On the Gap Conjecture concerning group growth

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

We discuss some new results concerning Gap Conjecture on group growth and present a reduction of it (and its ...-version) to several special classes of groups. Namely we show that its validity for the classes of simple groups and residually finite groups will imply the Gap Conjecture in full generality. A similar type reduction holds if the Conjecture is valid for residually polycyclic groups and just-infinite groups. The cases of residually solvable groups and right orderable groups are considered as well.

Bull. Math. Sci. (2014) 4:113128

DOI 10.1007/s13373-012-0029-4

Received: 4 April 2012 / Accepted: 17 September 2012 / Published online: 7 October 2012 The Author(s) 2012. This article is published with open access at SpringerLink.com

Abstract We discuss some new results concerning Gap Conjecture on group growth and present a reduction of it (and its -version) to several special classes of groups.

Namely we show that its validity for the classes of simple groups and residually nite groups will imply the Gap Conjecture in full generality. A similar type reduction holds if the Conjecture is valid for residually polycyclic groups and just-innite groups. The cases of residually solvable groups and right orderable groups are considered as well.

1 Introduction

Growth functions of nitely generated groups were introduced by Schvarz [37] and independently by Milnor [29], and remain popular subject of geometric group theory. Growth of a nitely generated group can be polynomial, exponential or intermediate between polynomial and exponential. The class of groups of polynomial growth coincides with the class of virtually nilpotent groups as was conjectured by Milnor and conrmed by Gromov [24]. Milnors problem on the existence of groups of intermediate growth was solved by the author in [12,13], where for any prime p an uncountable family of 2-generated torsion p-groups G(p) with different types of inter

mediate growth was constructed. Here is a parameter of construction taking values in the space of innite sequences over the alphabet on p + 1 letters. All groups G(p)

satisfy the following lower bound on growth function

Communicated by Em Zelmanov.

The author is partially supported by the Simons Foundation and by NSF grant DMS-1207699.

R. Grigorchuk (B)

Department of Mathematics, Mailstop 3368, Texas A&M University, College Station, TX 77843-3368, USAe-mail: [email protected]

On the Gap Conjecture concerning group growth

Rostislav Grigorchuk

123

114 R. Grigorchuk

G(n) e

n, (1.1)

where G(n) denotes the growth function of a group G and is a natural comparison

of growth functions (see the next section for denition). The inequality (1.1) just indicates that growth of a group is not less than the growth of the function en.

All groups from families G(p) are residually nite-p groups (i.e. are approximated

by nite p-groups). In [15] the author proved that the lower bound (1.1) is universal for all residually nite-p groups and this fact has a straightforward generalization to residually nilpotent groups, as it is indicated in [28].

The paper [13] also contains an example of a torsion free group of intermediate growth, which happened to be right orderable group, as was shown in [19]. For this group the lower bound (1.1) also holds.

In the ICM Kyoto paper [23] the author raised a question if the function en gives a universal lower bound for all groups of intermediate growth. Moreover, later he conjectured that indeed this is the case. The corresponding conjecture is now called the Gap Conjecture on group growth. In this note we collect known facts related to the Conjecture and present some new results. A recent paper [22] gives further information about the history and developments around the notion of growth in group theory.

The rst part of the note is introductory. The second part begins with the case of residually solvable groups where basically we present some of results of Wilson from [40,42] and a consequence from them. Then we consider the case of right orderable groups, and the nal part contains two reductions of the Conjecture (and its

-version) to the classes of residually nite groups and simple groups (Theorem 7.4), and to the class of just-innite groups, modulo its correctness for residually polycyclic groups (Theorem 7.3).

2 Preliminary facts

Let G be a nitely generated group with a system of generators A = {a1, a2, . . . , am}

(throughout the paper we consider only innite nitely generated groups and only nite systems of generators). The length |g| = |g|A of an element g G with respect

to A is the length n of the shortest presentation of g in the form

g = a1i1a1i2 a1in,

where aij are elements in A. It depends on the set of generators, but for any two systems of generators A and B there is a constant C

N such that the inequalities

|g|A C|g|B, |g|B C|g|A. (2.1)

hold.

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On the Gap Conjecture concerning group growth 115

The growth function of a group G with respect to the generating set A is the function

AG(n) =

[vextendsingle][vextendsingle]{g G : |g|A n}

where |E| denotes the cardinality of a set E, and n is a natural number.

If = (G, A) is the Cayley graph of a group G with respect to the generating

set A, then |g| is the combinatorial distance between vertices g and e (the identity

element in G), and AG(n) counts the number of vertices at combinatorial distance n

from e (i.e., it counts the number of elements in the ball of radius n with center at the identity element).

