# Liftable self-similar groups and scale groups

Rostislav Grigorchuk  Department of Mathematics
Texas A&M University
College Station, TX 77843-3368
<EMAIL_ADDRESS>Dmytro Savchuk  Department of Mathematics and
Statistics
University of South Florida
4202 E Fowler Ave
Tampa, FL 33620-5700
<EMAIL_ADDRESS>
###### Abstract

We canonically identify the groups of isometries and dilations of local fields
and their rings of integers with subgroups of the automorphism group of the
$(d+1)$-regular tree $\widetilde{T}_{d+1}$, where $d$ is the residual degree.
Then we introduce the class of liftable self-similar groups acting on a
$d$-regular rooted tree whose ascending HNN extensions act faithfully and
vertex transitively on $\widetilde{T}_{d+1}$ fixing one of the ends. The
closures of these extensions in $\mathop{\rm
Aut}\nolimits(\widetilde{T}_{d+1})$ are totally disconnected locally compact
group that belong to the class of scale groups as defined in [82]. We give
numerous examples of liftable groups coming from self-similar groups acting
essentially freely or groups admitting finite $L$-presentations. In
particular, we show that the finitely presented group constructed by the first
author in [37] and the finitely presented HNN extension of the Basilica group
constructed in [55] embed into the group $\mathcal{D}(\mathbb{Q}_{2})$ of
dilations of the field $\mathbb{Q}_{2}$ of $2$-adic numbers. These actions,
translated to $\widetilde{T}_{3}$, are 2-transitive on the punctured boundary
of $\widetilde{T}_{3}$. Also we explore scale-invariant groups studied in [70]
with the purpose of getting new examples of scale groups.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Rooted trees and self-similar groups
    2. 2.2 Mealy Automata
    3. 2.3 The boundary of a rooted tree as a topological and measure space
    4. 2.4 Historical remarks
  3. 3 Isometries and dilations of local fields and their rings of integers
    1. 3.1 Local fields
    2. 3.2 Actions on rooted and unrooted regular trees
    3. 3.3 Notational remarks
  4. 4 Liftable groups and actions of their HNN extensions on regular trees
    1. 4.1 Definition and main embedding result
    2. 4.2 Connection to the results of Willis on scale groups
  5. 5 Examples of liftable groups
    1. 5.1 Liftable groups coming from essentially free actions
    2. 5.2 Liftable groups admitting finite $L$-presentations
    3. 5.3 Transitivity of the scale group coming from liftable groups
    4. 5.4 GGS-groups
    5. 5.5 The group $\mathcal{G}_{(01)^{\infty}}$
  6. 6 Scale groups coming from Bass-Serre theory
  7. 7 Scale-invariant groups and Haagerup property

##  1 Introduction

The main goal of this article is to shed more light on the question about what
kind of groups can act by automorphisms and cocompactly on a locally finite
infinite tree. As a byproduct of our considerations we gain some information
about subgroups of groups of isometries (and more generally of dilations) of
local fields, and about the class of scale groups, an important subclass of
totally disconnected locally compact (TDLC) groups. Our focus is on subgroups
of automorphisms of a $(d+1)$-regular tree ($d\geq 2$) which we will denote by
$\widetilde{T}_{d+1}$ (keeping the notation $T_{d}$ for a $d$-regular rooted
tree that also will play an important role in the paper). Such trees are the
most symmetric objects among all trees and deserve special attention. The
group $\mathop{\rm Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$ of
automorphisms of $\widetilde{T}_{d+1}$ preserving an end $\omega$, is known
for a while, in particular in relation with amenability (of topological
groups), random walks, and harmonic analysis [25, 9, 83].

The study of groups acting on trees was initiated by J.-P. Serre and J. Tits
in 1960’s and the basics of the theory are known as the _Bass-Serre theory_
[76, 12, 29]. One of the key questions is which (abstract) countable groups
act on a tree faithfully and vertex transitively. There are many articles on
this topic (see, for example, [21, 13, 22]) but the question still is far from
getting the satisfactory answer. A special case distinguished by Nebbia [67]
(see also a more detailed exposition of this result in [72]) is when the group
$G$ does not contain a free group $F_{2}$ on two generators as a subgroup. In
this case any action of $G$ on a tree must preserve a vertex, an edge, an end
or a pair of ends (as a set). Amenable groups do not contain $F_{2}$ as a
subgroup, hence when acting on a tree, they must fix something. The case when
there is a global fixed vertex is orthogonal to co-compactness of the action
and is a matter of study of groups acting on rooted trees [43, 69]. This case
is quite understood and the groups in this class are residually finite [43,
22]. The case when the edge is fixed is not far from the previous one, and the
case when a group fixes a set of two ends is also well understood. For
instance, when it fixes both ends then the involved group is a semidirect
product of $\mathbb{Z}$ and a group acting level transitively on a rooted
tree.

Thus we are left with the case when $G$ fixes an end of the tree (i.e., a
point of the boundary of the tree) and we focus our attention to this case.

If a field is supplied with a natural metric then one could be interested in
the group of its isometries (or more generally dilations). A natural class of
fields to be considered is the class of non-Archimedian local fields which, by
the famous characterization (see, for example, Theorem 22 in [74] or Remark
7.49 in [62], or an earlier work [71]), are either finite extensions of the
field $\mathbb{Q}_{p}$ of $p$-adic numbers or are isomorphic to the field
$\mathbb{F}_{q}((x))$ of Laurent powers series over a finite field
$\mathbb{F}_{q}$.

Our result (in fact observation) given by Theorem A and numerous examples
coming from theory of groups acting on rooted trees show that the groups of
isometries and dilations of local fields contain a plethora of exotic
subgroups satisfying various conditions of transitivity, including those that
do not satisfy the famous Tits alternative that states that a finitely
generated linear group (i.e., a group generated by matrices over a field)
either contains a free subgroup $F_{2}$ or is virtually solvable, and hence
amenable. This includes groups of intermediate growth, finitely presented
amenable, but not elementary and not subexponentially amenable groups, non-
elementary just-infinite groups, groups of branch type, and finitely
constrained groups. Thus, the situation here is very different from the
situations with subgroups of linear groups.

While the theory of connected locally compact topological groups is well
developed, totally disconnected locally compact groups are very far from being
well understood. A mysterious connection of them to self-similar groups was
discovered recently in [24, 82] via the notion of _scale_ group, that is a
closed vertex transitive subgroup of $\mathop{\rm
Aut}\nolimits(\widetilde{T}_{d+1})$ fixing an end.

Our main result, Theorem B, gives a way to construct new examples of scale
groups realizing them as closures of the ascending HNN extensions of the so-
called liftable groups introduced in the article (see Definition 4.1).

The alternative approach of getting scale groups is via not finitely co-
hopfian groups, i.e., groups containing proper subgroups of finite index
isomorphic to them. Getting interesting examples of finitely generated groups
of this sort is a challenging problem which is closely related to the problem
of constructing _scale-invariant_ groups (see Definition 5.3) initiated in
[70]. The level of transitivity of the scale group is a matter of a special
interest and is discussed for instance in [82]. We give some insight on it as
well in Theorem C.

Scale groups are amenable groups in the category of topological groups and in
most of our examples they are closures of (non-elementary) amenable abstract
groups. Moreover, in some cases these amenable groups are finitely presented.

Observe that the closed amenable (in the category of topological groups)
subgroups of the group of automorphisms of a tree were characterized in
already mentioned article of Nebbia [67] and scale groups fit in one of four
classes of Nebbia characterization (groups preserving an end). In most of our
examples they are closures of amenable abstract groups. Moreover, in some
cases these amenable groups are finitely presented. Also we give also examples
of scale groups that are closures of non-amenable groups (even not satisfying
the Haagerup property).

Now let us pass more closely to the context of the article. We start from
establishing a connection between groups of isometries and dilations of local
fields and groups of automorphisms of regular trees. The following theorem
could be viewed as a folklore type statement, but it seems that it was never
stated in this form before. By dilation of a metric space we mean a
homeomorphism that uniformly expands distances by some constant $\lambda>0$
(see Definition 3.1).

######  Theorem A.

Let $F$ be a (non-Archimedean) local field with the ring of integers
$\mathcal{O}$ and the residue field of size $q$, endowed with the metric
induced by its valuation. Then

  * (i)

the group $\mathop{\rm Isom}\nolimits(\mathcal{O})$ of isometries of
$\mathcal{O}$ is isomorphic to the group $\mathop{\rm Aut}\nolimits(T_{q})$ of
automorphisms of $T_{q}$.

  * (ii)

the group $\mathop{\rm Isom}\nolimits(F)$ of isometries of $F$ is isomorphic
to the group $\mathop{\rm Aut}\nolimits_{0}(\widetilde{T}_{q+1})_{\omega}$
consisting of automorphisms of $\widetilde{T}_{q+1}$ that fix a distinguished
end $\omega$ pointwise (eventually fixing the vertices in the ray
corresponding to $\omega$).

  * (iii)

the group $\mathcal{D}(F)$ of dilations of $F$ is isomorphic to the group
$\mathop{\rm Aut}\nolimits(\widetilde{T}_{q+1})_{\omega}$ consisting of
automorphisms of $\widetilde{T}_{q+1}$ that fix an end $\omega$ as a point of
the boundary of $\widetilde{T}_{q+1}$ (but possibly moving the vertices in
rays corresponding to $\omega$ along those rays while preserving $\omega$ as a
point of the boundary).

The automorphism groups involved in the theorem were discussed before (see,
for example, [25, 9]) and we present some background about them in Subsection
3.3.

As was already mentioned one of the goal of the paper is to construct examples
of amenable totally disconnected locally compact (TDLC) groups coming from the
actions of self-similar groups on $d$-regular rooted tree $T_{d}$, $d\geq 2$.
Such groups are built as the closures of embeddings of the HNN extensions of
_liftable self-similar groups_ defined in the article (see Definition 4.1) and
scale-invariant groups (see Definition 5.3), in the group of automorphisms of
$\widetilde{T}_{d+1}$. Obtained groups belong to the class of scale groups.
Scale groups play an important role in theory of TDLC groups as shown in works
of P.-E. Caprace and G. Willis [24, 82]. We comment on the connection of our
results to the work of Willis [82] in Subsection 4.2.

The idea of lifting (the notion that will appear in the next Theorem B) has
its roots in the “trick” used in [37] to embed the group $\mathcal{G}$ of
intermediate growth constructed by the first author [34] into a finitely
presented group. The presentation of $\mathcal{G}$

$\mathcal{G}=\langle a,b,c,d\mid
a^{2},b^{2},c^{2},d^{2},bcd,\sigma^{i}((ad)^{4}),\sigma^{i}((adacac)^{4}),i\geq
0\rangle,$ (1.1)

where the substitution $\sigma$ is defined as

$\sigma\colon\left\\{\begin{array}[]{l}a\mapsto aca\\\ b\mapsto d\\\ c\mapsto
b\\\ d\mapsto c\end{array}\right.$ (1.2)

discovered by I. Lysionok in [59], leads to the embedding of $\mathcal{G}$
into the ascending HNN extension $\widetilde{\mathcal{G}}$

$\widetilde{\mathcal{G}}=\langle a,b,c,d,t\mid
a^{2},b^{2},c^{2},d^{2},bcd,(ad)^{4},(adacac)^{4},tst^{-1}\sigma(s)^{-1},s\in\\{a,b,c,d\\}\rangle.$
(1.3)

The latter is a finitely presented amenable, but not elementary amenable group
containing a subgroup of intermediate growth. The presentation (1.3) was later
simplified by L. Bartholdi (given in [28]) to the one with only 2 generators
and 4 relators of the combined length of 109:

$\widetilde{\mathcal{G}}\cong\left\langle a,t\mid
a^{2}=TaTatataTatataTataT=(Tata)^{8}=(T^{2}ataTat^{2}aTata)^{4}=1\right\rangle,$
(1.4)

where $T$ denotes $t^{-1}$.

The further observation made by the first author a long time ago and privately
reported to A. Valette is that $\widetilde{\mathcal{G}}$ embeds into the group
of automorphisms of $\widetilde{T}_{3}$ preserving one end of this tree. Hence
we have a good example illustrating the theorem of Nebbia [67] mentioned
earlier. Moreover, as recently shown by C. Bodart this group does not have
rational cross-sections [19], which is the first example of such a group in
the class of finitely presented groups with solvable word problem.

The presentation found by Lysionok attracted a lot of attention and led to the
definition (in different forms) of (finite) $L$-_presentation_ (also sometimes
called _endomorphic presentation_) [6, 3, 49]. It was shown in [38] that the
Schur multiplier $H_{2}(\mathcal{G},\mathbb{Z})$ of $\mathcal{G}$ is
isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{\infty}$ and that the relators in
presentation (1.1) are independent. This also gave an answer to G. Bumslag’s
question raised in [15].

The idea of self-similarity that is in the root of the construction of
$\mathcal{G}$ got remarkable developments in various directions, including
theory of automata groups [43, 20], iterated monodromy groups [69],
combinatorics [51], analysis on graphs [2, 50, 53, 49], computer science [43,
61, 60], cryptography [45, 66, 65, 32, 73] and coding theory [27, 51], random
walks [57, 7, 1]. It came also to the theory of TDLC groups as follows from
the articles [24, 82]. Self-similar groups related to the scale groups have an
additional property of being _self-replicating_ (see Definition 2.3). Such
groups appear quite often among self-similar groups as one can conclude from
classification result on groups generated by 3-state invertible automata
acting over a 2-letter alphabet [20].

The question of interest is which of self-replicating groups lead to examples
of scale groups. This question is studied from one angle in [82] and
represents one of the interests of this work. In Section 4.2 we make a comment
on the relation between results of this paper and of [82]. Our next result is
based on the notion of a liftable self-similar group and a homomorphism of
lifting $\sigma\colon G\to\mathop{\rm St}\nolimits_{G}(i)$, where $i$ is one
of the vertices of the first level of the tree on which the group $G$ acts,
and $\mathop{\rm St}\nolimits_{G}(i)$ is the stabilizer of this vertex in $G$.
An example of such lifting is given by substitution (1.2), which extends to
the injective endomorphism of $\mathcal{G}$. To unify the notation, we will
use the same letter $\sigma$ for the liftings in most of the groups that we
deal with, unless it is important to emphasize the group that we are dealing
with. In the result below $\Lambda$ denotes the “leftmost” vertex of level 0
in $\widetilde{T}_{d+1}$ (as shown in Figure 2) and
$\pi_{\Lambda}\colon\mathop{\rm St}\nolimits_{\mathop{\rm
Aut}\nolimits(\widetilde{T}_{d+1})}(\Lambda)\to\mathop{\rm
Aut}\nolimits(T_{d})$ is a homomorphism that sends each element to its section
(or projection) at the vertex $\Lambda$, that is an automorphism of a rooted
tree $T_{d}$ with the root $\Lambda$.

######  Theorem B.

Let $G$ be a liftable group acting on $T_{d}$ and $\sigma\colon
G\to\mathop{\rm St}\nolimits_{G}(i)$ be the corresponding lifting. Then

  * (i)

There is an embedding $\theta$ of the ascending HNN extension $\widetilde{G}$
of $G$ by $\sigma$

$\widetilde{G}=G*_{\sigma}=\langle G,t\mid\text{relations in}\
G,tgt^{-1}=\sigma(g)\ \text{for all}\ g\in G\rangle$ (1.5)

into $\mathop{\rm Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$ for the end
$\omega=0^{-\infty}$ of $\widetilde{T}_{d+1}$.

  * (ii)

If $G$ acts transitively on the first level of $T_{d}$, then $G$ is self-
replicating, $\theta(\widetilde{G})$ acts transitively on the set of vertices
of $\widetilde{T}_{d+1}$, and the closure $W$ of $\theta(\widetilde{G})$ in
$\mathop{\rm Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$ is a scale group.

  * (iii)

The projection $\pi_{\Lambda}$ maps $\mathop{\rm
St}\nolimits_{\theta(\widetilde{G})}(\Lambda)$ onto $G$ and $\mathop{\rm
St}\nolimits_{\theta(\widetilde{G})}(\Lambda)=\bigl{(}\ker\pi_{\Lambda}\cap\theta(\widetilde{G})\bigr{)}\rtimes\theta(G)$.

