arXiv:1802.08387v1 [math.DS] 23 Feb 2018

On the rigidity of rank gradient in a group of
intermediate growth
Rostislav Grigorchuk∗
Texas A&M University,
Rostyslav Kravchenko
Northwestern University
February 26, 2018

Dedicated to A.M. Samoilenko on the occasion of his 80th birthday
Abstract
We introduce and investigate the rigidity property of rank gradient
in the case of the group G of intermediate growth constructed by the
ﬁrst author in [10]. We show that G is normally (f, g)-RG rigid where
f (n) = log(n) and g(n) = log(log(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. Grunewald, 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
† To

author was partially supported by Simons Collaboration Grant # 527814
appear in Ukrainian Mathematical Journal v.70, no. 2 (2018)

1

The rank gradient of a ﬁnitely generated residually ﬁnite group G is
deﬁned as
d(H) − 1
(1)
[G : H]
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
whether 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, 9, 12].
The deﬁnition (1) can be modiﬁed in various directions. Instead of
inf one can consider sup, 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(G) = inf

rg(n) = RG(G, {Hn }) =

d(Hn ) − 1
[G : Hn ]

(2)

and its limit
lim rg(n),

n→∞

if it exists, or the upper and the lower limits
lim sup rg(n),
n→∞

lim inf rg(n)
n→∞

otherwise.
It is known [12, 1] 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 L = Z/2Z ≀ Z 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 (see for instance
[4]).
We suggest the following deﬁnition. Let f (n), g(n) be two increasing
functions of natural argument n taking values in N and having the limit
∞ when n → ∞.

2

Figure 1: Binary Tree

Definition 1.1. (a) A finitely generated group G is (f, g) − RG-rigid if
there is C ∈ N 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 ⊳ G of finite index.
For instance the free group Fr of rank r ≥ 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 G = ha, b, c, di 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]. G 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 T shown by Figure 1, as explained, for instance,
in [6].
Theorem 1.2. The group G is normally (f, g) − RG-rigid with
f (n) = log log(n), g(n) = log(n),
.
Moreover, there there is a constant D > 1 such that
1
log(d(H)) ≤ log log([G : H]) ≤ D log(d(H))
D
hold for every nontrivial normal subgroup H ⊳ G.

3

(3)

Conjecture 1.3. The group G is (f, g)−RG-rigid with the same functions
f, g as in previous theorem.
If the conjecture is true then we have an interesting rigidity property
concerning the rank and index of ﬁnite index subgroups. At least we
have this property for normal subgroups as shows the above theorem.
Recall that all normal subgroups in G have ﬁnite index because G is justinﬁnite (i.e. inﬁnite, but every proper quotients ﬁnite) as shown in [6].
Interestingly, the group G 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 G.
At the moment we are able only to conﬁrm the conjecture for important subclass of subgroups of ﬁnite index in G, namely for stabilizers of
vertices of the binary rooted tree T on which the group G acts. The vertices of T are in bijection with ﬁnite words over binary alphabet {0, 1}.
Let v be a vertex and stG (v) be its stabilizer which has index 2n in G if v
is a vertex of level n.
Theorem 1.4. There is a constant D such that the inequalities (3) hold
for all subgroups H = stG (v) where v run over the set of vertices in T . In
fact
d(H) − 1
n+3
=
[G : H]
2n
if v is a vertex of level n ≥ 2.
Observe that this result is announced in [4].

2

The group G

We recall some basic facts about the group G and its subgroups. G can
be deﬁned as a group of automorphisms of a rooted binary tree T shown
in Figure 1 (the root, corresponding to the empty word, is a ﬁxed point
for the action). The generators a, b, c, d of G are involutions, the elements
b, c, d commute and together with the identity element they constitute
the Klein group Z2 × Z2 . The stabilizer of the ﬁrst level H = stG (1) is
a subgroup of index 2 in G generated by elements b, c, d, ba , ca , da (where
xy = x−1 yx), and the restrictions of H on the left and right subtrees
T0 , T1 with the roots at vertices 0, 1 determine surjective homomorphisms
ϕ0 , ϕ1 : H → G. The direct product ψ = ϕ0 × ϕ1 of them determines the
embedding H → G × G and acts on generators as:
b → (a, c), c → (a, d), d → (1, b), ba → (c, a), ca → (d, a), da → (b, 1) (4)
Together with the information that the generator a permute the subtrees T0 , T1 (without extra action inside them), this uniquely determines
the group G.

