Table Of ContentProgress in Mathematics
Volume 184
Series Editors
Hyman Bass
Joseph Oesterle
Alan Weinstein
New Horizons
in pro-p Groups
Marcus du Sautoy
Dan Segal
Aner Shalev
Editors
Springer Science+Business Media, LLC
Marcus du Sautoy Dan Segal Aner Shalev
DPMMS All Souls College Institute of Mathematics
Centre for Mathematical Sciences Oxford OX1 4AL Hebrew University
Cambridge CB3 OWB United Kingdom Jerusalem 91904
United Kingdom Israel
Ubrary of Congress Cataloging-in-Publication Data
New horizons in pro-p groups 1 Marcus du Sautoy, Dan Segal, Aner Shalev, editors.
p. cm. - (Progress in mathematics ; v. 184)
Includes bibliographical references and index.
ISBN 978-1-4612-7122-2 ISBN 978-14612-1380-2 (eBook)
DOI 10.1007/978-1-4612-1380-2
1. Nilpotent groups. 2. p-adic groups. I. du Sautoy, Marcus. II. Segal, Daniel, Ph.D.
III. Shalev, Aner, 1958- IV. Progress in mathematics (Boston, Mass.) ; vol. 184.
QAI77.N48 2000
00-023027
CIP
AMS Subject Classifications: llR23, 11S15, 11S20, 20015, 20E6, 20E7, 20E8, 20E18, 20F05, 20FlO,
20F40,20F50,20J05,20M35,22E20
Printed on acid-free paper.
© 2000 Springer Science+Business Media New York
Originally published by Birkh!luser Boston in 2000
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ISBN 978-1-4612-7122-2 SPIN 10757748
Reformatted from editors' files by TEXniques, Inc., Cambridge, MA.
9 8 765 4 3 2 1
Contents
Introduction ............................................................... vii
Prerequisites and Notation .................................................. xi
Chapter 1. Lie Methods in the Theory of pro-p Groups
Aner Shalev ................................................................. 1
Chapter 2. On the Classification of p-groups and pro-p Groups
C.R. Leedham-Green and S. McKay ........................................... 55
Chapter 3. Pro-p Trees and Applications
Luis Ribes and Pavel Zalesskii ................................................ 75
Chapter 4. Just Infinite Branch Groups
R.I. Grigorchuk ............................................................ 121
Chapter 5. On Just Infinite Abstract and Profinite Groups
John S. Wilson ............................................................. 181
Chapter 6. The Nottingham Group
Rachel Cam ina ............................................................ 205
Chapter 7. On Groups Satisfying the Golod-Shafarevich Condition
E. Zelmanov ............................................................... 223
Chapter 8. Subgroup Growth in pro-p Groups
Avinoam Mann ............................................................. 233
Chapter 9. Zeta Functions of Groups
Marcus du Sautoy and Dan Segal ............................................ 249
Chapter 10. Where the Wild Things are:
Ramification Groups and the Nottingham Group
Marcus du Sautoy and Ivan Fesenko .......................................... 287
vi Contents
Chapter 11. p-adic Galois Representations and pro-p Galois Groups
Nigel Boston ............................................................... 329
Chapter 12. Cohomology of p-adic Analytic Groups
Peter Symonds and Thomas Weigel ........................................... 349
Appendix: Further Problems .............................................. 411
Index ..................................................................... 417
Introduction
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.
The first main impetus towards their study came from number theory, specifically in the
context of Galois cohomology, and the first discussion in book form of pro-p groups
was in Serre's influential Cohomologie galoisienne [6]. A burst of activity in the 1960s
produced substantial results on the cohomology of pro-p groups, and the classification
of Demushkin groups. A self-contained account of some of this theory is given in
Wilson's book [7]. The Demushkin groups are 'large', being rather close to free pro
p groups. The other major achievement of the 1960s was Lazard's theory of p-adic
analytic groups [3]; these are rather 'small' pro-p groups, having finite rank in the
sense of profinite group theory. The group theoretic aspect of Lazard's work was taken
up and extended in the 1980s by Lubotzky and Mann [5], who brought into prominence
the concept of powerful pro-p groups; this approach and developments arising out of it
are expounded in detail in the book Analytic pro-p groups [1]. In a sense, the present
book is a sequel to that one: it aims to cover those aspects of pro-p group theory that
lay outside the scope of [1].