  * (iv)

The projection $\pi_{\Lambda}$ maps $\mathop{\rm St}\nolimits_{W}(\Lambda)$
onto $\overline{G}$ and $\mathop{\rm
St}\nolimits_{W}(\Lambda)=\bigl{(}\ker\pi_{\Lambda}\cap
W\bigr{)}\rtimes\theta(\overline{G})$.

There are several sources of examples when this theorem is applicable. The
first one is the class of self-similar groups acting essentially freely on the
boundary of rooted tree. It includes such groups as the lamplighter group
$\mathbb{Z}_{2}\wr\mathbb{Z}$, some of Baumslag-Solitar groups, realizations
of free groups and the free products of cyclic groups as self-similar groups
[79, 80, 75]. All such groups generated by 3-state automata over 2-letter
alphabet are completely classified in [47].

Another source of examples consists of groups of branch type admitting finite
$L$-presentations satisfying an additional condition that the substitution
$\sigma$ involved in the presentation induces a lifting. This class includes
the group $\mathcal{G}$, Gupta-Sidki 3-group [56], Basilica group
$\mathcal{B}$ [54], etc. In some cases, like GGS-groups [6] and some of
$\mathcal{G}_{\omega}$ groups [35], liftability can be verified directly
without the knowledge of a group presentation. It follows from [54, 10] that
the Basilica group is an example of an amenable but not subexponentially
amenable group (i.e., whose amenability does not come in any way from groups
of subexponential growth). That was the first example of such a group, whose
existence was questioned in [36]. As pointed out in [55], using the simplified
$L$-presentation of $\mathcal{B}$ constructed by Bartholdi, the ascending HNN
extension $\widetilde{\mathcal{B}}$ of Basilica group $\mathcal{B}$ defined in
Example 5.12, similarly to $\widetilde{\mathcal{G}}$, is a finitely presented
group. Also in [55] it was shown that $\mathcal{B}$ (and hence
$\widetilde{\mathcal{B}}$) does not belong to the class $SG$ of
subexponentially amenable groups introduced in [37] and the question about
amenability of these groups was raised. It got a positive answer by Barthlodi
and Virag in [10]. More than a half of our examples belong to the class of
branch groups introduced by the first author in [40, 39].

As one of the main applications of Theorem B we establish the following
embedding result.

######  Theorem C.

The groups $\widetilde{\mathcal{G}}$ and $\widetilde{\mathcal{B}}$ embed into
$\mathcal{D}(\mathbb{Q}_{2})$. Moreover, when $\mathcal{D}(\mathbb{Q}_{2})$ is
identified with $\mathop{\rm Aut}\nolimits(\widetilde{T}_{3})_{\omega}$, the
images of these embeddings act transitively on $\widetilde{T}_{3}$ and their
closures are scale groups acting 2-transitively on the punctured boundary
$\partial_{\omega}\widetilde{T}_{3}$ of $\widetilde{T}_{3}$.

It is shown in [82] that every scale group acts transitively on the
corresponding punctured boundary $\partial_{\omega}\widetilde{T}_{d+1}$. The
2-transitivity condition in the above theorem is a stronger statement.

Finally, at the end of the article we discuss one more method of constructing
scale groups. It is based on the use of some facts from the Bass-Serre theory
of groups acting on trees and not finitely co-Hopfian groups, i.e., groups
that contain a subgroup of finite index isomorphic to the whole group. The
class of scale-invariant groups studied in [70] is suitable for our purposes
and allows us to construct more examples of scale groups. One such example is
obtained as a closure of a group without Haagerup property. Observe that the
closure of a group with Kazhdan’s property (T) cannot be a scale group since
this property implies Serre’s property (FA) [81], which means that every
action of a group on a tree has a global fixed point. Nevertheless we use
property (T) in Section 7 to construct groups without Haagerup property whose
closures are scale groups. Thus (and we emphasize it), the examples of scale
groups given in this article are of two kinds: obtained as the closures of
(elementary and non-elementary) amenable groups and as the closures of non-
amenable groups.

The paper is organized as follows. We begin with preliminaries in Section 2.
In Section 3 we describe the identification of the groups of isometries and
dilations of local fields and their rings of integers with groups of
automorphisms of regular rooted and unrooted trees. Section 4 contains the
definition of liftable groups and the main embedding result. In Section 5 we
provide many examples of liftable groups and analyze the actions of their HNN
extensions on corresponding regular trees. Section 6 offers an alternative
method of constructing of actions of ascending HNN extentions on regular trees
via the Bass-Serre theory (see Theorem 6.1). Finally, in Section 7 we use some
scale-invariant groups to provide new examples of a scale groups. Note that
the paper contains a rather extended bibliography as we tried to unite various
notation and approaches used in the literature.

Acknowledgements. The authors would like to thank A. Garrido, A. Minasyan, V.
Nekrashevych, A. Valette, and G. Willis for fruitful discussions that enhanced
the paper. The work on this project was partially conducted during the
authors’ visits to American Institute of Mathematics via the SQuaRE program;
authors thank the Institute for hospitality and support. The first author was
supported by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) – SFB-TRR 358/1 2023-491392403, Humboldt Foundation and expresses
his acknowledgement to the University of Bielefeld. Also he is supported by
Travel Support for Mathematicians grant MP-TSM-00002045 from Simons
Foundation. Finally, the authors appreciate the support from University of
Geneva (Swiss NSF grant 200020-200400), Texas A&M University and University of
South Florida during working on the paper.

##  2 Preliminaries

We start this section by introducing the notions and terminology of
automorphisms of regular rooted trees, self-similar groups, and
transformations generated by Mealy automata. For more details, the reader is
referred to [43, 69].

###  2.1 Rooted trees and self-similar groups

Let $X=\\{0,1,\ldots,d-1\\}$ be a finite alphabet with $d\geq 2$ elements
(called letters) and let $X^{*}$ denote the set of all finite words over $X$.
The set $X^{*}$ can be equipped with the structure of a rooted $d$-ary tree
$T(X)$ by declaring that $v$ is adjacent to $vx$ for every $v\in X^{*}$ and
$x\in X$. As a graph the tree $T(X)$ is isomorphic to $T_{d}$, but the
vertices of $T(X)$ are labelled by finite words over $X$. We will denote the
length of a word $w\in X^{*}$ by $|w|$. The empty word corresponds to the root
of the tree and for each positive integer $n$ the set $X^{n}$ corresponds to
the $n\,$th level of $T(X)$. Also the set $X^{\infty}$ of infinite (to the
right) words over $X$ can be identified with the _boundary_ $\partial T(X)$ of
the tree $T(X)$, which consists of all infinite paths in the tree, without
backtracking, initiating at the root. We consider automorphisms of the tree
$T(X)$ (that is, the bijections of $T(X)$ that preserve the root and the
adjacency of vertices). We sometimes denote the tree $T(X)$ as $T_{|X|}$ when
only the cardinality of $X$ is important. The group of all automorphisms of
$T(X)$ is denoted by $\mathop{\rm Aut}\nolimits(T(X))$. To operate with such
objects, we use alternatively the languages of sections and wreath recursions,
and of Mealy automata.

Let $g\in\mathop{\rm Aut}\nolimits(T(X))$ and $x\in X$. For any $v\in X^{*}$
we can write

$g(xv)=g(x)v^{\prime}$

for some $v^{\prime}\in X^{*}$. Then the map $g|_{x}\colon X^{*}\to X^{*}$
given by

$g|_{x}(v)=v^{\prime}$

defines an automorphism of $T(X)$ which we call the _section_ (or _state_) of
$g$ at vertex $x$. We can inductively extend the definition of a section at a
letter $x\in X$ to a section at any vertex $x_{1}x_{2}\ldots x_{n}\in X^{*}$
as follows:

$g|_{x_{1}x_{2}\ldots x_{n}}=g|_{x_{1}}|_{x_{2}}\cdots|_{x_{n}}.$

With this notation for any $g\in\mathop{\rm Aut}\nolimits(T(X))$, and any
words $v,w\in X^{*}$ we have

$g(vw)=g(v)g|_{v}(w).$ (2.1)

######  Definition 2.1.

A subgroup of $\mathop{\rm Aut}\nolimits(T(X))$ is called _self-similar_ if it
is closed under taking sections at the vertices of $T(X)$.

We note here that this standard definition of a self-similar group is
different from the one used in [82], where only the sections of elements in
the stabilizers of vertices at those vertices are required to remain in the
group.

We adopt the following convention throughout the paper.

######  Convention 2.2.

If $g$ and $h$ are elements of some group acting on a set $Y$ and $y\in Y$,
then

$gh(y)=h(g(y)).$

We work with the right actions partially because it corresponds to the
standard order of multiplication of permutations in `GAP`, the computer
algebra platform containing the package `AutomGrp` [64] dealing with self-
similar groups and semigroups and used in this paper for many routine
calculations. However, we due convenience in notation, we still write the
arguments of functions on the right. Note that knowledge of GAP is not
required for reading this paper or for verification of its results.

With the above convention, the state $g|_{v}$ at $v\in X^{*}$ of any product
$g=g_{1}g_{2}\cdots g_{n}$, where $g_{i}\in\mathop{\rm Aut}\nolimits(T(X))$
for $1\leq i\leq n$, can be computed as follows:

$g|_{v}=g_{1}|_{v}g_{2}|_{g_{1}(v)}\cdots g_{n}|_{g_{1}g_{2}\cdots
g_{n-1}(v)}.$

Also we use the language of wreath recursions. For each self-similar group $G$
there is a natural embedding

$G\hookrightarrow G\wr\mathop{\rm Sym}\nolimits(X),$

where $\wr$ is a symbol for the permutational wreath product, and $\mathop{\rm
Sym}\nolimits(X)$ denotes the symmetric group on $X$. This embedding is given
by

$G\ni g\mapsto(g_{0},g_{1},\ldots,g_{d-1})\sigma_{g}\in G\wr\mathop{\rm
Sym}\nolimits(X),$ (2.2)

where $g_{0},g_{1},\ldots,g_{d-1}$ are the states of $g$ at the vertices of
the first level, and $\sigma_{g}$ is the transformation of $X$ induced by the
action of $g$ on the first level of the tree. If $\sigma_{g}$ is the trivial
permutation, it is customary to omit it in (2.2). We call
$(g_{0},g_{1},\ldots,g_{d-1})\sigma_{g}$ the _decomposition of $g$ at the
first level_ (or the _wreath recursion of $g$_). When this does not cause
confusion, we identify $g$ with its wreath recursion and write simply

$g=(g_{0},g_{1},\ldots,g_{d-1})\sigma_{g}.$ (2.3)

The decomposition at the first level of all generators $\mathcal{A}_{q}$ of an
automaton group $\mathbb{G}(\mathcal{A})$ under the embedding (2.2) is called
the _wreath recursion defining the group_. Such a decomposition is especially
convenient for computing the states of group elements. Indeed, the products
and inverses of automorphisms can be found as follows. If
$g=(g_{0},g_{1},\ldots,g_{d-1})\sigma_{g}$ and
$h=(h_{0},h_{1},\ldots,h_{d-1})\sigma_{h}$ are two elements of $\mathop{\rm
Aut}\nolimits(T(X))$, then

$gh=\left(g_{0}h_{\sigma_{g}(0)},g_{1}h_{\sigma_{g}(1)},\ldots,g_{d-1}h_{\sigma_{g}(d-1)}\right)\sigma_{g}\sigma_{h}$

and the wreath recursion of $g^{-1}$ is

$g^{-1}=\left(g^{-1}_{\sigma_{g}^{-1}(0)},g^{-1}_{\sigma_{g}^{-1}(1)},\ldots,g^{-1}_{\sigma_{g}^{-1}(d-1)}\right)\sigma_{g}^{-1}.$

Note that the order of multiplication here is compatible with Convention 2.2.
Some authors choose to use the left actions and they write permutations on the
left side in wreath recursions. These approaches are equivalent and there is
an easy translation between them: every product has to be reversed everywhere.

Equation (2.1) for an automorphism $g$ given by decomposition (2.3), and for
$v=x\in X$ then can be written as:

$g(xw)=\sigma_{g}(x)g_{x}(w).$

For a group $G$ acting on $T(X)$ and $v\in X^{*}$ the stabilizer $\mathop{\rm
St}\nolimits_{G}(v)$ of vertex $v$ is the subgroup of $G$ consisting of all
elements that fix $v$. For each $m\geq 1$ the stabilizer $\mathop{\rm
St}\nolimits_{G}(X^{m})$ of level $m$ in $G$ is the subgroup of $G$ consisting
of all elements that fix all vertices of level $m$. Stabilizers of levels are
normal finite index subgroups of $G$ such that

$\cap_{m\geq 1}\mathop{\rm St}\nolimits_{G}(X^{m})=\\{1\\}.$

For each $v\in X^{*}$ the map

$\begin{array}[]{llll}\pi_{v}\colon&\mathop{\rm St}\nolimits_{\mathop{\rm
Aut}\nolimits(T(X))}(v)&\to&\mathop{\rm Aut}\nolimits(T(X))\\\
&g&\mapsto&g|_{v}\end{array}$ (2.4)

defines a homomorphism that we will call projection. For each subgroup $G$ of
$\mathop{\rm Aut}\nolimits(T(X))$ homomorphism $\pi_{v}$ restricts to a
homomorphism $\mathop{\rm St}\nolimits_{G}(v)\to\mathop{\rm
Aut}\nolimits(T(X))$. Moreover, if $G$ is a self-similar group, then since $G$
is closed under taking the sections, $\pi_{v}$ is a homomorphism from
$\mathop{\rm St}\nolimits_{G}(v)$ to $G$. In the case
$X=\\{0,1,\ldots,d-1\\}$, $\pi_{i}$, $i=0,1,\ldots,d-1$, denotes the
corresponding projection.

An important subclass of self-similar groups is the class of self-replicating
groups.

######  Definition 2.3.

A self-similar group is called _self-replicating_ if it acts transitively on
$X$ and for each $v\in X^{*}$ the map $\pi_{v}\colon\mathop{\rm
St}\nolimits_{G}(v)\to G$ is onto.

It is easy to see that in the above definition we can consider only
projections $\pi_{v}$ for $v\in X$.

It follows from the definition by induction on the level that self-replicating
groups act transitively on all levels of $T(X)$.

We can visualize the action of an automorphism $g\in\mathop{\rm
Aut}\nolimits(T(X))$ using the notion of the _portrait_ of $g$, denoted by
$\mathcal{P}(g)$. It is a labeled infinite rooted $|X|$-ary tree with the root
labeled by $g$ and each vertex $v$ labeled by $g|_{v}$. Under each vertex $v$,
we write the name of the mapping that $g|_{v}$ defines on the first level of
the subtree hanging from $v$. In the case of a rooted binary tree, we draw an
arc (called switch) connecting the two edges hanging down from $v$ if $h|_{v}$
acts nontrivially on the first level of the subtree hanging from $v$. If there
is no switch, it means the action is trivial. Four examples of portraits
corresponding to the generators of the group $\mathcal{G}$ are shown in Figure
6.

###  2.2 Mealy Automata

Another common language to describe self-similar groups is the language of
Mealy automata.

######  Definition 2.4.

A _Mealy automaton_ (or simply _automaton_) is a 4-tuple

$(Q,X,\delta,\lambda),$

where

  * •

$Q$ is a set of states

  * •

$X$ is a finite alphabet (not necessarily $\\{0,1,\ldots,d-1\\}$)

  * •

$\delta\colon Q\times X\to Q$ is the _transition function_

  * •

$\lambda\colon Q\times X\to X$ is the _output function_.

If the set of states $Q$ is finite, the automaton is called _finite_. If for
every state $q\in Q$ the output function $\lambda_{q}(x)=\lambda(q,x)$ induces
a permutation of $X$, the automaton $\mathcal{A}$ is called _invertible_.
Selecting a state $q\in Q$ produces an _initial automaton_ $\mathcal{A}_{q}$,
which formally is a $5$-tuple $(Q,X,\delta,\lambda,q)$.

Here we consider automata with the same input and output alphabets.

