Schreier Graphs for a Self-Similar Action of the Heisenberg Group
We construct a faithful self-similar action of the discrete Heisenberg group with the following properties: This action is self-replicating, finite-state, level-transitive, and noncontracting. Moreover, there exist orbital Schreier graphs of action on the boundary of the tree with different degrees...
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| author | Bondarenko, I. Kravchenko, R. Бондаренко, І. Кравченко, Р. |
| author_facet | Bondarenko, I. Kravchenko, R. Бондаренко, І. Кравченко, Р. |
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| description | We construct a faithful self-similar action of the discrete Heisenberg group with the following properties: This action is self-replicating, finite-state, level-transitive, and noncontracting. Moreover, there exist orbital Schreier graphs of action on the boundary of the tree with different degrees of growth. |
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UDC 512.54
I. Bondarenko (Kyiv. Nat. Taras Shevchenko Univ.),
R. Kravchenko (Univ. Chicago, USA)
SCHREIER GRAPHS FOR A SELF-SIMILAR ACTION
OF THE HEISENBERG GROUP
ГРАФИ ШРАЙЄРА САМОПОДIБНОЇ ДIЇ ГРУПИ ГЕЙЗЕНБЕРГА
We construct a faithful self-similar action of the discrete Heisenberg group with the following properties: This action is
self-replicating, finite-state, level-transitive, and noncontracting. Moreover, there exist orbital Schreier graphs of action on
the boundary of the tree with different degrees of growth.
Побудовано точну самоподiбну дiю дискретної групи Гейзенберга з наступними властивостями. Дiя є рекурентною,
скiнченностановою, сферично транзитивною, нестискуючою та iснують графи Шрайєра дiї групи на межi дерева з
рiзними степенями зростання.
1. Introduction. The celebrated theorem of Gromov shows that the Cayley graphs of a finitely
generated group have polynomial growth if and only if the group is virtually nilpotent. There are
groups of exponential and intermediate growth between polynomial and exponential, but it is still not
clear what are the possible growth rates of Cayley graphs.
In this paper we consider Schreier graphs of groups, which are generalization of Cayley graphs.
Schreier graphs model the action of a group on a set. There is no sense to ask the question about
possible growth rates of Schreier graphs: every connected regular graph of even degree is a Schreier
graph of a free group and one can realize any growth rate with natural restrictions. However this
question may be interesting in certain classes of groups and for some natural actions. We want to
ask the following question. Given a finitely generated group G and a nested sequence {Hn}n≥1 of
subgroups of finite index in G with trivial intersection ∩n≥1Hn = {e}, consider the action of G on
the coset tree of {Hn}n≥1.
Question. In what cases the orbital Schreier graphs of the action of G on the boundary of the
coset tree have polynomial growth?
It is hard to believe that there is an algebraic characterization of such groups, but rather certain
geometric characterization of the action on a tree.
One natural class of groups for the above question are self-similar groups. Every self-similar
group is given by its action on a regular rooted tree and we may consider the orbital Schreier graphs of
the action on the boundary of the tree. There are examples of self-similar groups with Schreier graphs
of polynomial growth with irrational degree, exponential growth, and even intermediate growth (see
[1, 2]). It is known that all orbital Schreier graphs have polynomial growth for every contracting
self-similar action (see [3], Proposition 2.13.8), where contracting property of the action corresponds
to the expanding property of the associated dynamical system. One may consider how far contracting
groups are from self-similar groups with polynomial Schreier graphs. Since nilpotent groups have
polynomial growth, all their Schreier graphs also have polynomial growth. Therefore it is interesting
to understand self-similar actions of nilpotent groups. If we do not add additional restrictions, then
one can construct a non-contracting action even of the infinite cyclic group with orbital Schreier
graphs of linear growth. But the question happens to be more interesting under additional assumption
c© I. BONDARENKO, R. KRAVCHENKO, 2013
1456 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
SCHREIER GRAPHS FOR A SELF-SIMILAR ACTION OF THE HEISENBERG GROUP 1457
that the action is finite-state, i.e., can be described by a finite automaton. The result of Nekrashevych
and Sidki [4] says that a faithful self-replicating finite-state action of the free abelian group Zn is
necessary contracting. This result does not hold for all nilpotent groups and the complete picture for
finitely generated torsion-free nilpotent groups was described by the authors in [5]. The following
problem seems to be interesting: characterize finitely generated finite-state self-similar groups (i. e.,
groups generated by finite automata), whose all orbital Schreier graphs have polynomial growth.
