Lectures on Profinite Topics in Group Theory

In this book, three authors introduce readers to strong approximation methods, analytic pro-p groups and zeta functions of groups. Each chapter illustrates connections between infinite group theory, number theory and Lie theory. The first explains how methods from linear algebraic groups can be utilised to study the finite images of linear groups. The second introduces the theory of compact p-adic analytic groups. The final chapter provides an overview of zeta functions associated to groups and rings. Derived from an LMS/EPSRC Short Course for graduate students, this book provides a concise introduction to a very active research area and assumes less prior knowledge than existing monographs or original research articles. Accessible to beginning graduate students in group theory, it will also appeal to researchers interested in infinite group theory and its interface with Lie theory and number theory.

Spis treści:
Preface
General introduction
Part I. Strong Approximation Methods
1. Introduction
2. Algebraic groups
3. Arithmetic groups and the congruence topology
4. The strong approximation theorem
5. The Nori–Weisfeiler theorem and Lubotzkys alternative
6. Some applications of Lubotzkys alternative
7. Theorem 1.5.1
8. Exercises
9. Bibliography
Part II. Analytic Pro-p Groups
10. Introduction, classifying finite p-groups
11. First lecture
12. Basic notions and facts from point-set topology
13. First set of exercises
14. Second lecture
15. Second set of exercises
16. Third lecture
Part III. Zeta Functions of Groups and Rings - A Newcomers Guide
17. Introduction
18. Local and global zeta functions of groups and rings
19. Variations on a theme
20. Open problems and conjectures
21. Exercises
22. Bibliography.