4

The action of G on T 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 ∈ {0, 1} , and {0, 1} denotes the set of all ﬁnite words over
the binary alphabet. The important property of the action of G on T is
level transitivity, i.e. transitivity of the action on each level Vn = {0, 1}n .
Additionally to the stabilizers stG (n) of levels n = 1, 2, . . . , an important descending series of normal subgroups is the series of rigid stabilizers
ristG (n) which are subgroups generated by rigid stabilizers ristG (v), v ∈
{0, 1}n of vertices of the nth level, and ristG (v) is a subgroup in G ﬁxing
vertex v and consisting of elements acting trivially outside the subtree
Tv in T 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 ristG (n) is a direct product of copies of the
same group (which may depend on the level n in general case). Observe
∞
that {stG (n)}∞
n=1 and {ristG (n)}n=1 are descending chains of normal subgroups of ﬁnite index with trivial intersection. The structure of groups
stG (n) and ristG (n) is well understood and described in [2].
Let B = hbiG be a normal closure of generator b and K = h(ab)2 iG . For
each n there is a natural embedding ψn : stG (n) → G ×· · ·×G into a direct
product of 2n copies of G which is the nth iteration of the embedding ψ
and has a geometric meaning of the attaching to the element g ∈ stG (n)
the 2n -tuple (g1 , . . . , g2n ) of its restrictions on the subtrees with the roots
at the nth level. Instead writing
ψn (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).
The important facts about groups stG (n) and ristG (n) are that the
ψn−3 image of stG (n) has the decomposition
stG (3) × stG (3) × · · · × stG (3)

(5)

n−3

(products of 2
copies of stG (3)) when n ≥ 4, and the ψn image of
ristG (n) has the decomposition
K × K × ··· × K
(6)
(product of 2n copies of K) when n ≥ 2 (see [2]). We will use later
the notations Kn for ristG (n) when n ≥ 2 and keep in mind the decompositions (5) and (6). We also denote by K1 a subgroup in stG (1) whose
ψ-image is K × K. The group G is regularly branched over K as K1 is a
subgroup of K, and [G : K1 ] < ∞. Thus G contains subgroups shown by
Figure 2. To each level n corresponds a group Kn that ﬁxes each vertex
v of this level and whose restriction to the subtree Tv with the root v is
K (if to identify Tv with T ). Moreover Kn is a direct product of these
projections.

5

G
K
K
K

K
K

K

K

K K K KK K K K
Figure 2: Structure of branching subgroups

3

Proof of Theorem 1.2

We begin this section with reminding that the group G is branch group
as deﬁned in [6], because it acts level transitive on the tree T and rigid
stabilizers ristG (n), n = 1, 2, . . . have ﬁnite index in G. The branch structure of G that we will use is given by the Figure 2 and was mentioned in
previous section. The proof of the theorem 1.2 is based on the following
Proposition 3.1. Let N ⊳ G be a nontrivial normal subgroup. Let n be a
smallest nonnegative integer such that N < stG (n) but N is not a subgroup
of stG (n + 1). Then
(a)
stG (n + 6) < N.
(b) If n ≥ 4 then
ristG (n + 3) < N < ristG (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 G is given in our case by the pair ({Ln }, {Hn })
where Ln = K and Hn = Kn for all n. The reader should keep in mind
the picture given by the ﬁgure 2.
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) shows
that if N < stG (n) but N is not a subgroup of stG (n + 1), then N contains
′
commutator subgroup Kn+1
= [Kn+1 , Kn+1 ] (recall that ristG (n) = Kn if
′
n > 1). By proposition 9 from [6] K ′ = K2 , hence Kn+1
= K ×K ×· · ·×K
n+3
′
(2
factors) so Kn+1 = Kn+3 , and N > Kn+3 , which gives the lower
inclusion in part (b).
Also by proposition 9 from [6] we have the inclusion K > stG (3) from
which, together with the fact presented by factorization (5) we conclude

