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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() | 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 group | 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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