Automata are often represented by their _Moore diagrams_. The Moore diagram of
automaton $\mathcal{A}=(Q,X,\delta,\lambda)$ is a directed graph in which the
vertices are in bijection with the states of $Q$ and the edges have the form
$q\stackrel{{\scriptstyle x|\lambda(q,x)}}{{\longrightarrow}}\delta(q,x)$ for
$q\in Q$ and $x\in X$. For example, Figure 1 shows the Moore diagram of the
automaton $\mathcal{A}$ that, as is explained later, generates the group
$\mathcal{G}$.

Figure 1: The automaton generating the group $\mathcal{G}$

Every invertible initial automaton $\mathcal{A}_{q}$ induces an automorphism
of $T(X)$, which is also denoted by $\mathcal{A}_{q}$, defined as follows.
Given a word $v=x_{1}x_{2}x_{3}\ldots x_{n}\in X^{*}$, it scans the first
letter $x_{1}$ and outputs $\lambda(q,x_{1})$. The rest of the word is handled
similarly by the initial automaton $\mathcal{A}_{\delta(q,x_{1})}$. So we can
actually extend the functions $\delta$ and $\lambda$ to $\delta\colon Q\times
X^{*}\to Q$ and $\lambda\colon Q\times X^{*}\to X^{*}$ via the equations

$\begin{array}[]{l}\delta(q,x_{1}x_{2}\ldots
x_{n})=\delta(\delta(q,x_{1}),x_{2}x_{3}\ldots x_{n}),\\\
\lambda(q,x_{1}x_{2}\ldots
x_{n})=\lambda(q,x_{1})\lambda(\delta(q,x_{1}),x_{2}x_{3}\ldots x_{n}).\\\
\end{array}$

######  Definition 2.5.

The group generated by all states of an invertible automaton $\mathcal{A}$
viewed as automorphisms of the rooted tree $T(X)$ under the operation of
composition is called an _automaton group_.

In the definition of the automaton, we do not require the set $Q$ of states to
be finite. With this convention, the notion of an automaton group is
equivalent to the notions of _self-similar group_ [69] and _state-closed
group_ [68]. However, most of the interesting examples of automaton
(semi)groups are finitely generated (semi)groups defined by finite automata.

###  2.3 The boundary of a rooted tree as a topological and measure space

The boundary $\partial T(X)$ of the rooted tree is endowed with a natural
topology in which two infinite words are close if they have a large common
prefix. Each automorphism of $T(X)$ naturally induces a continuous
transformation (respectively, homeomorphism) of $\partial T(X)$.

The boundary $\partial T(X)$ can be viewed as an (ultra)metric space in which
the distance between two points is equal to $\lambda_{k}$, where
$\\{\lambda_{n}\\}$ is any monotonically decreasing sequence converging to 0,
and $k$ is the length of the longest common beginning of these two points.
This metric defines a topology on $\partial T(X)$ that coincides with the
Tychonoff product topology on $X^{\mathbb{N}}$ (when using the discrete
topologies on each copy of $X$. In this topology $\partial T(X)$ is
homeomorphic to the Cantor set. Let $\mu$ be the uniform Bernoulli measure
defined by

$\mu(C_{v})=\frac{1}{|X|^{n}},$

where for a vertex $v\in X^{n}$ the cylindrical set $C_{v}\subset\partial
T(X)$ consists of all infinite paths in $\partial T(X)$ that go through $v$.
Such sets generate a $\sigma$-algebra of Borel sets and hence measure $\mu$ is
defined by its values on them. This measure is invariant with respect to the
action of $\mathop{\rm Aut}\nolimits(T(X))$. A action of a countable group
$G<\mathop{\rm Aut}\nolimits(T(X))$ on $(\partial T(X),\mu)$ by measure
preserving transformations is called essentially free if the measure of a set
of points in $X$ that have non-trivial stabilizers is equal to 0. The examples
of such groups will be given in Section 5.

It is a well known fact (see, for example, [43]) that if a group
$G<\mathop{\rm Aut}\nolimits(T(X))$ acts transitively on levels of $T(X)$,
then its action on $\partial T(X)$ is ergodic and its closure in $\mathop{\rm
Aut}\nolimits(T(X))$ acts transitively on $\partial T(X)$. More generally, the
orbit structure of the action of $G$ on the levels of $T(X)$ gives rise to the
decomposition into ergodic components of the action of $G$ on $\partial T(X)$
[48].

###  2.4 Historical remarks

The reader should be aware that during the period 1999–2003, when there was an
explosion of the activity on self-similar and branch groups, the terminology
associated with the groups acting on rooted trees and, in particular, with
self-similar groups was changing. Now it is more or less settled, but still
there are nuances. In the articles [38, 39, 42, 43] the terms “self-similar”,
“self-replicating”, “branch”, “regular branch”, etc. already appear, but
sometimes with different meaning. For instance, the term “self-similar” action
of a group was used in the past to denote what is now called “self-
replicating” (or “recurrent” in [69]). Currently most of the authors use the
definition of a self-similar group presented in [69] as used in this article,
which we will call _classical_ , while G. Willis uses a slightly different
definition in [82] calling a group self-similar if the projection of the
stabilizer of every vertex in the tree on the subtree rooted at that vertex
remains in the original group. By that definition any automorphism $g$ acting
on the binary rooted tree with the wreath recursion $g=(h,h^{-1})\varepsilon$
for arbitrary $h\in\mathop{\rm Aut}\nolimits(T_{2})$ would generate a self-
similar group isomorphic to $\mathbb{Z}/2\mathbb{Z}$ as the stabilizer of
every vertex is trivial. Clearly, this group would not be self-similar in the
classical sense if $h$ is not the identity or $g$. Note, that this small
difference does not usually play a significant role, but it is important to
understand it. Additionally, other terms have been used to denote the class of
self-similar groups, like “automaton groups” and “state-closed groups” (mostly
in the school of S. Sidki). In some papers, a group is called self-similar if
it is generated by all the states of a finite automaton, which is again
different from the classical sense since in the latter one there is no
condition of finiteness. In such papers sometimes classical self-similar
groups are called “weakly self-similar”. Also, the terms of “fractal” and
“semifractal” were used in the past to denote self-similar or self-replicating
groups. Finally, the terms “self-similar” and self-replicating were also used
in earlier papers to denote regular branch groups and branching subgroups
[42]. The reader should be aware of these differences and nuances.

##  3 Isometries and dilations of local fields and their rings of integers

In this section, for a (non-Archimedean) local field $F$ with a ring of
integers $\mathcal{O}$, we identify the groups $\mathop{\rm
Isom}\nolimits(\mathcal{O})$ of isometries of $\mathcal{O}$ and
$\mathcal{D}(F)$ of dilations of $F$ with groups of automorphisms of regular
trees.

###  3.1 Local fields

Recall that a discrete valuation $v$ on a commutative field $F$ is a
homomorphism from the multiplicative group $F^{*}$ onto $\mathbb{Z}$
satisfying $v(a+b)\geq\inf\\{v(a),v(b)\\}$ for all $a,b\in F$, taking into
account the convention $v(0)=\infty$. Let $\mathcal{O}=\\{a\in F\colon
v(a)\geq 0\\}$ be the _ring of integers_ of $F$ and $\mathfrak{m}$ be the
maximal ideal in $\mathcal{O}$. Then $F$ is called a _(non-archimedean) local
field_ if

  * (i)

the cardinality $q$ of the _residue field_ $\mathcal{O}/\mathfrak{m}$ is
finite,

  * (ii)

$F$ is complete when equipped with the metric $d_{v}(a,b)=q^{-v(a-b)}$.

Examples of local fields include the field $\mathbb{Q}_{p}$ of $p$-adic
numbers and its finite extensions and the field of formal Laurent series
$F_{q}((x))$ over a field $F_{q}$ of finite characteristic $p$.

Each element $\pi\in\mathfrak{m}$ with $v(\pi)=1$ generates $\mathfrak{m}$ as
an ideal in $\mathcal{O}$. Let $X\subset F$ be the set of representatives of
the residue field such that $0\in X$. Then each element $a\in F^{*}$ can be
uniquely written as a formal Laurent series

$a=\sum_{i=n}^{\infty}x_{i}\pi^{i},x_{i}\in X,x_{n}\neq 0,$ (3.1)

where $n=v(a)\in\mathbb{Z}$. This identification can be used to identify any
local field $F$ as a metric space with metric $d_{v}$ with the punctured
boundary
$\partial_{\omega}\widetilde{T}_{q+1}=\partial_{\omega}\widetilde{T}_{q+1}-\omega$
of the $(q+1)$-regular tree $\widetilde{T}_{q+1}$ for a fixed end $\omega$ of
the tree, equipped with the corresponding ultrametric. Under this
identification the ring of integers $\mathcal{O}$ is identified with the
boundary of a rooted tree $T_{q}$ that corresponds to the series of the form
(3.1) with $n=v(a)\geq 0$ (i.e., with no negative powers of $\pi$. We refer
the reader to [77, 25] for details. This tree is sometimes called Serre’s tree
and is a particular case of a Bruhat-Tits building [78]. The affine group
$\mathrm{Aff}(F)$, which is a subgroup of $\mathrm{GL}(2,F)$, acts on
$\widetilde{T}_{d+1}$ and is isomorphic to a closed subgroup of $\mathop{\rm
Aut}\nolimits(\widetilde{T}_{d+1})$.

###  3.2 Actions on rooted and unrooted regular trees

For the sake of simplicity of notation we will explicitly describe the
correspondence (3.1) on the example of the field $\mathbb{Q}_{p}$. However,
the same construction with minimal changes works in the general case. The ring
of $p$-adic integers $\mathbb{Z}_{p}$ is the ring of integers of
$\mathbb{Q}_{p}$ with respect to the $p$-adic metric $d_{p}$ induced by
$p$-adic valuation $v_{p}$. The maximal ideal $\mathfrak{m}=p\mathbb{Z}_{p}$
in $\mathbb{Z}_{p}$ is generated by a uniformizer $p=.010^{\infty}$ and
$X=\\{0,1,\ldots,p-1\\}$ is the set of representatives of $\mathfrak{m}$ in
$\mathbb{Z}_{p}$. Then every element $a$ of $\mathbb{Q}_{p}^{*}$ can be
uniquely written as

$a=\sum_{i=n}^{\infty}x_{i}p^{i},x_{i}\in X,x_{n}\neq 0,$

where $n=v_{p}(a)\in\mathbb{Z}$. This decomposition corresponds to the
standard representation of elements of $\mathbb{Q}_{p}$ as biinfinite
sequences over $X$ that has finite number of non-zero symbols at positions
with negative indexes.

######  Definition 3.1.

For a metric space $(\mathcal{Y},d_{\mathcal{Y}})$ a homeomorphism
$f\colon\mathcal{Y}\to\mathcal{Y}$ called a _dilation_ if there is $\lambda>0$
such that for all $x,y\in\mathcal{Y}$

$d_{\mathcal{Y}}\bigl{(}f(x),f(y)\bigr{)}=\lambda d_{\mathcal{Y}}(x,y).$

For an alphabet $X=\\{0,1,\ldots,p-1\\}$, the set $X^{n}$ of vertices of level
$n$ of $T(X)$ can be naturally identified with the ring
$\mathbb{Z}/(p^{n}\mathbb{Z})$ via

$X^{n}\ni x_{0}x_{1}\ldots x_{n-1}\longleftrightarrow x_{0}+p\cdot
x_{1}+\cdots+p^{n-1}\cdot x_{n-1}\in\mathbb{Z}/(p^{n}\mathbb{Z}).$

The boundary $\partial T(X)$ of $T(X)$ is then identified with the ring
$\mathbb{Z}_{p}$ of $p$-adic integers:

$\partial T(X)\ni
x_{0}x_{1}x_{2}\ldots\longleftrightarrow\sum_{i=0}^{\infty}x_{i}p^{i}\in\mathbb{Z}_{p}.$

Now for arbitrary $d\geq 2$ (not necessarily prime) we build an infinite
unrooted $(d+1)$-regular tree $\widetilde{T}_{d+1}$ with a selected end (point
of the boundary), whose punctured boundary (to be defined below) for the case
of prime $d=p$, will be identified with $\mathbb{Q}_{p}$. In our notation the
“tilde” is used to separate the rooted case from the unrooted one.

As a graph the tree $\widetilde{T}_{d+1}$ is an infinite regular (non-rooted)
$(d+1)$-ary tree. We build it as a direct limit of a sequence of graphs
$T_{d}^{(n)}$, $n\geq 0$ each of which is isomorphic to the rooted $d$-ary
tree $T_{d}=T(X)$, for $X=\\{0,1,\ldots,d-1\\}$. Namely, we define

$\widetilde{T}_{d+1}=\bigcup_{n=0}^{\infty}T_{d}^{(n)},$

where embedding of $T_{d}^{(n)}$ into $T_{d}^{(n+1)}$ is induced by the map

$v\mapsto 0v,\quad v\in V(T_{d})=X^{*}.$ (3.2)

and induced map on the set of edges. The reader should visualize
$\widetilde{T}_{d+1}$ as a tree shown in Figure 2 (with some of the notation
used in the figure explained later in the text).

If a vertex $v$ of $\widetilde{T}_{d+1}$ is at level $k$ of $T_{d}^{(n)}$ for
some $k,n\geq 0$, then we say that it belongs to level $k-n$ of
$\widetilde{T}_{d+1}$, where $k-n$ ranges over $\mathbb{Z}$. The children of a
vertex $v$ of $\widetilde{T}_{d+1}$ of level $l$ are defined to be the
neighbors of $v$ from level $l+1$. Each vertex in $\widetilde{T}_{d+1}$ has
exactly $d$ children.

Figure 2: $\mathbb{Z}_{p}$ and $\mathbb{Q}_{p}$ as boundaries of trees $T_{p}$
and $\widetilde{T}_{p+1}$

In order to define the punctured boundary of $\widetilde{T}_{d+1}$ we recall
that an _end_ of an infinite unrooted tree is an equivalence class of infinite
geodesic rays (one-sided geodesics) in $\widetilde{T}_{d+1}$, where two rays
are called equivalent if they coincide starting from some point. The
collection of all ends of $\widetilde{T}_{d+1}$ forms the boundary
$\partial\widetilde{T}_{d+1}$. Among all ends of the tree we select one end,
denoted by $0^{-\infty}$ (or $\omega$ in most of the cases), that contains a
ray consisting of the roots of rooted trees $T_{d}^{(n)}$, $n\geq 0$. This is
the end corresponding to an infinite path going up in Figure 2. All other ends
of $\widetilde{T}_{d+1}$ correspond to rays going down as shown again in
Figure 2.

######  Definition 3.2.

The punctured boundary $\partial_{\omega}\widetilde{T}_{d+1}$ of the tree
$\widetilde{T}_{d+1}$ is the subset
$\partial\widetilde{T}_{d+1}\setminus\\{\omega\\}$ obtained from
$\partial\widetilde{T}_{d+1}$ by deleting a singleton $\omega$.

Any two ends of $\widetilde{T}_{d+1}$ can be joined by the unique bi-infinite
geodesic and conversely, any bi-infinite geodesic joins two ends of
$\widetilde{T}_{d+1}$. Thus, the punctured boundary
$\partial_{\omega}\widetilde{T}_{d+1}$ can be identified with the set of all
bi-infinite geodesics in $\widetilde{T}_{d+1}$ initiating at $\omega$ (by
analogy with the boundary of a rooted tree that consists of geodesic rays
emanating from the root of the tree). Under this identification an end in
$\partial_{\omega}\widetilde{T}_{d+1}$ corresponds to a bi-infinite geodesic
connecting $\omega$ to it. Two points of the punctured boundary, denoted by
$\xi$ and $\eta$ are shown in Figure 2.

Define a metric on $\partial_{\omega}\widetilde{T}_{d+1}$ in a similar way as
the metric was defined on the boundary of a rooted $d$-ary tree $T_{d}$.
Namely, for any two elements $\xi_{1},\xi_{2}$ of
$\partial_{\omega}\widetilde{T}_{d+1}$ (viewed as bi-infinite geodesics) the
distance $d_{\partial_{\omega}\widetilde{T}_{d+1}}(\xi_{1},\xi_{2})$ between
them is defined as

$d_{\partial_{\omega}\widetilde{T}_{d+1}}(\xi_{1},\xi_{2})=d^{-l+1},$ (3.3)

where $l$ is the smallest level where $\xi_{1}$ differs from $\xi_{2}$. Under
this metric $\partial_{\omega}\widetilde{T}_{d+1}$ becomes an ultrametric
space.