6

that Kn+3 > stG (n + 6) and thus N > stG (n + 6). The part (a) is thus
established. To get the upper inclusion in part (b) under assumption that
n > 3 we observe that the inclusion K > stG (3) and factorization (5)
imply that Kn−3 > stG (n), therefore we are done.
The group G is not virtually cyclic (by many reasons, for instance
because it is ﬁnitely generated inﬁnite torsion group). Let N ⊳ G be a
nontrivial normal subgroup. It is automatically of ﬁnite index, as G is justinﬁnite 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 1.2 (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 G containing stG (5), their
ranks are ≥ 2, and so there is a constant D satisfying the condition of the
theorem 1.2. Therefore we can assume that n ≥ 4. Now apply part (b) of
the proposition.
The quotient ristG (n − 3)/ristG (n + 3) = Kn−3 /Kn+3 is isomorphic
to
n−3

A := (K/K6 )2

= K/K6 × · · · × K/K6 ,

n−3

(2
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 3·2n−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 3.2. 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) ≤ d(G). For G of class
c let γ1 (G) = G and γi+1 (G) = [G, γi (G)] i = 1, 2, . . . be the elements
of the lower central series. Suppose G is generated by set S. Each factor
γi (G)/γi+1 (G) is generated by the iterated commutators [s1 , [s2 , . . . ]] of
length i + 1, where sj ∈ S. Thus d(γi (G)/γi+1 (G)) ≤ d(G)i+1 . Denote
Hi = H ∩ γi (G). Then Hi γi+1 (G)/γi+1 (G) is an abelian group, and thus
d(Hi γi+1 (G)/γi+1 (G)) ≤ d(γi (G)/γi+1 (G)) ≤ d(G)i+1 . Note that
d(H) ≤ d(Hγ1 (G)/γ1 (G)) + d(H ∩ γ1 (G)) ≤ d(G) + d(H1 ).
Applying this iteratively we obtain
d(H) ≤ d(G) + d(H1 ) ≤ d(G) + d(H1 γ2 (G)/γ2 (G)) + d(H2 ) ≤
d(G) + d(γ1 (G)/γ2 (G)) + d(H2 ) ≤
d(G) + d(G)2 + d(H2 ) ≤ · · · ≤ d(G) + · · · + d(G)c−1 ≤ d(G)c

7

Using this lemma we get
d(N ) < (3 · 2n−3 )l+2 = 3l+2 2(n−3)(l+2) < a1 2a2 n
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 Kn+3
getting
′
′
′
Kn+3 /Kn+3
< N/Kn+3
< Kn−3 /Kn+3
′
The group Kn−3 /Kn+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 /Kn+3
also has nilpotency class s . As N̄ := N/Kn+3
is a
′
subgroup of Kn−3 /Kn+3 its class of nilpotency is ≤ s.
′
The group Kn+3 /Kn+3
is a direct product of 2n+3 copies of the group
′
′
K/K = K/K2 . Let t = d(K/K ′ ) (in fact t = 3). Then d(Kn+3 /Kn+3
)=
t2n+3 . Using lemma 3.2 we conclude
′
d(Kn+3 /Kn+3
) = t2n+3 ≤ (d(N ))s+2

from which we conclude that there are positive constants a3 , a4 such that
a3 2a4 n ≤ d(N )
Now using the part (a) of the Proposition 3.1 we provide upper and lower
bounds for the index [G : N ]. For this purpose we use the fact that
n−3

[G : stG (n)] = 25·2

+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
n

2a5 2

+a7

n

≤ [G : N ] ≤ 2a6 2

+a8

Taking the double logarithm with base 2 of this inequalities and applying
the same logarithm to the previously obtained inequalities
a3 2a4 n ≤ d(N ) ≤ a1 2a2 n
a simple calculus ﬁnishes the proof of the theorem.

