JOURNAL OF ALGEBRA 209, 402᎐426Ž. 1998 ARTICLE NO. JA987502

Asymptotic Growth of Finite Groups

Sarah Black

Institute of , Hebrew Uni¨ersity, Jerusalem 91904, Israel

Communicated by Efim Zelmano¨ View metadata, citation and similar papers at core.ac.uk brought to you by CORE

Received February 16, 1997 provided by Elsevier - Publisher Connector

We investigate word growth and structure of certain infinite families of finite groups. This work is motivated by results of Bass, Wolf, Milnor, Gromov, and Grigorchuk on the word growth and structure of infinite groups. We define word growth for families of finite groups, and prove structure theorems relating their growth types to their structures. Some results are analogous to the infinite cases. However, differences are also noted, as well as other results. ᮊ 1998 Aca- demic Press

INTRODUCTION

In this paper, we consider infinite families of finite groups, each gener- ated by a bounded sized set of generators, and study connections between their rates of growth on the one hand, and their group structures on the other. We seek to understand similarities and differences between the theory developed herein for finite groups, and a similar theory developed previously for infinite groups; seewx Ba, Wo, Mil1, Mil2, Gri1, Gro . The paper is organized as follows. Section 1 contains a survey of known results for infinite groups obtained by the above-mentioned authors, which will serve as a background for our results. We also define growth and growth types for infinite families of finite groups, namely, polynomial, exponential, or intermediate, and describe the problems to be tackled in subsequent sections. It turns out that in our ‘‘finite’’ case, growth type is, in general, generator dependent! Accordingly, in Section 2 we characterize those families that grow polynomially with respect to all generating sets, and discuss families of subexponential growth.

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0021-8693r98 $25.00 Copyright ᮊ 1998 by Academic Press All rights of reproduction in any form reserved. ASYMPTOTIC GROWTH OF FINITE GROUPS 403

In Section 3 we show that certain classes of groups cannot have intermediate growth with respect to any family of generating sets. Section 4 contains descriptions of various open problems and some connections with other fields of research. Notation is generally standard. GŽi. denotes the ith derived of a group G, and ␥iŽ.G is the ith member of its lower central series. When [ is used, it serves to define the left-hand side.

SECTION 1

A. Infinite Theory of Word Growth Let ⌫ be an infinite, finitely generatedŽ. f.g. group, with S a finite generating set. Define the length function of ⌫ with respect toŽ. w.r.t. S as

lgS Ž.[ minimal length of a word in the elements of S representing g.

Define the growth function of ⌫ w.r.t. S as

⌫ ␥SSŽ.n [ ࠻Ä4g g Gl< Ž. g F n .

⌫ 1 The function ␥ [ ␥S Ž.n is said to haveŽ. at least exponential growth if ⌫ n there exist constants c ) 1 and K ) 0 so that ␥S Ž.n G Kc for all n G 1. Otherwise, we say that ␥ is subexponential. ␥ growsŽ. at most polynomi- ⌫ d ally if there exist constants d, A ) 0 so that ␥S Ž.n F An for all n G 1. The minimal such d is called the degree of ␥ ŽŽsee Theorems 1.11 iii. and 1.5, where such a d can readily be seen to exist, being an integral function of the structure of the group ⌫.. A function that grows subexponentially but is not of polynomial growth is said to have intermediate growth. Two growth functions ␥ and ␭ are said to be equi¨alent if there exist constants P, Q so that ␭Ž.n F ␥ ŽPn .and ␥ Ž.n F ␭ ŽQn .for all n G 1. The following facts are easily verified: Ž.i Two growth functions of ⌫ Žwith respect to different generating sets. are equivalent.

1 Note that for any f.g. group G, the word growth is at most exponential; this follows, since for all g, h g G,

lghSSSŽ.F lg Ž.q lh Ž.« ␥ Žn q m .F ␥ Ž.Ž.n и ␥ m for all n, m g ގ,

n and in particular, ␥ Ž.n F ␥ Ž.1. Furthermore, it can be seen that lim ␥ Ž.n 1r n necessarily exists and is finite; see Mil1 . nªϱ wx Thus ␥ grows exponentially iff lim ␥ Ž.n 1r n 1, and subexponentially iff for all ␧ 0, nªϱ ) ) n there exists an N s.t. for all n ) N, ␥ Ž.n - Ž1 q ␧ .. 404 SARAH BLACK

Ž.ii Equivalent functions have the same ‘‘growth type’’ Ž namely, polynomial, intermediate, or exponential. , and, in the case of polynomial growth, they have the same degree. Ž.iii Let H be a subgroup of ⌫ of finite index. Then H and ⌫ have equivalent growth functions. For a group theoretic property X, we say a group ⌫ is ‘‘virtually-X ’’ Ž.Ž.resp., ‘‘X-by-finite’’ if ⌫ has a resp., H of finite index with property X.

THEOREM 1.1Ž.Ž. Milnor-Wolf . iwx Mil2-Wo Af. g. sol¨able group of subexponential growth is ¨irtually nilpotent. Ž.iiwx Wo Af. g. has a polynomial growth rate. Its degree, d, is gi¨en by the following result:

Ž.iiiwx Ba-Wo Let ⌫ ha¨e nilpotency class c, let ⌫ s ␥12) ␥ ) иии ) ␥ ␥ cc) q1 s 1 be its descending central series, and let ri be the torsion free Ž.␥ ␥ rank of the abelian group iir q1 for i s 1,...,c. Then the growth function of ⌫ is equi¨alent to a polynomial of degree d where

c

d s Ý iri . is1

This immediately yields the following.

COROLLARY 1.2. Af. g. sol¨able group ⌫ has either polynomial or exponential growth, and the growth is polynomial if and only if ⌫ is ¨irtually nilpotent. However, the above result is false in general, for Grigorchuk has constructed groups with intermediate growth.

THEOREM 1.3wx Gri1 . There exist two-generated torsion groups with growth nn␣ ␤ 1 rates between 2 and 2,where ␣ sy2 ␧ for any ␧ ) 0, and ␤ s log32 31. However, we do have

COROLLARY 1.4. Let ⌫ be a f. g. linear group. Then ⌫ has either polynomial or exponential growth, with polynomial growth if and only if ⌫ is ¨irtually nilpotent. This follows from the ‘‘Tits alternative’’wx Ti applied to f.g. linear groups; such groups either contain a noncyclic free groupŽ and hence have expo- nential growth. , or they are virtually solvable, in which case Theorem 1.1 applies. ASYMPTOTIC GROWTH OF FINITE GROUPS 405

The following celebrated result of Gromov characterizes all groups of polynomial growth:

THEOREM 1.5wx Gro . Af. g. group has a polynomial growth rate if and only if it is ¨irtually nilpotent.