It follows from (2.1) that growth functions AG(n), BG(n) satisfy the inequalities

AG(n) BG(Cn), BG(n) AG(Cn). (2.2)

The dependence of the growth function on generating set is inconvenience and it is customary to avoid it by using the following trick. Two functions on the naturals 1 and 2 are called equivalent (written 1 2) if there is a constant C

N such

that 1(n) C2(Cn), 2(n) C1(Cn) for all n 1. Then according to (2.2),

the growth functions constructed with respect to two different systems of generators are equivalent. The class of equivalence [ AG] of growth function is called degree of

growth, or rate of growth of G. It is an invariant not only up to isomorphism but also up to weaker equivalence relation called quasi-isometry [8].

We will also consider a preoder on the set of growth functions:

1(n) 2(n) (2.3)

if there is an integer C > 1 such that 1(n) 2(Cn) for all n 1. This converts the

set W of growth degrees of nitely generated groups into a partially ordered set. The

notation will be used in this article to indicate a strict inequality.

Let us remind some basic facts about growth rates that will be used in the paper.

The power functions n belong to different equivalence classes for different 0. The polynomial function Pd(n) = cdnd + + c1n + c0, where cd = 0 is

equivalent to the power function nd.

All exponential functions n, > 1 are equivalent and belong to the class [en]. All functions of intermediate type en, 0 < < 1 belong to different equivalence

classes.

This is not a complete list of rates of growth that a group may have. Much more is provided in [12] and [3].

It is easy to see that growth of a group coincides with the growth of a subgroup of nite index, and that growth of a group is not smaller than the growth of a nitely generated subgroup or of a factor group. Since a group with m generators can be presented as a quotient group of a free group of rank m, the growth of a nitely generated group cannot be faster than exponential (i.e., it can not be superexponential). Therefore we can split the growth types into three classes:

[vextendsingle][vextendsingle],

123

116 R. Grigorchuk

Polynomial growth. A group G has polynomial growth if there are constants C > 0

and d > 0 such that (n) < Cnd for all n 1. Minimal d with this property is

called the degree of polynomial growth.

Intermediate growth. A group G has intermediate growth if (n) grows faster than

any polynomial but slower than any exponent function n, > 1 (i.e. (n) en). Exponential growth. A group G has exponential growth if (n) is equivalent to en. The question on the existence of groups of intermediate growth was raised in 1968

by Milnor [30]. For many classes of groups (for instance for linear groups by Tits alternative [38], or for solvable groups by the results of Milnor [31] and Wolf [43]) intermediate growth is impossible. Milnors question was answered by author in 1983 [10,12,20], where it was shown that there are uncountably many 2-generated torsion groups of intermediate growth. Moreover, it was shown in [12,13,20] that for any prime p a partially ordered set Wp of growth degrees of nitely generated torsion

p-groups contains uncountable chain and contains uncountable anti-chain. The immediate consequence of this result is the existence of uncountably many quasi-isometry equivalence classes of nitely generated groups (in fact 2-generated groups) [12].

Below we will use several times the following lemma [24, page 59].

Lemma 2.1 (Splitting lemma) Let G be a nitely generated group of polynomial growth of degree d and H [triangleleft] G be a normal subgroup with quotient G/H being an innite cyclic group. Then H has polynomial growth of degree d 1.

3 Gap Conjecture and its modications

We will say that a group is virtually nilpotent (virtually solvable) if it contains nilpo-tent (solvable) subgroup of nite index. It was observed around 1968 by Milnor, Wolf, Hartly and Guivarch that a nilpotent group has polynomial growth and hence a virtually nilpotent group also has polynomial growth. In his remarkable paper [24], Gromov established the converse.

Theorem 3.1 (Gromov 1981) If a nitely generated group G has polynomial growth, then G contains a nilpotent subgroup of nite index.

In fact Gromov obtained stronger result about polynomial growth.

Theorem 3.2 For any positive integers d and k, there exist positive integers R, N and q with the following property. If a group G with a xed system of generators satises the inequality (n) knd for n = 1, 2, . . . , R then G contains a nilpotent subgroup

H of index at most q and whose degree of nilpotence is at most N.

The above theorem implies existence of a function growing faster than any polynomial and such that if G , then growth of G is polynomial.