4

Proof of Theorem 1.4

We begin the proof of Theorem 1.4. 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.
Let us introduce the elements t = (ab)2 , u = (bada)2 = (t, 1), v =
(abad)2 = (1, t). The direct computation gives the result of their conjugation by generators, together with the conjugation of (ac)4 , as shown in
the next subsections:

8

4.1

Conjugates of t, u, v
ta = t−1
tb = t−1
tc = t−1 v
d

t =v
t

dda

−1

=v

(7)

t

−1

tu

a

u =v
ub = u−1
uc = u−1

(8)

d

u =u
a

udd = u−1
va = u
v b = t−1 v −1 t
v c = t−1 vt
d

v =v

(9)

−1

a

v dd = v −1

4.2

Conjugates of (ac)4

Denote x0 = (ac)4
xa0 = x0
xb0 = (1, u)x0
xc0 = x0
xd0 = (1, u)x0
a

= (u, u)x0 .
xdd
0
Now let us introduce more elements and show the result of their conjugation.

4.3

Definition and conjugates of xm , um , vm

Recall that ristG (1m ) is the subgroup of G that ﬁxes the vertex 1m together with all vertices that do not start with 1m . It is easy to check
that (ac)4 , u and v belong to K, and that ψ(K) ⊃ K × K. It follows that
there are such xm , um and vm in ristG (1m ) that xm (1m w) = 1m (ac)4 (w),
um (1m w) = 1m u(w) and vm (1m w) = 1m v(w). Note that x0 = (ac)4 .
We compute their conjugates below (the proof is by induction, since
xm = (1, xm−1 ), um = (1, um−1 ) and vm = (1, vm−1 )):

if 3|m
 um+1 xm
b
xm
if 3|(m − 1)
(10)
xm =

um+1 xm
if 3|(m − 2)

9

xcm

xdm
We have that


 xm
um+1 xm
=

um+1 xm

 um+1 xm
um+1 xm
=

xm

if 3|m
if 3|(m − 1)
if 3|(m − 2)
if 3|m
if 3|(m − 1)
if 3|(m − 2)

a

xdd
= (u, u)x0 ,
0

(11)

(12)

and for m > 0
xdd
m

a


 um+1 xm
um+1 xm
= (1, xbm−1 ) =

xm

if 3|m
if 3|(m − 1)
if 3|(m − 2)

(13)

a

xx1 0 = (1, xdd
0 ) = (1, 1, u, u)(1, x0 ) = (1, 1, u, u)x1
When n + 1 < m we have,

xm
xxmn =
um+1 xm

if 3|(m − n)
if 3 6 |(m − n)

Here is the list of conjugates of um and vm :
 −1
if 3|m
 um
b
u−1
if 3|(m − 1)
um =
m

um
if 3|(m − 2)

if 3|m
 um
a
if 3|(m − 1)
u−1
udd
m
m =
 −1
um
if 3|(m − 2)
 −1 −1
if 3|m
 vm−1 vm vm−1
b
−1
vm
=
vm vm−1
if 3|(m − 1)
vm−1
 −1
if 3|(m − 2)
vm
 −1
if 3|m
 vm
dda
−1
−1
vm−1
vm
vm−1
if 3|(m − 1)
vm =
 −1
vm−1 vm vm−1
if 3|(m − 2)

(14)

(15)

(16)

(17)

(18)

In the next two subsections we introduce sequences of subgroups Rn and
Qn , n = 1, 2, . . . and prove some structural results about them.

4.4

Groups Rn

Let R1 = K = ht, u, vi. Let Rn = (K × Rn−1 ){1, (ac)4 } for n ≥ 2. Then
Proposition 4.1.
R2 = hx0 , u0 , u1 , v0 , (u, u)i,
Rn = hx0 , . . . , xn−2 , u0 , u1 , u2 , vn−2 , (u, u)i
for n ≥ 3.

10

Proof. Since R2 = (K × K){1, x0 }, it follows that
R2 = hx0 , (t, 1), (u, 1), (v, 1), (1, t), (1, u), (1, v)i.
a

