ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 20 (2021) 29–36
https://doi.org/10.26493/1855-3974.2338.5df
(Also available at http://amc-journal.eu)
A family of fractal non-contracting weakly
branch groups
Marialaura Noce *
Georg-August-Universität Göttingen, Mathematisches Institut
Received 20 May 2020, accepted 21 July 2020, published online 13 July 2021
Abstract
We construct a new example of an infinite family of groups acting on a d-adic tree, with
d ≥ 2 that is non-contracting and weakly regular branch over the derived subgroup.
Keywords: Groups of automorphisms of rooted trees, branch groups.
Math. Subj. Class. (2020): 20E08
1 Introduction
Weakly branch groups were first defined by Grigorchuk in 1997 as a generalization of the
famous p-groups constructed by Grigorchuk himself [4, 5], and Gupta and Sidki [6]. These
groups possess remarkable and exotic properties. For instance, the Grigorchuk group is the
first example of a group of intermediate word growth, and amenable but not elementary
amenable. Also, together with the Grigorchuk group, other subgroups of the group of
automorphisms of rooted trees like the Gupta-Sidki p-groups and many groups in the family
of the so-called Grigorchuk-Gupta-Sidki groups have been shown to be a counterexample
to the General Burnside Problem.
For these reasons, (weakly) branch groups spread great interest among group theo-
rists, who have actively investigated further properties of these in the recent years: just-
infiniteness, fractalness, maximal subgroups, or contraction.
Roughly speaking, a group is said to be contracting if the sections of every element are
“shorter” than the element itself, provided the element does not belong to a fixed finite set,
called the nucleus (see the exact definition in Section 2).
*The author wants to thank Laurent Bartholdi for pointing out the existence of [9], and Gustavo A. Fernández-
Alcober and Albert Garreta for useful discussions. The author thanks the anonymous referee for helpful com-
ments. The author is supported by EPSRC (grant number 1652316), and partially by the Spanish Government,
grant MTM2017-86802-P, partly with FEDER funds.
E-mail address: mnoce@unisa.it (Marialaura Noce)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
30 Ars Math. Contemp. 20 (2021) 29–36
Even though in the literature there are many examples of weakly branch contracting
groups, not much is known about weakly branch groups that are non-contracting. In 2005
Dahmani [2] provided the first example of a non-contracting weakly regular branch au-
tomaton group. Another example with similar properties was constructed by Mamaghani
in 2011 [7]. Both are examples of groups acting on the binary tree. We also point out that
in [9] Sidki and Wilson proved in particular that every group acting on the binary tree with
finite abelianization (including non-contracting groups) embeds in a branch group. This
provides more examples of non-contracting branch groups acting on the binary tree.
For d ≥ 3, the Hanoi Towers group H(d) ≤ Aut Td (which represents the famous game
of Hanoi Towers on d pegs) is non-contracting and only weakly branch. To the best of our
knowledge if d > 3 it is not known if these groups can be branch. For more information on
the topic, see [3] and [10].
In this paper we explicitly construct an example of an infinite family of non-contracting
weakly branch groups acting on d-adic trees for any d ≥ 2. This result gives a wealth of
examples of groups with these properties. In the following we denote with Aut Td the
group of automorphisms of a d-adic tree.
Theorem 1.1. For any d ≥ 2, there exists a group M(d) ≤ Aut Td that is weakly regular
branch over its derived subgroup, non-contracting and fractal.
1.1 Organization
In Section 2 we give some definitions of groups acting on regular rooted trees and of prop-
erties like fractalness, branchness and contraction. In Section 3 we introduce these groups
and we prove the main theorem together with some additional results regarding the order
of elements of M(d).
2 Preliminaries
In this section we fix some terminology regarding groups of automorphisms of d-adic
(rooted) trees. For further information on the topic, see [1] or [8].
Let d be a positive integer, and Td the d-adic tree. We denote with Aut Td the group of
automorphisms of Td. We let Ln be the nth level of Td, and L≥n the levels of the tree from
level n and below.
The stabilizer of a vertex u of the tree is denoted by st(u), and, more generally, the
nth level stabilizer st(n) is the subgroup of Aut Td that fixes every vertex of Ln. If
G ≤ Aut Td, we define the nth level stabilizer of G as stG(n) = st(n) ∩ G. Notice that
stabilizers are normal subgroups of the corresponding group. We let ψ be the isomorphism
ψ : st(1) −→ Aut Td ×
d· · · ×Aut Td
g 7−→ (gu)u∈L1 ,
where gu is the section of g at the vertex u, i.e. the action of g on the subtree Tu that
hangs from the vertex u. Let Sd be the symmetric group on d letters. An automorphism
a ∈ Aut Td is called rooted if there exists a permutation σ ∈ Sd such that a permutes
rigidly the vertices of the subtrees hanging from the first level of the tree according to the
permutation σ, i.e. if v = xu ∈ V (Td), with x ∈ L1, then a(xu) = σ(x)u. We usually
identify a and σ.
