Progress in Mathematics Volume 212
Series Editors H. Bass 1. Oesterle A. Weinstein Alexander Lubotzky Dan Segal Subgroup Growth
Birkhiiuser Verlag Basel· Boston· Berlin Authors:
Alexander Lubotzky Dan Segal Institute of Mathematics Mathematical Institute Hebrew Un iversity All Soul s College Jerusalem 9 1904 Oxford OX I 4AL Israel UK e-mail: [email protected] e-mail : [email protected]
2000 Mathematics Subject Classification 20E07
A CIP cata logue record for this book is available from the Library of Congress, Washington D.C., USA
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0 2003 Birkhauser Verlag, P.O. Box 133, CH-40 10 Basel, Switzerland Softcover reprint of the hardcover 1st edition 2003 Member of the BertelsmannSpringer Publishing Group Printed on acid-free paper produced of chlorine-free pulp. rCF ...,
987654321 www.birkhauser.ch Ferran Sunyer i Balaguer (19121967) was a self• taught Catalan mathematician who, in spite of a serious physical disability, was very active in research in classical mathematical analysis, an area in which he acquired international recognition. His heirs cre• ated the Fundacio Ferran Sunyer i Balaguer inside the Institut d'Estudis Catalans to honor the memory of I: http://www.crm.es/info/ffsb.htm This book has been awarded the Ferran Sunyer i Balaguer 2002 prize. The members of the scientific commitee ~ESTVDJSA of the 2002 prize were: f-> Hyman Bass r-- University of Michigan VOl Pilar Bayer -Z Universitat de Barcelona • MCMVrr Ie A ntunio Cordoba Universidad Autonoma de Madrid Paul Malliavin Universite de Paris VI Alan Weinstein University of California at Berkeley mESTVDISIC Ferran Sunyer i Balaguer Prize winners: > n 1992 Alexander Lubotzky f- Discrete Groups, Expanding Graphs and r-- Invariant Measures, PM 125 VOl 1993 Klaus Schmidt Z Dynamical Systems of Algebraic Origin, --MCMVIIIC PM 128 1994 The scientific committee decided not to award the prize 1995 As of this year, the prizes bear the year in which they are awarded, rather than the previous year in which they were announced 1996 V. Kumar Murty and M. Ram Murty Non-vanishing of L-Functions and Applications, PM 157 1997 A. Bottcher and Y.I. Karlovich Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators, PM 154 1998 Juan J. Morales-Ruiz Differential Galois Theory and Non-integrability of Hamiltonian Systems, PM 179 1999 Patrick Dehornoy Braids and Self-Distributivity, PM 192 2000 Juan-Pablo Ortega and Tudor Ratiu Hamiltonian Singular Reduction, (to be published) 2001 Martin Golubitsky and Ian Stewart The Symmetry Perspective, PM 200 Dedicated with love to my mother and father "p~n:nJ ilpJ111tJ"N - "nNJ1 ,n111J)J W")ilJ , "JNJ A.L. and to Julia, Joel, Paul and Amber D.S. Alexander Calder, Floating Wood Objects and Wire Spines, 1941 © 2002 Pro Litteris, CH-803:3 Zurich, Switzerland Contents Preface xv Notation. XXI o Introduction and Overview 1 0.1 Preliminary comments and definitions 2 0.2 Overview of the chapters 3 0.3 On CFSG .. 8 0.4 The windows 9 0.5 The 'notes' 9 1 Basic Techniques of Subgroup Counting 11 1.1 Permutation representations 12 l.2 Quotients and subgroups .. 13 l.3 Group extensions ...... 14 l.4 Nilpotent and soluble groups 20 l.5 Abelian groups T 22 1.6 Finite p-groups . 24 1. 7 Sylow's theorem 26 1.8 Restricting to soluble subgroups 28 l.9 Applications of the 'minimal index' 29 l.10 Abelian groups II 31 l.11 Growth types. 34 Notes 36 2 Free Groups. 37 2.1 The subgroup growth of free groups 39 2.2 Subnormal subgroups ...... 42 2.3 Counting d-generator finite groups . 43 Notes 50 3 Groups with Exponential Subgroup Growth . 51 3.1 Upper bounds 55 3.2 Lower bounds .. 58 3.3 Free pro-p groups 61 x Contents 3.4 Normal subgroups in free pro-p groups 63 3.5 Relations in p-groups and Lie algebras . 69 Notes 72 4 Pro-p Groups ...... 73 4.1 Pro-p groups with polynomial subgroup growth 74 4.2 Pro-p groups with slow subgroup growth 77 4.3 The groups SL~(lFp[[t]]) 80 4.4 A-perfect groups ...... 83 4.5 The Nottingham group .... . 86 4.6 Finitely presented pro-p groups. 86 Notes 90 5 Finitely Generated Groups with Polynomial Subgroup Growth 91 5.1 Preliminary observations 94 5.2 Linear groups with PSG 96 5.3 Upper chief factors ... 98 5.4 Groups of prosoluble type. 101 5.5 Groups of finite upper rank . 102 5.6 The degree of polynomial subgroup growth 104 Notes 108 6 Congruence Subgroups...... 111 6.1 The characteristic 0 case . . . . 115 6.2 The positive characteristic case. 121 6.3 Perfect Lie algebras ...... 125 6.4 Normal congruence subgroups . 128 Notes 132 7 The Generalized Congruence Subgroup Problem 133 7.1 The congruence subgroup problem. 136 7.2 Subgroup growth of lattices .. 141 7.3 Counting hyperbolic manifolds 149 Notes 151 8 Linear Groups ...... 