Now, by (7) tdd = v −1 tu, 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 × R2 ){1, x0 }, hence R3 is generated by
R3 = hx0 , (t, 1), (u, 1), (v, 1), x1 , u1 , u2 , v1 , (1, 1, u, u)i.
Where x1 = (1, x0 ), u1 = (1, u0 ), u2 = (1, u1 ) and v1 = (1, v0 ).
In the same way as for R2 we can discard (v, 1) and replace (u, 1) by
(u, u). By (14) xx1 0 = (1, 1, u, u)x1 , and so we can discard (1, 1, u, u), and
we are done.
Now suppose
Rn = hx0 , . . . , xn−2 , u0 , u1 , u2 , vn−2 , (u, u)i
Then from the formula Rn+1 = (K × Rn ){1, (ac)4 } we obtain that Rn+1
is generated by
hx0 , (t, 1), (u, 1), (v, 1), x1 , . . . , xn−1 , u1 , u2 , u3 , vn−1 , (1, 1, u, u)i.
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 xx2 0 = u3 x2 by (15), so u3 = [x0 , x2 ] and
hence we can discard it from the list.
For any group G deﬁne G(2) = G/G(X 2 ), where G(X 2 ) is the subgroup
generated by all squares of elements in G. Then G(2) is elementary abelian
2-group, and so dimF2 G(2) is deﬁned.
(2)

Proposition 4.2. dimF2 Rn
dimF2 K (2) = 3.

(2)

= n + 4 for n ≥ 3, dimF2 R2

= 5, and

Proof. The equality for K follows from the theory of the group G, see
[2, 6, 9].
To prove the rest, we need the following obvious lemma
Lemma 4.3. Suppose G = H ⋊ (Z/2Z). Let α : H (2) → H (2) be the
operator on H (2) induced by the action of Z/2Z. Then G(2) = H (2) /(1 +
α)(H (2) ) ⊕ Z/2Z.
Note that by (7), (8), (9), dda induces the following action on K (2) :
[t, u, v] 7→ [t + u + v, u, v]. Hence (1 + dda )(K (2) ) = {0, u + v}. Thus
(2)
dimF2 R2 = 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 ≥ 3 that um = [xm−3 , xm−1 ].
Also, from (17), (18), (1 + dda )(um ) = 0, (1 + dda )(vm ) = 0 for all
m ≥ 0, and (1 + dda )((u, u)) = 0.
(2)
(2)
Thus (1+dda )(R2 ) = {0, (u, u)}, and so dimF2 R3 = 2+5−1+1 = 7.
(2)
Analogously, (1 + dda )(Rn ) = h(u, u), u2 i for n ≥ 3, and so by induction
(2)
(2)
dimF2 Rn+1 = 2 + dimF2 Rn − 2 + 1 = n + 5.

11

Corollary 4.4. d(R1 ) = 3, d(R2 ) = 5, d(Rn ) = n + 4 for n ≥ 3.

4.5

Groups Qn

Let Q1 = B = hb, t, u, vi = K ⋊ {1, b} and Qn = (K × Rn−1 )hb, (ac)4 i for
n > 1.
Proposition 4.5. Qn = Rn ⋊ {1, b}.
Q2 = hb, x0 , u0 , v0 , (u, u)i.
Qn = hb, x0 , x1 , . . . , xn−2 , u0 , u2 , vn−2 , (u, u)i.
for n ≥ 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 = u1 x0 , hence u1 = xb0 x0 .
(2)

Proposition 4.6. dimF2 Qn
(2)
dimF2 Q1 = 4.

(2)

= n + 4 for n ≥ 3, dimF2 Q2

= 5, and

(2)

Proof. Note that to compute dimF2 Qn we may use the Lemma 4.3, since
Qn = Rn ⋊ {1, b}. Thus we need to compute the induced action of b on
(2)
Rn .
Note that (1 + b)(K (2) ) = 0, by (7), (8), (9). Also, (1 + b)(x0 ) =
u1 , (1 + b)(xm ) = 0 for m ≥ 1 by (10). (1 + b)(um ) = 0 for m ≥
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.
(2)
(2)
Hence (1 + b)(Rn ) = hu1 i, and thus dimF2 Q2 = 5 − 1 + 1 = 5, and
(2)
dimF2 Qn = n + 4 − 1 + 1 = n + 4.
Corollary 4.7. d(Q1 ) = 4, d(Q2 ) = 5, d(Qn ) = n + 4 for n ≥ 3.
Finally we introduce groups Pn .