Indeed, taking a sequence {ki, di}i=1 with ki and di when i and

the corresponding sequence {Ri}i=1, whose existence follows from Theorem 3.2, one

can build a function (n) which coincides with the polynomial kindi on the interval

[Ri1 + 1, Ri] and separates polynomial growth from intermediate. Therefore there

is a Gap in the scale of rates of growth of nitely generated groups and a big problem

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On the Gap Conjecture concerning group growth 117

is to nd the optimal function (or at least to provide good lower and upper bounds for it) which separates polynomial growth from intermediate. The best known result in this direction is the function n(loglogn)c (c some positive constant) which appeared recently in the paper of Shalom and Tao [36, Corollary 8.6].

The lower bound of the type en for all groups G(p) of intermediate growth estab

lished in [10,12,13,20] allowed the author to guess that equivalence class of function en could be a good candidate for a border between polynomial and exponential growth. This guess was further strengthened in 1988 when the author obtained the result published in [15] (see Theorem 5.1). For the rst time the Gap Conjecture was formulated in the form of a question in 1991 (see [23]).

Conjecture 1 (Gap Conjecture) If the growth function G(n) of a nitely generated group G is strictly bounded from above by en (i.e. if G(n) e

n), then growth of

The question of independent interest is whether there is a group, or more generally a cancellative semigroup, with growth equivalent to en (for the role of cancellative semigroups in growth business see [14]).

In [22] the author formulated a number of conjectures relevant to the main Conjecture discussed there and in this note. Let us recall some of them as they will play some role in what follow.

Conjecture 2 (Gap Conjecture with parameter , 0 < < 1). If the growth function G(n) of a nitely generated group G is strictly bounded from above by en (i.e. if (n) en ) then the growth of G is polynomial.

Thus the Gap Conjecture with parameter 1/2 is just the Gap Conjecture 1. If < 1/2 then the Gap Conjecture with parameter is weaker than the Gap Conjecture, and if > 1/2 then it is stronger than the Gap Conjecture.

Conjecture 3 (Weak Gap Conjecture). There is a , 0 < < 1 such that if G(n) en then the Gap Conjecture with parameter holds.

The gap type conjectures can be formulated for other asymptotic characteristics of groups like return probabilities P(n)e,e (e denotes the identity element) for a non degenerate random walk on a group, Flner function F(n), or spectral density N ().

There is a close relation between them and the Gap Conjecture on growth, which was mentioned in [22]. When writing this note the author realized that to understand better the relation between different forms of the gap type conjectures it is useful to consider in parallel to the Conjecture 2 [which we will denote C()] a stronger version of it, which we will denote C():

Conjecture 4 (Conjecture C()) If a group C is not virtually nilpotent then C(n)

en .

It is obvious that C() implies C() but the opposite is not clear. This is related to the fact that there are groups with incomparable growths [12] as the set W of

rates of growth of nitely generated groups is not linear ordered. The motivation for introducing a -version of the Gap Conjecture will be more clear when a second note

[21] of the author is submitted to the arXiv.

G is polynomial.

123

118 R. Grigorchuk

4 Growth and elementary amenable groups

Amenable groups were introduced by von Neumann in 1929 [39]. Now they play extremely important role in many branches of mathematics. Let AG denote the class of amenable groups. By a theorem of Adelson-Velskii [1], each nitely generated group of subexponential growth belongs to the class AG. This class contains nite groups and commutative groups and is closed under the following operations:(1) taking a subgroup,(2) taking a quotient group,(3) extensions,(4) unions (i.e. if for some net {}, G AG and G G if < then G AG).

Let EG be the class of elementary amenable groups i.e., the smallest class of groups containing nite groups, commutative groups which is closed with respect to the operations (1)(4). For instance, virtually nilpotent and, more generally, virtually solvable groups belong to the class EG. This concept dened by Day in [6] got further development in the article [5] of Chou who suggested the following approach to study of elementary amenable groups.

For each ordinal dene a subclass EG of EG in the following way. EG0 consists of nite groups and commutative groups. If is a limit ordinal then

EG =

[uniondisplay]

Further, EG+1 is dened as as the class of groups which are extensions of groups

from set EG by groups from the same set or are direct limits of a family of groups from set EG. It is known (and easy to check) that each of the classes EG is closed with respect to the operations (1) and (2) [5]. By the elementary complexity of a group G EG we call the smallest such that G EG.

It was shown in [5] that class EG does not contain groups of intermediate growth, groups of Burnside type (i.e. nitely generated innite torsion groups), and nitely generated innite simple groups. A further study of elementary groups and its generalizations was done by Osin [33].

A larger class SG of subexponentially amenable groups was (implicitly) introduced in [9], and explicitly in [16], and studied in [7] and other papers.

A useful fact about groups of intermediate growth which we will use is due to Rosset [35].

Theorem 4.1 If G is a nitely generated group which does not grow exponentially and H is a normal subgroup such that G/H is solvable, then H is nitely generated.