B. Finite Word Growth We study the asymptotic growth behavior of an infinite family of finite groups, each generated by a uniformly bounded set of generators. This is analogous to the infinite case, where, in general, finitely generated groups are considered.

EFINITIONS Ž. D . Let G s Giig I be a family of k-generated finite groups. Ž. Ž. Ž. Let S s Siig I be a family of respective generating sets for G s Giig I with <

␥ G n max ␥ G i n for all n ގ. S Ž.s Ž.Si Ž. g igI

Note that this definition makes sense, as S is a bounded sized generating family and so, for every n, max Ž␥ G i Ž..n is attained. In this ‘‘finite’’ case ig ISi also, it can be seen that lim ␥ G Ž.n 1r n exists and is finite,2 and so the nªϱ S definitions for exponential, subexponential, and polynomial growth types given above in A for a growth function of a group apply analogously here to a growth function of a group family.

C. Finite Versus Infinite The first remark to be made in this context is that in general, the growth type of a growth function for a family G is generator dependent. Substan- tiated by a theory that will be developed in the following sections, we shall see, for example, that there exist families that grow at an intermediate pace, and further, that any family G possessing a bounded sized generating family S w.r.t. which it grows at an intermediate pace, possesses another such family T w.r.t. that grows exponentially!

2 This is evident, since for every i, we have ␥ G iŽ.n Ž␥ G iŽ..1n . It therefore follows that SSiiF lim Ž␥ G iŽ..n 1r nG␥ iŽ.1 k for all i, giving lim Ž␥ GŽ..n 1r n k;whence nª ϱ SSiiF F nª ϱ S F lim Ž␥ GŽ..n 1r n is finite if it exists. Furthermore, arguing as in Mil1 , if ␣ n t 1, nªϱ S wxs wxr q def then, as in the infinite caseŽ see Mil1.Ž.Ž.Ž. , we have ␥ G n [ max ␥ G i n , whence ␥ G n wx S iSi S F max ŽŽ␥ G iŽ..t ␣ .Žmax Ž␥ G iŽ..t ␣ Ž␥ GŽ..t ␣, and the proof of the existence of the limit thus iSiis iS s S proceeds exactly as in the infinite case. 406 SARAH BLACK

This, in stark contrast to the infinite case, gives some indication of the fact that some of the corresponding methods of proof, and even results, might be different. We accordingly make the following definitions: Call a group family G a totally exponential familyŽ. resp., polynomial, intermediate if G grows exponentiallyŽ. resp., polynomially, at an intermediate pace w.r.t. e¨ery Ž.bounded sized generating family. If a family G is not of any of the above types, we call it mixed. The remarks above show that there do exist mixed families; by the same token, there are no totally intermediate families! In what follows, we study three central issues, namely: Ž.1 Is there a group-theoretic structure characterizing each of the above families? And, in particular, Ž.2 Do there exist classes of groups that cannot be mixed? Ž.3 Which analogies or differences can be derived between the infinite case and our case of an infinite family of finite groups? In connection with the third question, we remark that there are strong group-theoretic connections between residually finite groups on the one hand, and their finite quotients on the other. See, for example,w LM1, MS, Se1, Wix , where these connections have been exploited. Since the growth behavior of infinite groups is quite well understoodŽ see the results summarized in Section 1.A. , it therefore seems natural, given an infinite group family of finite groups, to consider a finitely generated residually finite group that is a subdirect product of members of the group family under question, and to then invoke the structure of that infinite group. Indeed, given a f.g. residually finite group ⌫ and finite generating set S, Ž. ⌫ consider the family G s Giig I , consisting of all finite quotients of , Ž. with respective generating sets S s Siig Ii, where, for each i, S is the epimorphic image of S under the mapping ⌫ ¸ Gi. Then it is easy to see ⌫ G that ␥S and ␥S represent the same function. However, if we consider Ž. Ž. ⌫ instead a subfamily H s Gjjg J : Ij, T s S jg J : I , where is merely a subdirect product of the family H, then on the face of it, we can only say H ⌫ that ␥T Ž.n F ␥S Ž.n for all n g ގ, and so this technique, although useful in some cases we will encounter, is not the key to solving all of the various problems we suggest.

SECTION 2

In this section we partially address the first question posed above. Specifically, we characterize totally polynomial families in terms of their group structures, and show that no totally intermediate families exist. ASYMPTOTIC GROWTH OF FINITE GROUPS 407

We introduce the following terminology: Ž.G, S grows exponentially Ž.resp., polynomially, at intermediate pace will mean that G is a family of groups that grows exponentiallyŽ. resp., polynomially, at intermediate pace w.r.t. a generating family S.

A. Polynomial Growth As remarked earlier, for families of finite groups, growth type is, in general, generator dependent. However, it turns out that a family G with a bounded sized generating family S, so that Ž.G, S grows polynomially,ischaracterized by this property.

THEOREM 2.1. Let G be a k-generator family of finite groups. Then the following conditions are equi¨alent:

Ž.1 There exist constants c, l ) 0 such that e¨ery Gi g G has a nilpo- tent normal subgroup of nilpotency class at most c and index in Gi at most l. Ž.2 There exists a generating family S for G such that Ž.G, S grows polynomially. Ž.3 ŽG, S .grows polynomially for all bounded sized generating families S for G. Comments. Call families G satisfying propertyŽ. 1 nil c-by-index-l- families. We thus see that totally polynomial families are precisely the nil c-by-index-l families. This result is then analogous to Theorems 1.1 and 1.5, namely, infinite f.g. groups with polynomial growth are precisely the nilpotent-by-finite ones.

Proof of Theorem 2.1. We shall prove the implicationsŽ. 1 « Ž.3 « Ž.2 « Ž.1. Ž.1 « Ž.3 : This part of the proof is indeed amenable to treatment using Ž. infinite . For, let G s Giig I be a family of finite groups that is nil-c-by-index-l for some constants c and l. Consider any generating Ž. family S s Siig I for G, of bounded cardinality. We shall show that Ž.G, S grows polynomially. Suppose that for all i, <

␾ ⌫ Set K iii[ ker ; then F g IiK s 1, and G i, rK ifor each i g I.In other words, ⌫ is a subdirect product of groups Gi g G, with S mapping onto Si for each i g I. We call the pair Ž.⌫, S thus constructed a co¨er for the family Ž.G, S .

