| 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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