On the rigidity of rank gradient in a group of intermediate growth
We introduce and investigate a rigidity property of rank gradient for an example of a group $\scr G$ of intermediate growth constructed by the first author in [Grigorcuk R. I. On Burnside’s problem on periodic groups // Funktsional. Anal. i Prilozhen. – 1980. – 14, № 1. – P. 53 – 54]. It is shown t...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507343997894656 |
|---|---|
| author | Grigorchuk, R. I. Kravchenko, R. Григорчук, Р. І. Кравченко, Р. |
| author_facet | Grigorchuk, R. I. Kravchenko, R. Григорчук, Р. І. Кравченко, Р. |
| author_sort | Grigorchuk, R. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:18:03Z |
| description | We introduce and investigate a rigidity property of rank gradient for an example of a group $\scr G$ of intermediate growth constructed by the first author in [Grigorcuk R. I. On Burnside’s problem on periodic groups // Funktsional. Anal. i
Prilozhen. – 1980. – 14, № 1. – P. 53 – 54]. It is shown that $\scr G$ is normally $(f, g)$-RG rigid, where$ f(n) = \mathrm{l}\mathrm{o}\mathrm{g}(n)$ and
$g(n) = \mathrm{l}\mathrm{o}\mathrm{g}(\mathrm{l}\mathrm{o}\mathrm{g}(n))$. |
| first_indexed | 2026-03-24T02:07:49Z |
| format | Article |
| fulltext |
UDC 512.5
R. Grigorchuk* (Texas A&M Univ., USA),
R. Kravchenko (Northwestern Univ., USA)
ON THE RIGIDITY OF RANK GRADIENT
IN A GROUP OF INTERMEDIATE GROWTH
ПРО ЖОРСТКIСТЬ ГРАДIЄНТНОГО РАНГУ
В ГРУПI ПРОМIЖНОГО ЗРОСТАННЯ
We introduce and investigate a rigidity property of rank gradient for an example of a group \scrG of intermediate growth
constructed by the first author in [Grigorčuk R. I. On Burnside’s problem on periodic groups // Funktsional. Anal. i
Prilozhen. – 1980. – 14, № 1. – P. 53 – 54]. It is shown that \scrG is normally (f, g)-RG rigid, where f(n) = \mathrm{l}\mathrm{o}\mathrm{g}(n) and
g(n) = \mathrm{l}\mathrm{o}\mathrm{g}(\mathrm{l}\mathrm{o}\mathrm{g}(n)).
Введено та вивчено властивiсть жорсткостi градiєнтного рангу для прикладу групи \scrG промiжного зростання, що
була введена першим автором в роботi [Grigorčuk R. I. On Burnside’s problem on periodic groups // Funktsional. Anal.
i Prilozhen. – 1980. – 14, № 1. – P. 53 – 54]. Показано, що \scrG є нормально (f, g)-RG жорсткою, де f(n) = \mathrm{l}\mathrm{o}\mathrm{g}(n) та
g(n) = \mathrm{l}\mathrm{o}\mathrm{g}(\mathrm{l}\mathrm{o}\mathrm{g}(n)).
1. Introduction. A group is said to be residually finite if it has sufficiently many subgroups of finite
index so that the intersection of them is trivial. This is important class of groups studied throughout
more than eight decades by various tools and means. Residually finite groups are the reach source
of examples in group theory. In particular they are often used in the three main branches of modern
group theory: geometric group theory, asymptotic group theory and measured group theory. Such
groups have realization by actions on spherically homogeneous rooted trees as indicated in [6, 8],
which, in many cases, gives a possibility to study them and their subgroup structure using the structure
of the tree. They are also closely connected to the theory of profinite groups.
A very important invariant of residually finite group is a subgroup growth introduced by F. Grune-
wald, D. Segal and G. Smith [11] and studied by many researches (see a comprehensive book [13]
and the literature therein on this subject). Recently another asymptotic characteristics of residually
finite groups were introduced with the focus on the notion of rank gradient.
The rank gradient of a finitely generated residually finite group G is defined as
RG(G) = \mathrm{i}\mathrm{n}\mathrm{f}
d(H) - 1
[G : H]
, (1)
where the infimum is taken over all subgroups H in G of finite index and d(H) is the rank of H
(i.e., the minimal number of generators of H ). It is a finite number because a subgroup of finite index
in a finitely generated group is finitely generated, and the first question that arises is RG(G) = 0 or
not.
This notion, as well as the notion of the rank gradient relative to the descending chain of subgroups
(defined by (2)), were introduced for the first time by M. Lackenby [12] with motivation from 3-
dimensional topology. Since that the rank gradient and its variations were intensively studied [1, 4,
7, 8, 12].
* The author was partially supported by Simons Collaboration Grant No. 527814.
c\bigcirc R. GRIGORCHUK, R. KRAVCHENKO, 2018
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 2 165
166 R. GRIGORCHUK, R. KRAVCHENKO
The definition (1) can be modified in various directions. Instead of \mathrm{i}\mathrm{n}\mathrm{f} one can consider \mathrm{s}\mathrm{u}\mathrm{p},
instead of all subgroups one can consider only normal subgroups, or subgroups with index a power
of a prime number p, etc. Another direction of modifications is to consider descending sequence Hn,
n = 1, 2, . . . , of subgroups of finite index and associated sequence rg(n) of numbers defined as
rg(n) = RG(G, \{ Hn\} ) =
d(Hn) - 1
[G : Hn]
(2)
and its limit
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
rg(n)
if it exists, or the upper and the lower limits
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
rg(n),
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
rg(n)
otherwise.