153 8.1 Subgroup growth, characteristic 0 155 8.2 Residually nilpotent groups .... 156 8.3 Subgroup growth, characteristic p 156 8.4 Normal subgroup growth 158 Notes 160 9 Soluble Groups ...... 161 9.1 Metabelian groups .... 162 9.2 Residually nilpotent groups. 164 9.3 Some finitely presented metabelian groups 167 9.4 Normal subgroup growth in metabelian groups 172 Notes 175 Contents Xl 10 Profinite Groups with Polynomial Subgroup Growth 177 10.1 Upper rank ...... 180 10.2 Profinite groups with wPSG: structure ... 181 10.3 Quasi-semisimple groups ...... 187 10.4 Profinite groups with wPSG: characterization. 194 10.5 Weak PSG = PSG. 197 Notes 200 11 Probabilistic Methods 201 11.1 The probability measure 205 11.2 Generation probabilities. 207 11.3 Maximal subgroups 209 11.4 Further applications 211 11.5 Pro-p groups ... 214 Notes 217 12 Other Growth Conditions 219 12.1 Rank and bounded generation 225 12.2 Adelic groups ...... 226 12.3 The structure of finite linear groups 229 12.4 Composition factors ...... 230 12.5 BG, PIG and subgroup growth. 233 12.6 Residually nilpotent groups. . . 234 12.7 Arithmetic groups and the CSP 236 12.8 Examples .... 237 Notes 241 13 The Growth Spectrum . 243 13.1 Products of alternating groups 244 13.2 Some finitely generated permutation groups 248 13.3 Some profinite groups with restricted composition factors 255 13.4 Automorphisms of rooted trees . 260 Notes 266 14 Explicit Formulas and Asymptotics . 269 14.1 Free groups and the modular group 269 14.2 Free products of finite groups. 271 14.3 Modular subgroup arithmetic. 274 14.4 Surface groups ...... 277 Notes 283 15 Zeta Functions I: Nilpotent Groups. 285 15.1 Local zeta functions as p-adic integrals 290 15.2 Alternative methods...... 297 15.3 The zeta function of a nilpotent group. 304 Notes 307 xu Contents 16 Zeta Functions II: p-adic Analytic Groups ...... 309 16.1 Integration on pro-p groups ...... 312 16.2 Counting subgroups in a p-adic analytic group 313 16.3 Counting orbits .. 315 16.4 Counting p-groups . 316 Notes 318 Windows 1 Finite Group Theory ...... 319 1 Hall subgroups and Sylow bases 319 2 Carter subgroups ...... 320 3 The Fitting subgroup ...... 320 4 The generalized Fitting subgroup 322 5 Tate's theorem . . 322 6 Rank and p-rank . 323 7 Schur multiplier 323 8 Powerful p-groups 324 9 GLn and Sym(n) . 325 2 Finite Simple Groups 329 1 The list .. 330 2 Generators . . 331 3 Subgroups .. 332 4 Representations 332 5 A utomorphisms 334 6 Schur multipliers . 335 7 An elementary proof. 335 3 Permutation Groups ..... 337 1 Primitive groups . . . 337 2 Groups with restricted sections . 338 3 Subgroups of alternating groups 343 4 Profinite Groups ...... 349 1 Completions . . . . . 350 2 Free profinite groups. 352 3 Profinite presentations 353 5 Pro-p Groups ...... 357 1 Generators and relations 357 2 Pro-p groups of finite rank 359 3 Linear pro-p groups over local fields 361 4 Automorphisms of finite p-groups 363 5 Hall's enumeration principle . . . . 364 Contents Xlll 6 Soluble Groups ...... 367 1 Nilpotent groups ...... 367 2 Soluble groups of finite rank 369 3 Finitely generated met abelian groups 372 7 Linear Groups ...... 375 1 Soluble groups . . 375 2 Jordan's theorem 376 3 Monomial groups 376 4 Finitely generated groups 376 5 Lang's theorem...... 377 8 Linearity Conditions for Infinite Groups 379 1 Variations on Marcev's local theorem 379 2 Groups that are residually of bounded rank. 383 3 Applications of Ado's theorem ...... 385 9 Strong Approximation for Linear Groups ...... 389 1 A variant of the Strong Approximation Theorem . 390 2 Subgroups of SLn(lFp) ...... 397 3 The 'Lubotzky alternative' ...... 400 4 Strong approximation in positive characteristic. 406 10 Primes ...... 409 1 The Prime Number Theorem .. 409 2 Arithmetic progressions and the Bombieri-Vinogradov theorem . 411 3 Global fields and Chebotarev's theorem 413 11 Probability ...... 415 12 p-adic Integrals and Logic 419 1 Results ...... 419 2 A peek inside the black box. 421 Open Problems ...... 425 1 'Growth spectrum' ...... 42.5 2 Normal subgroup growth in pro-p groups and met abelian groups ...... 428 3 The degree of f.g. nilpotent groups . 428 4 Finite extensions. 429 5 Soluble groups ...... 429 6 Isospectral groups ...... 429 7 Congruence subgroups, lattices in Lie groups 430 8 Other growth conditions 430 9 Zeta functions 431 Bibliography 433 Index .... 