We propose the following generalization of this result.

Theorem 4.2 Let G be a nitely generated group with no free subsemigroup on two generators and let the quotient G/N be an elementary amenable group. Then the kernel N is a nitely generated group.

The latter two statements and the chain of further statements of the same spirit that appeared in the literature were initiated by the following lemma of Milnor [31]: if G

EG.

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On the Gap Conjecture concerning group growth 119

is a nitely generated group with subexponential growth, and if x, y G, then the

group generated by the set of conjugates y, xyx1, x2yx2, . . . is nitely generated.

Proof For the proof of the Theorem 4.2 we will apply induction on elementary complexity of the quotient group H = G/N. If complexity is 0 then the group is either

nite or abelian. In the rst case N is nitely generated for obvious reason. In the second case we apply the following statements from the paper of Longobardi and Rhemtulla [27, Lemmas 1,2].

Lemma 4.3 If G has no free subsemigroups, then for all a, b G the subgroup abn, n

is nitely generated.

Lemma 4.4 Let G be a nitely generated group. If N [triangleleftequal] G, G/N is cyclic, and

abn, n

is nitely generated for all a, b G, then N is nitely generated. Assume that the statement of the theorem is correct for quotients H = G/H with

complexity 1 for some ordinal , 1. The group H, being nitely

generated, allows a short exact sequence

{1} A H B {1},

where A, B EG1. Let : G G/N be the canonical homomorphism and

M = 1(A). Then M is a normal subgroup in G and G/M G/N/M/N H/A

B. By the inductive assumption M is nitely generated and has no free subsemigroup on two generators. As M/N A, again by induction, N is nitely generated and we

are done.

We will discuss just-innite groups in detail in the last section. But let us prove now a preliminary result which will be used later. Recall that a group is called just-innite if it is innite, but every proper quotient is nite (i.e. every nontrivial normal subgroup is of nite index). A group G is called hereditary just-innite if it is residually nite and every subgroup H < G of nite index (including G itself) is just innite. Observe that a subgroup of nite index of a hereditary just-innite group is hereditary just-innite.

We learned the following result from de Cournulier. A proof is provided here as there is no one in the literature.

Theorem 4.5 Let G be a nitely generated hereditary just-innite group, and suppose that G belongs to the class EG of elementary amenable groups. Then G is isomorphic either to the innite cyclic group Z or to the innite dihedral group D.

Proof If G EG0 then G is abelian and hence G

Z. Assume that the statement is correct for all groups from classes EG, < for some ordinal . Let us prove it for . Assume G EG and is smallest with this property. can not be a limit

ordinal because G is nitely generated. Therefore G is the extension of a group A by a group B = G/A, where A, B EG1. In fact B is a nite group (as G is

just-innite). As a subgroup of nite index in a hereditary just-innite group, A is hereditary just-innite and moreover nitely generated (as a subgroup of nite index in a nitely generated group). By inductive assumption A is isomorphic either to the innite cyclic group Z or to the innite dihedral group D. In particular G has a

normal subgroup H of nite index isomorphic to Z.

Z

Z

123

120 R. Grigorchuk

Let G act on H by conjugation. Then we get a homomorphism : G Aut(H)

Z2. If (G) = {1}, then H is a central subgroup. It is a standard fact in group theory

(see for instance [25, Proposition 2.4.4]) that if there is a central subgroup of nite index in G then the commutator subgroup G is nite. But as G is just-innite, G = {1}

and so G is abelian, hence G

Z in this case.

If (G) = Aut(H) then N = ker is a centralizer CG(H) of H in G. Subgroup

N has index 2 in G, is just-innite and hence by the same reason as above N = {1},

so N is abelian. Being nitely generated and just innite implies N

N. The element x acts on N by conjugation mapping each element

to its inverse. In particular, x1(x2)x = x2, so (x2)2 = 1. But x2 N. Since N is

torsion free x2 = 1. Therefore

G = x, N = x, y : x2 = 1, x1yx = y1 D,

where y is a generator of N.

5 Gap Conjecture for residually solvable groups

Recall that a group G is said to be a residually nite-p group (sometimes also called residually nite p-group) if it is approximated by nite p-groups, i.e., for any g G

there is a nite p-group H and a homomorphism : G H with (g) = 1. This

class is, of course, smaller than the class of residually nite groups, but it is pretty large. For instance, Golod-Shafarevich groups, p-groups G from [12,13], and many

other groups belong to this class.

Theorem 5.1 [15] Let G be a nitely generated residually nite-p group. If G(n)

en then G has polynomial growth.

