A pro-p group is the inverse limit of some system of finite p-groups, that is, of groups of prime-power order where the prime - conventionally denoted p - is fixed. Thus from one point of view, to study a pro-p group is the same as studying an infinite family of finite groups; but a pro-p group is also a compact topological group, and the compactness works its usual magic to bring 'infinite' problems down to manageable proportions. The p-adic integers appeared about a century ago, but the systematic study of pro-p groups in general is a fairly recent development. Although much has been dis covered, many avenues remain to be explored; the purpose of this book is to present a coherent account of the considerable achievements of the last several years, and to point the way forward. Thus our aim is both to stimulate research and to provide the comprehensive background on which that research must be based. The chapters cover a wide range. In order to ensure the most authoritative account, we have arranged for each chapter to be written by a leading contributor (or contributors) to the topic in question. Pro-p groups appear in several different, though sometimes overlapping, contexts.

Autorentext
Aner Shalev wurde 1958 im Kibbuz Kinneret geboren. Er studierte Mathematik und Philosophie an der Hebrew University of Jerusalem, an der er heute lehrt. Marcus du Sautoy ist Professor für Mathematik an der Universität von Oxford und Research Fellow der Royal Society. Seine in der Times erscheinenden und von der BBC ausgestrahlten Beiträge über mathematische Fragen erfreuen sich großer Beliebtheit.

Klappentext

The impetus for current research in pro-p groups comes from four main directions: from new applications in number theory, which continue to be a source of deep and challenging problems; from the traditional problem of classifying finite p-groups; from questions arising in infinite group theory; and finally, from the younger subject of 'profinite group theory'. A correspondingly diverse range of mathematical techniques is being successfully applied, leading to new results and pointing to exciting new directions of research. In this work important theoretical developments are carefully presented by leading mathematicians in the field, bringing the reader to the cutting edge of current research. With a systematic emphasis on the construction and examination of many classes of examples, the book presents a clear picture of the rich universe of pro-p groups, in its unity and diversity. Thirty open problems are discussed in the appendix. For graduate students and researchers in group theory, number theory, and algebra, this work will be an indispensable reference text and a rich source of promising avenues for further exploration.

Inhalt

1. Lie Methods in the Theory of pro-p Groups.- 2. On the Classification of p-groups and pro-p Groups.- 3. Pro-p Trees and Applications.- 4. Just Infinite Branch Groups.- 5. On Just Infinite Abstract and Profinite Groups.- 6. The Nottingham Group.- 7. On Groups Satisfying the GolodShafarevich Condition.- 8. Subgroup Growth in pro-p Groups.- 9. Zeta Functions of Groups.- 10. Where the Wild Things are: Ramification Groups and the Nottingham Group.- 11. p-adic Galois Representations and pro-p Galois Groups.- 12. Cohomology of p-adic Analytic Groups.- Appendix: Further Problems.