In this paper we consider the self-similar action of the discrete Heisenberg group constructed
by the authors in [5] and prove that this action provides an example of a self-replicating finite-state
and non-contracting self-similar action with orbital Schreier graphs of polynomial growth. We also
compute the degree of growth for every orbital Schreier graph of this action. It happens that while the
action is level-transitive, there are orbital Schreier graphs with different growth degrees. This seems
to be the first example of a self-similar group with such properties.
2. Self-similar actions and Schreier graphs. Let X be a finite set with at least two elements.
Let X∗ = {x1x2 . . . xn : xi ∈ X,n ≥ 0} be the free monoid freely generated by X with empty word
denoted by ∅. The set Xn of words of length n is called the n-th level. We will also consider the set
XN of all infinite words x1x2 . . . , xi ∈ X.
Self-similar actions. Self-similar group actions are specific actions of a group on the spaces
X∗ and XN. We are not going to give definition of a self-similar action, but rather define how one
can construct every self-similar action of a group with transitive action on X (see [3] for more
information about self-similar groups).
Let G be a group, H a subgroup of finite index in G, and φ : H → G a homomorphism. Let D be
a set of coset representatives for H in G and let X be in bijection with D so that |X| = |D| = [G : H]
and D = {dx : x ∈ X}. The self-similar action of the group G on the spaces X∗ and XN associated
to the triple (G,φ,D) is constructed as follows: for every x ∈ X and v ∈ X∗∪XN define the action
of an element g ∈ G recursively by the rule
g(∅) = ∅ and g(xv) = yh(v) with h = φ
(
d−1y gdx
)
, (1)
where y ∈ X is the unique element such that d−1y gdx ∈ H. The action may be not faithful. The
kernel of the action is equal to the maximal normal φ-invariant subgroup of G called the φ-core ([3],
Proposition 2.7.5). Every self-similar action of a group G with transitive action on X corresponds to
a certain triple (G,φ,D) as above (see [3], Chapter 2).
Note that each self-similar action preserves the length of words and one can restrict the action
to every level Xn. The action is called level-transitive if it is transitive on each level. A self-similar
action is called self-replicating if it is transitive on X and φ is surjective.
Finite-state actions. The element h from equation (1) is called the state of g at x and is
denoted by g|x; iteratively one can define the state of g at every finite word by the rule g|x1x2...xn =
= g|x1 |x2 . . . |xn . A self-similar action (G,X∗) is called finite-state if for every g ∈ G the set{
g|v : v ∈ X∗
}
is finite. A finite-state self-similar action of a finitely generated group can be given
by a finite graph (automaton): the vertices are the generators of the group and all their states, and for
every state s we have an arrow s→ s|x labeled by x|s(x) for each x ∈ X.
A self-similar action is called contracting if there exists a finite set N ⊂ G with the property that
for every g ∈ G there exists n ∈ N such that g|v ∈ N for all words v ∈ X∗ of length ≥ n. Every
contracting action is finite-state.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
1458 I. BONDARENKO, R. KRAVCHENKO
Schreier graphs. Let G be a group with a finite generating set S and H a subgroup of G. The
Schreier coset graph for H in G with respect to S is the graph whose vertices are the left cosets
G/H = {gH : g ∈ G} and the edges are (gH, sgH) for s ∈ S ∪ S−1. Let G be acting on a set M
from the left. The (simplicial) Schreier graph Γ(G,S,M) of the action is the graph with the set of
vertices M and two points u, v ∈ M are connected by an edge if s(u) = v for some s ∈ S ∪ S−1.