As was established by the author in a discussion with Lubotzky and Mann during the conference on pronite groups in Oberwolfach in 1990, the same arguments as given in [12] combined with the following lemma from [28].

Lemma 5.2 (Lemma 1.7, [28]) Let G be a nitely generated residually nilpotent group. Assume that for every prime p the pro-p-closure G

p of G is p-adic analytic.

allows one to prove a stronger version of the above theorem (see the Remark after Theorem 1.8 in [28]):

Theorem 5.3 Let G be a residually nilpotent nitely generated group. If G(n) e

then G has polynomial growth.

To be linear means to be isomorphic to a subgroup of the linear group GLn(K) for some eld K. By Tits alternative [38] every nitely generated linear group either contains a free subgroup on two generators or is virtually solvable. Hence the above lemma immediately reduces Theorem 5.3 to Theorem 5.1.

The latter two theorems (where the rst one is the corresponding statement from [15] while the second one is a corrected form of what is stated in Remark on page

Z.

Let x G, x /

Then G is linear.

n

123

On the Gap Conjecture concerning group growth 121

527 in [28]) show that Gap Conjecture C(1/2) holds for the class of residually nite-p groups and more generally for the class of residually nilpotent groups. In fact, arguments provided in [15,28] allow to prove stronger conjecture C(1/2) for these classes of groups.

Let p be a prime and a(p)n be the n-th coefcient of the power series given by

n=0

[summationdisplay] a(p)nzn =

n holds. Moreover if a group G is a residually nite-p group and is not virtually nilpotent then for any system of generators A

AG(n) a(p)n, n = 1, 2, . . .

(see the relation (23) and Lemma 8 in [15]). Observe that the latter statement is valid not only in the case when A is a system of elements that generate G as a group but even in a more general case when A is a generating set for the group G considered as a semigroup. In fact, growth function of any group is bounded from below by a sequence of coefcients of Hilbert-Poincar series of the universal p-enveloping algebra of the restricted Lie p-algebra associated with the group using the factors of the lower p-central series [15].

Theorem 1.8 from [28] contains an interesting approach to polynomial growth type theorems in the case of residually nilpotent groups. Moreover, as is mentioned in [28] in the remark after the theorem, the proof provided there yields the same conclusion

under a weaker assumption: G(n) 22

log2 n .

Surprisingly, in his rst paper on the gap type problem [42] Wilson used a similar

upper bound G(n) ee(1/2)

ln n to measure size of a gap for residually solvable groups.

Wilsons approach is quite different from those that were used before and is based on exploring self-centralizing chief factors in nite solvable groups.

Recall that a chief factor of a group G is a (nontrivial) minimal normal subgroup of some quotient G/N, and that L/M is a self-centralizing chief factor of a group G if M is normal in G, L/M is a minimal normal subgroup of G/M, and L/M =

CG/M(L/M). One of the results in [42] is

Theorem 5.4 (Wilson) Let G be a residually solvable group of subexponential growth whose nite self-centralizing chief factors all have rank at most k. Then G has a residually nilpotent normal subgroup whose index is nite and bounded in terms of k and G(n).

If, in addition G(n) e

n, then G has a nilpotent normal subgroup whose index is nite and bounded in terms of k and G(n).

The proof of this result is based on the following lemma the proof of which uses ultraproducts.

Lemma 5.5 (Lemma 2.1, [42]) Let k be a positive integer and :

R+ a function

such that (n)/n 0 as n . Suppose that G is a nite solvable group having (i)

n=1

[productdisplay] 1 z pn 1 zn

.

Then the lower bound a(p)n e

N

123

122 R. Grigorchuk

a self-centralizing minimal normal subgroup V of rank at most k and (ii) a generating set A such that AG(n) e(n) for all n. Then |G/V | is bounded in terms of k and

alone.

One of the almost immediate corollaries of the technique developed in [42] are the facts stated below in Theorems 5.6 and 5.7.

Recall that a group is called supersolvable if it has a nite normal descending chain of subgroups with cyclic quotients. Every nitely generated nilpotent group is supersolvable [34], and the symmetric group Sym(4) is the simplest example of a solvable but not supersolvable group.

Theorem 5.6 The Gap Conjecture holds for residually supersolvable groups. Moreover, the conjecture C(1/2) holds for residually supersolvable groups.

Developing his technique and using the known facts about maximal primitive solvable subgroups of GLn(p) (p prime) Wilson in [40] proved that the Gap Conjecture with parameter 1/6 holds for residually solvable groups. In fact what follows from arguments in [42], combined with arguments from [15,28] and with what was written above, can be formulated as

Theorem 5.7 The conjecture C(1/6) holds for residually solvable groups.