For each i, let Hi be the specified nilpotent normal subgroup of Gi of nilpotency class at most c and index in Gi at most l. y1 Then we note that the subgroup H [ Fii␾ Ž.H iof ⌫ is nilpotent of class c, and of finite indexŽ. a function of l and k in ⌫. Now the results of Bass and WolfŽ.Ž. Theorem 1.1 show that ⌫, S has ⌫ d polynomial growth, say ␥S Ž.n F Kn for some K and d; it is then clear that for every i I, ␥ G i Ž.n Knd,so␥ G Ž.n Knd and Ž.G, S grows g Si F S F polynomially. Ž.2 « Ž.1 : This implication follows from the following corollary to Gro- mov’s theorem.Ž Seewx Gro , Effective Version of the Main Theorem, pp. 71᎐72. : ‘‘For any positive integers d and k, there exist positive integers R, N, and q with the following properties: If a group ⌫ with a fixed system of ⌫ d generators satisfies the inequality ␥S Ž.n F kn , for n s 1, 2, . . . , R, then ⌫ contains a nilpotent subgroup of index at most q whose nilpotency class is at most N.’’ The proof is now complete, for we clearly haveŽ. 3 « Ž.2.

Notes.Ž. 1 The implications Ž. 1 « Ž.3 can also be proved directly by using commutator collectionŽ seewx AB. , where it is shown that a nilpotent 2Ž kc.c group of class c and generating set S of size k has ␥SŽ.n F n for n large enough. Ž.2 Whereas finite nil-c-by-index-l families are totally polynomial families, the following exampleŽ. due to Prof. A. Mann shows that the degree of the polynomial growth of such families is actually generator dependent; thus we have a departure from the infinite case. Ž. Indeed, consider the family G [ Giig I of cyclic groups of orders Ž.respectively piiи q , where p iand q iare coprime and of the same order of magnitudeŽ. say qiis p q 1 . Then with respect to a one element generat- ing family, the growth is linear, whereas with respect to the family ŽÄ4. xiii, y g Ii, where x is of order p iiiand y of order q , the elements of the lm group expressible as xyii for <

B. Intermediate Growth

PROPOSITION 2.2Ž. cf. Theorem 1.3 . There exist families of finite groups with generating families of intermediate growth.

Proof. The examples we will cite are even p-groups. We make use of Grigorchuk’s constructions. Let ⌫ be an infinite f.g. torsion group of intermediate growth of the type constructed by Grigorchuk inwx Gri1 . ⌫ is a residually finite p group.

Let G s Ž.Gi be the family consisting of all of the finite quotients of ⌫, Ž. and let S s Siig I be the generating family consisting of the respective images in Gi of a finite generating set S for ⌫. Then, as remarked earlier, G ⌫ ␥S s ␥S , and so the intermediate growth of Ž.G, S follows from that of ⌫. Furthermore, the fact that ⌫ is a p group implies that G is a family of p groups. Thus we have in fact found a nilpotent family of intermediate growth! Needless to say, though, the nilpotency classes of members of such a family are unbounded.

We now show that there are no totally intermediate families.

PROPOSITION 2.3. Let G be a group family such that Ž.G, S grows subexponentially for all bounded sized generating sets, S. Then G is nil-c-by-index-l.

The proof is a straightforward application of a result of Shalevwx Sh1 , and Zelmanov’s solution to the restricted Burnside problemwx Z , both of whose proofs implicitly apply the classification of finite simple groups. First we give some definitionsŽ seewx Sh1. . m Let G be a group, and S : G a subset of G. Let

Ž.i exp ŽGrN .divides fn Ž .and Ž.ii every d-generated subgroup of N has class at most gnŽ., d .’’ 410 SARAH BLACK

Ž. Proof of Proposition 2.3. Let G s Giig I be a group family satisfying the hypotheses of the proposition. We will show that there exists an

N g ގ so that every Gi g G is N-collapsing. Since G is boundedly generated, an application of Zelmanov’s positive solution to the restricted Burnside problemwx Z to Shalev’s theorem above yields the desired result. Suppose, to the contrary, that there is no such N. Then there exists a sequence n ϱ, a subfamily Ž.G of G and pairs Ä4x , y G , iijjª jg J : Iijijg ij n n such that <Ä4x , y ij< 2ini j G for each i J I. ij ijjs i j g : Ž. Now choose a bounded sized generating family S s Siig I with the property that, for each i J, S Ä4x , y . But then we have ␥ G Ž.n jijiig > jj S i jG G n n ␥ i jjŽ.n <Ä4x , y i < 2i j ; and Ž.G, S grows exponentially, contrary to SiijG ii jjs j Ž. hypothesis. We thus conclude that the family G s Giig I is indeed N-collapsing for some N g ގ, and, in view of the comments above, the proof is complete.

COROLLARY 2.4. Let S be a generating family for a group family G such that Ž.G, S grows subexponentially. Then either G is nil-c-by-index-l Ž and then G is a totally polynomial family., or else there exists a generating family T such that Ž.G, T grows exponentiallyŽ and then G is a mixed family..

COROLLARY 2.5. Any group family G that is boundedly generated has either Ž.1 a bounded sized generating family S such that ŽG, S . grows polynomially or Ž.2 a bounded sized generating family S such that ŽG, S . grows exponentially, and case Ž.1 occurs iff G is nil-c-by-index-l for some c and l. Proof. This follows from Theorem 2.1 and Proposition 2.3. Note that Corollary 2.5 gives us a weak dichotomy for growth types of finite groups that does not exist in general for infinite groups. However, in view of the fact that growth type is generator dependent in general, Corollary 2.5 is not a satisfactory characterization for our pur- poses, and we seek sharper results where possible.

SECTION 3

In this section we show that certain classes of group families cannot be ‘‘mixed,’’ thus giving some answers to question 2.

A. Sol¨able Groups

THEOREM 3.1. Let G s Ž.Gbeaki -generator family of finite sol¨able groups, with uniformly bounded deri¨ed length. Then G is a totally polyno- mial family, or a totally exponential family, polynomial iff G is nil-c-by-in- dex-l for some c and l. ASYMPTOTIC GROWTH OF FINITE GROUPS 411

Comments.Ž. 1 Theorem 3.1 is analogous to Corollary 1.2, namely, an infinite f.g. solvable group has polynomial or exponential growth, polyno- mial iff it is virtually nilpotent. Ž.2 We stress that we cannot improve the result by relinquishing the bound on the derived length, as the family Ž.G, S constructed in the proof in Proposition 2.2 is solvableŽ. even nilpotent and is of intermediate growth. Ž. Proof of Theorem 3.1. Let G s Giig I be a family of finite solvable groups of derived lengths at most d.If G is not a totally exponential family, then G has a bounded sized generating family S such that Ž.G, S grows subexponentially. We shall prove that in that case, G is nil c-by- index-l, and accordingly, Theorem 2.1 will show that G is a totally polynomial family. The breakdown of the proof is similar to that of the infinite case.