It is known [1, 12] that if G is amenable (the notion of amenable group was introduced by von
Neumann [14] and by Bogolyubov in topological case [3], and \{ Hn\} is a sequence of subgroups of
finite index satisfying some technical condition (sometimes called the Farber condition, it is equivalent
to the essential freeness of the action of the group on the boundary of coset tree [7]), then the limit
(2) exists and is equal to 0. The example of the lamplighter group \scrL = \BbbZ /2\BbbZ \wr \BbbZ show that rg(n)
may have arbitrary fast decay, just because the group has a subgroup of index 2 isomorphic to itself,
so iterating this fact one gets a descending sequence of groups of growing index power of 2 but a
fixed rank = 2 [4].
We suggest the following definition. Let f(n), g(n) be two increasing functions of natural
argument n taking values in \BbbN and caring the limit \infty when n\rightarrow \infty .
Definition 1.1. (a) A finitely generated group G is (f, g) - RG-rigid if there is C \in \BbbN such
that for every subgroup H < G of finite index
g(d(H)) < Cf(C[G : H]) and f([G : H]) < Cg(Cd(H)).
(b) G is normally (f, g) - RG-rigid if previous inequalities hold for each normal subgroup
H \triangleright G of finite index.
For instance the free group Fr of rank r \geq 1 is (f, g) - RG-rigid where f(n) = g(n) = n as in
this case the ratio (d(H) - 1)/([Fr : H]) is constant and equal to r - 1. Also (n, n) - RG-rigid are
all groups with RG(G) > 0. For finitely presented groups this hold if and only if G is “large” in the
sense of S. Pride, i.e., contains a subgroup of finite index that surjects onto a noncommutative free
group, as shown in [12].
We present the following two results. Let \scrG = \langle a, b, c, d\rangle be an infinite 2-group constructed by
the first author in [10]. Recall that it has intermediate growth between polynomial and exponential
and has many other interesting properties [5, 6, 9]. \scrG has many other ways to be defined but for us
it will be important that it has a natural action by automorphisms of a rooted binary tree \scrT shown by
Fig. 1, as explained, for instance, in [6].
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 2
ON THE RIGIDITY OF RANK GRADIENT IN A GROUP OF INTERMEDIATE GROWTH 167
Fig. 1. Binary tree.
Theorem 1.1. The group \scrG is normally (f, g) - RG-rigid with
f(n) = \mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g}(n), g(n) = \mathrm{l}\mathrm{o}\mathrm{g}(n).
Moreover, there there is a constant D > 1 such that
1
D
\mathrm{l}\mathrm{o}\mathrm{g}(d(H)) \leq \mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g}([\scrG : H]) \leq D \mathrm{l}\mathrm{o}\mathrm{g}(d(H)) (3)
hold for every nontrivial normal subgroup H \lhd \scrG .
Conjecture 1.1. The group \scrG is (f, g) - RG-rigid with the same functions f, g as in previous
theorem.
If the conjecture is true then we have interesting rigidity property concerning the rank and index
of subgroups of finite index. At least we have this property for normal subgroups as shows the above
theorem. Recall that all normal subgroups in \scrG have finite index because \scrG is just-infinite (i.e.,
infinite, but every proper quotients finite) as shown in [6]. Interestingly, the group \scrG was used by
M. Lackenby in [12] to demonstrate some phenomenon that may hold for the rank gradient. The
present article develops the observation made by M. Lackenby concerning the rank gradient in \scrG .
At the moment we are able only to confirm the conjecture for important subclass of subgroups of
finite index in \scrG , namely for stabilizers of vertices of the binary rooted tree \scrT on which the group
\scrG acts. The vertices of \scrT are in bijection with finite words over binary alphabet \{ 0, 1\} . Let v be a
vertex and \mathrm{s}\mathrm{t}\scrG (v) be its stabilizer which has index 2n in \scrG if v is a vertex of level n.
Theorem 1.2. There is a constant D such that the inequalities (3) hold for all subgroups
H = \mathrm{s}\mathrm{t}\scrG (v), where v run over the set of vertices in \scrT . In fact
d(H) - 1
[\scrG : H]
=
n+ 3
2n
if v is a vertex of level n \geq 2.
Observe that this result is announced in [4].