M. Noce: A family of fractal non-contracting weakly branch groups 31
Notice that if g ∈ st(1) with ψ(g) = (g1, . . . , gd), and σ is a rooted automorphism,
then,
ψ(gσ) =
(
gσ−1(1), . . . , gσ−1(d)
)
. (2.1)
Any element g ∈ G can be written uniquely in the form g = hσ, where h ∈ st(1) and σ is
a rooted automorphism.
Notice also that the decomposition g = hσ, together with the action (2.1), yields iso-
morphisms
Aut Td ∼= st(1)⋊ Sd ∼=
(
Aut Td × d. . .×Aut Td
)
⋊ Sd
∼= Aut Td ≀ Sd ∼= ((· · · ≀ Sd) ≀ Sd) ≀ Sd.
(2.2)
Throughout the paper, we will use the following shorthand notation: let f ∈ Aut T of
the form f = gh, where g ∈ stG(1) and h is the rooted automorphism corresponding to
the permutation σ ∈ Sd. If ψ(g) = (g1, . . . , gd), we write f = (g1, . . . , gd)σ.
Definition 2.1. Let G ≤ Aut Td, and let V (Td) be the set of vertices of Td. Then:
(a) The group G is said to be self-similar if for any g ∈ G we have
{gu | g ∈ G, u ∈ V (Td)} ⊆ G.
In other words, the sections of g at any vertex are still elements of G. For example,
Aut Td is self-similar.
(b) A self-similar group G is said to be fractal if ψu(stG(u)) = G for all u ∈ V (Td),
where ψu is the homomorphism sending g ∈ st(u) to its section gu.
To prove that a group is self-similar it suffices to show that the condition above is
satisfied by the vertices of the first level of the tree (see [3, Proposition 3.1]). The situation
is similar in the case of fractal groups. More precisely, using Lemma 2.2, we deduce that
to show that a group G is fractal, it is enough to check the vertices in the first level of Td.
We recall that G is said to be level transitive if it acts transitively on every level of the tree.
Lemma 2.2 ([11, Lemma 2.7]). If G ≤ Aut Td is transitive on the first level and
ψx(stG(x)) = G for some x ∈ L1, then G is fractal and level transitive.
Here we present a family of non-contracting weakly branch groups. To this end, in the
following, we recall the corresponding two definitions.
Definition 2.3. A self-similar group G ≤ Aut Td is contracting if there exists a finite
subset F ⊆ G such that for every g ∈ G there is n such that gv belongs to F for all
vertices v of L≥n. Note that if you take two finite sets F1 and F2 satisfying the condition
on the sections above, then also F1∩F2 will satisfy the condition. For this reason, one can
consider the set that is intersection of such sets. This is called the nucleus of G and it is
denoted by N .
Definition 2.4. Let G be a self-similar subgroup of Aut Td. We say that G is weakly
regular branch over a subgroup K ≤ G if G is level transitive and we have
ψ(K ∩ stG(1)) ≥ K × · · · ×K.
If, additionally, K is of finite index in G, then G is said to be regular branch over K.
32 Ars Math. Contemp. 20 (2021) 29–36
3 The groups M(d)
Let d ≥ 2, and let Td be the d-adic tree. The group M(d) ≤ Aut Td is generated by d
elements m1, . . . ,md, where m1, . . . ,md are defined recursively as follows:
m1 = (1, . . . , 1,m1)(1 . . . d)
m2 = (1, . . . , 1,m2, 1)(1 . . . d− 1)
m3 = (1, . . . ,m3, 1, 1)(1 . . . d− 2)
...
md−1 = (1,md−1, 1, . . . , 1)(1 2)
md = (m1, . . . ,md).
For example, for d = 3, we have M(3) = ⟨m1,m2,m3⟩, where
m1 = (1, 1,m1)(1 2 3), m2 = (1,m2, 1)(1 2), m3 = (m1,m2,m3).
m1 :
(1 2 3)
(1 2 3)
(1 2 3)
...
m2 :
(1 2)
(1 2)
(1 2)
...
m3 :
1
(1 2 3) (1 2) 1
(1 2 3) (1 2)
1
...