446 Preface Let G be a finitely generated group, and for n E N let an (G) denote the number of subgroups of index n in G. By the 'subgroup growth' of G one means the asymptotic behaviour of the sequence (an (G)). The first main theme of this book is the relationship between the subgroup growth of a group and its algebraic structure. This may be viewed as a new chapter in the theory of finiteness conditions in infinite groups, originated early in the last century by the Russian school of O.J. Schmidt and largely associated with the names of Kurosh and P. Hall. This studied questions of the following sort: let P be a property that is common to all finite groups F, for example 'there exist Tn and n such that F is generated by Tn elements and every element x of F satisfies xn = 1'. Now let G be an arbitrary group having property P; does it follow that G is finite? In the case of the above example this is the Burnside problem. In other cases one would only expect to deduce that G is virtually soluble, perhaps, e.g. when P is the property of having finite rank: there exists Tn such that every (finitely generated) subgroup can be generated by Tn elements. Numerous positive results were obtained in the middle of the century, pertaining to special classes of groups such as linear groups. However, many of the natural conjectures resisted all attempts at a general proof. The reason for this became clear in the 1970s, when Olshanskii and Rips constructed the so-called 'Tarski monsters': these are infinite groups G such that every proper subgroup of G is cyclic of (a fixed) prime order. Such a group G satisfies any reasonable finiteness condition and is a counter-example to any reasonable conjecture, such as the Burnside problem. On the other hand, in the '80s and '90s it gradually appeared that if one takes the old conjectures and adds the hypothesis that G is residually finite, then the conjectures indeed become theorems. The most famous example is the positive solution of the restricted Burnside problem, which may be interpreted as saying that every finitely generated residually finite group of finite exponent is finite - this earned Zelmanov a Fields medal. Other examples, closer to the spirit of this book, are the proof by Lubotzky and Mann that every finitely generated residually finite group of finite rank is virtually soluble, and results of Wilson and Zelmanov about residually finite Engel groups. These may all be seen as wide generalizations of the earlier results about finitely generated linear groups, since all such groups are indeed residually finite. XVI Preface For a group to be residually finite means that it has many subgroups of finite index: enough so that their intersection is trivial. It is entirely natural, then, to ask the question "how many subgroups of each finite index?" From the most naive point of view, the study of subgroup growth may be seen as the project of arranging residually finite groups in a spectrum, from the 'very residually finite' - with fast subgroup growth - at one end to the 'only just residually finite' - having very slow subgroup growth - at the other. The new developments mentioned above rest on three main planks: (i) the classification of the finite simple groups, (ii) Lie algebra methods applied to finite p-groups, and (iii) 'linearization techniques' via p-adic analytic groups. We shall see how all three are applied throughout this book, to derive information about the algebraic structure of a group, now assumed to be residually finite, when its subgroup growth is restricted: by specifying the type of subgroup growth we obtain a whole spectrum of refined finiteness conditions. Thus subgroup growth puts a new slant on a long tradition in infinite group theory. However, it can also be seen in a quite different light. As well as the asymptotic behaviour, the arithmetic of the sequence an(G) may be of interest; this is conveniently encoded in the Dirichlet series (0( s) = Lan (G)n -8, known as the 'zeta function of the group G' (the definition parallels that of the Dedekind zeta function of a number field, which encodes in the same way the number of ideals of each index in a ring of algebraic integers). One can now begin to develop a branch of 'non-commutative analytic number theory' that relates the analytic properties of the function (0 (s) to the structure of the group G. This is the second main theme of the book. Let us mention some highlights, beginning with growth. A well-established theory of 'growth of groups' relates to the 'word growth', that is the nature of the sequence (b~ (G)) where b~ (G) denotes the number of elements of G that can be expressed as words of length at most n in some (fixed) finite generating set S. While the precise values of the b~ (G) depend on the choice of generating set S, their growth (i.e., asymptotic behaviour) does not. For example, free groups have exponential growth, while abelian, and more generally nilpotent, groups have polynomial growth. In fact a celebrated theorem of Gromov characterizes the groups of polynomial growth precisely as those which are virtually nilpotent. The situation with subgroup growth is similar in broad