There is a hope that eventually the Gap Conjecture and its -version will be proved

for residually solvable groups, or at least for residually polycyclic groups (which is the same as to prove it for groups approximated by nite solvable groups, because polycyclic groups are residually nite [34]). If the latter is done, then we will have complete reduction of the Gap Conjecture to just-innite groups (more on this in the last section).

6 Gap Conjecture for right orderable groups

Recall that a group is called right orderable if there is a linear order on the set of its elements invariant with respect to multiplication on the right. In a similar way are dened left orderable groups. A group is bi-orderable (or totally orderable) if there is a linear order invariant with respect to multiplication on the left and on the right. Every right orderable group is left orderable and vise versa but there are right orderable groups which are not totally orderable (see [26] for examples). As was shown by Machi and the author the class of nitely generated right orderable groups of intermediate growth is nonempty [19]. The corresponding group was earlier constructed in [16] as

an example of a torsion free group of intermediate growth. It was implicitly observed in [19] that the class of countable right orderable groups coincides with the class of groups acting faithfully by homeomorphisms on the line R (or, what is the same, on the interval [0, 1]). Recently Erschler and Bartholdi managed to compute the growth

of which happens to be elog(n)n0 where 0 = log 2/ log(2/) 0.7674, and is

the real root of the polynomial x3 + x2 + x 2. The question if there exists a nitely

generated, totally orderable group of intermediate growth is still open.

The Gap Conjecture and it modications stated in Sect. 3 are interesting problems even for the class of right orderable groups. Our next result makes some contribution to

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On the Gap Conjecture concerning group growth 123

this topic. The result of Wilson combined with theorems of Morris [32] and Rosset [35] can be used to prove the following statement.

Theorem 6.1 (i) The Gap Conjecture with parameter 1/6, and, moreover, the conjecture C(1/6) hold for right orderable groups.

(ii) The Gap Conjecture C(1/2) [or its -version C(1/2)] holds for right orderable

groups if it [or its -version C(1/2)] holds for residually polycyclic groups.

Proof (i) Let G be a nitely generated right orderable group with growth en1/6. In

[32] Morris proved that every nitely generated right orderable amenable group is indicable (i.e. can be mapped onto Z). As by Adelson-Velskii theorem [1] a group of intermediate growth is amenable, we conclude that the abelianization Gab = G/[G, G] is

innite and hence has a decomposition Gab = Gab G+ab where Gab

Zd, d 1

is a torsion free part of an abelian group and G+ab is a torsion part. Let N [triangleleft] G be a normal subgroup such that G/N = Gab. Since the commutator subgroup of a group is

a characteristic group and the torsion free part of abelian group also is a characteristic subgroup we conclude that N is a characteristic subgroup of G. By Theorem 4.1 N is a nitely generated group. Therefore we can proceed with N as we did with G. This allows us to get a descending chain

G > G1 > G2 > (6.1)

(where G1 = N etc) of characteristic subgroups with the property that Gi/Gi+1

if Gi+1 = {1}, for some sequence di

N, i = 1, 2, . . . .

If the chain (6.1) terminates after nitely many steps then G is solvable and by the results of Milnor and Wolf [31,43] G is virtually nilpotent in this case.

Suppose that chain (6.1) is innite and consider the intersection G =

If G = {1}, then the group G is residually solvable (in fact residually polycyclic),

and, because of restriction on growth, by Theorem 5.7, G is virtually nilpotent and hence has polynomial growth of some degree d. But this contradicts Splitting Lemma2.1. Therefore G = {1}. G/G is residually polycyclic, has growth not greater

than the growth of G and by previous argument is virtually nilpotent. If the degree of polynomial growth of G/G is l then again by Splitting Lemma the length of the chain (6.1) can not be larger than l, and we get a contradiction. The part (i) of the theorem is proven.

Now the proof of part (ii) follows immediately. If we assume that G has growth

e

n and that the Gap Conjecture holds for the class of residually polycyclic groups then the arguments from previous part (i) are applicable in the same manner. The only difference is that instead of Theorem 5.7 one should use the assumption that the Gap Conjecture holds for residually polyciclic groups. The same argument works in the case of conjecture C(1/2).

7 Gap Conjecture and just-innite groups

There is a strong evidence based on considerations presented below that the Gap Conjecture can be reduced to three classes of groups: simple groups, branch groups and

Zdi

[intersectiontext]

i=1 Gi.

123

124 R. Grigorchuk

hereditary just-innite groups. These three types of groups appear in a natural partition of the class of just-innite groups into three subclasses described in Theorem 7.3. The following statement is an easy application of Zorns lemma.