The connected component of Γ(G,S,M) around a point w ∈ M is called the orbital Schreier
graph Γw(G,S). The graph Γw(G,S) is the Schreier coset graph of G with respect to the stabilizer
StG(w). Now to every self-similar action of the group G it is associated an uncountable family of
orbital Schreier graphs Γw(G,S) for w ∈ XN. The graphs Γw(G,S) bring important information
about the group action and were studied in relation to such topics as spectrum, growth, amenability,
etc. (see [1, 2] and the reference therein).
3. Our results. Let G be the discrete Heisenberg group, which consists of upper unitriangular
matrices of dimension 3 with integer coefficients. We use notation
(x, y, z) =
1 x z
0 1 y
0 0 1
,
so that G =
{
(x, y, z) : x, y, z ∈ Z
}
. The multiplication of the group elements written in this form
can be performed by the rule
(x1, y1, z1)(x2, y2, z2) = (x1 + x2, y1 + y2, z1 + z2 + x1y2).
It is easy to see that the elements (1, 0, 0) and (0, 1, 0) are generators of the group G, and the element
(0, 0, 1) is a generator of the center Z(G) =
{
(0, 0, z) : z ∈ Z
}
of the group.
Consider the subgroup H =
{
(x, 2y, 2z) : x, y, z ∈ Z
}
of index four in G and the map
φ : H → G, φ(x, y, z) = (x, y/2, z/2).
Proposition 1. The map φ is an isomorphism with trivial φ-core.
Proof. It is clear that the map φ is bijective. Let us show that it is a homomorphism
φ(x1, y1, z1)φ(x2, y2, z2) = (x1, y1/2, z1/2)(x2, y2/2, z2/2) =
= (x1 + x2, y1/2 + y2/2, z1/2 + z2/2 + x1y2/2) =
= φ
(
(x1 + x2, y1 + y2, z1 + z2 + x1y2)
)
=
= φ
(
(x1, y1, z1)(x2, y2, z2)
)
.
It is left to compute the φ-core K. If (x, y, z) ∈ K then (x, y/2k, z/2k) ∈ K for all k ∈ N and
therefore y = z = 0. Conjugating by (0, 1, 0) we get
(0, 1, 0)−1(x, 0, 0)(0, 1, 0) = (x, 0, x) ∈ K
and hence x = 0. We have proved that the φ-core is trivial.
Proposition 1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
SCHREIER GRAPHS FOR A SELF-SIMILAR ACTION OF THE HEISENBERG GROUP 1459
Corollary 1. For every choice of coset representatives D for H in G, every self-similar action
associated to the triple (G,φ,D) is faithful.
We choose the set of coset representatives D = {(0, 0, 0), (0, 1, 0), (0, 0, 1), (0, 1, 1)} for H in
G and construct the self-similar action (G,X∗) of the group G over the alphabet X = {1, 2, 3, 4}
associated to the triple (G,φ,D). The action of the generators a = (1, 0, 0) and b = (0, 1, 0) of the
group satisfies the following recursions:
a(1v) = 1a(v), a(2v) = 4a(v), a(3v) = 3a(v), a(4v) = 2(b−1ab)(v),
b(1v) = 2v, b(2v) = 1b(v), b(3v) = 4v, b(4v) = 3b(v),
for v ∈ X∗; or in wreath recursion notation
a = (a, a, a, b−1ab)(2, 4), b = (e, b, e, b)(1, 2)(3, 4).
Theorem 1. The constructed above self-similar action (G,X∗) of the Heisenberg group is faith-
ful, level-transitive, self-replicating, finite-state, and non-contracting.
Proof. The action is self-replicating, because it is transitive on X and the virtual endomorphism
φ is surjective. Every self-replicating action is level-transitive by [3] (Proposition 2.8.2).
The elements a and b are finite-state, namely the computations
b−1ab =
(
b−1ab, a, b−2ab2, a
)
(1, 3),
b−2ab2 =
(
b−1ab, b−1ab, b−1ab, b−2ab2
)
(2, 4)
show that the states of a are a, b−1ab, b−2ab2 and the states of b are e, b. Hence the action (G,X∗)
is finite-state.