outline, but differs in interesting ways. Again, the fastest growth occurs for free groups, but now it is of type n!, which is like en log n, that is, slightly faster than exponential. At the other end of the spectrum, the PSG Theorem (Theorem 5.1 in the book) characterizes the (finitely generated) groups with polynomial subgroup growth: these are precisely the groups G such that G / R( G) is virtually soluble of finite rank - here R( G) denotes the intersection of all subgroups of finite index in G. (Of course the numbers an (G) can only contain information about the quotient G / R( G), so when studying subgroup growth it is natural to assume throughout that R( G) = 1, that is, G is residually finite.) Preface XVll While the PSG theorem and Gromov's theorem are logically quite indepen• dent, they share a number of features. To begin with, both classify the groups of polynomial growth as (finite extensions of) groups in a well-understood subclass of the soluble groups. Moreover, both are proved following a similar pattern, centred on a reduction via topological groups to the special case of linear groups. Gromov developed new methods in geometric group theory in order to embed his group in a topological group, at which point he was in a position to apply the solution to Hilbert's 5th problem (the characterization of topological groups having the structure of a real Lie group). In the case of polynomial subgroup growth, the classification of the finite simple groups is invoked along the way to embedding the group in a pro-p group, at which point one is in a position to appeal to Lazard's solution of the p-adic version of Hilbert's 5th problem (the proof that we give in Chapter 5 actually avoids p-adic Lie groups but is based on the same ideas). The characterization of linear groups with polynomial word growth depends on the 'Tits alternative': a finitely generated linear group either is virtually soluble or else contains a non-abelian free subgroup. The presence of a free subgroup in G implies that G has exponential word growth, but tells us nothing about the subgroup growth. To deal with linear groups of polynomial subgroup growth, a different dichotomy had to be established: now sometimes known as the 'Lubotzky alternative', this asserts that a finitely generated linear group either is virtually soluble or else has a subgroup of finite index whose profinite completion maps onto the congruence completion of some semisimple arithmetic group. This reduces the problem to the question of counting congruence subgroups in arithmetic groups. The counting of congruence subgroups may be seen as a form of 'non-com• mutative number theory'. The proof of the PSG theorem is completed by an application of the Prime Number Theorem; but the precise estimation of 'con• gruence subgroup growth' in arithmetic groups is of interest in itself, and this involves both some serious group theory and deep number theory, such as the Bombieri-Vinogradov theorem on the 'Riemann hypothesis in the average'. Another highlight in the story of word growth was the construction by Grig• orchuk of groups of intermediate growth, strictly between polynomial and expo• nential. While (continuously) many such growth types have been realized, they all lie between evn and en, and the nature of the 'spectrum' of possible growth types is still very much a mystery. In contrast, the spectrum of subgroup growth types is known to be essentially complete: explicit constructions that demonstrate this are given in Chapter 13. On the other hand, as is the case for word growth, there are definite 'gaps' in the subgroup growth spectrum when one restricts to special classes of groups such as linear groups. The class of finitely generated nilpotent groups plays a special role both in word growth and subgroup growth. As mentioned above, these are (virtually) just the groups of polynomial word growth, and an exact formula for the minimal degree of a bounding polynomial was given by Bass; it takes integral values and depends on simple structural invariants of the group. Finitely generated nilpotent groups also have polynomial subgroup growth; but there is no known way to determine XVlll Preface the degree of polynomial subgroup growth in terms of the structure of the group, and it is not always an integer. It is known, however, that this degree is a rational number. This brings us to our second theme, the 'arithmetic of subgroup growth'. If G is a finitely generated nilpotent group, its degree in the above sense is denoted 0:( G), and is equal to the abscissa of convergence of the zeta function (0 (8). Such zeta functions share some of the properties of the more traditional zeta functions of number theory: for example they enjoy an Euler product