Theory

Cover......Page 1
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
Editor\'s introduction......Page 13
1 Introduction......Page 19
2.1 Nilpotent groups......Page 22
2.2 Finite p-groups......Page 23
2.3 Lie rings......Page 24
2.4 Applying Lie methods to groups......Page 25
2.5 Absolute values......Page 27
2.6 p-adic numbers......Page 28
2.7 p-adic integers......Page 29
2.8 Preview: p-adic analytic pro-p groups......Page 30
3 Basic notions and facts from point-set topology......Page 31
4 First series of exercises......Page 33
5.1 Powerful finite p-groups......Page 37
5.2 Profinite groups as Galois groups......Page 40
5.3 Profinite groups as inverse limits......Page 41
5.4 Profinite groups as profinite completions......Page 42
5.5 Profinite groups as topological groups......Page 43
5.6 Pro-p groups......Page 44
5.7 Powerful pro-p groups......Page 45
5.8 Pro-p groups of finite rank - summary ofcharacterisations......Page 46
6 Second series of exercises......Page 47
7.1 Uniformly powerful pro-p groups......Page 51
7.2 Associated additive structure......Page 52
7.3 Associated Lie structure......Page 53
7.4 The Hausdorff formula......Page 54
7.5 Applying the Hausdorff formula......Page 55
8.1 The group GLd(Zp) − an example......Page 56
8.2 Just-infinite pro-p groups......Page 58
8.3 Potent filtrations and saturable pro-p groups......Page 59
8.4 Lie correspondence......Page 60
9 Third series of exercises......Page 61
10.1 Representation growth and Kirillov\'s orbit method......Page 65
10.2 The orbit method for saturable pro-p groups......Page 66
10.3 An application of the orbit method......Page 68
References for Chapter I......Page 69
1 Introduction......Page 75
2.1 The Zariski topology on Kn......Page 76
2.2 Linear algebraic groups as closed subgroups of GLn(K)......Page 78
Basic examples......Page 79
Basic properties of Algebraic groups......Page 80
Fields of definition and restriction of scalars......Page 81
The Lie algebra of G......Page 83
Connection with Lie algebras of locally compact topological groups......Page 84
2.3 Semisimple algebraic groups: the classification ofsimply connected algebraic groups over K......Page 85
2.4 Reductive groups......Page 88
3 Arithmetic groups and the congruence topology......Page 89
3.2 The congruence topology on GLn(k) and GLn(O)......Page 90
Valuations of k......Page 91
3.3 Arithmetic groups......Page 92
4 The strong approximation theorem......Page 94
4.1 An aside: Serre\'s conjecture......Page 96
5 Lubotzky\'s alternative......Page 97
6.2 Refinements......Page 99
6.4 Representations, sieves and expanders......Page 101
7 The Nori--Weisfeiler theorem......Page 102
7.1 Unipotently generated subgroups of algebraic groups over finite fields......Page 104
8 Exercises......Page 105
References for Chapter II......Page 107
1.1 Zeta functions of groups......Page 111
1.2 Zeta functions of rings......Page 113
1.3 Linearisation......Page 115
1.4 Organisation of the chapter......Page 116
2 Local and global zeta functions......Page 117
2.1 Rationality and variation with the prime......Page 118
2.2 Flag varieties and Coxeter groups......Page 120
2.3 Counting with p-adic integrals......Page 122
2.4 Linear homogeneous diophantine equations......Page 126
2.5 Local functional equations......Page 128
2.6 A class of examples: 3-dimensional p-adic anti-symmetric algebras......Page 137
2.7 Global zeta functions of groups and rings......Page 138
3.1 Normal subgroups and ideals......Page 139
3.2 Representations......Page 141
T-groups......Page 142
Semisimple arithmetic groups......Page 146
Compact p-adic analytic groups......Page 147
Finite p-groups......Page 149
4.1 Subring and subgroup zeta functions......Page 150
4.2 Representation zeta functions......Page 151
5 Exercises......Page 152
References for Chapter III......Page 153
Index......Page 157