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Publisher:

Cambridge University Press

Publication Date:

2011

Number of Pages:

147

Format:

Paperback

Series:

London Mathematical Society Student Texts 77

Price:

45.00

ISBN:

9780521183017

Category:

Monograph

[Reviewed by , on ]

Fernando Q. Gouvêa

08/17/2011

This is really three short books on closely related subjects. The overarching goal is to study infinite groups, and the dominant theme is the use of profinite methods. The first section, by Klopsch, is a useful introduction to the related concepts of p-adic Lie groups and pro-p groups, with the main focus being on group-theoretic characterizations. The second, by Nikolov, focuses on the Strong Approximation Theorem, hence has a more number-theoretic flavor and includes lots of material on algebraic and arithmetic groups. The final section, by Voll, introduces the notion of the zeta cunction of a group. All three are pitched to an audience well-versed in group theory but perhaps not in number theory and algebraic geometry.

Preface

Editor's introduction

Part I. An Introduction to Compact p-adic Lie Groups: 1. Introduction

2. From finite p-groups to compact p-adic Lie groups

3. Basic notions and facts from point-set topology

4. First series of exercises

5. Powerful groups, profinite groups and pro-p groups

6. Second series of exercises

7. Uniformly powerful pro-p groups and Zp-Lie lattices

8. The group GLd(Zp), just-infinite pro-p groups and the Lie correspondence for saturable pro-p groups

9. Third series of exercises

10. Representations of compact p-adic Lie groups

References for Part I

Part II. Strong Approximation Methods: 11. Introduction

12. Algebraic groups

13. Arithmetic groups and the congruence topology

14. The strong approximation theorem

15. Lubotzky's alternative

16. Applications of Lubotzky's alternative

17. The Nori–Weisfeiler theorem

18. Exercises

References for Part II

Part III. A Newcomer's Guide to Zeta Functions of Groups and Rings: 19. Introduction

20. Local and global zeta functions of groups and rings

21. Variations on a theme

22. Open problems and conjectures

23. Exercises

References for Part III

Index.

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