PROPOSITION A. Let Ž.G, S be a family of finite sol¨able groups of deri¨ed lengths at most d, with S a bounded sized generating family, such that Ž.G, S grows subexponentially. Then any co¨er Ž.⌫, S for the family Ž.G, S is a polycyclic group. Note.Ž. 1 The concept of cover was defined in the proof of Theorem 2.1 in Section 2. Ž.2 We cannot invoke Wolf and Milnor’s results directly here, for, although ⌫ has group structure similar to that of G, G does not constitute a complete set of finite quotients for ⌫, so we cannot conclude that ⌫ grows subexponentially from the fact that Ž.G, S does. Ž. Ž. Ž . Proof of Proposition A. Let G s Giig Iii, S s S g I , so that G, S grows subexponentially. We proceed by induction on d, and keep the notation introduced in previous sections. The case d s 1 is the case in which ⌫ is abelian and f.g. Such a group is clearly polycyclic. Suppose, inductively, that the result is true for d y 1, and consider the family Ž⌫ ⌫Ždy1. .ŽŽ dy1.. r K iig Iiisomorphic to the family H s G rGiig I of derived Ž. Ž. length d y 1 , with generating family T s Tiig Ii, where, for each i, T is Ž dy1. the image of Siiunder the epimorphism G ¸ GirGi. Denoting by T Ž dy1. the image of S under the epimorphism ⌫ ª ⌫rFii⌫ K , we see that Ždy1. Ž⌫rFii⌫ K , T .Žis a cover for the family H, T .. Furthermore, it is easy to see that Ž.H, T grows subexponentiallyŽ with growth type at most that of Ž..G, S , and so the induction hypothesis applied to Ž.H, T implies that Ždy1. ⌫rFii⌫ K is a polycyclic group. Ždy1. Furthermore, note that A [ Fii⌫ K is an abelian group. By a theorem of P. HallŽ see, for example,wx Ro, 15.3.1. , f.g. abelian by polycyclic groups satisfy the maximum condition for normal . Since Ae⌫, 412 SARAH BLACK

it follows that there exist a finite number of elements a1,...,an g ⌫,so ⌫ that A s ²:a1,...,an . We claim further that A is finitely generated as a group. In view of the fact that finitely generated abelian groups are polycyclic, and polycyclicity is extension closed, this will complete the proof of Proposition A. ⌫ ⌫ ⌫ ⌫ Indeed, we have A s ²:a1 иии an s ²²a12 : , ²:a ,...,²:an :.Soitis ⌫ sufficient to show that for any aii, i s 1,...,n, the group ²:a is finitely generated. ⌫rA is polycyclic, and so is a product of a finite number of cyclic groups, say ⌫rA s ²:c1 иии ²:cAr . ⌫ 1 Since A is abelian, we have for any a g A that ²:a s ²tattAy < is a Ž.left coset representative of A in ⌫ :. So we need only show that for i1 i2 i3 ir c c c ...cr a g A, the group ²ai123 < j g ޚ, j s 1,...,r: is f.g. To this end, we interrupt the proof of Proposition A to state and prove Lemma B, which we will subsequently invoke.

LEMMA B. Let Ž.⌫, Sbeaco¨er for a family ŽG, S .of subexponential growth. Suppose that Ae⌫ is an abelian subgroup of ⌫; let a g A, and ␻ g ⌫. Then there exists a monic polynomial fŽ. x with constant term "1 fŽw. Ž.dependent on a and w , such that w satisfies the equation a s 1. Ž In additi¨e notation, considering²: a as ޚw²:wx module, this means that there exists a monic polynomial fŽ. x with constant term "1, such that f Ž w .g AnnŽ²a :.. . Proof. We may assume, without loss of generality, that w has infinite n orderŽ. modulo A , for otherwise w g A for some n, and then fxŽ.s n x y 1 is such a polynomial. Consider the subgroup ²:a, w of ⌫. Set L s maxŽŽ.laSS, lw Ž... If, for i g I, ai Žresp., w i .denotes the image of a Ž.resp., w under the epimorphism ⌫ G , then we clearly have maxŽŽ.la, ¸ iSi i lwŽ.. L for all i I. It follows that, for each subgroup ²a , w :of a , Sii F g ii i we have ␥ ² ai , w i: n ␥ G i Ln .1 Äaii, w 4 Ž.F Si Ž . Ž.

Now Ž.G, S grows subexponentially; thus there exists an n0 such that for every i g I, and for every n G n0 , n ␥ G i n - 2.1r2 L 2 Si Ž. Ž . Ž. Ž.1 and Ž. 2 imply that

␥ ² ai , w i: 2n ␥ G i 2 Ln - 2 n for all n n and for all i I.3 Äaii, w 4 Ž.F Si Ž . G 0 g Ž.

Following general ideas developed by Milnorwx Mil2 , consider, in Gi, words of the form ␧ ␧ ␧ 0 1 ny 1 awawiiiiиии aw i i with ␧jg Ä40, 1 for j s 0,...,n y 1.Ž. 4 ASYMPTOTIC GROWTH OF FINITE GROUPS 413

There are 2n possibilities for words of the formŽ. 4 , and each such word is of Ä4aii, w -length at most 2n. InequalityŽ. 3 above implies that if n G n0 , they cannot all be distinct.

Let n s n0 , and rewriteŽ. 4 in the form

␧12␧␧n 1 ␧ 0 y12y2 Žny1. yŽny1. y n awawiŽ.Ž.Ž iii waw i ii иии wawi ii .и wi Ž.4Ј or

␧ ␧ y1 ␧ y2 ␧ yŽ ny1. 0q 1w i q 2 w i q ... ny 1w i n aiiи w ,

n g yn n where ␧j are as above, and a denotes g ag for n g ޚ, g g ⌫. Now, if two such words coincide in Gi, we have

␧ ␧ y1 ␧ y2 ␧ yŽ ny1. y1 yŽ ny1. 0q 1w i q 2 w i q ... ny 1w i n f 0qf 1w i q ... f ny 1w i n aiiiiи w s a и w for sets

␧jj, f g Ä40,1 , j s 1,...,n y 1

Canceling, rearranging, and multiplying throughout by wŽ ny1. gives a monic polynomial fx␯ Ž.of degree less than n0 with coefficients 0, 1, or n0y2 f␯Žw i. f␯Žw. Ž.Ž.2и3 Ž. y1, so that aii' 1; then a ' 1 mod K . Set fx[ Ł␯s1 fx␯ , where the product is taken over all 2 и 3n0y2 possible monic polynomials of degrees less than n0 in x and with coefficients 0, 1, y1. We claim that fxŽ.has the required properties. For fxŽ.is clearly monic and has constant term "1, factors of fxŽ.commute with each other in their action on a, and, by the discussion above, at least one fw␯ Ž.annihilates a modulo fŽw. fŽw. K ii, for each K , whence a ' 1Ž. mod K ifor all i,ora s 1, as Fig IiK s 1. This completes the proof of Lemma B. Returning to the proof of Proposition A, consider the subgroup of ⌫ generated by a and c11, for a g A, c g ⌫. Lemma B implies that there fŽc1. exists a monic polynomial fxŽ.of degree m1, say, such that a s 1, with fŽ.0 s "1. Rearranging, this gives