For a prime $p$, we now identify $\partial_{\omega}\widetilde{T}_{p+1}$ with
the field $\mathbb{Q}_{p}$ of $p$-adic numbers by producing an appropriate
labelling of its elements. Note that a similar identification can be made for
arbitrary $d\geq 2$ between $\partial_{\omega}\widetilde{T}_{d+1}$ with the
ring $\mathbb{Q}_{d}$ of $d$-adic numbers, but we will not consider this case
separately as all the technical details are identical.

First, there is a unique bi-infinite geodesic from
$\partial_{\omega}\widetilde{T}_{d+1}$ consisting of the roots of the trees
$T_{d}^{(n)}$, $n\geq 0$ and vertices $0^{m}$, $m>0$ in $T_{d}^{(0)}$, called
the _spine_. It is shown in Figure 2 as the path connecting $0^{-\infty}$ to
$0^{\infty}$. This element will represent zero in $\mathbb{Q}_{p}$.

By definition of $\partial_{\omega}\widetilde{T}_{p+1}$ for each
$\xi\in\partial_{\omega}\widetilde{T}_{p+1}$ there is $N\in\mathbb{Z}$ (not
unique) such that $\xi$ follows the spine outside $T_{p}^{(N)}$. In other
words, $\xi$ branches out of the spine inside $T_{p}^{(N)}$. We will then
label each $\xi$ as a bi-infinite word $\ldots
x_{-2}x_{-1}x_{0}.x_{1}x_{2}x_{3}\ldots$ over $X$, where $x_{i}=0$ for
$i\leq-N$ and $x_{i}$ is the $(i+N)$-th letter of the label of the restriction
of $\xi$ to $T_{d}^{(N)}$ for $i>-N$, and the “dot” symbol is placed between
the letters at positions 0 and 1. In other words, $x_{i}$ indicates what
happens with $\xi$ right below level $i$. For example, the labels of $\xi$ and
$\eta$ in Figure 2 are $\ldots 000.010\ldots$ and $\ldots 001.101\ldots$ that
we will usually abbreviate as $.010\ldots$ and $1.101\ldots$, respectively, by
omitting the beginning 0’s before the first nonzero digit or the dot symbol.
The spine of $\widetilde{T}_{d+1}$ is then labelled by $\ldots
000.000\ldots=0^{-\infty}.0^{\infty}$ or just by $.0^{\infty}$.

Under the above labelling, the distance formula (3.3) can be rewritten as

$d_{\partial_{\omega}\widetilde{T}_{d+1}}(\ldots
x_{-2}x_{-1}x_{0}.x_{1}x_{2}x_{3}\ldots,\ldots
y_{-2}y_{-1}y_{0}.y_{1}y_{2}y_{3}\ldots)=d^{-\min\\{i\in\mathbb{Z}\colon
x_{i}\neq y_{i}\\}+1}.$ (3.4)

Now define a natural bijection
$\varphi\colon\partial_{\omega}\widetilde{T}_{p+1}\to\mathbb{Q}_{p}$ by

$\varphi(\ldots
x_{-3}x_{-2}x_{-1}x_{0}.x_{1}x_{2}x_{3}\ldots)=\sum_{i=-\infty}^{\infty}x_{i}p^{i},$

which is an isometry by the definition of a $p$-adic metric on
$\mathbb{Q}_{p}$.

The subset

$A=\\{\ldots
x_{-2}x_{-1}x_{0}.x_{1}x_{2}x_{3}\ldots\in\partial_{\omega}\widetilde{T}_{p+1}\colon
x_{i}=0\ \text{for}\ i\leq 0\\}$

of $\partial_{\omega}\widetilde{T}_{p+1}$ is naturally isometric under
$\varphi$ to the ring $\mathbb{Z}_{p}$ of $p$-adic integers, and also to the
boundary $\partial T_{p}^{(0)}=\partial T_{p}$ of the rooted $p$-ary tree
supplied with the standard metric.

The identification $\varphi$ also gives rise to a natural labelling by
(equivalence classes of) pairs $(\xi,n)$,
$\xi\in\partial_{\omega}\widetilde{T}_{d+1}$ and $n\in\mathbb{Z}$, of the
vertices of $\widetilde{T}_{d+1}$. Indeed, every vertex of level
$n\in\mathbb{Z}$ can be identified (non-uniquely) with a pair $(\xi,n)$, where
$\xi$ is a bi-infinite word over $X$ with the dot symbol representing a bi-
infinite geodesic connecting $\omega$ to another end and passing through $v$.
Clearly, if $\xi^{\prime}$ coincides with $\xi$ at all positions $m\leq n$,
then $(\xi,n)$ and $(\xi^{\prime},n)$ correspond to the same vertex of
$\widetilde{T}_{d+1}$. This defines an equivalence relation on the set of all
such pairs
$\\{(\xi,n)\colon\xi\in\partial_{\omega}\widetilde{T}_{d+1},n\in\mathbb{Z}\\}$.
Then the equivalence class of a pair $(\xi,n)$, denoted by $[\xi]_{n}$,
uniquely identifies a vertex of $\widetilde{T}_{d+1}$. We denote the vertices
$[0^{-\infty}.0^{\infty}]_{n}$ of the spine by $0^{n}$, where
$n\in\mathbb{Z}$. Finally, we denote the root $0^{0}$ of $T^{(0)}_{d}$ by
$\Lambda$ and color it in the red color in Figures 2 and 3.

The group $\mathop{\rm Aut}\nolimits(\widetilde{T}_{p+1})$ of all
automorphisms of $\widetilde{T}_{p+1}$ is a locally compact group, whose
elements are either elliptic (fixing a vertex of $T_{p+1}$), inversions
(fixing an edge but not a vertex of $T_{d+1}$) or hyperbolic (the rest). For a
fixed end $\omega\in\partial_{\omega}\widetilde{T}_{p+1}$ let $\mathop{\rm
Aut}\nolimits(\widetilde{T}_{p+1})_{\omega}$ be a subgroup of $\mathop{\rm
Aut}\nolimits(\widetilde{T}_{p+1})$ consisting of automorphisms that preserve
an end $\omega$ of $\widetilde{T}_{p+1}$ (i.e., such automorphism
$g\in\mathop{\rm Aut}\nolimits(\widetilde{T}_{p+1})$ for which there is
$N,k\in\mathbb{Z}$ such that $g(0^{-n})=0^{-n+k}$ for all $n\geq N$). Let also
$\mathop{\rm Aut}\nolimits_{0}(\widetilde{T}_{p+1})_{\omega}$ be a subgroup of
$\mathop{\rm Aut}\nolimits(\widetilde{T}_{p+1})_{\omega}$ consisting of
automorphisms that eventually preserve the vertices of an end $\omega$ of
$\widetilde{T}_{p+1}$ (i.e., automorphisms $g\in\mathop{\rm
Aut}\nolimits(\widetilde{T}_{p+1})$ for which there is $N\in\mathbb{Z}$ such
that $g(0^{-n})=0^{-n}$ for all $n\geq N$).

The groups $\mathop{\rm Aut}\nolimits(\widetilde{T}_{p+1})_{\omega}$ and
$\mathop{\rm Aut}\nolimits_{0}(\widetilde{T}_{p+1})_{\omega}$ were studied
before in [25] under the names of _affine_ and _horocyclic_ groups of the tree
$\widetilde{T}_{d+1}$, respectively. The levels of $\widetilde{T}_{p+1}$ are
often called horocycles. In [9] the affine group of $\widetilde{T}_{d+1}$ is
defined as the group consisting of Busemann isometries. It is well known that
$\mathop{\rm Aut}\nolimits(\widetilde{T}_{p+1})_{\omega}$ is amenable as a
topological group (we again quote Nebbia [67] or [83, Lemma 12.14]). Also, we
note the group of automorphisms of a tree $D_{\overline{d}}$, which is the
induced subgraph in $\widetilde{T}_{d+1}$ spanned by all the vertices of non-
positive levels, was studied in [18] in the context of Sylow $p$-subgroups in
the finitary symmetric group $FS_{\mathbb{N}}$.

The subgroup $\mathop{\rm Aut}\nolimits(\widetilde{T}_{p+1})_{\omega}$ of
$\mathop{\rm Aut}\nolimits(\widetilde{T}_{p+1})$ consists only of elliptic and
hyperbolic elements. For an end $\omega=0^{-\infty}$, one particularly
important for us element of $\mathop{\rm
Aut}\nolimits(\widetilde{T}_{p+1})_{\omega}$ is a shift $\tau\in\mathop{\rm
Aut}\nolimits(\widetilde{T}_{p+1})_{\omega}$ that acts on each
$v\in\widetilde{T}_{d+1}$ by moving it in the direction of $O^{-\infty}$ as
shown in Figure 4. In terms of the action on the boundary
$\partial_{\omega}\widetilde{T}_{d+1}$, it moves the dot in the expression of
each element $\xi\in\partial_{\omega}\widetilde{T}_{d+1}$, viewed as a bi-
infinite string over $X$ that have $0^{-\infty}$ as a prefix, by one position
to the right, that is corresponds to the left shift map $s$ on
$X^{\mathbb{Z}}$:

$\theta(t)(\ldots x_{-2}x_{-1}x_{0}.x_{1}x_{2}x_{3}\ldots)=\ldots
x_{-1}x_{0}x_{1}.x_{2}x_{3}x_{4}\ldots$

The following theorem stated in the introduction connects the groups of
automorphisms of trees with the groups of isometries and dilations of local
fields and their rings of integers.

######  Theorem A.

Let $F$ be a (non-Archimedean) local field with the ring of integers
$\mathcal{O}$ and the residue field of size $q$, endowed with the metric
induced by its valuation. Then

  * (i)

the group $\mathop{\rm Isom}\nolimits(\mathcal{O})$ of isometries of
$\mathcal{O}$ is isomorphic to the group $\mathop{\rm Aut}\nolimits(T_{q})$ of
automorphisms of $T_{q}$.

  * (ii)

the group $\mathop{\rm Isom}\nolimits(F)$ of isometries of $F$ is isomorphic
to the group $\mathop{\rm Aut}\nolimits_{0}(\widetilde{T}_{q+1})_{\omega}$
consisting of automorphisms of $\widetilde{T}_{q+1}$ that fix a distinguished
end $\omega$ pointwise (fixing the vertices in $\omega$).

  * (iii)

the group $\mathcal{D}(F)$ of dilations of $F$ is isomorphic to the group
$\mathop{\rm Aut}\nolimits(\widetilde{T}_{q+1})_{\omega}$ consisting of
automorphisms of $\widetilde{T}_{q+1}$ that fix an end $\omega$ as a point of
the boundary of $\widetilde{T}_{q+1}$ (but possibly moving the vertices in
rays corresponding to $\omega$ along those rays while preserving $\omega$ as a
point of the boundary).

###### Proof.

Since $\mathcal{O}$ and $F$ are isometric as metric spaces to the boundaries
$\partial T_{q}$ and $\partial_{\omega}\widetilde{T}_{q+1}$ of trees $T_{q}$
and $\widetilde{T}_{q+1}$, their groups of isometries are isomorphic to the
groups of isometries of these boundaries. Therefore, we need to show that
these groups of isometries are isomorphic to the respective automorphism
groups.

For $T_{q}$ this is shown in Proposition 3.7 in [43]. For the case of
$\widetilde{T}_{q+1}$ the proof is similar. Clearly, every automorphism of
$\widetilde{T}_{q+1}$ fixing $0^{-\infty}$ induces an isometry of
$\partial_{\omega}\widetilde{T}_{q+1}$ as the action on the set of vertices of
$\widetilde{T}_{q+1}$ induces the action on the boundary. Conversely, suppose
$h\in\mathop{\rm Isom}\nolimits(\partial_{\omega}\widetilde{T}_{q+1})$ is an
isometry. We will extend the action of $h$ to $V(\widetilde{T}_{q+1})$ by
defining

$h([\xi]_{n})=[h(\xi)]_{n}$

for each vertex $[\xi]_{n}$ of $\widetilde{T}_{q+1}$. Since $h$ is an isometry
on $\partial_{\omega}\widetilde{T}_{q+1}$, this induced action on
$V(\widetilde{T}_{q+1})$ is an automorphism preserving the end $0^{-\infty}$
pointwise.

Finally, for the third item the proof is analogous with the only difference
that the dilations of $\partial_{\omega}\widetilde{T}_{q+1}$ will move all
vertices of $\widetilde{T}_{q+1}$ up or down depending on the dilation
coefficient. Suppose that $h\in D(F)$ is a dilation. Since the distances
between any two points in $F$ are powers of $q$, the dilation coefficient also
must be of the form $q^{m}$ for some $m\in\mathbb{Z}$ (that is,
$d(h(\xi),h(\xi^{\prime}))=q^{m}d(\xi,\xi^{\prime})$ for all
$\xi,\xi^{\prime}$ in $F$). Then we extend the action of $h$ to
$V(\widetilde{T}_{q+1})$ by defining

$h([\xi]_{n})=[h(\xi)]_{n+m}$

for each vertex $[\xi]_{n}$ of $\widetilde{T}_{q+1}$. ∎

Similarly to sections of automorphisms of $T_{d}$, we define the sections of
automorphisms from $\mathop{\rm
Aut}\nolimits_{0}(\widetilde{T}_{d+1})_{\omega}$ and $\mathop{\rm
Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$. However, with the lack of the
root in $\widetilde{T}_{d+1}$ the notation become a little more involved.
Suppose $g\in\mathop{\rm Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$ and
$v=[\xi]_{n}$ is a vertex of $\widetilde{T}_{d+1}$ that is moved by $g$ to a
vertex $w=[\eta]_{m}$. Then the rooted $d$-ary tree $T_{v}$ hanging down at
vertex $v$ in $\widetilde{T}_{d+1}$ is mapped bijectively to the tree $T_{w}$
hanging down at vertex $w$. Since both of this trees are canonically
isomorphic to $T_{d}$, this induces the unique automorphism $g|_{v}$ of
$T_{d}$ that we will call the _section of $g$ at $v$_. Formally, we define it
as follows. If $u\in X^{*}$ is a vertex of $T_{d}$, then
$g|_{v}(u)=u^{\prime}$ if and only if

$g\left(\bigl{[}[\xi]_{n}u0^{\infty}\bigr{]}_{n+|u|}\right)=\bigl{[}[\eta]_{m}u^{\prime}0^{\infty}\bigr{]}_{m+|u^{\prime}|},$

where $[\xi]_{n}u0^{\infty}$ denotes the bi-infinite word obtained from $\xi$
by replacing the tail of $\xi$ starting from position $n+1$ by $u0^{\infty}$,
and similarly for $[\eta]_{m}u^{\prime}0^{\infty}$.

Similarly to the definition of a projection homomorphism (2.4), for any vertex
$v\in V(\widetilde{T}_{d+1})$ we define a natural homomorphism

$\begin{array}[]{llll}\pi_{v}\colon&\mathop{\rm St}\nolimits_{\mathop{\rm
Aut}\nolimits(T(X))}(v)&\to&\mathop{\rm Aut}\nolimits(T(X))\\\
&g&\mapsto&g|_{v}\end{array}$ (3.5)

Since for each subgroup $G$ of $\mathop{\rm
Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$ we have $\mathop{\rm
St}\nolimits_{G}(v)<\mathop{\rm St}\nolimits_{\mathop{\rm
Aut}\nolimits(T(X))}(v)$, the homomorphism $\pi_{v}$ naturally restricts to a
homomorphism $\mathop{\rm St}\nolimits_{G}(v)\to\mathop{\rm
Aut}\nolimits(T(X))$.

Also, similarly to the case of rooted trees, we can visualize the action of an
automorphism $g\in\mathop{\rm Aut}\nolimits_{0}(\widetilde{T}_{d+1})_{\omega}$
using the notion of the _portrait_ of $g$, also denoted by $\mathcal{P}(g)$.
It is a labeled infinite unrooted $(|X|+1)$-ary tree with the vertex $\Lambda$
(the root of $T^{(0)}_{d}$) labeled by $g$ and each vertex $v$ labeled by
$g|_{v}$. Under each vertex $v$, we write the name of the mapping that
$g|_{v}$ defines on the first level of the subtree hanging from $v$. In the
case $d=2$, we draw an arc (called switch) connecting the two edges hanging
down from $v$ if $h|_{v}$ acts nontrivially on the first level of the subtree
hanging from $v$. If there is no switch, it means the action on the first
level is trivial. Examples of portraits corresponding to the generators of the
finitely presented group $\widetilde{\mathcal{G}}$ that will be discussed
below are shown in Figures 7 and 8.