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 2
168 R. GRIGORCHUK, R. KRAVCHENKO
2. The group \bfscrG . We recall some basic facts about the group \scrG and its subgroups. \scrG can be
defined as a group of automorphisms of a rooted binary tree \scrT shown in Fig. 1 (the root, corresponding
to the empty word, is a fixed point for the action). The generators a, b, c, d of \scrG are involutions,
the elements b, c, d commute and together with the identity element they constitute the Klein group
\BbbZ 2 \times \BbbZ 2. The stabilizer of the first level H = \mathrm{s}\mathrm{t}\scrG (1) is a subgroup of index 2 in \scrG generated
by elements b, c, d, ba, ca, da (where xy = x - 1yx), and the restrictions of H on the left and
right subtrees \scrT 0, \scrT 1 with the roots at vertices 0, 1 determine surjective homomorphisms \varphi 0, \varphi 1 :
H \rightarrow \scrG . The direct product \psi = \varphi 0 \times \varphi 1 of them determines the embedding H \rightarrow \scrG \times \scrG and acts
on generators as:
b\rightarrow (a, c), c\rightarrow (a, d), d\rightarrow (1, b), ba \rightarrow (c, a), ca \rightarrow (d, a), da \rightarrow (b, 1).
(4)
Together with the information that the generator a permute the subtrees \scrT 0, \scrT 1 (without extra
action inside them), this uniquely determines the group \scrG .
The action of \scrG on \scrT also can be described by the following recursive rules:
a(0w) = 1w, a(1w) = 0w,
b(0w) = 0a(w), b(1w) = 1c(w),
c(0w) = 0a(w), c(1w) = 1d(w),
d(0w) = 0w, d(1w) = 1b(w),
where w \in \{ 0, 1\} \ast , and \{ 0, 1\} \ast denotes the set of all finite words over the binary alphabet. The
important property of the action of \scrG on \scrT is level transitivity, i.e., transitivity of the action on each
level Vn = \{ 0, 1\} n.
Additionally to the stabilizers \mathrm{s}\mathrm{t}\scrG (n) of levels n = 1, 2, . . . , an important descending series of
normal subgroups is the series of rigid stabilizers \mathrm{r}\mathrm{i}\mathrm{s}\mathrm{t}\scrG (n) which are subgroups generated by rigid
stabilizers \mathrm{r}\mathrm{i}\mathrm{s}\mathrm{t}\scrG (v), v \in \{ 0, 1\} n, of vertices of the nth level and \mathrm{r}\mathrm{i}\mathrm{s}\mathrm{t}\scrG (v) is a subgroup in \scrG fixing
vertex v and consisting of elements acting trivially outside the subtree \scrT v in \scrT with a root at v.
The rigid stabilizers of distinct vertices of the same level commute and are conjugate (because of the
level transitivity). Thus algebraically the \mathrm{r}\mathrm{i}\mathrm{s}\mathrm{t}\scrG (n) is a direct product of copies of the same group
(which may depend on the level n in general case). Observe that \{ \mathrm{s}\mathrm{t}\scrG (n)\} \infty n=1 and \{ \mathrm{r}\mathrm{i}\mathrm{s}\mathrm{t}\scrG (n)\} \infty n=1
are descending chains of normal subgroups of finite index with trivial intersection. The structure of
groups \mathrm{s}\mathrm{t}\scrG (n) and \mathrm{r}\mathrm{i}\mathrm{s}\mathrm{t}\scrG (n) is well understood and described in [2].
Let B = \langle b\rangle \scrG be a normal closure of generator b and K = \langle (ab)2\rangle \scrG . For each n there is a natural
embedding \psi n : \mathrm{s}\mathrm{t}\scrG (n) \rightarrow \scrG \times . . . \times \scrG into a direct product of 2n copies of \scrG which is the nth
iteration of the embedding \psi and has a geometric meaning of the attaching to the element g \in \mathrm{s}\mathrm{t}\scrG (n)
the 2n-tuple (g1, . . . , g2n) of its restrictions on the subtrees with the roots at the nth level. Instead
writing
\psi n(g) = (g1, . . . , g2n)
we will write
g = (g1, . . . , g2n).
In particular the relations (4) can be rewritten as b = (a, c), c = (a, d), d = (1, b), ba = (c, a),
ca = (d, a), da = (b, 1).
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 2
ON THE RIGIDITY OF RANK GRADIENT IN A GROUP OF INTERMEDIATE GROWTH 169
G
K
K K
K K K K
K K K KK K K K
Fig. 2. Structure of branching subgroups.
The important facts about groups \mathrm{s}\mathrm{t}\scrG (n) and \mathrm{r}\mathrm{i}\mathrm{s}\mathrm{t}\scrG (n) are that their \psi n - 3 images have the
decompositions:
\mathrm{s}\mathrm{t}\scrG (3)\times \mathrm{s}\mathrm{t}\scrG (3)\times . . .\times \mathrm{s}\mathrm{t}\scrG (3) (5)
(products of 2n - 3 copies of \mathrm{s}\mathrm{t}\scrG (3)) when n \geq 4, and
K \times K \times . . .\times K (6)
(product of 2n copies of K ) when n \geq 2, respectively [2]. We will use later the notations Kn for
\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{t}\scrG (n) when n \geq 2 and keep in mind their decompositions (5) and (6). We also denote by K1 a
subgroup in \mathrm{s}\mathrm{t}\scrG (n) whose \psi -image is K \times K. The group \scrG is regularly branched over K as K1 is
a subgroup of K, and [\scrG : K1] < \infty . Thus \scrG contains subgroups shown by Fig. 2. To each level n
corresponds a group Kn that fixes each vertex v of this level and whose restriction to the subtree \scrT v
with the root v is K (if to identify \scrT v with \scrT ). Moreover Kn is a direct product of this projections.