Proposition 7.1 Let G be a nitely generated innite group. Then G has a just-innite quotient.

Corollary 7.2 Let P be a group theoretical property preserved under taking quotients.

If there is a nitely generated group satisfying the property P then there is a just-innite

group satisfying this property.

Although the property of a group to have intermediate growth is not preserved when passing to a quotient group (the image may have polynomial growth), by theorems of Gromov [24] and Rosset [35], if the quotient G/H of a group G of intermediate growth has polynomial growth then H is a nitely generated group (of intermediate growth, as the extension of a virtually nilpotent group by a virtually nilpotent group is an elementary amenable group and therefore can not have intermediate growth), and one may look for a just-innite quotient of H and iterate this process in order to represent G as a consecutive extension of a chain of groups that are virtually nilpotent or just-innite groups. This observation was used in the previous section for the proof of Theorem 6.1 and is the base of the arguments for Theorems 7.4 and 7.5.

Recall that hereditary just-innite groups were already dened in Sect. 4. We call a just innite group near simple if it contains a subgroup of nite index which is a direct product of nitely many copies of a simple group.

Branch groups are groups that have a faithful level transitive action on an innite spherically homogeneous rooted tree T m dened by a sequence {mn}n=1 of natural

numbers mn 2 (determining the branching number for vertices of level n) with the

property that the rigid stabilizer ristG(n) has nite index in G for each n 1. Here by

ristG(n) we mean a subgroup [producttext]vVn ristG(vn) which is a product of rigid stabilizers

ristG(vn) of vertices vn taken over the set Vn of all vertices of level n, and ristG(v) is a subgroup of G consisting of elements xing the vertex v and acting trivially outside the full subtree with the root at v. For a more detailed discussion of this notion see [4,18]. This is a geometric denition. It follows immediately from the denition that branch groups are innite. The denition of an algebraically branch group can be found in [4,17]. Every geometrically branch group is algebraically branch but not vice versa. If G is algebraically branch then it has a quotient G/N which is geometrically branch. The difference between two versions of the denitions is not large but still there is no complete understanding how much the two classes differ (it is not clear what can be said about the kernel N, it is believed that it should be central in G). For just-innite branch groups the algebraic and geometric denitions are equivalent. Not every branch group is just-innite, but every proper quotient of a branch group is virtually abelian [18]. Therefore branch groups are almost just-innite and most of known nitely generated branch groups are just-innite. Observe that a nitely generated virtually nilpotent group is not branch. This follows for instance from the fact that a nitely generated nilpotent group satises a minimal condition for normal subgroups while a branch group not.

123

On the Gap Conjecture concerning group growth 125

The next theorem was derived by the author from a result of Wilson [41].

Theorem 7.3 [18] The class of just-innite groups naturally splits into three subclasses: (B) branch just-innite groups, (H) hereditary just-innite groups, and (S) near-simple just-innite groups.

It is already known that there are nitely generated branch groups of intermediate growth. For instance, groups G of intermediate growth from the articles [11,13] are of

this type. In fact, all known examples of groups of intermediate growth are of branch type or are reconstructions on the base of groups of branch type. The question about existence of amenable but non-elementary amenable hereditary just-innite group is still open (remind that by Theorem 4.5 the only elementary amenable hereditary just-innite groups are Z and D).

Problem 1 Are there nitely generated hereditary just-innite groups of intermediate growth?

Problem 2 Are there nitely generated simple groups of intermediate growth?

The next theorem is a straightforward corollary of the main result of Bajorska and Makedonska from [2] (observe that it was not stated in [2]). Here we suggest a different proof which is adapted to the needs of the proof of the main Theorem 7.5.

Theorem 7.4 If the Gap Conjecture or conjecture C(1/2) holds for the classes of residually nite groups and simple groups, then the corresponding conjecture holds for the class of all groups.

Proof Assume that the Gap Conjecture is correct for residually nite groups and for simple groups. Let G be a nitely generated group with growth e

n. By Proposition7.1 it has just-innite quotient G = G/N, which belongs to one of the three types of

groups listed in the statement of the Theorem 7.3. The rate of growth of G is not greater

than the rate of growth of en. The group G can not be near simple because in this

case it will have a subgroup H of nite index with innite nitely generated simple quotient whose rate of growth is e

n. This is impossible as a virtually nilpotent group can not be innite simple.

The group G also can not be branch as branch groups are residually nite and

nitely generated virtually nilpotent groups are not branch. So we can assume that

G is hereditary just innite and hence residually nite. Using the assumption of the theorem we conclude that G is virtually nilpotent, and therefore elementary amenable.