Since the action is faithful and the group G is torsion-free, all powers a2n are different. Since
a2n =
(
a2n, (ab−1ab)n, a2n, (b−1aba)n
)
,
we get a2n|1 = a2n and therefore the action is not contracting.
Theorem 1 is proved.
Since nilpotent groups have polynomial growth, all their Schreier graphs also have polynomial
growth. Hence the constructed above self-similar action of the Heisenberg group provides an example
of a self-replicating finite-state and non-contracting self-similar action with orbital Schreier graphs
Γw of polynomial growth.
For every faithful level-transitive action, the action on every orbit on XN is faithful. In the next
theorem we describe orbits, where the action of the group G is free, and compute the growth of every
orbital Schreier graph.
Theorem 2. Let Ω be the collection of all pre-periodic sequences from XN together with all
sequences obtained from those by arbitrary changes of letters 1↔ 3 and 2↔ 4. Then the stabilizer
StG(w) for w ∈ Ω is the infinite cyclic group and the stabilizer StG(w) for w ∈ XN \ Ω is trivial.
Proof. Let us describe which elements (x, y, z) ∈ H have a fixed letter depending on the parity
of x, y, z. We use notation g · x = y · h instead of equality in equation (1). By direct computation
we get
(2x+ 1, 2y, 2z + 1) · 2 = 2 · (2x+ 1, y, z + x+ 1),
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
1460 I. BONDARENKO, R. KRAVCHENKO
(2x+ 1, 2y, 2z + 1) · 4 = 4 · (2x+ 1, y, z + x+ 1),
(x, 2y, 2z) · 1 = 1 · (x, y, z),
(x, 2y, 2z) · 3 = 3 · (x, y, z),
(2x, 2y, 2z) · 2 = 2 · (2x, y, z + x),
(2x, 2y, 2z) · 4 = 4 · (2x, y, z + x),
while the other triples do not have a fixed letter. Notice that in the above equations the states at 2 and
4 are equal, and the same holds for 1 and 3. That explains why one can make arbitrary letter changes
2↔ 4 and 1↔ 3 considering fixed sequences.
Let g = (x, y, z) ∈ G fix an infinite sequence w = x1x2 . . . ∈ XN. If y 6= 0 then the equations
above show that for some n the y-component of g|x1x2...xn will be odd and this element doesn’t have
a fixed letter. Hence y = 0. Also notice that the x-component of each g|x1x2...xn is equal to x. We
will trace what happens with the component zn of g|x1x2...xn = (x, 0, zn). The sequence {zn}n≥1
satisfies the recurrence
z0 = z, zn+1 =
zn/2, xn = 1, 3,
(zn + x)/2, xn = 2, 4.
(2)
One can express zn in terms of the word v = x1x2 . . . xn as
zn =
z + αvx
2n
, where αv = α1 + α22 + . . .+ αn2n−1 and αi =
0, xn = 1, 3,
1, xn = 2, 4.
(3)
Since the action is finite-state, the sequence {zn}n≥1 assumes a finite number of values. Let
the value t = zn1 be assumed infinitely many times and present the sequence w in the form w =
= v0v1v2 . . . with the shortest possible nonempty words vi ∈ X∗ so that
(x, 0, z) · v0 = v0 · (x, 0, t) and (x, 0, t) · vi = vi · (x, 0, t)
for all i ∈ N. Applying expression (3) to the element (x, 0, t) and the word vi we get
t =
t+ αvix
2|vi|
⇒
(
2|vi| − 1
)
t = αvix ⇒
t
x
=
αvi
2|vi| − 1
, (4)
where |v| denotes the length of the word v.
If x = 0 then t = 0 and (x, 0, t) is the identity element.
Let x 6= 0. Let us prove that for i ≥ 1 all words vi have the same length, all numbers αvi are
equal, and hence all words vi coincide up to arbitrary changes 1 ↔ 3 and 2 ↔ 4. Consider two
words vi and vj and let ni = |vi| ≤ |vj | = nj . Then we have
αvi(2
nj − 1) = αvj (2
ni − 1) ⇒ 2ni(2nj−niαvi − αvj ) = αvi − αvj .