3. Proof of Theorem 1.1. We begin this section with reminding that the group \scrG is branch
group as defined in [2], because it acts level transitive on the tree \scrT and rigid stabilizers \mathrm{r}\mathrm{i}\mathrm{s}\mathrm{t}\scrG (n),
n = 1, 2, . . . , have finite index in \scrG . The branch structure of \scrG that we will use is given by the Fig. 2
and was mentioned in previous section. The proof of the Theorem 1.1 is based on the following
proposition.
Proposition 3.1. Let N \lhd \scrG be a nontrivial normal subgroup. Let n be a smallest nonnegative
integer such that N < \mathrm{s}\mathrm{t}\scrG (n) but N is not a subgroup of \mathrm{s}\mathrm{t}\scrG (n+ 1). Then
(a) \mathrm{s}\mathrm{t}\scrG (n+ 5) < N.
(b) If n \geq 4, then
\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{t}\scrG (n+ 3) < N < \mathrm{r}\mathrm{i}\mathrm{s}\mathrm{t}\scrG (n - 3).
Proof. For the part (a) we first address the reader to the proof of the Theorem 4 given in [6] as
we will follow the same line in our arguments. First, let us make a comparison of notations used in
[6] and here. The branch structure for the group \scrG is given in our case by the pair
\bigl(
\{ Ln\} , \{ Hn\}
\bigr)
,
where Ln = K and Hn = Kn for all n. The reader should keep in mind the picture given by the
Fig. 2.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 2
170 R. GRIGORCHUK, R. KRAVCHENKO
Also, the normal subgroup in the statement of Theorem 4 is denoted by P while in the proposition
under consideration it is denoted N. The proof of Theorem 4 from [6] (modulo of the change of
notations) show that if N < \mathrm{s}\mathrm{t}\scrG (n) but N is not a subgroup of \mathrm{s}\mathrm{t}\scrG (n + 1), then N contains
commutator subgroup K \prime
n+1 = [Kn+1,Kn+1] (recall that \mathrm{r}\mathrm{i}\mathrm{s}\mathrm{t}\scrG (n) = Kn if n > 1). By Proposition 9
from [6] K \prime = K2, hence K \prime
n+1 = K \times K \times . . . \times K (2n+3 factors) so K \prime
n+1 = Kn+3, and
N > Kn+3.
Also by Proposition 9 from [6] we have the inclusion K > \mathrm{s}\mathrm{t}\scrG (3) from which, together with the
fact presented by factorization (5), we conclude that N > \mathrm{s}\mathrm{t}\scrG (n+ 5). The part (a) has been proven
as well as the lower bound in part (b). To get part (b) under assumption that n > 3 we observe that
the inclusion K > \mathrm{s}\mathrm{t}\scrG (3) and factorization (5) imply that Kn - 3 > \mathrm{s}\mathrm{t}\scrG (n), so we are done.
Proposition 3.1 is proved.
The group \scrG is not virtually cyclic (by many reasons, for instance because it is finitely generated
infinite torsion group). Let N\lhd G be a nontrivial normal subgroup. It is automatically of finite index,
as \scrG is just-infinite group [6] (i.e., infinite group with every proper quotient finite). In fact we can
assume from the beginning that N is a normal subgroup of finite index when proving Theorem 1.1
(so the just-infiniteness property is not needed). If n in the statement of Proposition 3.1 is less than
4, then there are only finitely many subgroups in \scrG containing \mathrm{s}\mathrm{t}\scrG (5), their ranks are \geq 2, and so
there is a constant D satisfying the condition of the Theorem 1.1. Therefore we can assume that
n \geq 4. Now apply part (b) of the proposition.
The quotient \mathrm{r}\mathrm{i}\mathrm{s}\mathrm{t}\scrG (n - 3)/ \mathrm{r}\mathrm{i}\mathrm{s}\mathrm{t}\scrG (n+ 3) = Kn - 3/Kn+3 is isomorphic to
A := (K/K6)
2n - 3
= K/K6 \times . . .\times K/K6
(2n - 3 factors). The group K/K6 is a finite 2-group of certain nilpotency class l (for us it is not
important the exact value of l). Therefore A is nilpotent of the class l as well and it is generated by
not more than 3 \cdot 2n - 3 elements as K is 3-generated group [6]. It is well known that a subgroup
of finitely generated nilpotent group is finitely generated and there is a universal upper bound on the
ranks of subgroups of nilpotent group G in terms of d(G) and class of nilpotency c of G.
We use the most simple upper bound given by the following lemma.
Lemma 3.1. Let G be a finitely generated nilpotent group of class c. Then for every subgroup
H < G the upper bound
d(H) < d(G)c
holds.
