Analytic Pro-P Groups

Cover......Page 1
Cambridge Studies in Advanced Mathematics 61......Page 2
Analytic Pro-p groups (Second Edition)......Page 4
0521542189......Page 5
Contents......Page 6
Preface......Page 10
Introduction......Page 11
0.1 Commutators......Page 20
0.2 Nilpotent groups......Page 21
0.3 Stability group theory......Page 23
0.5 Prattini subgroup......Page 24
0.6 Group algebras......Page 25
0.7 Topology......Page 26
0.8 Lie algebras......Page 28
0.9 p-adic numbers......Page 29
Part I: Pro-p groups......Page 32
1.1 Profinite groups......Page 34
1.2 Pro-p groups......Page 41
1.3 Procyclic groups......Page 48
Exercises......Page 50
2 Powerful p-groups......Page 56
Exercises......Page 64
3.1 Powerful pro-p groups......Page 67
3.2 Pro-p groups of finite rank......Page 70
3.3 Characterisations of finite rank......Page 71
Exercises......Page 77
4.1 Uniform groups......Page 80
4.2 Multiplicative structure......Page 83
4.3 Additive structure......Page 84
4.4 On the structure of powerful pro-p groups......Page 89
4.5 The Lie algebra......Page 94
4.6 Generators and relations......Page 97
Exercises......Page 102
5.1 The group GL_d(\mathbb{Z}_p)......Page 106
5.2 The automorphism group of a profinite group......Page 108
5.3 Automorphism groups of pro-p groups......Page 110
5.4 Finite extensions......Page 111
Exercises......Page 115
Interlude A - 'Fascicule de résultats': pro-p groups of finite rank......Page 116
Part II: Analytic groups......Page 118
6.1 Normed rings......Page 120
6.2 Sequences and series......Page 123
6.3 Strictly analytic functions......Page 127
6.4 Commuting indeterminates......Page 136
6.5 The Campbell-Hausdorff formula......Page 141
6.6 Power series over pro-p rings......Page 148
Exercises......Page 153
7.1 The norm......Page 157
7.2 The Lie algebra......Page 167
7.3 Linear representations......Page 172
7.4 The completed group algebra......Page 174
Exercises......Page 185
Interlude B - Linearity criteria......Page 190
8.1 p-adic analytic manifolds......Page 197
8.2 p-adic analytic groups......Page 204
8.3 Uniform pro-p groups......Page 208
8.4 Standard groups......Page 212
Exercises......Page 222
Interlude C - Finitely generated groups, p-adic analytic groups and Poincaré series......Page 225
9.1 Powers......Page 232
9.2 Analytic structures......Page 236
9.3 Subgroups, quotients, extensions......Page 239
9.4 Powerful Lie algebras......Page 240
9.5 Analytic groups and their Lie algebras......Page 247
Exercises......Page 254
Part III: Further topics......Page 260
10 Pro-p groups of finite coclass......Page 262
10.1 Coclass and rank......Page 264
10.2 The case p = 2......Page 267
10.3 The dimension......Page 269
10.4 Solubility......Page 275
10.5 Two theorems about Lie algebras......Page 281
Exercises......Page 286
11.1 Modular dimension subgroups......Page 289
11.2 Commutator identities......Page 292
11.3 The main results......Page 301
11.4 Index growth......Page 304
Exercises......Page 308
12 Some graded algebras......Page 310
12.1 Restricted Lie algebras......Page 311
12.2 Theorems of Jennings and Lazard......Page 317
12.3 Poincaré series: 'l'alternative des gocha'......Page 325
Exercises......Page 329
Interlude D - The Golod-Shafarevich inequality......Page 330
Interlude E - Groups of sub-exponential growth......Page 338
13.1 Analytic manifolds and analytic groups......Page 341
13.2 Standard groups......Page 349
13.3 The Lie algebra......Page 358
13.4 The graded Lie algebra......Page 362
13.5 R-perfect groups......Page 363
13.6 On the concept of an analytic function......Page 366
Exercises......Page 369
Appendix A The Hall—Petrescu formula......Page 374
Appendix B Topological groups......Page 377
Bibliography......Page 381
Index......Page 385