###  3.3 Notational remarks

In [9] the notation $\mathrm{Aff}(\widetilde{T}_{d+1})$ is suggested for
$\mathop{\rm Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$ but we think that
the prefix “$\mathrm{Aff}$” may confuse the reader because of its connection
to linearity. The group $\mathrm{Aff}(F)$ of all affine transformations of the
field $F$, which is embedded in $\mathop{\rm
Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$ as a proper subgroup, is a linear
group satisfying the Tits alternative, and hence the exotic groups like groups
of Burnside type, of intermediate growth, or of branch type avoid it. At the
same time the group $\mathop{\rm Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$
is full of such exotic groups as explained in this article.

##  4 Liftable groups and actions of their HNN extensions on regular trees

In this section we embed certain ascending HNN extensions of liftable self-
similar groups into the group $\mathop{\rm
Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$.

###  4.1 Definition and main embedding result

######  Definition 4.1.

A self-similar group $G$ acting on a tree $T_{d}=T(X)$ is called _liftable_ if
there exists some $i\in X$ and a homomorphism $\sigma\colon G\to\mathop{\rm
St}\nolimits_{G}(i)$, called the _lifting_ , that is the right inverse of the
projection map $\pi_{i}$ defined by (2.4) (i.e., such that
$\pi_{i}\circ\sigma$ is the identity on $G$).

Note that it follows immediately from the definition that the lifting $\sigma$
must be a monomorphism.

An equivalent definition of a liftability of a group is that for some $i\in
X=\\{0,1,\ldots,d-1\\}$ the sequence

$1\to\ker\pi_{i}\hookrightarrow\mathop{\rm
St}\nolimits_{G}(i)\stackrel{{\scriptstyle\pi_{i}}}{{\twoheadrightarrow}}G\to
1$

is a short exact sequence defining a split extension, and hence the stabilizer
$\mathop{\rm St}\nolimits_{G}(i)$ of vertex $i$ decomposes as a semidirect
product:

$\mathop{\rm St}\nolimits_{G}(i)=\ker\pi_{i}\rtimes G.$

It follows from the definition of a liftable group that the projection map
$\pi_{i}\colon\mathop{\rm St}\nolimits_{G}(i)\to G$ must be surjective. This
is always the case when $G$ is self-replicating. However, the fact that
$\pi_{i}$ is onto for some $i$ does not automatically imply the existence of a
lifting. On the other hand we do not know of any example of a self-replicating
non-liftable group.

######  Question 4.2.

Is there a self-replicating non-liftable group?

We proceed to the main theorem of this section stated in the introduction.
Recall that $\Lambda=0^{0}$ denotes the leftmost vertex of level 0 in
$\widetilde{T}_{d+1}$ that is also the root of the tree $T_{d}^{(0)}$.

######  Theorem B.

Let $G$ be a liftable group acting on $T_{d}$ and $\sigma\colon
G\to\mathop{\rm St}\nolimits_{G}(i)$ be the corresponding lifting. Then

  * (i)

There is an embedding $\theta$ of the ascending HNN extension $\widetilde{G}$
of $G$ by $\sigma$

$\widetilde{G}=\langle G,t\mid\text{relations in}\ G,tgt^{-1}=\sigma(g)\
\text{for all}\ g\in G\rangle$

into $\mathop{\rm Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$ for an end
$\omega$ of $\widetilde{T}_{d+1}$.

  * (ii)

If $G$ acts transitively on the first level of $T_{d}$, then $G$ is self-
replicating, $\theta(\widetilde{G})$ acts transitively on the set of vertices
of $\widetilde{T}_{d+1}$, and the closure $W$ of $\theta(\widetilde{G})$ in
$\mathop{\rm Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$ is a scale group.

  * (iii)

The projection $\pi_{\Lambda}$ maps $\mathop{\rm
St}\nolimits_{\theta(\widetilde{G})}(\Lambda)$ onto $G$ and $\mathop{\rm
St}\nolimits_{\theta(\widetilde{G})}(\Lambda)\cong\bigl{(}\ker\pi_{\Lambda}\cap\theta(\widetilde{G})\bigr{)}\rtimes\theta(G)$.

  * (iv)

The projection $\pi_{\Lambda}$ maps $\mathop{\rm St}\nolimits_{W}(\Lambda)$
onto $\overline{G}$ and $\mathop{\rm
St}\nolimits_{W}(\Lambda)\cong\bigl{(}\ker\pi_{\Lambda}\cap
W\bigr{)}\rtimes\theta(\overline{G})$.

###### Proof.

By relabelling of $X$ we may assume that $X=\\{0,1,\ldots,d-1\\}$ for $d=|X|$
and that $i=0$. We will then set $\omega=0^{-\infty}$ and build an embedding
$\theta\colon\widetilde{G}\to\mathop{\rm
Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$ as follows. Given $g\in G$ we
define $\theta(g)\in\mathop{\rm
Aut}\nolimits_{0}(\widetilde{T}_{d+1})_{\omega}$ so that its section at vertex
$0^{-n}$ is $\sigma^{n}(g)$ for $n\geq 0$, as shown in Figure 3. This is a
well-defined embedding of $G$ into $\mathop{\rm
Aut}\nolimits_{0}(\widetilde{T}_{d+1})_{\omega}$ (which we will denote by
$\theta|_{G}$ for now and extend to $\theta\colon\widetilde{G}\to\mathop{\rm
Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$ below) as follows from the fact
that $\sigma$ is a monomorphism with the image inside $\mathop{\rm
St}\nolimits_{G}(0)$ and such that $\pi_{0}\circ\sigma$ is the identity
endomorphism on $G$.

Figure 3: Embedding of a self-similar group into $\mathop{\rm
Aut}\nolimits_{0}(\widetilde{T}_{d+1})_{\omega}$

Alternatively, it can be explained in the following way. The map
$\theta|_{G}\colon G\to\mathop{\rm
Aut}\nolimits_{0}(\widetilde{T}_{d+1})_{\omega}$ induces a map

$\begin{array}[]{llll}\phi\colon&G&\to&\prod_{-\infty}^{0}G\\\\[2.84526pt]
&g&\mapsto&\bigl{(}\ldots,\sigma^{n}(g),\ldots,\sigma(g),g\bigr{)},\end{array}$

where the component of $\phi(g)$ at coordinate $-i$ is the section of
$\theta(g)$ at vertex $0^{-i}$. Since $\theta|_{G}$ is a monomorphism and
$\sigma(G)<\mathop{\rm St}\nolimits_{G}(0)$, the map $\phi$ is also a
monomorphism. Clearly $\theta(G)\cong\phi(G)\cong G$.

The element $\theta(t)$ is defined as a hyperbolic automorphism
$\tau\in\mathop{\rm Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$ that
represents the right shift along the geodesic ray
$\alpha=(0^{-\infty},0^{\infty})$ defined in Section 3. Its action on the
$\widetilde{T}_{p+1}$ is shown in Figure 4.

Figure 4: Embedding of the stable letter of the HNN extension $\widetilde{G}$
into $\mathop{\rm Aut}\nolimits_{0}(\widetilde{T}_{d+1})_{\omega}$

We claim that so defined $\theta$ extends to a monomorphism from
$\widetilde{G}$ to $\mathop{\rm Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$.
To justify that $\theta$ is a homomorphism it is enough to check that the
family of relations $tgt^{-1}=\sigma(g)$, $g\in G$ agrees with $\theta$:

$\theta(t)\theta(g)\theta(t)^{-1}=\theta(\sigma(g))$

or, equivalently,

$\phi(t)\phi(g)\phi(t)^{-1}=\phi(\sigma(g)).$

By constructions of $\theta(g)$ we have

$\phi(\sigma(g))=\bigl{(}\ldots,\sigma^{n+1}(g),\ldots,\sigma^{2}(g),\sigma(g)\bigr{)}.$

On the other hand, the portrait of $\theta(t)\theta(g)\theta(t)^{-1}$ is
obtained from the portrait of $\theta(g)$ by shifting all the labels towards
the end $0^{-\infty}$, thus

$\phi(t)\phi(g)\phi(t)^{-1}=\bigl{(}\ldots,\sigma^{n+1}(g),\ldots,\sigma^{2}(g),\sigma(g)\bigr{)}$

as well, so $\theta$ is a homomorphism.

Now we claim that $\theta$ is injective. Let $H=\theta(G)\cong G$ and
$\eta\colon H\to\theta(\widetilde{G})$ be a conjugation monomorphism

$\eta(h)=\theta(t)h\theta(t)^{-1}.$

We claim that the image of $\eta$ is in $H$. Indeed, if $h=\theta(g)\in H$
then

$\eta(h)=\theta(t)\theta(g)\theta(t)^{-1}=\theta(tgt^{-1})=\theta(\sigma(g))\in
H$

since $\sigma(g)\in G$. So $\eta\colon H\to H$ is a monomorphism (since
$\sigma$ and $\theta|_{G}$ are monomorphisms) and the diagram

${G}$${G}$${H}$${H}$$\scriptstyle{\sigma}$$\scriptstyle{\theta|_{G}}$$\scriptstyle{\theta|_{G}}$$\scriptstyle{\eta}$

is commutative.

Hence, $\widetilde{H}=\langle H,\theta(t)\rangle=\langle
H,\theta(t)\mid\theta(t)h\theta(t)^{-1}=\eta(h),h\in H\rangle$ is an ascending
HNN extension isomorphic to $\widetilde{G}$ and the embedding
$\theta\colon\widetilde{G}\to\mathop{\rm
Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$ is justified, which completes the
proof of item (i).

To prove the transitivity claim from item (ii), we use the well-known (and
easily proved) fact that if a self-similar group $G$ satisfies
$\pi_{i}(\mathop{\rm St}\nolimits_{G}(i))=G$ for some vertex $i$ of the first
level, and $G$ acts transitively on the first level, then $G$ is self-
replicating group and it acts transitively on all levels of the tree.

Given two vertices $u,v\in V(\widetilde{T}_{d+1})$ we can find
$i,j\in\mathbb{Z}$ such that for $\tau=\theta(t)$: $\tau^{i}(u)$ and
$\tau^{j}(v)$ are vertices at the same level of $T_{d}^{(0)}$. Then, using
level transitivity of $G$ (and hence of $\theta(G)$ on levels of
$T_{d}^{(0)}$), we can map $\tau^{i}(u)$ to $\tau^{j}(v)$ by an element of
$\theta(G)$.

Now the fact that the closure $\overline{\theta(\widetilde{G})}$ in
$\mathop{\rm Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$ is a scale group
fixing $\omega=0^{-\infty}$ is obvious. Moreover,
$\overline{\theta(\widetilde{G})}$ has the structure of the semidirect product
$\overline{\theta(G)}\rtimes\langle\tau\rangle$.

For item (iii) we observe that $\theta(G)<\mathop{\rm
St}\nolimits_{\theta(\widetilde{G})}(\Lambda)$ and
$\pi_{\Lambda}(\theta(g))=g$ for all $g\in G$, so $G<\pi_{\Lambda}(\mathop{\rm
St}\nolimits_{\theta(\widetilde{G})}(\Lambda))$. On the other hand, since all
sections of $\theta(t)$ are trivial, all sections of elements of
$\theta(\widetilde{G})$ at all vertices are in $G$, therefore
$\pi_{\Lambda}(\mathop{\rm St}\nolimits_{\theta(\widetilde{G})}(\Lambda))<G$.

To justify the semidirect product structure of $\mathop{\rm
St}\nolimits_{\theta(\widetilde{G})}(\Lambda)$ we note that since the action
of $\theta(G)$ on the subtree $T_{\Lambda}$ is faithful (because it coincides
with the action of $G$ on $T_{d}$),

$\theta(G)\cap\bigl{(}\ker\pi_{\Lambda}\cap\theta(\widetilde{G})\bigr{)}=\\{1\\},$
(4.1)

On the other hand,

$\mathop{\rm
St}\nolimits_{\theta(\widetilde{G})}(\Lambda)=\bigl{(}\ker\pi_{\Lambda}\cap\theta(\widetilde{G})\bigr{)}\cdot\theta(G).$
(4.2)

Indeed, since all sections of the shift $\theta(t)$ are trivial, for each
$s\in\mathop{\rm St}\nolimits_{\theta(\widetilde{G})}(\Lambda)$ we have
$g:=s|_{\Lambda}\in G$, so
$h=s\cdot\theta(g)^{-1}\in\ker\pi_{\Lambda}\cap\theta(\widetilde{G})$.
Therefore,
$s=h\cdot\theta(g)\in\bigl{(}\ker\pi_{\Lambda}\cap\theta(\widetilde{G})\bigr{)}\cdot\theta(G)$.
Equations (4.1) and (4.2) yield that $\mathop{\rm
St}\nolimits_{\theta(\widetilde{G})}(\Lambda)=\ker\pi_{\Lambda}\rtimes\theta(G)$.

Item (iv) is proved analogously to item (iii).

∎

###  4.2 Connection to the results of Willis on scale groups

Here we briefly discuss the relation between Theorem B and recent results of
George Willis, who noticed a connection between the classes of self-
replicating and scale groups [82]. Willis showed in Proposition 3.8 that,
given a scale group acting on $\mathop{\rm
Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$, the stabilizers of vertices of
$\widetilde{T}_{d+1}$ in this group project to the same subgroup of $T_{d}$
that is self-replicating. Conversely, in Proposition 3.10 he proved that given
a closed self-replicating group $G$ acting on $T_{d}$, there is a scale group
$\widetilde{G}<\mathop{\rm Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$ such
that $\pi_{\Lambda}(\mathop{\rm St}\nolimits_{\widetilde{G}}(\Lambda))=G$. The
group $\widetilde{G}$ is constructed as a subgroup of $\mathop{\rm
Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$ consisting of all automorphisms
of $\widetilde{T}_{d+1}$ whose sections at all vertices of that tree belong to
$G$. This part of the article does not address the question of when one can
embed a closed self-replicating group into a scale groups, but realizes the
former as a section of the latter. Theorem B, on the other hand, provides
explicit way to embed many self-replicating groups (not necessarily closed)
and their HNN extensions into scale groups under condition of liftability.
Moreover, it yields many examples of countable discrete transitive subgroups
of $\mathop{\rm Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$, whose closures
are scale groups, as described in Section 5 below.

##  5 Examples of liftable groups

As was mentioned in the introduction, there are several main sources of groups
satisfying the conditions of Theorem B.

###  5.1 Liftable groups coming from essentially free actions

The first class is the class of self-replicating groups acting _essentially
freely_ on the boundary of $T_{d}$ and the second type are the groups with a
suitable $L$-presentation. Recall that the action of a countable group $G$ on
a probability measure space $(X,\mu)$ by measure preserving transformations is
called essentially free if the measure of a set of points in $X$ that have
non-trivial stabilizers is equal to 0 (or, equivalently, for each $g\in
G,g\neq 1$, $\mu(\mathrm{Fix}(g))=0$, where $\mathrm{Fix}(g)$ is the set of
fixed points of element $g$). In our case $X=\partial T_{d}$ and $\mu$ is the
uniform Bernoulli measure defined in Section 2. The groups acting essentially
freely on the boundaries of rooted trees have been studied in [41, 11, 58, 47,
70]. Two examples in this class are the lamplighter group
$\mathcal{L}=\mathbb{Z}_{2}\wr\mathbb{Z}$ and the Baumslag-Solitar group
$BS(1,3)$ (see [47]) that are generated by automata shown in the left and
right sides of Figure 5 respectfully. We study them in detail below.

Figure 5: The automata generating the lamplighter group (left) and Baumslag-
Solitar group $BS(1,3)$ (right)

######  Proposition 5.1.

If a group $G<\mathop{\rm Aut}\nolimits(T_{d})$ acts on $\partial T_{d}$
essentially freely and is self-replicating, then it is liftable.