Proof. If c = 1 then G is Abelian and thus d(H) \leq d(G). For G of class c let \gamma 1(G) = G
and \gamma i+1(G) = [G, \gamma i(G)], i = 1, 2, . . . , be the elements of the lower central series. Suppose G is
generated by set S. Each factor \gamma i(G)/\gamma i+1(G) is generated by the iterated commutators [s1, [s2, . . .]]
of length i+ 1, where sj \in S. Thus d(\gamma i(G)/\gamma i+1(G)) \leq d(G)i+1. Denote Hi = H \cap \gamma i(G). Then
Hi\gamma i+1(G)/\gamma i+1(G) is an Abelian group, and thus d(Hi\gamma i+1(G)/\gamma i+1(G)) \leq d(\gamma i(G)/\gamma i+1(G)) \leq
\leq d(G)i+1. Note that
d(H) \leq d(H\gamma 1(G)/\gamma 1(G)) + d(H \cap \gamma 1(G)) \leq d(G) + d(H1).
Applying this iteratively we obtain
d(H) \leq d(G) + d(H1) \leq d(G) + d(H1\gamma 2(G)/\gamma 2(G)) + d(H2) \leq
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 2
ON THE RIGIDITY OF RANK GRADIENT IN A GROUP OF INTERMEDIATE GROWTH 171
\leq d(G) + d(\gamma 1(G)/\gamma 2(G)) + d(H2) \leq
\leq d(G) + d(G)2 + d(H2) \leq . . . \leq d(G) + . . .+ d(G)c - 1 \leq d(G)c.
Lemma 3.1 is proved.
Using this lemma we get
d(N) < (3 \cdot 2n - 3)l+2 = 3l+22(n - 3)(l+2) < a12
a2n
for some positive constants a1, a2.
Now we are going to give a lower bound for d(N). We factorize the inclusions Kn+3 < N <
< Kn - 3 by K \prime
n+3 getting
Kn+3/K
\prime
n+3 < N/K \prime
n+3 < Kn - 3/K
\prime
n+3.
The group Kn - 3/K
\prime
n+3 is a direct product of 2n - 3 copies of the group K/K8 as K = K2.
K/K8 is a finite 2-group. Let s be its nilpotency class, so Kn - 3/K
\prime
n+3 also has nilpotency class s.
As \=N := N/K \prime
n+3 is a subgroup of Kn - 3/K
\prime
n+3 its class of nilpotency is \leq s.
The group Kn+3/K
\prime
n+3 is a direct product of 2n+3 copies of the group K/K \prime = K/K2. Let
t = d(K/K \prime ) (in fact t = 3). Then d(Kn+3/K
\prime
n+3) = t2n+3. Using Lemma 3.1 we conclude
d
\bigl(
Kn+3/K
\prime
n+3
\bigr)
= t2n+3 \leq
\bigl(
d(N)
\bigr) s+2
from which we conclude that there are positive constants a3, a4 such that
a32
a4n \leq d(N).
Now using the part (a) of the Proposition 3.1 we provide an upper and lower bounds for the index
[\scrG : N ]. For this purpose we use the fact that\bigl[
\scrG : \mathrm{s}\mathrm{t}\scrG (n)
\bigr]
= 25\cdot 2
n - 3+2
as shown at the end of the proof of Theorem 14 from [6]. Hence part (a) of the proposition lead us
to the existence of positive constants a5, a6 and constants a7, a8 such that
2a52
n+a7 \leq [\scrG : N ] \leq 2a62
n+a8 .
Taking the double logarithm with base 2 of this inequalities and applying the same logarithm to
the previously obtained inequalities
a32
a4n \leq d(N) \leq a12
a2n
a simple calculus lead us to the proving of the theorem.
4. Proof of Theorem 1.2. We begin the proof of Theorem 1.2. The groups Rn, Qn and Pn
which are defined below are the finite index analogs of groups R, Q, P introduced and studied in
[2]. The recursive relations between them are analogous to the corresponding relations between R,
Q, P given in [2] by Theorems 4.4 and 4.5. The proof is based on a number of computations that
we split in propositions and lemmas.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 2
172 R. GRIGORCHUK, R. KRAVCHENKO
Let us introduce the elements t = (ab)2, u = (bada)2 = (t, 1), v = (abad)2 = (1, t). The direct
computation shows the result of their conjugation and of element (ac)4 by generators as shown in
the next subsections:
4.1. Conjugates of \bfitt , \bfitu , \bfitv :
ta = t - 1,
tb = t - 1,
tc = t - 1v,
td = v - 1t,
tdd
a
= v - 1tu;
(7)
ua = v,
ub = u - 1,
uc = u - 1,
ud = u,
udd
a
= u - 1;
(8)
va = u,
vb = t - 1v - 1t,
vc = t - 1vt,
vd = v - 1,
vdd
a
= v - 1.
(9)
4.2. Conjugates of (\bfita \bfitc )\bffour . Denote
x0 = (ac)4,
xa0 = x0,
xb0 = (1, u)x0,
xc0 = x0,
xd0 = (1, u)x0,
xdd
a
0 = (u, u)x0.
Now let us introduce more elements and show the result of their conjugation.