Figure 1: The generators of M(3).
3.1 Proof of the main theorem
In this section we prove the main result of the paper. In order to ease notation, and unless
it is strictly necessary, we will simply write M to denote an arbitrary group M(d).
Proposition 3.1. The group M is fractal and level transitive.
M. Noce: A family of fractal non-contracting weakly branch groups 33
Proof. Notice that the group is transitive on the first level because the rooted part of the
generator m1 is (1 2 . . . d). Also, it is straightforward to see that the group is self-similar,
since the sections of every generator at the first level are generators of M. To see that M is
fractal, note that
md1 = (m1, . . . ,m1)
m
md−21
d = (m
m1
3 , . . . ,m2)
...
m
m21
d = (m
m1
d−1, . . . ,md−2)
mm1d = (m
m1
d , . . . ,md−1)
md = (m1, . . . ,md).
Then in the last component of the elements above we obtain all the generators of M. Using
Lemma 2.2, we conclude that M is level transitive and fractal.
Proposition 3.2. Let d ≥ 2. Then the group M(d) is weakly regular branch over its derived
subgroup M′(d).
Proof. We will distinguish the case d = 2, and d ≥ 3 separately. Let d = 2. The element
[m1,m2] is non-trivial since
[m1,m2] = (m
−1
1 m
−1
2 m
2
1,m
−1
1 m2),
and m−11 m2 /∈ stM(1). Then M(2)′ is non-trivial, and we have
[m21,m2] = (1, [m1,m2]). (3.1)
From (3.1) and since M(2)′ = ⟨[m1,m2]⟩M(2), we obtain that {1} ×M(2)′ ≤ ψ(M(2)′).
As M(2) is level transitive, we conclude that M(2)′ ×M(2)′ ≤ ψ(M(2)′), as desired.
Let d ≥ 3, and write M for M(d). First we show that M′ is non-trivial. Let us denote
σ = (1 2 . . . d) and τ = (1 2 . . . d− 1). We have
[m1,m2] = σ
−1(1, . . . , 1,m−11 )τ
−1(1, . . . , 1,m−12 ,m1)σ(1, . . . , 1,m2, 1)τ
= (1, . . . , 1,m−11 )
σ(1, . . . , 1,m−12 ,m1)
τσ(1, . . . , 1,m2, 1)
τσ [σ, τ ]
= (m−11 , 1, . . . , 1)(m1,m
−1
2 , 1, . . . , 1)(1, . . . , 1,m2)(1 2 d).
Hence, we obtain that
[m1,m2] = (1,m
−1
2 , 1, . . . , 1,m2)(1 2 d). (3.2)
By (3.2), we have [m1,m2] /∈ stM(1), thus M′ is non-trivial.
Now, for i = 1, . . . , d− 2, and j = i+ 1, . . . , d− 1, we have
[md+1−ii ,mj ]
md−11 = (1, . . . , 1, [mi,mj ]). (3.3)
Then in order to prove that {1} × · · · × {1} ×M′ ≤ ψ(M′ ∩ stM(1)), it only remains to
show that for any i = 1, . . . , d− 1, there exists x(i) ∈ M′ ∩ stM(1) such that
x(i) = (1, . . . , 1, [mi,md]).
34 Ars Math. Contemp. 20 (2021) 29–36
To find such x(i), we first observe that
[(md+1−ii )
mi−11 ,md] = (1, i. . ., 1, [mi,mi+1], . . . , [mi,md−1], [mi,md]).
In order to cancel all these commutators above except for the last component, we use (3.3),
and we observe that since M is level transitive, if we conjugate with a suitable power of
m1, we get [mi,mi+1]−1, . . . , [mi,md−1]−1 in each component. For example, if i = 2,
we have
[(md−12 )
m1 ,md] = (1, 1, [m2,m3], [m2,m4], . . . , [m2,md]).
By using the considerations above, we obtain that x(2) must be of the form
x(2) = [m3,m
d−1
2 ]
m21 [m4,m
d−1
2 ]
m31 . . . [md−1,m
d−1
2 ]
md−21 [(md−12 )
m1 ,md]
= (1, . . . , 1, [m2,md]).
To prove last part of the main theorem (that M(d) is non-contracting), we need some
preliminary tools. Namely, we show some results regarding the order of elements of M(d).