###### Proof.

We identify $T_{d}$ with $T(X)$ for an alphabet $X=\\{0,1,\ldots,d-1\\}$.
Since $G$ acts essentially freely, the projection homomorphism
$\pi_{0}\colon\mathop{\rm St}\nolimits_{G}(0)\to G$ for a letter $0\in X$ must
be injective. Indeed, if there is a $g\in\mathop{\rm St}\nolimits_{G}(0)$,
$g\neq 1$, such that $g|_{0}$ is trivial, then this $g$ would be in the
stabilizers of all points from $\partial T_{d}$ that start with 0. Clearly,
the set of such points has a non-zero measure, thus contradicting to the fact
that the action of $G$ is essentially free. Therefore, since $G$ is self-
replicating and, hence, $\pi_{0}(\mathop{\rm St}\nolimits_{G}(0))=G$, we can
uniquely define the right inverse $\sigma$ of the projection $\pi_{0}$ that
will be a monomorphism as required by Theorem B. ∎

######  Corollary 5.2.

Suppose a group $G<\mathop{\rm Aut}\nolimits(T_{d})$ acts on $\partial T_{d}$
essentially freely and is self-replicating. Then the ascending HNN extension
of $G$ given by the lifting from Proposition 5.1 embeds into the group
$\mathop{\rm Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$,
$\omega=0^{-\infty}$ with the image acting transitively on the set of vertices
of $\widetilde{T}_{d+1})$.

There is an interesting connection here to the class of scale-invariant
groups. We recall the definition below.

######  Definition 5.3.

A finitely generated group $G$ is called _B-scale-invariant_ if there exists a
nested sequence of subgroups of finite index in $G$, each isomorphic to $G$,
and whose intersection is a finite group. If the latter intersection is
trivial, then $G$ is called _scale-invariant_.

This class was introduced by Benjamini (this is why we add “B” in front of
“scale-invariant”) who conjectured that every such group is virtually
nilpotent. A counterexample to this conjecture, the lamplighter group
$\mathcal{L}$ discussed in Examples 5.4 and 6.3, was provided implicitly in
[53] (the paper was printed before the conjecture was stated) and explicitly
in [70], where many other examples where produced. It was shown in [47,
Proposition 3.13] that every self-replicating group acting essentially freely
on $\partial T_{d}$ is scale-invariant. Therefore, by Proposition 5.1, there
are examples of liftable scale-invariant groups, some of which are given
below.

######  Example 5.4.

Lamplighter group $\mathcal{L}\cong\mathbb{Z}_{2}\wr\mathbb{Z}$ is a
2-generated group that can be realized as a self-similar group defined by the
following wreath recursion corresponding to the automaton representation given
by automaton on the left side of Figure 5 [43, 53]:

$\begin{array}[]{rcl}a&=&(a,b)\varepsilon\\\ b&=&(a,b)\end{array}$

The self-replicating action of $\mathcal{L}$ on $T_{2}$ has been studied in
[53, 70, 33]. In particular, it was shown in [47] that this group acts
essentially freely on $\partial T_{2}$ so $\mathcal{L}$ satisfies the
conditions of Proposition 5.1 and there exist monomorphisms
$\sigma_{0}\colon\mathcal{L}\to\mathop{\rm St}\nolimits_{\mathcal{L}}(0)$ and
$\sigma_{1}\colon\mathcal{L}\to\mathop{\rm St}\nolimits_{\mathcal{L}}(1)$
given by:

$\sigma_{0}\colon\left\\{\begin{array}[]{l}a\mapsto b\\\ b\mapsto
b^{a}\end{array}\right.\qquad\qquad\sigma_{1}\colon\left\\{\begin{array}[]{l}a\mapsto
b^{a}\\\ b\mapsto b\end{array}\right.$ (5.1)

with $\pi_{i}\sigma_{i}=id_{\mathcal{L}}$, $i=0,1$, because $b=(a,b)$ and
$b^{a}=(b,a)$. In other words, both $\sigma_{0}$ and $\sigma_{1}$ are liftings
of $\mathcal{L}$. Hence, we can apply Theorem B and embed corresponding HNN
extensions in $\mathop{\rm
Aut}\nolimits(\widetilde{T}_{3})_{\omega}\cong\mathcal{D}(\mathbb{Q}_{2})$.

The lamplighter group will appear again in the last section of the paper.

######  Example 5.5.

The Baumslag-Solitar group $BS(1,3)=\langle s,r\mid rsr^{-1}=s^{3}\rangle$ can
be realized as a 3-generated self-similar group defined by the following
wreath recursion corresponding to the automaton representation given by
automaton on the right side of Figure 5 (see, for example, [11, 20]):

$\begin{array}[]{rcl}a&=&(c,b)\varepsilon\\\ b&=&(a,c)\\\
c&=&(b,a)\end{array}$

With respect to this generating set it has the following presentation:

$BS(1,3)\cong\langle a,b,c\mid c=ab^{-1}a,bab^{-1}=ab^{-1}ab^{-1}a\rangle,$

where the isomorphism between the two presentations given above is induced by
$s\mapsto ab^{-1}$ and $r\mapsto b$ (see [20], where the automaton groups act
on the left and thus all words in generators have to be reversed to align with
the notation of the present paper).

It acts essentially freely on $\partial T_{2}$ [47], is self-replicating, and
satisfies the conditions of Proposition 5.1. The monomorphism $\sigma$ in this
case is

$\sigma\colon\left\\{\begin{array}[]{l}a\mapsto b\\\ b\mapsto c\\\ c\mapsto
bc^{-1}b\end{array}\right..$

since $b=(a,c)$, $c=(b,a)$, and $bc^{-1}b=(ab^{-1}a,ca^{-1}c)=(c,ca^{-1}c)$.

We note that there are self-similar groups acting on $\partial T_{d}$
essentially freely that are not self-replicating. These examples include the
free group $F_{3}$ and the free product
$(\mathbb{Z}/2\mathbb{Z})*(\mathbb{Z}/2\mathbb{Z})*(\mathbb{Z}/2\mathbb{Z})$
[47, 20]. Since the stabilizer of any vertex projects onto a proper subgroup
of $G$, so no lifting could exist. We finish this section with the following
question.

######  Question 5.6.

Are there non-amenable liftable groups?

###  5.2 Liftable groups admitting finite $L$-presentations

Now let us switch to groups admitting $L$-presentations. We refer the reader
to [3] for more details on this type of presentations by generators and
relators. We give here somewhat restricted definition compared to the one
given in [3] that will be useful in our context.

######  Definition 5.7.

A group $G$ admits an _$L$ -presentation_ if there is a generating set $S$ for
$G$, subsets $Q$ and $R$ of a free group $F_{S}$ on $S$ and an endomorphism
$\phi\colon F_{S}\to F_{S}$ such that

$G\cong\left\langle S\mid Q,\phi^{n}(R),n\geq 0\right\rangle.$ (5.2)

An $L$-presentation (5.2) is called _finite_ if $S,Q,R$ are finite and
_ascending_ if $Q$ is empty. In the case of ascending $L$-presentation the
endomorphism $\phi$ induces the endomorphism $\bar{\phi}\colon G\to G$. We
call $L$-presentation (5.2) _injective_ if it is ascending and $\bar{\phi}$ is
injective. In this case we can associate with $G$ an ascending HNN extension

$\widetilde{G}=G*_{\bar{\phi}}=\langle S,t\mid R,s^{t}=\phi(s),s\in S\rangle.$

If a self-similar group $G$ acting on $T(X)$ admits an injective
$L$-presentation

$G\cong\left\langle S\mid\phi^{n}(R),n\geq 0\right\rangle,$ (5.3)

such that $\mathrm{Im}\bar{\phi}\subset\mathop{\rm St}\nolimits_{G}(i)$ for
some $i\in X$ and $\bar{\phi}$ is a lifting according to Definition 4.1, we
call presentation (5.3) _liftable_. Observe that if (5.3) is liftable and
finite then $\widetilde{G}$ is a finitely presented group.

As the main example of this type we now explicitly build an embedding of the
finitely presented group $\widetilde{\mathcal{G}}$ from [37] into
$\mathcal{D}(\mathbb{Q}_{2})$ and study in detail some of its properties.

######  Example 5.8.

(Finitely presented amenable but not elementary amenable group
$\widetilde{\mathcal{G}}$).

Recall that the group $\mathcal{G}=\langle a,b,c,d\rangle$ was defined in [34]
as an example of an infinite finitely generated torsion group. Later it was
shown to be the first example of a group of intermediate growth in [35] and
that it has many other unusual properties [44] like being branch and just-
infinite. It acts on the rooted binary tree $T_{2}$ and can be generated by a
Mealy automaton shown in Figure 1 or, equivalently, by the following wreath
recursion:

$\begin{array}[]{rcl}a&=&(1,1)\varepsilon,\\\ b&=&(a,c),\\\ c&=&(a,d),\\\
d&=&(1,b),\end{array}$ (5.4)

where $\varepsilon$ denotes the nontrivial permutation of $X=\\{0,1\\}$.

The group $\mathcal{G}$ has a finite $L$-presentation (1.1), which formally is
not ascending, but has an advantage of being minimal, i.e., no relators from
(1.1) can be dropped without changing $\mathcal{G}$ [38]. A slight
modification of (1.1) to

$\mathcal{G}=\langle
a,b,c,d\mid\sigma^{i}(a^{2}),\sigma^{i}(b^{2}),\sigma^{i}(c^{2}),\sigma^{i}(d^{2}),\sigma^{i}(bcd),\sigma^{i}((ad)^{4}),\sigma^{i}((adacac)^{4}),i\geq
0\rangle$ (5.5)

allows to associate with $\mathcal{G}$ a finite ascending $L$-presentation,
which is liftable with the lifting $\sigma\colon\mathcal{G}\to\mathop{\rm
St}\nolimits_{G}(1)$ induced by substitution (1.2) as $\sigma$ is the right
inverse of the projection map $\pi_{1}$. This follows from the wreath
recursion (5.4) and a relation $aca=(d,a)$. Then the corresponding HNN
extension $\widetilde{\mathcal{G}}$ is a finitely presented group given by
presentation (1.3) (or by a more sophisticated but shorter presentation
(1.4)). By Theorem B the group $\widetilde{\mathcal{G}}$ embeds into
$\mathop{\rm Aut}\nolimits(\widetilde{T}_{3})_{\omega}$ as a vertex transitive
group. Below we will describe in detail the images under the embedding
$\theta$ of the generators $a,b,c,d$ of $\mathcal{G}$ in $\mathop{\rm
Aut}\nolimits_{0}(\widetilde{T}_{3})$. Since $\sigma$ is the right inverse of
$\pi_{1}$, and not of $\pi_{0}$, we will construct the action on the tree
isomorphic to $\widetilde{T}_{3}$ for which the end $1^{-\infty}$ is fixed (so
the tree “grows up to the left” as shown in Figure 7). We will also denote
this tree $\widetilde{T}_{3}$, but one has to keep in mind the difference.
With this convention, in particular, $\theta(t)$ will move vertices of
$\widetilde{T}_{3}$ along the end $1^{-\infty}$ up and to the left.

Figure 6: Portraits of the generators of the Grigorchuk group

The action of the generators $a,b,c,d$ of $\mathcal{G}$ on $T_{2}$ is given by
the portraits shown in Figure 6. Recall that portraits for elements of
$\mathop{\rm Aut}\nolimits(T_{d})$ were defined in Section 2 and for elements
of $\mathop{\rm Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$ in Section 3. For
the generators $b,c,d$ the portraits are 3-periodic. Hence for $\sigma$ we get
the following:

$\sigma^{n}(b)=\left\\{\begin{array}[]{l}b,\ n\equiv 0\ (\mathrm{mod}\ 3),\\\
d,\ n\equiv 1\ (\mathrm{mod}\ 3),\\\ c,\ n\equiv 2\ (\mathrm{mod}\
3),\end{array}\right.\sigma^{n}(c)=\left\\{\begin{array}[]{l}c,\ n\equiv 0\
(\mathrm{mod}\ 3),\\\ b,\ n\equiv 1\ (\mathrm{mod}\ 3),\\\ d,\ n\equiv 2\
(\mathrm{mod}\
3),\end{array}\right.\sigma^{n}(d)=\left\\{\begin{array}[]{l}d,\ n\equiv 0\
(\mathrm{mod}\ 3),\\\ c,\ n\equiv 1\ (\mathrm{mod}\ 3),\\\ b,\ n\equiv 2\
(\mathrm{mod}\ 3).\end{array}\right.$

Therefore, the portraits of $\theta(b)$, $\theta(c)$, and $\theta(c)$ on
$\widetilde{T}_{3}$ will be periodic extensions upwards of the portraits of
$b$, $c$, and $d$ on $T_{2}$, as shown in Figure 7.

Figure 7: Portraits of the $\theta(b)$, $\theta(c)$, and $\theta(d)$ on
$\widetilde{T}_{3}$

To describe $\theta(a)$ we calculate the iterations $\sigma^{n}(a)$.

######  Proposition 5.9 ([46]).

For each $n\geq 0$ we have $\sigma^{n}(a)=P_{n}$, where the sequence
$\\{P_{n}\\}_{n\geq 0}$ consists of palindromic words defined recursively as
follows:

$P_{0}=a,\quad P_{n+1}=P_{n}q_{n}P_{n},n\geq 1,$

and

$q_{n}=\left\\{\begin{array}[]{l}c,\ n\equiv 0\ (\mathrm{mod}\ 3),\\\ b,\
n\equiv 1\ (\mathrm{mod}\ 3),\\\ d,\ n\equiv 2\ (\mathrm{mod}\
3).\end{array}\right.$

###### Proof.

Follows by induction on $n$. Clearly the base cases $n=0$ and $n=1$ are
trivially satisfied as $\sigma(a)=aca=P_{0}q_{0}P_{0}$. Now assume that $n>1$
and $\sigma^{i}(a)=P_{i}$ for all $i\leq n$. Then

$P_{n+1}=\sigma^{n}(a)=\sigma(P_{n})=\sigma(P_{n-1}q_{n-1}P_{n-1})=\sigma(P_{n-1})\sigma(q_{n-1})\sigma(P_{n-1})=P_{n}q_{n}P_{n},$

where the last equality follows from the induction assumption and the
definition of $q_{n}$. ∎

Observe that the sequence of palindromes $P_{n}$ play an important role in the
study of spectra of Schreier graphs associated with $\mathcal{G}$ [46].

We complete the description of the portrait of $\theta(a)$ by describing the
sections of $\sigma^{n}(a)$ at vertex 0 of the first level of the rooted
binary tree $T_{2}$ for all $n\geq 1$. Note, that these are also the sections
of $\theta(a)$ at vertices $1^{-n}0$ of the unrooted ternary tree
$\widetilde{T}_{3}$. For $n=1$ we have

$\sigma(a)|_{0}=(aca)|_{0}=d.$

Define an endomorphisms $\alpha_{n}\colon\mathcal{G}\to\mathcal{G},n\geq 1$ as
$\alpha_{n}=\pi_{0}\circ\sigma^{n}$, that is, for each $g\in\mathcal{G}$
$\alpha_{n}(g)=\sigma^{n}(g)|_{0}$. Visually, applying $\sigma^{n}$ to an
element $g\in\mathcal{G}$ corresponds to going $n$ steps in the North-West
direction in the portrait of $\theta(g)$ (along the rightmost bi-infinite
path) from the vertex $\Lambda$ labelled by $g$. Applying $\alpha_{n}$
corresponds to going $n$ steps in the North-West direction, and then one step
down in the South-West direction. So in the notation of Proposition 5.9 we
have

$\alpha_{n}(a)=\sigma^{n}(a)|_{0}=\sigma(\sigma^{n-1}(a))|_{0}=\sigma(P_{n-1})|_{0}=\alpha_{1}(P_{n-1}).$
(5.6)

######  Proposition 5.10.

For each $n\geq 1$ we have

$\alpha_{n}(a)=\left\\{\begin{array}[]{l}d,\ n=1,\\\ dad,\ n=2,\\\ 1,\ n\equiv
0\ (\mathrm{mod}\ 3),n\geq 3\\\ a,\ n\equiv 1\ (\mathrm{mod}\ 3)\ \text{or}\
n\equiv 2\ (\mathrm{mod}\ 3),n\geq 3.\end{array}\right.$

###### Proof.

Induction on $n$. For $n\in\\{1,2,3\\}$ we have

$\alpha_{1}(a)=\sigma(a)|_{0}=(aca)|_{0}=d,$
$\alpha_{2}(a)=\sigma^{2}(a)|_{0}=(aca\cdot b\cdot aca)|_{0}=dad,$
$\alpha_{3}(a)=\sigma^{3}(a)|_{0}=(acabaca\cdot d\cdot acabaca)|_{0}=dad\cdot
1\cdot dad=1,$

Now assume that the statement is true for some $n\geq 3$. Then we calculate,
using equality (5.6):

$\alpha_{n+1}(a)=\alpha_{1}(P_{n})=\alpha_{1}(P_{n-1}q_{n-1}P_{n-1})=\alpha_{1}(P_{n-1})\alpha_{1}(q_{n-1})\alpha_{1}(P_{n-1})=\alpha_{n}(a)\alpha_{1}(q_{n-1})\alpha_{n}(a).$

Now, considering three cases for $n$ $\mathrm{mod}\ 3$, we use the inductive
assumption to find the values of $\alpha_{n}(a)\in\\{1,a\\}$ and Proposition
5.9 to find $q_{n-1}\in\\{b,c,d\\}$. Finally, evaluating $\alpha_{1}(b)=1$,
$\alpha_{1}(c)=a$, $\alpha_{1}(d)=a$ and using the fact that both $a$ and $d$
are involutions completes the proof. ∎

Understanding of $\alpha_{n}(a)$ helps us to describe the portrait of
$\theta(a)$. Namely, the portrait of $\theta(a)$ on $\widetilde{T}_{3}$ is
shown in Figure 8. The original binary rooted tree on which the group
$\mathcal{G}$ acts is marked in the end of the right branch by an arc. As
desired by construction, $\theta(a)$ acts as $a$ on that subtree.