␣ m 1 ␣ m 2 ccmm11y1 11y cm 1y2 y c␣1 ␣ a 11s Ž.Ž.aa 1 иии Ž.aa10

with ␣i g ޚ for 0 F i F m10y 1, and ␣ s "1.Ž. 5

ConjugatingŽ. 5 by c1 and substitutingŽ. 5 in the resultant expression gives c m1q1 c c m1y1 a 1 as a word in the elements Äa, a 1,...,a 1 4 Ž.) . Conjugating successively by c1 gives inductively that for all i ) 0, c i elements of the form a 1 can be written as words in the elements Ž.) . yi Similarly, successively conjugatingŽ. 5 by ci1 s 1, . . . , and rearranging 414 SARAH BLACK shows that all powers ciyi Ž.) 0 can be expressed as words in the i m1y1 Ž. ²c1 :² c1 c : elements ) . Thus a < ig ޚ s a, a ,...,a is finitely generated. At the next stage, note that

c i1c i2 c i c c i c m1y1c i c i1c i2 ² 1 2 < :²2 1 2 1 2 < :²1 2 < : a i12, i g ޚ s a , a ,...,a ig ޚ s a i21g ޚ i s0,... m 1y1 , the first equality following from the discussion above. So, applying Lemma c i c i c i2 B to each of the m pairs Äa 1, c 4 gives that each ²a 1 2 < ޚ : is 12is0, . . . m1y1 i2 g c i1, c i2 ² 1 2 < : f.g., and, accordingly, so is a i12, i g ޚ . Since there are a finite number of cyclic factors, this process, applied inductively to all of the factors ²:ci , ⌫ terminates, giving that A s ²:a1 иии ar is finitely generated, as claimed, and ⌫ is a polycyclic group. Passing to the family of quotients, we obtain the following.

COROLLARY. If G s Ž.Gi is a family of soluble groups of deri¨ed length d with a bounded sized generating family S, such that Ž.G, S grows subexpo- nentially, then each Gi is polycyclic of polycyclic length at most l for some constant l s lŽ.G .

Reduction to Metabelian Groups Suppose we know that Theorem 3.1 is true for metabelian groups. We Ž. wish to show that this implies the result in general. Indeed, let G s Giig I be a family of solvable groups of bounded derived lengths, with S a bounded sized generating family so that Ž.G, S grows subexponentially. So ⌫ Gii, rK , with F ig IiK s 1 for all i. Ž.⌫ Ž. Let , S be a cover for G, S . Thus Fig IiK s 1 for all i. Since polycyclic groups are virtually nilpotent by abelianŽ see, for examplew Ro 15.1.6x. , ⌫ has a subgroup H of finite index so that HЈ is nilpotent. H is Ž. Ž. f.g. by a generating set T, say. Set H s Hiig Iii, T s T g Ii, where H Ž.resp., Tiiis the image of H Ž.resp., T in the quotient G . Note that Ž.H, T grows subexponentially, for, if L max ltŽ., then ␥ H i Ž.n ␥ G i Ž.Ln . s t g TS TiiF S So we may as well replace ⌫ by H, and prove that H is virtually nilpotent. So without loss of generality, Ž.⌫, S is a cover for ŽG, S ., and ⌫Ј is nilpotent, say, of class b. ⌫Љ ⌫ Set D s Fig IiK , and consider the metabelian quotient group rD, with generators Ä sD< s g S4. Then ŽŽ⌫rD, S mod D ..is a cover for the Ž. Ž Y .ŽŽ metabelian family K, T , where K s GiiirG g Iiand T s S mod Y .. Ž. Giig I . Moreover, K, T grows subexponentially as it is a family of quotients of Ž.G, S . Applying Theorem 3.1 for metabelian groups provides the existence of Y Y constants c and l, so that each GiirG has a nilpotent subgroup NiirG of Y index in GiirG at most l and of nilpotency class at most c. Without loss of ASYMPTOTIC GROWTH OF FINITE GROUPS 415

generalityŽ. since Giiare all k-generated , we may assume N to be normal X in Gii. Furthermore, G is nilpotent of class at most b, so, by Fitting’s XY theoremŽ see, for example,wx Ro 5.2.8. , NGiirG iis a nilpotent normal Y subgroup of GiirG of index at most l and nilpotency class at most b q c. Now a theorem of P. HallŽ see, for example,wx Ro, Section 5.2.10. implies X that there exists a function f s fcŽ., b such that each NGiiis nilpotent of class at most f, and of index in Gi at most l. The reduction to metabelian groups is now complete. Thus, to deduce Theorem 3.1, all that now remains for us to prove is the following:

PROPOSITION C. Let Ž.⌫, S be a metabelian polycyclic co¨er for a metabelian family Ž.G, S with Ž.G, S of subexponential growth. Then ⌫ is ¨irtually nilpotent. Proof. We adapt the method employed in Bass’s proofwx Ba that subex- ponential polycyclic groups are virtually nilpotent. We proceed by induction on ⌫ [ hrŽ.⌫ , the Hirsch rank of ⌫.If r s 0, then ⌫ is finite and the result is clear. Similarly, if r s 1, then ⌫ is either cyclic by finite or finite by cyclic, and since finite by cyclic groups are cyclic by finite, the result follows. Assume the result for r y 1, and let ⌫ be a group of Hirsch rank r. ThenŽ seewx Se2, Chap. 1, Proposition 2. there exists a subnormal series иии ⌫ 1 s H01e H e e Hrre H q1 s , ⌫ so that, for 1 F i F r, Hiy1 e Hiii, H rH y1 is infinite cyclic, and rHr is finite. By earlier reasoning, we may replace ⌫ by Hr and prove that it is virtually nilpotent.

Note additionally that H [ Hry1 is a , say H s ²:T for a finite set T. Then Ž.H, T is a cover for the family Ž.H, T , where, for each i, Hiiis the image of H in G , Tiiis the image of T in G , Ž. Ž. Ž. H [ Hiig Iii, and T [ T g I . Furthermore, H, T grows subexponen- tially, and hrŽ. H s r y 1. Thus the induction hypothesis applied to H implies that H has a normal nilpotent subgroup N of finite index with g nilpotency class, say, c. Replacing N by Fg g ⌫ N , this group is still of finite index in H and is normal in ⌫. Let t g ⌫ map onto a generator of the infinite ⌫rH. Since <⌫ : Nt²:<

so that, for every introduction Njq1 - Liji- N , L is of infinite index in N j. Note that this process must terminate, since for every introduction of such an additional ⌫-invariant subgroup Li, we increase the number of infinite factors in the series, whose number, in turn, cannot exceed r y 1. We thus reach a stage where the resulting ⌫-invariant series

иии ²: ⌫ Ž.Ž.)) 1 s N¨¨- N y11- - N s N - Nt s has either finite factors or infinite factors NjjrN q1, which are torsion free ޚ ⌫ ޑ and rationally irreducible as wxmodules. Thus if Mjjj[ N rN q1 mޚ , then Mj is either zero or irreducible as ޑwx⌫ module, for every j. Note иии Ž.Ž. that N acts trivially on every factor NjjrNjq1 s 1 ¨ y 1of )). In what follows, we shall show thatŽ. as in Bass’s proof there exists a s power s of t such that t centralizes every factor NjjrN q1. This will complete the proof, as it will then follow that N и ²t s: is a nilpotent subgroup of finite index in ⌫. We set Nk [ ␥ 2Ž.⌫ and examine three possible types of factors NjjrN q1: Ž. sj a NjjrN q1 is finite. In this case we clearly have t centralizes NjjrN q1 for some power sj.