The impetus for current research on pro-p groups could be broadly described as
coming from four directions. The first is number theory, which continues to be a source
of deep and challenging problems. Chapter 10 gives an account of the higher ramifica
tion theory of local fields and the corresponding pro-p Galois groups of totally rami
fied extensions. Chapter 11, on the other hand, explores exciting developments in the
viii Introduction
theory of p-adic Galois representations in the context of algebraic number fields, re
cently brought into prominence by Wiles's proof of Fermat's Last Theorem.
The second source is the problem of classifying finite p-groups. Traditionally
considered a hopeless undertaking, this problem acquired an entirely different aspect in
1980 thanks to the brilliant insight of Leedham-Green and Newman [4], who proposed
classifying p-groups by coc/ass, and conjectured (1) that almost all finite p-groups of
a given coclass r can be obtained in a natural way from infinite pro-p groups of coclass
r, and (2) that infinite pro-p groups of finite coclass rare p-adic space groups, and
moreover, that there are only finitely many of them for each r. With the subsequent
successful proof of the coclass conjectures, the pro-p method has come to dominate
this part of finite group theory. Attention has now turned to the problem of classifying
p-groups by width, a broader concept than coclass; this project is still in its early stages
and promises to be extremely challenging. It is explored in Chapter 2; some special
aspects are also discussed in Section 6 of Chapter 1. New results on groups of fixed
coclass are discussed in Section 7 of Chapter 9 (for a self-contained account of the
n.
'pro-p' part of the coclass theory, see Chapter 10 of [1
The third source is the theory of infinite groups: the formation of profinite, and in
particular pro-p, completions is a natural procedure that casts certain kinds of group
theoretic problem in a new light, and can lead to significant insights - a paradigm
is 'Lubotzky's linearity criterion' [3] (see [1], Interlude B). Sections 5-7 of Chapter 3
apply this method to the study of subgroups in free groups, while Section 1 of Chapter 9
illustrates the approach in the context of subgroup growth.
Finally, there is the 'umbrella' subject of 'profinite group theory': the whole range
of questions that arise when one begins to investigate profinite groups as groups of
interest in their own right. The theory of pro-p groups is only a part of this, but a par
ticularly well developed part. Of course, all the above-mentioned topics also fit under
this umbrella, in the sense that problems originating elsewhere are tackled by turning
them into problems about pro-p groups; and whatever the original motivation, most
of the chapters focus most of the time on the structure, properties and classification of
pro-p groups.
The crucial requirement for the healthy development of a subject still in its in
fancy is to have lots of examples: only by studying these can we hope to eventually
build up a reliable picture of the universe of pro-p groups. A large part of this book
is devoted to the construction of interesting families of groups, mainly on the 'large'
side; the 'small', that is the p-adic analytic, pro-p groups being already well served
by existing literature: [1] for their general properties, [2] for the construction of many
examples. In particular, Chapter 3 develops the theory of groups acting on pro-p trees,
analogous to the Bass-Serre theory of groups acting on trees; here we find the free
pro-p groups and more generally pro-p groups obtained by 'free constructions', such
as free pro-p products. Going 'one step down in size', Chapter 4 describes the con
struction of branch groups, commonly known as groups of 'Grigorchuk type'; these
are also groups of tree automorphisms, with many interesting properties. They tend to
be fairly 'small from above', some of them for example having finite width, but 'large
from below', containing all countably based pro-p groups as subgroups. Another group
having these two properties is the Nottingham group N(F p): the group of normalised
automorphisms of the power series ring F p[[t]]. From the point of view of classifica-
Introduction ix
tion, NCF p) appears in many ways to be the 'smallest' of the non-soluble, non-p-adic
analytic pro-p groups; it also arises naturally as a sort of universal Galois group in the
theory of totally ramified extensions of local fields. Chapter 6 gives the construction
and elementary properties of this group, while Chapter 10 explores its role in higher
ramification theory of local fields, as well as giving two proofs of Cam ina 's theorem
that N(fp ) contains all countably based pro-p groups.