4.3. Definition and conjugates of \bfitx \bfitm , \bfitu \bfitm , \bfitv \bfitm . Let xm \in \mathrm{r}\mathrm{i}\mathrm{s}\mathrm{t}(1m), such that xm(1mw) =
= 1m(ac)4(w). Note that x0 = (ac)4. Analogously, um, vm \in \mathrm{r}\mathrm{i}\mathrm{s}\mathrm{t}(1m), um(1mw) = 1mu(w),
vm(1mw) = 1mv(w). Then
\bigl(
proof by induction since xm = (1, xm - 1), um = (1, um - 1)
\bigr)
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 2
ON THE RIGIDITY OF RANK GRADIENT IN A GROUP OF INTERMEDIATE GROWTH 173
xbm =
\left\{
um+1xm, if 3| m,
xm, if 3| (m - 1),
um+1xm, if 3| (m - 2),
(10)
xcm =
\left\{
xm, if 3| m,
um+1xm, if 3| (m - 1),
um+1xm, if 3| (m - 2),
xdm =
\left\{
um+1xm, if 3| m,
um+1xm, if 3| (m - 1),
xm, if 3| (m - 2),
(11)
xdd
a
0 = (u, u)x0, (12)
and if m > 0, then
xdd
a
m = (1, xbm - 1) =
\left\{
um+1xm, if 3| m,
um+1xm, if 3| (m - 1),
xm, if 3| (m - 2),
(13)
xx0
1 = (1, xdd
a
0 ) = (1, 1, u, u)(1, x0) = (1, 1, u, u)x1. (14)
And if n+ 1 < m, then
xxn
m = (. . . , xbm - n - 2) =
\left\{ xm, if 3| (m - n),
um+1xm, if 3 \not | (m - n).
(15)
Here is the list of conjugates of um and vm :
ubm =
\left\{
u - 1
m , if 3| m,
u - 1
m , if 3| (m - 1),
um, if 3| (m - 2),
(16)
udd
a
m =
\left\{
um, if 3| m,
u - 1
m , if 3| (m - 1),
u - 1
m , if 3| (m - 2),
(17)
vbm =
\left\{
v - 1
m - 1v
- 1
m vm - 1, if 3| m,
v - 1
m - 1vmvm - 1, if 3| (m - 1),
v - 1
m , if 3| (m - 2),
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 2
174 R. GRIGORCHUK, R. KRAVCHENKO
vdd
a
m =
\left\{
v - 1
m , if 3| m,
v - 1
m - 1v
- 1
m vm - 1, if 3| (m - 1),
v - 1
m - 1vmvm - 1, if 3| (m - 2).
(18)
In the next two subsections we introduce sequences of subgroups Rn and Qn, n = 1, 2, . . . ,
and prove some structural results about them as well as ranks (or dimension) of associated quotients
(spaces).
4.4. Groups \bfitR \bfitn . Let R1 = K = \langle t, u, v\rangle . Let Rn = (K \times Rn - 1)\{ 1, (ac)4\} for n \geq 2. Then
the following proposition is true.
Proposition 4.1.
R2 =
\bigl\langle
x0, u0, u1, v0, (u, u)
\bigr\rangle
,
Rn =
\bigl\langle
x0, . . . , xn - 2, u0, u1, u2, vn - 2, (u, u)
\bigr\rangle
for n \geq 3.
Proof. Since R2 = (K \times K)\{ 1, x0\} , it follows that
R2 =
\bigl\langle
x0, (t, 1), (u, 1), (v, 1), (1, t), (1, u), (1, v)
\bigr\rangle
.
Now, by (7) tdd
a
= v - 1tu, and since x0 = (dda, dda), we have that (t, 1)x0 = (v, 1) - 1(t, 1)(u, 1),
and the same for (1, t). So we can discard (v, 1), (1, v) from the list. Notice also that multiplying
(u, 1) by (1, u) we can replace (u, 1) with (u, u). Since u0 = u = (t, 1), u1 = (1, u), and
v0 = v = (1, t), we obtain the generators for R2.
We have R3 = (K \times R2)\{ 1, x0\} , hence R3 is generated by
R3 =
\Bigl\langle
x0, (t, 1), (u, 1), (v, 1), x1 = (1, x0), u1 = (1, u0), u2 = (1, u1), v1 = (1, v0), (1, 1, u, u)
\Bigr\rangle
.
In the same way as for R2 we can discard (v, 1) and replace (u, 1) by (u, u). By (14) xx0
1 =
= (1, 1, u, u)x1, and so we can discard (1, 1, u, u), and we are done.
Now suppose
Rn =
\bigl\langle
x0, . . . , xn - 2, u0, u1, u2, vn - 2, (u, u)
\bigr\rangle
.
Then from the formula Rn+1 = (K \times Rn)\{ 1, (ac)4\} we obtain that Rn+1 is generated by\Bigl\langle
x0, (t, 1), (u, 1), (v, 1), x1, . . . , xn - 1, u1, u2, u3, vn - 1, (1, 1, u, u)
\Bigr\rangle
.
As above, we can discard (v, 1) and (1, 1, u, u) and replace (u, 1) with (u, u). It is left to note that
xx0
2 = u3x2 by (15), so u3 = [x0, x2] and hence we can discard it from the list.
Proposition 4.1 is proved.
For any group G define G(2) = G/G(X2), where G(X2) is the subgroup generated by all
squares of elements in G. Then G(2) is elementary abelian 2-group, and so \mathrm{d}\mathrm{i}\mathrm{m}\BbbF 2 G
(2) is defined.