Thus, without loss of generality, NjjrN q1 is infinite. Denote the generator ޑ ⌫ of NjjrN q1 as wxmodule by xNjjq1. Ž. ⌫ b j - k: Then t acts tri¨ially on NjjrN q1 as wxwxx jj, t ; N , ; ␥ Ž.⌫ 2 s NkjF N q1.

Ž.c j G k: In this case, x jjkg N : N s ⌫Ј,an abelian normal sub- group of ⌫. We may thus invoke Lemma B on the group Ž.⌫, S , the abelian normal subgroup ⌫Ј, and the pair x j g ⌫Ј, t g ⌫, to conclude that there fŽt. exists a monic polynomial fxŽ.g ޚ Ž.x so that x j s 1. A fortiori, ftŽ. ␣ acts trivially on NjjrN q1. Denoting by the automorphism of NjjrN q1 mޚ ޑ ޑ induced by t, the irreducibility of the module NjjrN q1 mޚ yields that Ž.␣ Ž ޑ. ޑ ␣ ޑŽ.␣ f ' 0 g End NjjrN q1 m , and further, that wxis a field, . ޑŽ.␣ is a finite field extension of ޑ, of degree F deg fŽ.␣ , and so ޑ Ž.␣ embeds into ރ. The fact that fŽ.␣ s 0 implies that ␣ is an algebraic integer. We claim, as in Bass’s proof, that ␣ is a root of unity, or, in other sj words, that there exists an integer sjjso that T centralizes N rNjq1. For, assuming to the contrary, that ␣ is not a root of unity, we can find a conjugate ␤ of ␣ in ރ of absolute value different from 1. Now embed ޑŽ.␣ into ރ according to the embedding ␣ ¬ ␤. Since <<␤ / 1, we can m m m find an integer m so that < ␤ <<

Ž.␤ 2и3 ny 1 Ž.␤ Ž.␤ 0, 1, or y1. Thus f s Ł js1 fjjs 0 m f s 0 for at least Ž. nn␧ y1 ␧ ny2 иии ␧ ␧ one j. But if fxjns x q y1 x q ny210x q x q with ␧ Ä4 iis1,...,n g 0, 1, y1 , then ␤ ␤ n ␧␤ny1 иии ␧␤ ␧ fjnŽ.s 0 m syŽ.y110q q <<␤ n 1 nny1 y m <<␤ F <<␤ q иии <<␤ q 1 s <<␤ y 1 n - <<␤ y 1as<<␤ ) 2. This contradiction proves the assertion that ␣ is a root of unity. Thus, in sj this case too, there exists an sjjso that t centralizes N rNjq1. Now set ¨y1 ²: ⌫ s [ Ł js1 sj. Then Nt is a nilpotent subgroup of finite index in , and Proposition C has been proved. With this, we have completed Theorem 3.1 and thus the first result of this section, namely, that solvable families of finite groups of bounded derived lengths cannot be mixed families.

B. Groups of Bounded Rank We start with a definition.

DEFINITION. For a finite group G, define the rank of G Ždenoted by rkŽ.. G as rk Ž. G [ maxÄdHŽ.< His a subgroup of G4, where dHŽ.is the minimal number of generators for the group H.

HEOREM Ž. T 3.2. Let G s Gbeakiig I -generator family of finite groups of uniformly bounded rank. Then G is either a totally polynomial or a totally exponential family, polynomial if and only if G is nil-c-by-index-l for some constants c and l. Comments.Ž. 1 Finite versus infinite. We can view Theorem 3.2 as an analogy to either of the following results A or B: Ž.A A f.g. infinite residually finite group of finite rank has polyno- mial or exponential growth, polynomial iff it is virtually nilpotent.Ž This follows from a result ofŽ. LM1 : A f.g. residually finite group of finite rank is virtually solvable.. Ž.B A f.g. infinite residually finite group of finite upper rank has polynomial or exponential growth, polynomial iff it is virtually nilpotent. ŽThis, in turn, follows from a result ofwx MS : A f.g. residually finite group of finite upper rank is virtually soluble minimax.. ŽŽRecall that the upper rank of a group ⌫ is defined as ur ⌫.[ supÄrkŽ. G< G is a finite quotient of ⌫4.. 418 SARAH BLACK

On the other hand, since it is clear that groups like those in Theorem 3.2 are not in general soluble, we can view Theorem 3.2 as a strengthening ofŽ. B above. Ž.2 The bound on the ranks cannot be relinquished, as Grigorchuk’s examples show. For it is not difficult to see that they are not residually of bounded rankŽ see, for example,wx Se3 to deduce this. . Proof of Theorem 3.2. We prove this theorem in two parts.

EMMA Ž. L D. Let G s Giig ގ be a family of finite groups. Suppose there exists a constant r ) 0 so that rkŽ. Gi F r for all i g I, and a bounded sized Ž. Ž. family of generators S s Siig I such that G, S grows subexponentially. Then there exists a function g [ gŽ. r so that e¨ery Gi g G has a soluble subgroup Hii of index in G at most g. Proof. We first describe the structure of finite groups of bounded rank due to ShalevŽ.Ž subject to the classification of finite simple groups see wxSh2, Proposition 3.6. and prove a general claim. ‘‘Let G be a finite group of rank r. Then there exists a chain

1eG21eG eG of characteristic subgroups of G such that

Ž.1 <

Ž.2 G12rG , S 1= иии = Ski, where 0 F k F r, and for 1 F i F k, S XpŽ ei . is a simple group of Lie type such that n , e are r-bounded, say s nii ii nii, e F frŽ.for all 1 F i F k.

Ž.3 G2 is solvable.’’ Claim. Let H be a subgroup of a group G with <

g s x1 иии xk for k ) l.