By Theorem 4.5 G is isomorphic either to the innite cyclic group or to the innite

dihedral group D. By Theorem 4.1 kernel N is nitely generated. As the rate of

growth of N is less than en we can apply to N the same arguments as for G in order to get a surjective homomorphism either onto Z or onto D.

If G/N

Z, then we repeat the rst step of the proof of Theorem 6.1 replacing N by a nitely generated characteristic subgroup N1 [triangleleft] G with quotient G/N1

for some d1 1. If G/N1 D then we slightly modify the rst step. Namely, in

this case G has indicable subgroup H of index 2. Let H1 be the intersection of groups H, Aut(G). As there are only nitely many subgroups of index 2 in G this

intersection involves only nitely many groups and H1 is a characteristic subgroup in

Zd1

123

126 R. Grigorchuk

G of nite index of type 2t for some t

N. Moreover, G/H1

Zt2 as the quotient

G/H1 is isomorphic to a subgroup of a direct product of nitely many copies of group

Z2 of order 2. The subgroup H1, being a subgroup of index 2t1 in H, is indicable and we can apply the argument of the rst step of the proof of Theorem 6.1 getting a nitely generated subgroup H2 [triangleleftequal] H1 characteristic in G with quotient H1/H2

N.

Let G1 [triangleleft] G be a subgroup N, H1 or H2 depending on the case. Proceed with G1 in a similar fashion as we did with G, etc. We get a descending chain {Gi}i1 of nitely

generated subgroups characteristic in G. There are two possibilities.

(1) After nitely many steps we get a group Gi which is hereditary just-innite and elementary amenable, and hence innite cyclic or D (Theorem 4.5). In this case

G is polycyclic and we are done in view of the result of Milnor and Wolf on growth of solvable groups.(2) The process of construction of the chain of subgroups will continue forever. In this case we get a chain with the property that Gi/Gi+1 is isomorphic either to

(i) Zdi , di

for some d1

N. Moreover, each step of type (ii) is immediately followed by a step of type (i).

Let us show that this is impossible. Let G be the intersection [intersectiontext]i1 Gi. Then

G/G is residually polycyclic and hence residually nite as every polycyclic group is residually nite [34]. Growth of G/G is less than en. Hence by the assumption of the theorem the group G/G is virtually nilpotent with the rate of polynomial growth of degree d for some d

N. But this contradicts the splitting lemma as for innitely many i the quotients Gi/Gi+1 are isomorphic to Zdi . This proves the conjecture

C(1/2).

In the case of the conjecture C(1/2) we proceed in a similar fashion. Only at the beginning we assume that the conjecture C(1/2) holds for residually nite groups and for simple groups and that G is a nitely generated group of intermediate growth whose growth does not satisfy inequality (n) en1/2.

Now we state and prove our main result.

Theorem 7.5 (i) If the Gap Conjecture with parameter 1/6 or its -version C(1/6)

holds for just-innite groups then the corresponding conjecture holds for all groups.

(ii) If the Gap Conjecture or its -version C(1/2) holds for residually polycyclic

groups and for just-innite groups then the corresponding conjecture holds for all groups.

Proof (i) The proof follows the same strategy as the proof of Theorem 7.4. Let G be a nitely generated group with growth en1/6. There can be two possibilities.

(1) G has a nite descending chain {Gi}ki=1 of nitely generated characteristic in G

groups with consecutive quotients Gi/Gi+1

Zdi or Gi/Gi+1

and Gk = {1}. In this case G is polycyclic and hence virtually nilpotent

Zd1

N or to

(ii) Zti2 , ti

Zti2 , for i < k

123

On the Gap Conjecture concerning group growth 127

(2) G has an innite descending chain {Gi}i=1, with the property that Gi/Gi+1

Zdi+1. The group

[intersectiontext]i1 Gi, is residually polycyclic with growth en1/6.

Apply in this case the result of Wilson stated in Theorem 5.4 concluding that G/G is virtually nilpotent which is impossible by the splitting lemma.

(ii) Proceed as in (i) with the only difference that in the subcase (2) we apply the assumption that the Gap Conjecture holds for residually polycyclic groups to conclude that this subcase is impossible.

These are arguments for C(1/2) version. The arguments for -version C(1/2) are

similar.

Acknowledgments This work was completed during visit of the author to the Institute Mittag-Lefer (Djursholm, Sweden) associated with the program Geometric and Analytic Aspects of Group Theory. The author acknowledges organizers of this program. Also the author would like to thank A. Mann for indication of the article [2], and I. Bondarenko and E. Zelmanov for numerous valuable remarks concerning the rst draft of this note.

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

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