Proposition 4.2. \mathrm{d}\mathrm{i}\mathrm{m}\BbbF 2 R
(2)
n = n+ 4 for n \geq 3, \mathrm{d}\mathrm{i}\mathrm{m}\BbbF 2 R
(2)
2 = 5, \mathrm{d}\mathrm{i}\mathrm{m}\BbbF 2 K
(2) = 3.
Proof. The equality for K follows from the theory of the group \langle a, b, c, d\rangle .
To prove the rest, we need the following obvious lemma.
Lemma 4.1. Suppose G = H\rtimes (\BbbZ /2\BbbZ ). Let \alpha : H(2) \rightarrow H(2) be the operator on H(2) induced
by the action of \BbbZ /2\BbbZ . Then G(2) = H(2)/(1 + \alpha )(H(2))\oplus \BbbZ /2\BbbZ .
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 2
ON THE RIGIDITY OF RANK GRADIENT IN A GROUP OF INTERMEDIATE GROWTH 175
Note that by (7), (8), (9), dda induces the following action on K(2) : [t, u, v] \mapsto \rightarrow [t+ u+ v, u, v].
Hence (1 + dda)(K(2)) = \{ 0, u+ v\} . Thus \mathrm{d}\mathrm{i}\mathrm{m}\BbbF 2 R
(2)
2 = 2 + 2 + 1 = 5.
Now, it follows from (12), (13) that (1 + dda)(x0) = (u, u), (1 + dda)(x1) = u2 and (1 +
+ dda)(xm) = 0 for m > 1, since from (15) we have for m \geq 3 that um = [xm - 3, xm - 1].
Also, from (17), (18), (1 + dda)(um) = 0, (1 + dda)(vm) = 0 for all m \geq 0, and (1 +
+ dda)((u, u)) = 0.
Thus (1 + dda)
\bigl(
R
(2)
2
\bigr)
= \{ 0, (u, u)\} , and so \mathrm{d}\mathrm{i}\mathrm{m}\BbbF 2 R
(2)
3 = 2 + 5 - 1 + 1 = 7. Analogously,
(1+dda)(R
(2)
n ) = \langle (u, u), u2\rangle for n \geq 3, and so by induction \mathrm{d}\mathrm{i}\mathrm{m}\BbbF 2 R
(2)
n+1 = 2+\mathrm{d}\mathrm{i}\mathrm{m}\BbbF 2 R
(2)
n - 2+1 =
= n+ 5.
Lemma 4.1 is proved.
Corollary 4.1. d(R1) = 3, d(R2) = 5, d(Rn) = n+ 4 for n \geq 3.
4.5. Groups \bfitQ \bfitn . Let Q1 = B = \langle b, t, u, v\rangle = K \rtimes \{ 1, b\} and Qn = (K \times Rn - 1)\langle b, (ac)4\rangle for
n > 1.
Proposition 4.3. Qn = Rn \rtimes \{ 1, b\} ,
Q2 =
\bigl\langle
b, x0, u0, v0, (u, u)
\bigr\rangle
,
Qn =
\bigl\langle
b, x0, x1, . . . , xn - 2, u0, u2, vn - 2, (u, u)
\bigr\rangle
for n \geq 3.
Proof. The proof is analogous to the proof of Proposition 4.1, using lists of generators for Rn
from the statement of Proposition 4.1, and the additional fact that xb0 = u1x0, hence u1 = xb0x0.
Proposition 4.4. \mathrm{d}\mathrm{i}\mathrm{m}\BbbF 2 Q
(2)
n = n+ 4 for n \geq 3, \mathrm{d}\mathrm{i}\mathrm{m}\BbbF 2 Q
(2)
2 = 5, and \mathrm{d}\mathrm{i}\mathrm{m}\BbbF 2 Q
(2)
1 = 4.
Proof. Note that to compute \mathrm{d}\mathrm{i}\mathrm{m}\BbbF 2 Q
(2)
n we may use the Lemma 4.1, since Qn = Rn \rtimes \{ 1, b\} .
Thus we need to compute the induced action of b on R(2)
n .
Note that (1+b)(K(2)) = 0, by (7), (8), (9). Also, (1+b)(x0) = u1, (1+b)(xm) = 0 for m \geq 1
by (10). (1+ b)(um) = 0 for m \geq 0 by (16). Finally, (1+ b)((u, u)) = (u, u) + (v, 1)+ (1, u), and
since (t, 1)x0 = (v, 1) - 1(t, 1)(u, 1), it follows that (v, 1) + (u, 1) = 0, and thus (1 + b)((u, u)) =
= (u, u) + (v, 1) + (1, u) = 0.
Hence (1 + b)(R
(2)
n ) = \langle u1\rangle , and thus \mathrm{d}\mathrm{i}\mathrm{m}\BbbF 2 Q
(2)
2 = 5 - 1 + 1 = 5, and \mathrm{d}\mathrm{i}\mathrm{m}\BbbF 2 Q
(2)
n =
= n+ 4 - 1 + 1 = n+ 4.
Corollary 4.2. d(Q1) = 4, d(Q2) = 5, d(Qn) = n+ 4 for n \geq 3.