Figure 8: Portrait of the $\theta(a)$ on $\widetilde{T}_{3}$

######  Example 5.11.

(Finitely presented extension of $\mathop{\rm IMG}\nolimits(z^{2}+i)$).
Another example of the branch type group for which Theorem B is applicable is
the iterated monodromy group $\Gamma=\mathop{\rm IMG}\nolimits(z^{2}+i)$
(detailed exposition of iterated monodromy groups can be found in [69, 4]).
This group was studied by K.-U. Bux and R. Perez in [23], who proved that
$\Gamma$ (as well as $\mathcal{G}$) has intermediate growth, and Z. Šuníc and
the authors studied in [49] its spectral properties. It is generated by a
4-state automaton acting on a binary tree shown in Figure 9. The corresponding
wreath recursion defining $\Gamma$ is

$a=(1,1)\varepsilon,\quad b=(a,c),\quad c=(b,1),$ (5.7)

and $\Gamma$ has the following finite ascending $L$-presentation [49]:

$\Gamma\cong\bigl{\langle}a,b,c\bigm{|}\sigma^{n}(a^{2}),\sigma^{n}\bigl{(}(ac)^{4}\bigr{)},\sigma^{n}([c,ab]^{2}),\sigma^{n}([c,bab]^{2}),\\\
\sigma^{n}([c,ababa]^{2}),\sigma^{n}([c,ababab]^{2}),\sigma^{n}([c,bababab]^{2}),n\geq
0\bigr{\rangle},$ (5.8)

where $\sigma$ is the substitution defined by

$\sigma\colon\begin{cases}a\to b,\\\ b\to c,\\\ c\to aba.\\\ \end{cases}$

It extends to a monomorphism $\sigma\colon\Gamma\to\mathop{\rm
St}\nolimits_{\Gamma}(1)$ which is a lifting for $\Gamma$: wreath recursion
(5.7) and the relation $aba=(c,a)$ make $\sigma$ a right inverse of $\pi_{0}$.
This makes the presentation (5.8) liftable. Thus, by Theorem B the HNN
extension $\widetilde{\Gamma}$ of $\Gamma$ via the homomorphism $\sigma$ is a
finitely presented group with presentation

$\widetilde{\Gamma}=\langle a,b,c,t\mid
a^{2},(ac)^{4},[c,ab]^{2},[c,bab]^{2},[c,ababa]^{2},[c,ababab]^{2},[c,bababab]^{2},\\\
tat^{-1}=b,tbt^{-1}=c,tct^{-1}=aba\rangle,$

that embeds into $\mathop{\rm Aut}\nolimits(\widetilde{T}_{3})_{\omega}$. The
group $\widetilde{\Gamma}$ is another example of a finitely presented amenable
but not elementary amenable group and it also has exponential growth (as well
as $\widetilde{\mathcal{G}}$ has).

Figure 9: Automaton generating the group $\Gamma=\mathop{\rm
IMG}\nolimits(z^{2}+i)$

######  Example 5.12.

(Finitely presented amenable but not subexponentially amenable group
$\widetilde{\mathcal{B}}$). Our final example in this section is the Basilica
group $\mathcal{B}$ first studied in [54, 55]. It was the first example of a
finitely generated amenable group that is not subexponentially amenable. The
group $\mathcal{B}$ is generated by a 3-state automaton acting on a binary
tree and defined by the following wreath recursion:

$a=(b,1)\varepsilon,\quad b=(a,1).\quad$ (5.9)

It was shown in [55, Proposition 2] that $\mathcal{B}$ has the following
presentation:

$\mathcal{B}\cong\langle
a,b\mid\sigma_{\mathcal{B}}^{i}\big{(}[b,b^{a}]\big{)},i=0,1,\ldots\rangle,$
(5.10)

where $\sigma_{\mathcal{B}}$ is the substitution defined by

$\sigma_{\mathcal{B}}\colon\left\\{\begin{array}[]{l}a\mapsto b\\\ b\mapsto
a^{2}\end{array}\right.$ (5.11)

By Proposition 10 in [54] the map $\sigma_{\mathcal{B}}$ extends to a
monomorphism $\sigma_{\mathcal{B}}\colon\mathcal{B}\to\mathop{\rm
St}\nolimits_{\mathcal{B}}(0)$ and since $b=(a,1),a^{2}=(b,b)$, this
monomorphism is a lifting for $\mathcal{B}$. Thus, by Theorem B the HNN
extension $\widetilde{\mathcal{B}}$ of $\mathcal{B}$ via the homomorphism
$\sigma$ embeds into $\mathcal{D}(\mathbb{Q}_{2})$. The group
$\widetilde{\mathcal{B}}$ is an example of a finitely presented amenable but
not subexponentially amenable group of exponential growth.

###  5.3 Transitivity of the scale group coming from liftable groups

We now study the closures of $\theta(\widetilde{\mathcal{G}})$ and
$\theta(\widetilde{\mathcal{B}})$ from Examples 5.8 and 5.12 in $\mathop{\rm
Aut}\nolimits(\widetilde{T}_{3})_{\omega}$. The next result completes the
proof of Theorem C from the Introduction. In this section $\sigma$ and
$\sigma_{\mathcal{B}}$ denote the substitutions defined in (1.2) and (5.11),
respectively.

######  Theorem 5.13.

The groups $\overline{\theta(\widetilde{\mathcal{G}})}$ and
$\overline{\theta(\widetilde{\mathcal{B}})}$ are scale groups that act
2-transitively on the punctured boundary $\partial_{\omega}\widetilde{T}_{3}$
of $\widetilde{T}_{3}$.

The proof of this theorem is based on the $2$-transitivity property of the
actions of $\mathcal{G}$ on the levels of $T_{2}$ that was discovered in [5]
(see also Appendix by Bekka, Grigorchuk, de la Harpe in [17]), and a similar
result for the Basilica group $\mathcal{B}$.

###### Proof.

Since $\mathcal{G}$ and $\mathcal{B}$ satisfy the conditions of Theorem B and
act transitively on the first level, we immediately obtain
$\theta(\widetilde{\mathcal{G}})$ and $\theta(\widetilde{\mathcal{B}})$ act
transitively on the set of vertices of $T_{3}$ and that their closures
$\overline{\theta(\widetilde{\mathcal{G}})}$ and
$\overline{\theta(\widetilde{\mathcal{B}})}$ are scale groups acting
transitively on the punctured boundary $\partial_{\omega}\widetilde{T}_{3}$ of
$\widetilde{T}_{3}$ by Proposition 2.4 in [82].

We start from proving $2$-transitivity of the action of
$\overline{\theta(\widetilde{\mathcal{G}})}$ on
$\partial_{\omega}\widetilde{T}_{3}$. Suppose $(\xi_{1},\xi_{2})$ and
$(\xi^{\prime}_{1},\xi^{\prime}_{2})$ are two pairs of distinct points of
$\partial_{\omega}\widetilde{T}_{3}$. Let $n$ (resp., $n^{\prime}$) be the
largest level at which $\xi_{1},\xi_{2}$ (resp.,
$\xi^{\prime}_{1},\xi^{\prime}_{2}$) agree. Then elements $\theta(t)^{-n}$ and
$\theta(t)^{-n^{\prime}}$ of $\theta(\widetilde{\mathcal{G}})$ will move them
to pairs $(\theta(t)^{-n}(\xi_{1}),\theta(t)^{-n}(\xi_{2}))$ and
$(\theta(t)^{-n^{\prime}}(\xi^{\prime}_{1}),\theta(t)^{-n^{\prime}}(\xi^{\prime}_{2}))$
such that all of the 4 involved elements of
$\partial_{\omega}\widetilde{T}_{3}$ pass through the root $\Lambda$ of
$T_{2}^{(0)}$ and such that $\theta(t)^{-n}(\xi_{1})$ and
$\theta(t)^{-n}(\xi_{2})$ (resp., $\theta(t)^{-n^{\prime}}(\xi^{\prime}_{1})$
and $\theta(t)^{-n^{\prime}}(\xi^{\prime}_{2})$) do not agree on level 1.

Since $\mathcal{G}$ (and thus
$\theta(\mathcal{G})<\theta(\widetilde{\mathcal{G}})$) acts transitively on
the levels of $T^{(0)}_{2}$, for each level $l>0$ there are elements
$g_{l},g^{\prime}_{l}\in\mathcal{G}$ such that

$[\theta(g_{l})\bigl{(}\theta(t)^{-n}(\xi_{1})\bigr{)}]_{l}=[1^{-\infty}.1^{\infty}]_{l}$
(5.12)

and

$[\theta(g^{\prime}_{l})\bigl{(}\theta(t)^{-n^{\prime}}(\xi^{\prime}_{1})\bigr{)}]_{l}=[1^{-\infty}.1^{\infty}]_{l}$
(5.13)

Automorphisms $\theta(g_{l})$ and $\theta(g_{l}^{\prime})$ preserve levels of
$\widetilde{T}_{3}$. Since $\theta(t)^{-n}(\xi_{1})$ and
$\theta(t)^{-n}(\xi_{2})$ (resp., $\theta(t)^{-n^{\prime}}(\xi^{\prime}_{1})$
and $\theta(t)^{-n^{\prime}}(\xi^{\prime}_{2})$) do not agree on level 1,
their images under $\theta(g_{l})$ and $\theta(g_{l}^{\prime})$ also must not
agree on level 1. Therefore, by (5.12) and (5.13) we have

$[\theta(g_{l})\bigl{(}\theta(t)^{-n}(\xi_{2})\bigr{)}]_{l}=[1^{-\infty}.0w]_{l}$

and

$[\theta(g^{\prime}_{l})\bigl{(}\theta(t)^{-n^{\prime}}(\xi^{\prime}_{2})\bigr{)}]_{l}=[1^{-\infty}.0w^{\prime}]_{l}$

for some infinite words $w,w^{\prime}$.

Now we use the second part of Lemma 9.10 in [5], which states, in particular,
that the stabilizer $P_{l}$ of the vertex $1^{l}$ in $\mathcal{G}$ has
$0X^{l-1}$ as one of the orbits. It implies that there is $p_{l}\in
P_{l}<\mathcal{G}$ such that

$[\theta(p_{l})\left(\theta(g_{l})\bigl{(}\theta(t)^{-n}(\xi_{2})\bigr{)}\right)]_{l}=[\theta(g^{\prime}_{l})\bigl{(}\theta(t)^{-n^{\prime}}(\xi^{\prime}_{2})\bigr{)}]_{l}.$

We can rewrite the last equality as

$[\theta((g^{\prime}_{l})^{-1}p_{l}g_{l})\bigl{(}\theta(t)^{-n}(\xi_{2})\bigr{)}]_{l}=[\bigl{(}\theta(t)^{-n^{\prime}}(\xi^{\prime}_{2})\bigr{)}]_{l}.$

Since the sequence of elements
$\left(\theta((g^{\prime}_{l})^{-1}p_{l}g_{l})\right)_{l\geq 1}$ belongs to
the compact stabilizer of the root $\Lambda$ of $T^{(0)}_{2}$ in $\mathop{\rm
Aut}\nolimits_{0}(\widetilde{T}_{3})_{\omega}$, it must contain a convergent
subsequence
$\left(\theta((g^{\prime}_{l_{j}})^{-1}p_{l_{j}}g_{l_{j}})\right)_{j\geq 1}$
such that

$\lim_{j\to\infty}\theta((g^{\prime}_{l_{j}})^{-1}p_{l_{j}}g_{l_{j}})=g\in\overline{\theta(\widetilde{\mathcal{G}})}.$

By construction we have

$g(\theta(t)^{-n}(\xi_{1}))=\theta(t)^{-n^{\prime}}(\xi^{\prime}_{1})$

and

$g(\theta(t)^{-n}(\xi_{2}))=\theta(t)^{-n^{\prime}}(\xi^{\prime}_{2}).$

The last two equalities imply that the element
$\theta(t)^{n^{\prime}}g\theta(t)^{-n}\in\overline{\theta(\widetilde{\mathcal{G}})}$
moves $(\xi_{1},\xi_{2})$ to $(\xi^{\prime}_{1},\xi^{\prime}_{2})$.

The proof that $\overline{\theta(\widetilde{\mathcal{B}})}$ acts
2-transitively on $\partial_{\omega}\widetilde{T}_{3}$ is similar to the above
proof for $\overline{\theta(\widetilde{\mathcal{G}})}$. We only need to
establish the analog of Lemma 9.10 from [5] for Basilica group. Namely, we
will prove that $\mathop{\rm St}\nolimits_{\mathcal{B}}(1^{l})$ acts
transitively on $0X^{l-1}$ for each $l\geq 1$. It is proved in [54] that the
group $\mathcal{B}$ is weakly regular branch over its commutator subgroup
$\mathcal{B}^{\prime}$, i.e.,

$\mathcal{B}^{\prime}\succ\mathcal{B}^{\prime}\times\mathcal{B}^{\prime}.$
(5.14)

We show by induction on $l$ that $\mathcal{B}^{\prime}$ acts transitively on
$0X^{l-1}$ and $1X^{l-1}$. Induction base is trivial for $l=1$. Assume that
the statement is true for a fixed value of $l$. Then equation 5.14 immediately
implies that $\mathcal{B}^{\prime}$ acts transitively on $00X^{l-1}$,
$01X^{l-1}$, $10X^{l-1}$, and $11X^{l-1}$. Since
$[a,b]=(a,b^{-1}a^{-1}b)\in\mathcal{B}^{\prime}$ is such that both $a$ and
$b^{-1}a^{-1}b$ act transitively on the first level, we obtain that
$\mathcal{B}^{\prime}$ acts transitively on $0X^{l}$ and $1X^{l}$.

To complete the proof we observe that since $\mathcal{B}^{\prime}\times
1<\mathop{\rm St}\nolimits_{\mathcal{B}}(1^{l})$ for each $l$, the above
argument implies that $\mathop{\rm St}\nolimits_{\mathcal{B}}(1^{l})$ acts
transitively on $00X^{l-2}$ and $01X^{l-2}$. Finally, since
$b=(a,1)\in\mathop{\rm St}\nolimits_{\mathcal{B}}(1^{l})$ is such that
$b|_{0}=a$ acts transitively on the first level, we deduce that $\mathop{\rm
St}\nolimits_{\mathcal{B}}(1^{l})$ acts transitively on $0X^{l-1}$ for all
$l\geq 1$. ∎

###  5.4 GGS-groups

Important situation where one can prove liftability is when the projections of
$\sigma(G)$ on the vertices of the first level, other than the distinguished
vertex $i\in X$, happen to be in a finite subgroup of $G$. This is somewhat
intermediate between the examples of groups acting essentially freely on the
boundary of the tree (where the mentioned subgroup is trivial) and the
examples of groups with finite $L$-presentation (where one needs to find the
corresponding $L$-presentation). This situation is different from the case of
groups acting essentially freely on the boundary of the tree that was
discussed above. While the class of such groups is larger, it requires more
work to check that the substitution $\sigma$ defined on the generators of $G$
extends to an endomorphism of $G$. In this section we demonstrate this
situation on a large subclass of GGS-groups [6, 31]. The class of GGS-groups
(Grigorchuk-Gupta-Sidki groups, the term coined to G. Baumslag) includes the
group $\mathcal{G}$ (a second group from [34]) and all Gupta-Sidki $p$-groups
constructed in [56]. It was shown in [31] that all GGS $p$-groups are regular
branch, so they act totally non-freely on $\partial T_{p}$, which makes them
very different from groups acting essentially freely on the boundaries of
trees considered above.