Denoting by pjŽ.s x1,..., x j,1F i F k, since <

Accordingly, for all i, denote by G1iithe subgroup of G of r-bounded index described above, and let G2 iibe the solvable radical of G . Choose a ASYMPTOTIC GROWTH OF FINITE GROUPS 419

set Ti of Schreier generatorsŽ seewx Hall M, p. 88. for G1i with respect to Si, in the following manner. Recall that a Schreier generator is a member y1 of the set SRSiii l G1ii, where R is a transversal to G1iiin G . Since <

taking as generators the respective images of Si g S under the epimor- phisms. , and seeing as we wish to prove Gi have solvable subgroups of r-bounded index, we may assume that G2 i s 1 for all i g I. Thus, without Ž. loss of generality, G s Giig I is a family of semisimple groups of the formŽ. 2 above. Since subexponentiality is preserved when passing to quotients, we may Ž. further assume that G s Giig I is a family of simple groups of Lie type, each member of which has Chevalley rank F frŽ., possessing a bounded Ž. Ž . sized family of generators S [ Siig I , so that G, S grows subexponen- tially. NowŽ. as in the proof of Theorem 2.1 pick any finitely generated cover Ž.⌫, S for ŽG, S .. Thus ⌫ is a subdirect product of linear groups of dimensions at most frŽ.. It follows from a result of Wilsonw Wi, Section 4.2x that ⌫ satisfies the Tits alternative. We claim that ⌫ has no noncyclic free subgroup.

For, assume to the contrary, that Ž.␣, ␤ , F2 is a free subgroup of ⌫ of rank 2. Set L [ maxŽŽlss␣ ., l Ž␤ ... Then if, for i g I, ␣ii, ␤ are the respec- tive images of ␣, ␤ in G , we have ␥ ² ␣ i, ␤ i:Ž.n ␥ G i Ž.Ln , and so, as in the i Ä␣ ii, ␤ 4 F Si proof of Lemma B, we see that there exists an N such that for every Gi, there are fewer than 2 N distinct words of the form ␧ ␧ ␧ 0 1 Ny 1 ␣␤␣␤iiiiиии ␣␤ i i with ␧jg Ä40,1 i s 0,...,N y 1. ␧ ␧ ␧ ␧ N 0 1 2 Ny 1 Listing all 2 words of the form ␣␤␣␤␣␤иии ␣ , ␤, ␧j g Ä40, 1 as y1 N w12,...,w N in some fixed order, set wl␮ [ wwl ␮ for 1 F l - ␮ F 2 . N 1 N Now form w, the left normed commutator of allŽ 2y Ž 2 y 1.. words of the form wl␮ in some prescribed order. Then w is a nontrivial word. Moreover, ⌫ is a subdirect product of the groups Gi, and from the above it follows that for every G we have w Ž.␣ , ␤ 1 for at least one word w . il␮ iis l, ␮ Consequently, we have wŽ.␣, ␤ ' 1, and obtain a contradiction to the freeness of ²:␣, ␤ . Thus, Wilson’s result implies that ⌫ is virtually solvable. But if H is a solvable normal subgroup of finite index in ⌫, then

by the simplicity of each Gi, we obtain that H has a trivial image in each Gii. Thus for all i, <

Now replace the family G in Theorem 3.2 by a family of finite soluble groups of r-bounded ranks. Our aim is to be able to invoke Theorem 3.1. However, since we as yet know nothing about the derived lengths of these groups, we must first show the following.

ROPOSITION Ž. P E. Let G s Giig I be a family of finite soluble groups of bounded rank r, with S a bounded sized family of generating sets so that Ž.G, S grows subexponentially. Then there exists a constant d such that e¨ery group Gi has a deri¨ed length of at most d. Proof. We need the following resultŽ. due to D. Segal . THEOREM wxSe3 . ‘‘Let G be a finitely generated group, and suppose that G is residually finite-soluble-of-bounded-rank. Then G has a nilpotent normal subgroup Q such that GrQ is a subdirect product of finitely many linear groups o¨er fields.’’ Now construct Ž.⌫, S , a cover for the family ŽG, S .. Then ⌫ satisfies the hypotheses of Segal’s theorem, and hence its conclusions. Accordingly, let Q be the nilpotent normal subgroup of ⌫ such that ⌫rQ is a subdirect product of finitely many linear groups over fields. Then ⌫rQ satisfies Tits’ alternative. It thus follows that ⌫rQ must be virtually soluble, for if ⌫rQ were to contain a noncyclic free subgroup, then so would ⌫, and this is impossible, by reasoning similar to that in Lemma D. We conclude, therefore, that ⌫ is virtually soluble, and, by surjection, each soluble quotient Gi has a soluble subgroup of bounded derived length of bounded index in Gii, and thus each G in turn has bounded derived length. Finally, Lemma D and Proposition E enable us to deduce that a family G with a bounded sized generating family S such that Ž.G, S grows subexponentially, satisfying the hypotheses of Theorem 3.2, in fact satisfies the hypotheses of Theorem 3.1, and so is a totally polynomial family. Thus Theorem 3.2 has been proved. We now mention some corollaries that follow from the previous two theorems and their proofs:

COROLLARY 3.3. Any family of finite groups, which, when ¨iewed as linear groups, ha¨e bounded dimension, is either a totally polynomial family, or a totally exponential family.

COROLLARY 3.4. Let ⌫ be a f. g. linear group. Then any family G of finite quotients for ⌫ such that ⌫ is a subdirect product of members of G is either a totally polynomial or a totally exponential family, and is totally polynomial iff ⌫ is ¨irtually nilpotent. This motivates us to ask the following question: In Corollaries 3.3 and 3.4, and indeed in Theorems 3.1 and 3.2, we see examples of families of finite groups that have growth types the same as any infinite f.g. group that ASYMPTOTIC GROWTH OF FINITE GROUPS 421

‘‘covers’’ it. We ask whether or not this is true for all families of finite groups. In particular, does there exist a family of finite groups with a bounded sized generating family of subexponential type, with a f.g. cover of exponential growth?

SECTION 4. SOME PARTIAL RESULTS AND OPEN QUESTIONS

In Section 2 we characterized totally polynomial families, and showed that totally intermediate families do not exist. Regarding totally exponen- tial families, although results from Section 3 provide some sufficient conditions, it would be helpful to have some more precise information in this direction. Accordingly, we discuss some related conjectures, and give partial re- sults where possible. Conjecture 1. Every family of finite nonabelian simple groups is a totally exponential family. We have seenŽ. see Section 3, Corollary 3.3 that Conjecture 1 is true for nonabelian simple groups of bounded Chevalley rank. It would be interesting to be able to tackle the general case. Although we knowŽ. see Proposition 2.3 and Corollary 2.5 of Section 2 that any family of finite nonabelian simple groups has an exponential generating family, we mention the following result mainly for any possible applications it may have.

