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 introduces the theory of compact p-adic Lie groups. The second explains how methods from linear algebraic groups can be utilised to study the finite images of linear 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.
Benjamin Klopsch is a Senior Lecturer in the Department of Mathematics at Royal Holloway, University of London. Nikolay Nikolov is a Reader in Mathematics in the Department of Mathematics at Imperial College London. Christopher Voll is a Lecturer in the School of Mathematics at the University of Southampton. Dan Segal is a Senior Research Fellow at All Souls College, Oxford.
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.