We will handle the case d = 2, and d > 2 separately. More precisely, we first prove
that M(2) is torsion-free, and then, for d > 2, we show that the groups M(d) are neither
torsion-free nor torsion, contrary to the case d = 2.
The following Remark 3.3 and Lemma 3.4 are key steps to prove that M(2) is torsion-
free. We write M for M(2).
Remark 3.3. Let h ∈ M′ with h = (h1, h2). Then h1h2 ∈ M′.
Proof. Consider the following map ρ:
ρ : stM(1) → M → M/M′
(h1, h2) 7→ h1h2 7→ h1h2.
Note that ρ is a homomorphism of groups since M/M′ is abelian. As stM(1)/Ker ρ is
abelian, M′ ≤ Ker ρ. This concludes the proof.
In the proof of next lemma, for a prime p we denote with νp(m) the p-adic valuation
of m, that is the highest power of p that divides m.
Lemma 3.4. We have M′ = (M′ ×M′)⟨[m1,m2]⟩. Furthermore
M/M′ ∼= ⟨m1M⟩ × ⟨m2M⟩ ∼= Z× Z.
Proof. Since M is weakly regular branch over M′ by Proposition 3.2, and
[m1,m2] = ([m1,m2]m
−1
2 m1,m
−1
1 m2),
we deduce that (m−12 m1,m
−1
1 m2) is an element of M
′. Furthermore, we claim that the
elements [m1,m2]y where y ∈ {m1,m2,m−11 ,m
−1
2 } are in ⟨[m1,m2]⟩ modulo M′ ×M′.
Indeed, we have
[m1,m2]
m1 = (m1
−2m2m1,m1
−1m−12 m1
2)
= ([m1
2,m2
−1]m2m1
−1, [m1,m2]m2
−1m1)
≡ (m1−1m2,m2−1m1) (mod M′ ×M′),
M. Noce: A family of fractal non-contracting weakly branch groups 35
and similarly for the other commutators. Thus M′ = (M′ ×M′)⟨[m1,m2]⟩, as required.
Now we claim that m1 is of infinite order. By way of contradiction suppose that, for
some k, m1 has order n = 2k, as m1 has order 2 modulo the first level stabilizer. We have
mn1 = (m
k
1 ,m
k
1) = (1, 1),
which yields a contradiction as k < n. This concludes the proof of the claim and implies
that also m2 is of infinite order, since m2 = (m1,m2).
Now we want to show that if mi1m
j
2 ∈ M′, then necessarily i = j = 0. As mi1m
j
2 ∈
M′ ≤ stM(1), then i must be even. By way of contradiction, we choose the element
mi1m
j
2 ∈ M′ subject to the condition that i is divisible by the least possible positive power
of 2, say 2a, for some a. In other words, ν2(i) = a. Then if mr1m
s
2 ∈ M′, necessarily
2a | r. Note that it cannot happen that r = 0 and s ̸= 0 as m2 is of infinite order. Now,
writing i = 2i1 for some i1, we have
mi1m
j
2 = (m1
i1+j ,m1
i1m2
j) ≡ [mk1 ,mk2 ] ≡ (m1km2−k,m1−km2k) (mod M′ ×M′).
This implies that m1i1+j−km2k ∈ M′ and m1i1+km2j−k ∈ M′. As 2a | i1 + j − k and
2a | i1 + k, then 2a divides also j. This is because 2a | 2i1 + j = i+ j and by hypothesis
2a | i. Finally, we also have m1i1+km2j−k ∈ M′, from which we get
m1
i1+km2
j−k =
(
m
i1+k
2 +j−k
1 ,m
i1+k
2
1 m
j−k
2
)
.
By Remark 3.3, we have mi1+j1 m
j−k
2 ∈ M′ which implies that 2a | i1 + j. As ν2(i1) =
a − 1 and 2a | j, then ν2(i1 + j) = a − 1, a contradiction as 2a | i1 + j. This completes
the proof.
As a consequence, we prove the following.
Proposition 3.5. The group M(2) is torsion-free.
Proof. Suppose by way of contradiction that there exists an element of finite order in M.
Since M/M′ ∼= Z × Z by Lemma 3.4, then this element must lie in M′ ≤ stM(1).
Suppose that among all elements of finite order, we take the element g that lies in
stM(n) \ stM(n + 1), with n minimum with this property. Write g = (g1, g2). As g is of
finite order, then also g1, g2 must be of finite order. By our minimality assumption of n, the
elements g1, g2 must lie at least in stM(n). This implies that g = (g1, g2) ∈ stM(n+ 1), a
contradiction to the fact that g ∈ stM(n) \ stM(n+ 1).