Having found, or constructed, a healthy spread of types of pro-p group, what we
also need are some general principles of classification. This issue is discussed in Chap
ter 2; but many results in the other chapters may also be construed as 'classification
principles'. Let us mention here two specific ones. The Nottingham group, and many
of the branch pro-p groups, have the property of being just infinite: they are infinite but
have no proper infinite quotients. In an obvious sense, the just infinite pro-p groups
are the basic building blocks for all pro-p groups, and an understanding of these must
be a prerequisite for understanding the rest. Chapter 5 examines just infinite groups in
general, and establishes a fundamental dichotomy: the profinite just infinite groups fall
into two disjoint classes, the branch groups and the hereditarily just infinite groups.
This is also discussed in Section 6 of Chapter 4. Next, a pro-p group is said to have a
'small set of relations' if it has a presentation by generators and relations that violates
the Golod-Shafarevich inequality. The deep theorem of Zelmanov asserts that every
finitely generated pro-p group having a small set of relations contains a non-abelian
free pro-p subgroup. This is proved for the first time in Chapter 7.
Several chapters concentrate on particular methodological approaches to pro-p
groups. Chapter 1 is a comprehensive survey of the role of Lie algebras, and the related
concept of analytic groups, in the theory. Here both structural and classification issues
are discussed; indeed, this chapter also serves as an introduction to many of the prob
lem areas in Pro-p group theory currently receiving attention. As mentioned above,
Chapter 3 explains the use ofpro-p trees, in what might be called the geometrical ap
proach to combinatorial pro-p group theory. In Chapter 9 the concept of integration on
p-adic analytic pro-p groups is explained, and applied to the study of certain Poincare
series associated to these groups; this has applications both to questions of subgroup
growth and to the classification of finite p-groups.
The specific question of subgroup growth in Pro-p groups is discussed in Chap
ter 8, where among others probabilistic methods are introduced.
Two chapters are devoted primarily to number-theoretic matters. Chapter 10 is
concerned with totally ramified extensions of local fields, both in characteristic zero
and in characteristic p; a key issue is the recognition in group-theoretic terms of the
arithmetically defined higher ramification groups inside a pro-p Galois group. A self
contained proof of Sen's theorem on p-adic analytic Galois extensions illustrates the
subtle interplay between arithmetic and group-theoretic information; and an applica
tion of the field-theoretic construction of the 'field of norms' is made to give a proof of
Camina's theorem on the Nottingham group, mentioned above.
Chapter 11, on the other hand, gives a survey of applications of pro-p group theory
in algebraic number fields; here, number-theoretic problems, such as the Fontaine
Mazur conjecture, give rise to interesting new questions about pro-p groups. These are
mostly still wide open, and present a lively challenge to the algebraic community.
Finally, Chapter 12 gives a unified treatment of Lazard's theorems on the cohomol-
x Introduction
ogy of p-adic analytic pro-p groups. This chapter is not intended as an introduction to
the topic (such as may be found in Chapters 9-11 of Wilson's book [7]). It presents a
new approach based on the introduction of a suitable module category and the system
atic use of spectral sequences, and should serve as the standard reference for further
research in this area.
A large number of open problems are presented throughout the text; a selection
of these, with additional problems suggested by many of the authors, are collected
together in the Appendix.
Acknowledgements
This book would not exist but for the dedication and effort of the individual chapter
authors, who all responded magnificently to our request for a contribution. Several
of the authors also contributed significantly during the process of mutual reviewing of
chapters, as did a small number of conscientious referees who remain anonymous.
The arduous task of translating chapters written in various dialects of TEX into a
uniform format was skillfully undertaken by Aaron Wilson. Chapter 4 was translated
from the original Russian by Olga Tabachnikova and Geoff Smith, who also created
the numerous pictures in that chapter. We thank all three for their indispensable contri
bution.
We have greatly appreciated the encouragement and practical help of Ann Kostant
of Birkhiiuser Boston, and the patience and skill of Elizabeth Loew of TEXniques, Inc.,
who made an excellent job of the final layout and production while accommodating the
last-minute whims of a gang of authors and editors.