decomposition, and for each prime p the local factor at p, denoted (0,p(8), is a rational function of p-s. The proof applies a beautiful rationality theorem for p-adic integrals due to Denef, based on considerations in p-adic model theory. However, in contrast to classical zeta functions, the global behaviour of (0(8) is erratic; for example, (0 (8) does not (usually) have analytic continuation to Attempts to answer the simple question: 'how many subgroups of index n does a group possess?' have thus encompassed a surprisingly broad sweep of math• ematics. The full range will become apparent on looking through this book; it in• cludes methods and results from the theories of finite simple groups, permutation groups, linear groups, algebraic and arithmetic groups, p-adic Lie groups, analytic and algebraic number theory, algebraic geometry, probability and logic. In many cases it has not been sufficient to quote results "off the peg" , and new results have been obtained that have nothing to do with subgroup growth as such. Among these are criteria for an infinite group to be linear, the 'Lubotzky alternative' mentioned above, and new theorems about finite permutation groups. There have also been applications outside subgroup growth: a group-theoretic characterization of arith• metic groups with the congruence subgroup property, estimates for the number of Preface XiX hyperbolic manifolds with given volume, and the results mentioned above on the enumeration and classification of finite p-groups. Our aim in this book is not to present a completed theory: the subject is still very young. Indeed, while some of the core results (such as the PSG theorem) have been around for a few years, many were discovered even as we wrote, and are still unpublished (around 44% of those labeled 'Theorem' in the main body of the book had not been published by the end of the 20th century). This book is an attempt to present the state of the art as we reach what is perhaps the end of the 'foundational stage'. The broad outlines of a rich theory have begun to emerge; it is ripe for deeper investigation and new discoveries, and we hope that this book will encourage more mathematicians to explore an intriguing new field. The present healthy state of the subject is largely due to the efforts and in• sight of a small number of colleagues (and friends): Avi no am Mann, Aner Shalev, Laci Pyber, Fritz Grunewald, Marcus du Sautoy and Thomas Muller. They have all materially contributed to the book by giving us access to their latest unpub• lished work, as have Andrei Jaikin-Zapirain, Benjamin Klopsch, Attila Mar6ti and Nikolay Nikolov. We thank them all most heartily. We are also most indebted to Efi Gelman, Dorian Goldfeld, Michael Larsen, Richard Lyons, A vinoam Mann and Laci Pyber for kindly reading various parts of the text and suggesting numerous corrections and improvements. Some of the material is based on the first author's lectures at Groups St. Andrews/Galway, 1993, and at Yale, Columbia, Rice and the Hebrew Universities in the intervening years; he is grateful for much helpful feedback from the audiences on these occasions, and to the BSF, NSF and ISF for several research grants over the years. The second author is grateful to Columbia University, Yale University and the Hebrew University of Jerusalem for hospitality during some of the writing, and to All Souls College for contributing travel expenses. Notation Number theory f rv 9 if f(n)/g(n) ----) 1 as n ----) 00 f = O(g) if there exists a > 0 such that f(n)/g(n) <; a for all large n f = o(g) if f(n)/g(n) ----) 0 as n ----) 00 f ;:: 9 if f = O(g) and 9 = OU)· log x = log2 x lnx = loge x [x]: greatest integer <; x Ixl: least integer 2: x Group theory H <; G, H GIS: the permutational wreath product of G with (the finite permutation group) S d( G): minimal size of a generating set for G d(H): H S G and d(H) < 00 (G an abstract group) rk ( G ) = sup { d(H) : H So G (G a profinite group) ur( G) = sup {rk( Q) : Q a finite quotient of G} rp(G) = rk(P) where P is a Sylow p-subgroup of G (G a finite group) urp( G) = sup {rp( Q) : Q a finite quotient of G} o(x): the order of the element x in a given group The Fitting length (or height) of a soluble group G is the minimal length of a chain 1 = No < Nl < ... < Nk = G of normal subgroups such that Ni/Ni- 1 is nilpotent for each i en: cyclic group of order n Sym(n), Alt(n): symmetric, alternating group of degree n an (G): the number of subgroups (or open subgroups) of index n in G Sn (G): the number of subgroups (or open subgroups) of index at most n in G s( G): the number of subgroups in the finite group G mn (G), a;{ (G), a;{ equivalently, if for each element x -I- 1 of G there is an epimorphism () : G ----> H where HEX and (}(x) -I- 1.