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Lectures on Profinite Topics in Group Theory

Benjamin Klopsch, Nikolay Nikolov, and Christopher Voll
Cambridge University Press
Publication Date: 
Number of Pages: 
London Mathematical Society Student Texts 77
[Reviewed by
Fernando Q. Gouvêa
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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.

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