Hence αvi ≡ αvj mod 2ni . It follows that the word vi is a prefix of vj up to arbitrary changes 1↔ 3
and 2↔ 4. Then (x, 0, t) · u = u · (x, 0, t) for the prefix u of vj of length ni. Hence ni = nj by the
minimality of vj in the presentation of w. The claim is proved.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
SCHREIER GRAPHS FOR A SELF-SIMILAR ACTION OF THE HEISENBERG GROUP 1461
We have proved that if the sequence w has a non-trivial stabilizer then it is pre-periodic up to
changes 1 ↔ 3 and 2 ↔ 4. Hence w ∈ Ω. Let us compute the stabilizer of such a sequence w.
We can assume that w = uvv . . . . Take the triple (x, 0, z) ∈ StG(vv . . .) with the smallest positive
x. The proportion in (4) shows that every element in StG(vv . . .) is of the form (mx, 0,mz). Since
(mx, 0,mz) = (x, 0, z)m, the stabilizer StG(vv . . .) is the infinite cyclic group generated by (x, 0, z).
The stabilizer StG(w) is the preimage of StG(vv . . .) under the isomorphism φ|u|, and therefore it is
an infinite cyclic group too.
Conversely, let w coincide with uvv . . . up to changes 1 ↔ 3 and 2 ↔ 4. There exist x, z ∈ Z,
x 6= 0, such that
(
2|v| − 1
)
z = αvx and z, x are divisible by 2|v|. Then (x, 0, z) · v = v · (x, 0, z).
Since the action is self-replicating, there exists g ∈ φ−|u|(G) such that φ|u|(g) = (x, 0, z). Then g
fixes the sequence w. The statement is proved.
Corollary 2. The growth of orbital Schreier graph Γw for w ∈ XN \ Ω is polynomial of degree
4. The growth of orbital Schreier graph Γw for w ∈ Ω is polynomial of degree 3.
Proof. The Heisenberg group G has polynomial growth of degree 4. Since StG(w) is trivial
for w ∈ XN \ Ω, the Schreier graph Γw is just the Cayley graph of the group and therefore it has
polynomial growth of degree 4.
Let w ∈ Ω. Then the stabilizerH = StG(w) is the infinite cyclic group generated by h = (x, 0, z)
with x > 0. We consider the orbital Schreier graph Γw as the Schreier coset graph for H in G with the
generating set S =
{
(±1, 0, 0), (0,±1, 0)
}
. Let g1, g2, . . . , gm be coset representatives of length ≤ n
for each coset in the ball of radius n in Γw around the vertex H. Each product gihk for k = 1, . . . , n
is a group element of length ≤ 2n. Since the number of such products is mn and the group G has
polynomial growth of degree 4, the growth degree of the graph Γw is not greater than 3.
For the converse, consider the products gihk for |k| ≤ 2n. Let f ∈ G be an element of length
≤ n. There exists gi such that giH = fH and thus f−1gi = hl for some l ∈ Z. Note that the element
hl = (lx, 0, lz) expressed as a word in S has length ≥ |l|. Therefore |l| ≤ 2n. We have proved that
the products gihk for |k| ≤ 2n cover the ball of radius n in G. Hence the Schreier graph Γw has
growth degree not less than 3 and the statement is proved.
A few remarks about the recurrence from equation (2). If x is odd and we want the sequence
{zn}n≥1 to assume only integer values, then the recurrence can be written in the form
zn+1 =
zn/2, zn is even,
(zn + x)/2, zn is odd,
independently on the word x1x2 . . . , i. e., it is iteration of a single map. Interestingly, the maps of this
form were studied in relation to 3x+1 conjecture (see [6, 7]). For even x we have a freedom to choose
between zn/2 and (zn +x)/2. Let us encode our choice in the sequence w = x1x2 . . . ∈ {0, 1}N and
write
zn+1 =
zn/2, xn = 0,
(zn + x)/2, xn = 1.
Then the problem that we solved in the proof on Theorem 2 is the classification of all x, z ∈ Z
and w ∈ {0, 1}N such that the corresponding sequence {zn}n≥1 with z1 = z assumes only integers
(even) values.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
1462 I. BONDARENKO, R. KRAVCHENKO
1. Bartholdi L., Grigorchuk R. On the spectrum of Hecke type operators related to some fractal groups // Proc. Steklov
Inst. Math. – 2000. – 231. – P. 1 – 41.