In the following we determine the order of some elements of M(d), for d > 2.
Proposition 3.6. Let d > 2. Then the group M(d) is neither torsion-free nor torsion.
Proof. For ease of notation we write M for M(d). We start by proving that the given
generators of M are of infinite order. Consider m1, and suppose by way of contradiction
that its order is n. Then if mn1 = 1, we obtain that m
n
1 must lie in stM(1). Also, its order
must be a multiple of d, say n = dk for some k, since m1 has order d modulo the first level
stabilizer. Since m1 = (1, . . . , 1,m1)(1 2 . . . d), we obtain
mn1 = (m
k
1 , . . . ,m
k
1) = (1, . . . , 1).
36 Ars Math. Contemp. 20 (2021) 29–36
This yields a contradiction since mk1 = 1 and k < n. Similar arguments can be used for
the generators m2, . . . ,md−1, and md has infinite order because md = (m1, . . . ,md).
Furthermore, by (3.2), we have
[m1,m2] = (1,m
−1
2 , 1, . . . , 1,m2)(1 2 d).
Thus it follows readily that [m1,m2]3 = 1. Hence M is not torsion-free.
We conclude the paper by proving the remaining part of the main theorem.
Proposition 3.7. The group M is non-contracting.
Proof. Suppose by way of contradiction that M is contracting with nucleus N . Notice that
the element mm1d stabilizes the vertex 1. As a consequence, by induction, m
m1
d fixes all the
vertices of the path v = 1 n. . .1 for all n ≥ 1. Also, (mm1d )v = m
m1
d . Clearly, this implies
that mm1d lies in N . Consider now a power k of m
m1
d . Arguing as before, we obtain again
that (mm1d )
k fixes v and its section at v is (mm1d )
k. Thus, (mm1d )
k ∈ N for any k ≥ 1.
This concludes the proof since mm1d has infinite order.
References
[1] L. Bartholdi, R. I. Grigorchuk and Z. Šunić, Branch groups, in: M. Hazewinkel (ed.),
Handbook of Algebra, Volume 3, North-Holland, Amsterdam, pp. 989–1112, 2003, doi:
10.1016/s1570-7954(03)80078-5.
[2] F. Dahmani, An example of non-contracting weakly branch automaton group, Contemp. Math.
372 (2005), 219–224, doi:10.1090/conm/372/06887.
[3] R. Grigorchuk and Z. Šunić, Self-similarity and branching in group theory, in: C. M. Camp-
bell, M. R. Quick, E. F. Robertson and G. C. Smith (eds.), Groups St. Andrews 2005, Volume
1, Cambridge University Press, Cambridge, volume 339 of London Mathematical Society Lec-
ture Note Series, pp. 36–95, 2007, doi:10.1017/cbo9780511721212.003, Proceedings of the
conference held at the University of St. Andrews, St. Andrews, July 30 – August 6, 2005.
[4] R. I. Grigorchuk, Degrees of growth of finitely generated groups, and the theory of invariant
means, Math. USSR Izvestiya 25 (1985), 259–300, doi:10.1070/im1985v025n02abeh001281.
[5] R. I. Grigorchuk, On the growth degrees of p-groups and torsion-free groups, Math. USSR
Sbornik 54 (1986), 185–205, doi:10.1070/sm1986v054n01abeh002967.
[6] N. Gupta and S. Sidki, On the Burnside problem for periodic groups, Math. Z. 182 (1983),
385–388, doi:10.1007/bf01179757.
[7] M. J. Mamaghani, A fractal non-contracting class of automata groups, Bull. Iranian Math. Soc.
29 (2011), 51–64, http://bims.iranjournals.ir/article_123.html.
[8] V. Nekrashevych, Self-Similar Groups, volume 117 of Mathematical Surveys and Monographs,
American Mathematical Society, 2005, doi:10.1090/surv/117.
[9] S. Sidki and J. Wilson, Free subgroups of branch groups, Arch. Math. (Basel) 80 (2003), 458–
463, doi:10.1007/s00013-003-0812-2.
[10] R. Skipper, On a Generalization of the Hanoi Towers Group, Ph.D. thesis, Binghamton Uni-
versity, 2018, https://orb.binghamton.edu/dissertation_and_theses/60.
[11] J. Uria-Albizuri, On the concept of fractality for groups of automorphisms of a regular rooted
tree, Reports@SCM 2 (2016), 33–44, doi:10.2436/20.2002.02.9.