DEFINITION. Let C be a subset of Sn, the of degree n. Define the support of C, SuppŽ.C [ ࠻Ä x : ᭚␴ g C : ␴ Ž.x / x4. ROPOSITION Ž. Ž P 4.1. Let G s Giig I be a family of perfect so, for example, simple. transiti¨e permutation groups of degrees nii, n g ގ. Suppose Ž. T [ Tiig I is a family of respecti¨e generating sets for G with the following property: There exist elements tiig T , such that suppŽTii_ Ä4t .Ž.s on i. Then Ž.G, T grows exponentially. Proof. Suppose, to the contrary, that Ž.G, T grows subexponentially. G i Since ²:TiiG , the perfectness of Giimplies that Giii ²T Ä4t : j s s _ s t jg ޚ G i ² i < :²Ä4: xiiiix iiig T _Ät 4 , for otherwise the quotient G r T _ t would be proper and abelian. Now, applying techniques similar to those in Lemma B, we conclude that there exists a natural number k such that, for all i, j ²: ²ti < Ä4: < Ž.< Giis V ,whereV i[ xx iig T _ t i, < j< F ki.NowSuppV F k

We thus obtain that< SuppŽ.Vii<

XAMPLE Ž. E . Let G s Snng ގ the family of symmetric groups of degrees Ž. n. Although G is not perfect, we see that if T s Tnng ގ, where Tn s Sn ÄŽ.Ž12, 1,2,...,n .4, then indeed²Ž 12 .: s Sn, and so we may conclude that Ž.G, T has exponential growth. At this point we mention briefly some connections between growth of finite groups and diameter of Cayley graphs of finite groupsŽ both with respect to specified generating sets. , on which considerable research has been doneŽ seewx BHKLS for a survey article, and the references therein. . First some definitions: Let G be a finite group, and let S be a generating set for G. The Cayley graph X Ž.G, S of G with respect to S is the graph whose vertices correspond to elements in G, and two vertices representing g and h are adjacent iff g s hs for some s g S. The diameter, d, of a graph is the maximum length of the shortest path between any two vertices in the G ŽŽ.. graph. Thus it is easily checked that we have dSgs max g Gslg, where G G dSSdenotes the diameter of X Ž.G, S , whence ␥ Ž.d s <

EMMA Ž. L 4.2. Let G s Giig I be a nil-c-by-index-l family of finite groups, Ž. and let S s Siig I be any bounded sized generating family. Then the corresponding family of Cayley graphs has a polynomial diameter. ASYMPTOTIC GROWTH OF FINITE GROUPS 423

Proof. By Theorem 2.1, there exist constants K and ␣ such that max Ž␥ G i Ž..n Kn␣, whence, in particular, if d denotes the diameter ig ISi F i of X Ž.Gii, S , we have

1r␣ <

On the other hand, the group family G s Ä4Gppprime, where Gpps C i AutŽ.Cp , has polynomial diameter w.r.t. all bounded sized generating << Ž .␪ ŽŽ .2 . families. Indeed, Aut Cpp( C y1; thus Gp s ppy 1 s p y 1 , and Aut Cppp( G rC , whence if Spdenotes the epimorphic image in Aut Cpof a bounded sized generating set S for G , then dG ppdAut C ⍀Ž p pp SSppG s y 1r ␣ 2r ␣ 1. s ⍀<

Gp is, in fact, even centerless! Thus polynomial diameter is not a strong enough invariant to detect ‘‘virtual nilpotency.’’ Further, we have

EMMA Ž. L 4.3. Let G s Giig I be a family of finite groups with respecti¨e Ž. generating sets S s Siig I of bounded cardinality. Suppose G has logarith- mic diameter w.r.t. S. Then Ž.G, S has exponential growth.

Proof. By hypothesis, there exists a K ) 0 with dG i K log<

K such that every Gi g G has a soluble subgroup of index at most K. Passing to infinite groups, this translates to a slightly weaker version: Conjecture 2Ј. A f.g. residually finite group of subexponential growth is virtually residually soluble. 424 SARAH BLACK

We remark that our results can be used to show something in this direction. Indeed, the following can be proved:

LEMMA 4.4. Let ⌫ be a f. g. residually finite group of subexponential growth. Suppose thatÄ rkŽ. M< M is a nonabelian upper chief factor of ⌫4 is bounded. Then ⌫ is ¨irtually residually soluble.

Sketch of Proof. Let ⌫ be a f.g. residually finite group satisfying the hypotheses of the lemma, with r the bound on the ranks of the upper chief factors. For any finite subdirect factor G, the subgroup H s FM perfect chief factor of GGCMŽ.is soluble, and GrH is a subdirect product of subgroups of AutŽ.M , where M ranges over the perfect chief factors of G. By Shalev’s theorem againwx Sh2, Proposition 3.6 , M is isomorphic to at most r copies of either a simple group of r-bounded order, or of a simple e group S s XpnŽ .Žof Lie type, where n and e are r-bounded. Aut S.,in turn, has faithful linear representation of r-bounded degreeŽ see, for examplewxwx Sh2, Lemma 3.4 together with We, Lemma 2.3. ; consequently, so, too, does AutŽ.M , Aut Ž.S wrSr . Ž.⌫ ⌫ Ž Thus, if rK iig I is the collection of finite subdirect factors of so Fi IiK s 1.Ž with HirK ii .I the Ž corresponding . above-mentioned solu- g g ˜ ble subgroups, the above discussion implies that H [ Fig IiH is residu- ally finite-soluble, and that ⌫rH˜ is f.g. residually linear of r-bounded degree. Applyingwx Wi, Section 4.2 again, since ⌫ has subexponential growth, we deduce that ⌫rH˜ is virtually soluble, and, accordingly, ⌫ is as claimed.

One might even wonder whether Grigorchuk’s constructions of groups with intermediate growth, are, in a sense, typical of all such groups; indeed, we ask, are all f.g. residually finite groups of intermediate growth virtually residually nilpotent? This would be especially interesting, for we already know quite a bit about ‘‘gaps’’ in the possible growths of residually nilpotent groups. Indeed, results of Grigorchukwx Gri2 and Lubotzky and Mannwx LM2 together imply the following:

THEOREM. Let ⌫ be a f. g. residually nilpotent group of growth - 02Ž.'n . Then ⌫ is ¨irtually nilpotent and thus has polynomial growth.

At any rate, we mention the following conjecture:

Conjecture 3. Let ⌫ be a f.g. residually finite group. Then either there 'n exist constants K ) 0, c ) 1, such that ␥SŽ.n G Kc ,or⌫ is virtually nilpotent. ASYMPTOTIC GROWTH OF FINITE GROUPS 425

ACKNOWLEDGMENTS

It is my pleasure to thank my supervisor, Prof. Alex Lubotzky, for his guidance during the preparation of this work, which formed part of my doctoral thesis. I also thank Prof. D. Segal for reading this paper and for providing some useful suggestions as to its layout.

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