2. Bondarenko I. Growth of Schreier graphs of automaton groups // Math. Ann. – 2012. – 354, № 2. – P. 765 – 785.
3. Nekrashevych V. Self-similar groups // Math. Surv. and Monogr. – Providence: Amer. Math. Soc., 2005. – 117.
4. Nekrashevych V., Sidki S. Automorphisms of the binary tree: state-closed subgroups and dynamics of 1/2-
endomorphisms // Groups: Topological, Combinatorial and Arithmetric Aspects. – 2004. – 311. – P. 375 – 404.
5. Bondarenko I., Kravchenko R. Finite-state self-similar actions of nilpotent groups // Geom. dedic. – 2013. – 163,
№ 1. – P. 339 – 348.
6. Anderson S. Struggling with the 3x+ 1 problem // Math. Gaz. – 1987. – 71. – P. 271 – 274.
7. Lagarias J. C. The ultimate challenge: The 3x+ 1 problem. – Providence, RI: Amer. Math. Soc., 2010.
Received 06.11.12
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
|
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| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:25:06Z |
| publishDate | 2013 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/25/e36b43f036f26a98a7598fcd78544825.pdf |
| spelling | umjimathkievua-article-25252020-03-18T19:25:49Z Schreier Graphs for a Self-Similar Action of the Heisenberg Group Графи Шрайєра самоподібної дiї групи Гейзенберга Bondarenko, I. Kravchenko, R. Бондаренко, І. Кравченко, Р. We construct a faithful self-similar action of the discrete Heisenberg group with the following properties: This action is self-replicating, finite-state, level-transitive, and noncontracting. Moreover, there exist orbital Schreier graphs of action on the boundary of the tree with different degrees of growth. Побудовано точну самоподiбну дію дискретної групи Гейзенберга з наступними властивостями. Дія є рекурентною, скінченностановою, сферично транзитивною, нестискуючою та існують графи Шрайєра дії групи на межі дерева з різними степенями зростання. Institute of Mathematics, NAS of Ukraine 2013-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2525 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 11 (2013); 1456–1462 Український математичний журнал; Том 65 № 11 (2013); 1456–1462 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2525/1811 https://umj.imath.kiev.ua/index.php/umj/article/view/2525/1812 Copyright (c) 2013 Bondarenko I.; Kravchenko R. |
| spellingShingle | Bondarenko, I. Kravchenko, R. Бондаренко, І. Кравченко, Р. Schreier Graphs for a Self-Similar Action of the Heisenberg Group |
| title | Schreier Graphs for a Self-Similar Action of the Heisenberg Group |
| title_alt | Графи Шрайєра самоподібної дiї групи Гейзенберга |
| title_full | Schreier Graphs for a Self-Similar Action of the Heisenberg Group |
| title_fullStr | Schreier Graphs for a Self-Similar Action of the Heisenberg Group |
| title_full_unstemmed | Schreier Graphs for a Self-Similar Action of the Heisenberg Group |
| title_short | Schreier Graphs for a Self-Similar Action of the Heisenberg Group |
| title_sort | schreier graphs for a self-similar action of the heisenberg group |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2525 |
| work_keys_str_mv | AT bondarenkoi schreiergraphsforaselfsimilaractionoftheheisenberggroup AT kravchenkor schreiergraphsforaselfsimilaractionoftheheisenberggroup AT bondarenkoí schreiergraphsforaselfsimilaractionoftheheisenberggroup AT kravčenkor schreiergraphsforaselfsimilaractionoftheheisenberggroup AT bondarenkoi grafišrajêrasamopodíbnoídiígrupigejzenberga AT kravchenkor grafišrajêrasamopodíbnoídiígrupigejzenberga AT bondarenkoí grafišrajêrasamopodíbnoídiígrupigejzenberga AT kravčenkor grafišrajêrasamopodíbnoídiígrupigejzenberga |