Finally we introduce groups Pn.
4.6. Groups \bfitP \bfitn . Let Pn = \mathrm{s}\mathrm{t}\scrG (1
n). Then P1 = \mathrm{s}\mathrm{t}\scrG (1), and Pn = (K \times Qn - 1)\langle c, (ac)4\rangle for
n > 1.
Proposition 4.5. Pn = Rn \rtimes \{ 1, b, c, d\} and
Pn =
\bigl\langle
c, d, x0, x1, . . . , xn - 2, u0, vn - 2, (u, u)
\bigr\rangle
for n \geq 2.
Proof. Analogous to the proofs of Propositions 4.1 and 4.3, using the additional fact that
xd1 = u2x1 by (11).
Proposition 4.6.
\mathrm{d}\mathrm{i}\mathrm{m}\BbbF 2P
(2)
n = n+ 4 (19)
for n \geq 2, and \mathrm{d}\mathrm{i}\mathrm{m}\BbbF 2 P
(2)
1 = 4.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 2
176 R. GRIGORCHUK, R. KRAVCHENKO
Proof. For n = 1 it follows from P1 = \mathrm{s}\mathrm{t}(1) = \langle d, c, da, ca\rangle . For n \geq 2 it follows from
Pn = Rn\rtimes \{ 1, b, c, d\} and a slight generalization of the Lemma 4.1: P (2)
n =
\bigl(
R
(2)
n /Vn
\bigr)
\oplus \BbbF 2
2, where
Vn = (1 + b)R
(n)
n + (1 + c)R
(n)
n .
Now we are ready to prove Theorem 1.2. As \scrG acts level transitive on \scrT , for each vertex v
of the level n the group \mathrm{s}\mathrm{t}\scrG (v) is conjugate in \scrG to Pn. From Proposition 4.5 it follows that the
minimum number of generators of Pn is \leq n+4. The last proposition shows that it is exactly n+4
when n \geq 2. As index of Pn in \scrG is 2n (because of the level transitivity of \scrG ) the ratio
d(H) - 1
[\scrG : H]
=
n+ 3
2n
for H = \mathrm{s}\mathrm{t}\scrG (v), where vertex v belongs to the level n \geq 2. If v is a vertex of the first level then
H = \mathrm{s}\mathrm{t}\scrG (1) is a subgroup of index 2 and is 4-generated (by elements b, c, ba, ca). Taking all this
into account we get the conclusion of the Theorem 1.2.
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Received 10.01.18
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 2
|
| id | umjimathkievua-article-1548 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:07:49Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/12/d8b7197bc482440d3164ebd483991f12.pdf |
| spelling | umjimathkievua-article-15482019-12-05T09:18:03Z On the rigidity of rank gradient in a group of intermediate growth Про жорсткiсть градiєнтного рангу в групi промiжного зростання Grigorchuk, R. I. Kravchenko, R. Григорчук, Р. І. Кравченко, Р. We introduce and investigate a rigidity property of rank gradient for an example of a group $\scr G$ of intermediate growth constructed by the first author in [Grigorcuk R. I. On Burnside’s problem on periodic groups // Funktsional. Anal. i Prilozhen. – 1980. – 14, № 1. – P. 53 – 54]. It is shown that $\scr G$ is normally $(f, g)$-RG rigid, where$ f(n) = \mathrm{l}\mathrm{o}\mathrm{g}(n)$ and $g(n) = \mathrm{l}\mathrm{o}\mathrm{g}(\mathrm{l}\mathrm{o}\mathrm{g}(n))$. Введено та вивчено властивiсть жорсткостi градiєнтного рангу для прикладу групи $\scr G$ промiжного зростання, що була введена першим автором в роботi [Grigorˇcuk R. I. On Burnside’s problem on periodic groups // Funktsional. Anal. i Prilozhen. – 1980. – 14, № 1. – P. 53 – 54]. Показано, що $\scr G$ є нормально $(f, g)$-RG жорсткою, де $f(n) = \mathrm{l}\mathrm{o}\mathrm{g}(n)$ та $g(n) = \mathrm{l}\mathrm{o}\mathrm{g}(\mathrm{l}\mathrm{o}\mathrm{g}(n))$. Institute of Mathematics, NAS of Ukraine 2018-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1548 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 2 (2018); 165-176 Український математичний журнал; Том 70 № 2 (2018); 165-176 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1548/530 Copyright (c) 2018 Grigorchuk R. I.; Kravchenko R. |
| spellingShingle | Grigorchuk, R. I. Kravchenko, R. Григорчук, Р. І. Кравченко, Р. On the rigidity of rank gradient in a group of intermediate growth |
| title | On the rigidity of rank gradient in a group of intermediate growth |
| title_alt | Про жорсткiсть градiєнтного рангу в групi промiжного зростання |
| title_full | On the rigidity of rank gradient in a group of intermediate growth |
| title_fullStr | On the rigidity of rank gradient in a group of intermediate growth |
| title_full_unstemmed | On the rigidity of rank gradient in a group of intermediate growth |
| title_short | On the rigidity of rank gradient in a group of intermediate growth |
| title_sort | on the rigidity of rank gradient in a group of intermediate growth |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1548 |
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