4.6

Groups Pn

Let Pn = stG (1n ). Then P1 = stG (1), and Pn = (K × Qn−1 )hc, (ac)4 i for
n > 1.
Proposition 4.8. Pn = Rn ⋊ {1, b, c, d} and
Pn = hc, d, x0 , x1 , . . . , xn−2 , u0 , vn−2 , (u, u)i,
for n ≥ 2.
Proof. Analogous to the proofs of Proposition 4.1 and Proposition 4.5,
using the additional fact that xd1 = u2 x1 by (11).
Proposition 4.9.
dimF2 Pn(2) = n + 4
for n ≥ 2, and

(2)
dimF2 P1

= 4.

12

(19)

Proof. For n = 1 it follows from P1 = st(1) = hd, c, da , ca i. For n ≥ 2 it
follows from Pn = Rn ⋊{1, b, c, d} and a slight generalization of the Lemma
(n)
(n)
(2)
(2)
4.3: Pn = (Rn /Vn ) ⊕ F22 , where Vn = (1 + b)Rn + (1 + c)Rn .
Now we are ready to prove theorem 1.4. As G acts level transitive on
T , for each vertex v of the level n the group stG (v) is conjugate in G to Pn .
From Proposition 4.8 it follows that the minimum number of generators
of Pn is ≤ n + 4. The last proposition shows that it is exactly n + 4 when
n ≥ 2. As index of Pn in G is 2n (because of the level transitivity of G)
the ratio
d(H) − 1
n+3
=
[G : H]
2n
for H = stG (v) where vertex v belongs to the level n ≥ 2. If v is a
vertex of the ﬁrst level then H = stG (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 1.4.

References
[1] Miklós Abért and Nikolay Nikolov. Rank gradient, cost of groups
and the rank versus Heegaard genus problem. J. Eur. Math. Soc.
(JEMS), 14(5):1657–1677, 2012.
[2] Laurent Bartholdi and Rostislav I. Grigorchuk. On parabolic subgroups and Hecke algebras of some fractal groups. Serdica Math. J.,
28(1):47–90, 2002.
[3] N.N. Bogolyubov. On some ergodic properties of continious groups of
transformations. Nauk. Zap. Kiïv Derzh. Univ. im. T.G.Shevchenka,
4(5):45–52, 1939.
[4] R. Grigorchuk and R. Kravchenko. On the lattice of subgroups of
the lamplighter group. Internat. J. Algebra Comput., 24(6):837–877,
2014.
[5] R. I. Grigorchuk. Degrees of growth of ﬁnitely generated groups and
the theory of invariant means. Izv. Akad. Nauk SSSR Ser. Mat.,
48(5):939–985, 1984.
[6] R. I. Grigorchuk. Just inﬁnite branch groups. In New horizons in
pro-p groups, volume 184 of Progr. Math., pages 121–179. Birkhäuser
Boston, Boston, MA, 2000.
[7] R. I. Grigorchuk. Some problems of the dynamics of group actions
on rooted trees. Tr. Mat. Inst. Steklova, 273(Sovremennye Problemy
Matematiki):72–191, 2011.
[8] R. I. Grigorchuk, V. V. Nekrashevich, and V. I. Sushchanskiı̆. Automata, dynamical systems, and groups. Tr. Mat. Inst. Steklova,
231(Din. Sist., Avtom. i Beskon. Gruppy):134–214, 2000.
[9] Rostislav Grigorchuk. Solved and unsolved problems around one
group. In Infinite groups: geometric, combinatorial and dynamical aspects, volume 248 of Progr. Math., pages 117–218. Birkhäuser, Basel,
2005.

13

[10] R. I. Grigorčuk. On Burnside’s problem on periodic groups. Funktsional. Anal. i Prilozhen., 14(1):53–54, 1980.
[11] F. J. Grunewald, D. Segal, and G. C. Smith. Subgroups of ﬁnite
index in nilpotent groups. Invent. Math., 93(1):185–223, 1988.
[12] Marc Lackenby. Expanders, rank and graphs of groups. Israel J.
Math., 146:357–370, 2005.
[13] Alexander Lubotzky and Dan Segal. Subgroup growth, volume 212 of
Progress in Mathematics. Birkhäuser Verlag, Basel, 2003.
[14] John von Neumann.
Zurr allgemeinen theorie des masses.
Fund.Math., 13:73–116, 1929.

14