For a fixed $d\geq 2$ every GGS-group is defined by a non-zero vector
$\mathrm{e}=(e_{1},e_{2},\ldots,e_{d-1})\in(\mathbb{Z}/d\mathbb{Z})^{d-1}$ as
the group acting on the regular rooted tree $T_{d}=T(X)$ for
$X=\mathbb{Z}/d\mathbb{Z}$ with the following wreath recursion:

$\begin{array}[]{rcl}a&=&(1,1,\ldots,1,1)\varepsilon,\\\
b&=&(a^{e_{1}},a^{e_{2}},\ldots,a^{e_{d-1}},b),\end{array}$

where $\varepsilon=(0,1,\ldots,d-1)$ denotes a long cycle in $\mathop{\rm
Sym}\nolimits(X)$. We will denote the corresponding group as
$G_{\mathrm{e}}=\langle a,b\rangle$.

######  Theorem 5.14.

Let $p\geq 3$ be a prime and $G_{\mathrm{e}}$ be a GGS-group defined by a non-
symmetric vector
$\mathrm{e}=(e_{1},e_{2},\ldots,e_{p-1})\in(\mathbb{Z}/p\mathbb{Z})^{p-1}$
with the property that there is $j\in\\{1,2,\ldots,p-1\\}$ such that
$e_{j}\neq 0$ and $e_{p-j}=0$. Then $G_{\mathrm{e}}$ is a liftable group.

###### Proof.

We first observe that both generators $a$ and $b$ of $G_{\mathrm{e}}$ have
order $p$. Moreover, it is shown in [31] (Theorem 2.1) that the abelianization
$G_{\mathrm{e}}/G_{\mathrm{e}}^{\prime}$ is isomorphic to
$(\mathbb{Z}/p\mathbb{Z})^{2}$.

Let $f=e_{j}^{-1}$, so that $(a^{e_{j}})^{f}=a$. Then we have

$\begin{array}[]{rcllllllllll}b^{a}&=&(b,&a^{e_{1}},&\ldots,&a^{e_{p-j-1}},&1,&a^{e_{p-j+1}},&\ldots,&a^{e_{p-2}},&a^{e_{p-1}}&),\\\
(b^{a^{1-j}})^{f}&=&(a,&a^{e_{j+1}f},&\ldots,&a^{e_{p-1}f},&b^{f},&a^{e_{1}f},&\ldots,&a^{e_{j-2}f},&a^{e_{j-1}f}&),\\\\[2.84526pt]
\text{positions}&&\phantom{(}0&1&\ldots&p-j-1&p-j&p-j+1&\ldots&p-2&p-1&\\\
\end{array}$

where the last row indicates the positions of corresponding coordinates of the
vectors in the first two rows. Define a substitution $\sigma$ as

$\sigma\colon\left\\{\begin{array}[]{l}a\mapsto(b^{a^{1-j}})^{f}\\\ b\mapsto
b^{a}\\\ \end{array}\right.$ (5.15)

Then $\sigma$ extends to an injective endomorphism of $G_{\mathrm{e}}$.
Indeed, suppose $w=w(a,b)$ is a word in the free group $F(a,b)$ that is a
relator in $G_{\mathrm{e}}$. Then

$\pi_{i}\circ\sigma(w)=\pi_{i}\bigl{(}w((b^{a^{1-j}})^{f},b^{a})\bigr{)}=\left\\{\begin{array}[]{ll}w(a,b)=_{G_{\mathrm{e}}}1,&\text{if
}\ i=0,\\\ w(1,b^{f}),&\text{if }\ i=p-j+1,\\\
w(a^{e_{i}},a^{j+i}),&\text{otherwise.}\end{array}\right.$

Since
$G_{\mathrm{e}}/G_{\mathrm{e}}^{\prime}\cong(\mathbb{Z}/p\mathbb{Z})^{2}$ and
$w=_{G_{\mathrm{e}}}1$, we must have that the total exponents of $a$ and $b$
in $w$ are both equal to 0 modulo $p$. This implies that both $w(1,b^{f})$ and
$w(a^{e_{i}},a^{j+i})$ are also trivial in $G_{\mathrm{e}}$ since both $a$ and
$b$ have order $p$. Here we use the fact that $\pi_{i}\circ\sigma(G)$ is a
subgroup of either $\langle b\rangle$ or $\langle a\rangle$ for $i\neq 0$,
which are both finite subgroups of $G$, which makes the verification that
$\sigma$ is an endomorphism possible.

Finally, by construction we have $\pi_{0}\circ\sigma$, so $\sigma$ is a
lifting of $G_{\mathrm{e}}$ and thus $G_{\mathrm{e}}$ is liftable. In
particular, Theorem B applies for $G_{\mathrm{e}}$. ∎

######  Corollary 5.15.

Gupta-Sidki $p$-groups $GS_{p}$ are liftable for all prime $p\geq 5$.

###### Proof.

The Gupta-Sidki $p$-group constructed in [56] is a GGS-groups defined by the
following vector $(1,-1,0,\ldots,0)\in(\mathbb{Z}/p\mathbb{Z})^{p-1}$, $p\geq
3$. This vector satisfies the condition from Theorem 5.14 for $j=1$ for $p\geq
5$. ∎

It is not known whether Gupta-Sidki 3-group $G_{3}$ is liftable. This is
related to the question 4.10 from [3] about the existence of a finite
ascending $L$-presentation for $GS_{3}$.

###  5.5 The group $\mathcal{G}_{(01)^{\infty}}$

Now we consider the group $\mathcal{G}_{(01)^{\infty}}$ studied by A. Erschler
in [30], who proved that it has intermediate growth with the growth function
growing faster than $\exp(n^{\alpha})$ for each $0<\alpha<1$ and also provided
quite sharp upper and lower bounds on the growth function. This group belongs
to the family $\mathcal{G}_{\omega},\omega\in\\{0,1,2\\}^{\infty}$ constructed
by the first author in [35]. Observe that the group $\mathcal{G}$ from Example
5.8 is isomorphic to $\mathcal{G}_{(012)^{\infty}}$. The group
$\mathcal{G}_{(01)^{\infty}}$ is determined by a periodic sequence
$(01)^{\infty}$, acts on the binary rooted tree, and is generated by the
following wreath recursion:

$\begin{array}[]{rcl}a&=&(1,1)\varepsilon\\\ b&=&(a,c)\\\ c&=&(1,b)\\\
d&=&(a,d)\\\ \end{array},$ (5.16)

Similarly to $\mathcal{G}$ and the GGS-groups studied in Subsection 5.4, the
group $\mathcal{G}_{(01)^{\infty}}$ is just-infinite branch and thus acts
totally non-freely on $\partial T_{2}$.

######  Proposition 5.16.

The group $\mathcal{G}_{(01)^{\infty}}$ is liftable.

Observe that it is proved by Bartholdi and Nekrashevych in [8] that
$\mathcal{G}_{(01)^{\infty}}$ is an iterated monodromy group of a quadratic
polynomial and is finitely $L$-presented. One can check that the presentation
that comes from this result is based on a lifting and Theorem B can be
applied. Instead we demonstrate another (practical) approach based on the use
of the GAP package working when no presentation of a group is known.

We introduce some groups related to $\mathcal{G}_{(01)^{\infty}}$ and use
`AutomGrp` GAP package [64] to perform our calculations. This is why we
separated this example from others considered in other sections. All of our
calculations can be verified by hands if needed. First, we observe that the
following relations in $\mathcal{G}_{(01)^{\infty}}$ hold:

$a^{2}=b^{2}=c^{2}=d^{2}=bcd=(ac)^{4}=1.$

Therefore, the group $C=\langle a,c\rangle$ is a quotient of the dihedral
group $D_{4}$. On the other hand, the quotient of $C$ by its stabilizer of the
third level has size 8:

    gap> G:=AutomatonGroup("a=(1,1)(1,2),b=(a,c),c=(1,b),d=(a,d)");
    < a, b, c, d >
    gap> Size(PermGroupOnLevel(Group([a,c]),3));
    8

Therefore, $C\cong D_{4}$. Further, let $B=\langle
b\rangle^{\mathcal{G}_{(01)^{\infty}}}$ denote the normal closure of $b$ in
$\mathcal{G}_{(01)^{\infty}}$. Then, $\mathcal{G}_{(01)^{\infty}}/B\cong
C\cong D_{4}$ and $G=B\rtimes C$.

Let $H=\mathop{\rm St}\nolimits_{\mathcal{G}_{(01)^{\infty}}}(1)$ be the
stabilizer of the first level in $\mathcal{G}_{(01)^{\infty}}$. Then
$[\mathcal{G}_{(01)^{\infty}}:H]=2$ and there is a homomorphism

$\begin{array}[]{llll}\psi\colon&H&\to&\mathcal{G}_{(01)^{\infty}}\times\mathcal{G}_{(01)^{\infty}}\\\
&h&\mapsto&(\pi_{0}(h),\pi_{1}(h))\end{array}$ (5.17)

The wreath recursion (5.16) implies that $aba=(c,a)$, so we define the
substitution $\sigma$ on the generators of $\mathcal{G}_{(01)^{\infty}}$ as

$\sigma\colon\left\\{\begin{array}[]{llll}a&\mapsto&aba&=(c,a)\\\
b&\mapsto&c&=(1,b)\\\ c&\mapsto&b&=(a,c)\\\
d&\mapsto&d&=(a,d)\end{array}\right.$ (5.18)

We will show below that $\sigma$ extends to an injective endomorphism of
$\mathcal{G}_{(01)^{\infty}}$ (whose image is obviously a subgroup of
$\mathop{\rm St}\nolimits_{\mathcal{G}_{(01)^{\infty}}}(1)$).

Define two subgroups $C_{0}=\langle(a,c),(c,a)\rangle$ and
$\widetilde{C}_{0}=\langle(1,a),(1,c)\rangle$ of
$\mathcal{G}_{(01)^{\infty}}\times\mathcal{G}_{(01)^{\infty}}$. Since
$C=\langle a,c\rangle\cong D_{4}$ and the map $a\mapsto c,c\mapsto a$ induces
an automorphism of $C$, we have that $C_{0}\cong\widetilde{C}_{0}\cong D_{4}$
as well.

######  Lemma 5.17.

The subgroup
$\psi(H)<\mathcal{G}_{(01)^{\infty}}\times\mathcal{G}_{(01)^{\infty}}$ is the
semidirect product $(B\times B)\rtimes C_{0}$. The subgroup
$\widetilde{C}_{0}$ is complementary to $\psi(H)$, so that
$\mathcal{G}_{(01)^{\infty}}\times\mathcal{G}_{(01)^{\infty}}=\psi(H)\rtimes\widetilde{C}_{0}$
and
$[(\mathcal{G}_{(01)^{\infty}}\times\mathcal{G}_{(01)^{\infty}}):\psi(H)]=8$.

###### Proof.

First, we observe that the stabilizer $H$ is generated by
$\\{b,c,d,b^{a},c^{a},d^{a}\\}$, so

$\psi(H)=\langle(1,b),(a,c),(a,d),(b,1),(c,a),(d,a)\rangle=\langle(1,b),(b,1),(a,c),(c,a)\rangle=(B\times
B)\cdot C_{0}.$

To confirm that $\psi(H)=(B\times B)\rtimes C_{0}$ we only need to check that
$(B\times B)\cap C_{0}=\\{1\\}$, which follows from the fact that
$\mathcal{G}_{(01)^{\infty}}=B\rtimes C$ and that the projection of $C_{0}$ on
both coordinates is equal to $C$.

Finally, since
$\mathcal{G}_{(01)^{\infty}}\times\mathcal{G}_{(01)^{\infty}}=\langle\psi(H),(1,a),(1,c)\rangle=\psi(H)\cdot\widetilde{C}_{0}$,
to prove that
$\mathcal{G}_{(01)^{\infty}}\times\mathcal{G}_{(01)^{\infty}}=\psi(H)\rtimes\widetilde{C}_{0}$
we only need to check that $\psi(H)\cap\widetilde{C}_{0}=\\{1\\}$. This can be
verified by considering the quotients of these groups by the stabilizers of
the fourth level.

    gap> G:=AutomatonGroup("a=(1,1)(1,2),b=(a,c),c=(1,b),d=(a,d)");
    < a, b, c, d >
    gap> AG_UseRewritingSystem(G);
    gap> H:=StabilizerOfFirstLevel(G);
    < b, c, d, a*b*a, a*c*a, a*d*a >
    gap> A:=TreeAutomorphism([1,a],());
    (1, a)
    gap> C:=TreeAutomorphism([1,c],());
    (1, c)
    gap> Intersection(PermGroupOnLevel(H,4),PermGroupOnLevel(Group([A,C]),4));
    Group(())

∎

###### Proof of Proposition 5.16.

Let $w$ be a word in the generators of $\mathcal{G}_{(01)^{\infty}}$ such that
$w=_{\mathcal{G}_{(01)^{\infty}}}1$. Then

$\psi(\sigma(w))=(U,W)=_{\mathcal{G}_{(01)^{\infty}}\times\mathcal{G}_{(01)^{\infty}}}(U,1),$

where $U$ is a word over $\\{a,c,a^{-1},c^{-1}\\}$. Then

$\psi(a\sigma(w)a)=(W,U)=_{\mathcal{G}_{(01)^{\infty}}\times\mathcal{G}_{(01)^{\infty}}}(1,U)\in\widetilde{C}_{0}.$

But since by Lemma 5.17 $\psi(H)\cap\widetilde{C}_{0}=\\{1\\}$, we get
$U=_{\mathcal{G}_{(01)^{\infty}}}1$ and so
$\sigma(w)=_{\mathcal{G}_{(01)^{\infty}}}1$. Thus, $\sigma$ extends to a
monomorphism $\mathcal{G}_{(01)^{\infty}}\to\mathcal{G}_{(01)^{\infty}}$. ∎

##  6 Scale groups coming from Bass-Serre theory

One more way to construct an action of a group on $\widetilde{T}_{d+1}$ is via
using Bass-Serre Theory. Recall, that if for a finitely generated group $G$
there is an injective endomorphism $\sigma\colon G\to H$ onto a subgroup $H$
of $G$ of finite index $d$, then the corresponding ascending HNN extension
$\widetilde{G}=G*_{\sigma}$, whose presentation is given by equation (1.5),
acts on its Bass-Serre tree $T\cong\widetilde{T}_{d+1}$ by automorphisms
preserving one of its ends. The set of vertices of $T$ is the set $\\{xG\colon
x\in\widetilde{G}\\}$ of left cosets of $G$ in $\widetilde{G}$, where vertices
$xG$ and $yG$ are connected by an oriented edge (going from $xG$ to $yG$) if
and only if $yG=xt^{-1}G$. In this tree a vertex $xG$ has one outgoing edge
connecting it to $xt^{-1}G$ and $d$ incoming edges starting at vertices
$xg_{s}tG$, $i=1,\ldots,d$, where $\\{g_{1},\ldots,g_{d}\\}$ is a left
transversal of $H$ in $G$ (for our purposes we do not need to talk about the
inversions of edges). The group $\widetilde{G}$ acts on $T$ by multiplication
on left, so this action is vertex transitive. It also preserves the end
$\omega$ of the tree $T$ corresponding to the sequence of vertices of the form
$t^{-n}G$, $n\geq 0$ as follows from the normal form for the elements of
ascending HNN extensions. This situation is discussed by Minasyan and Osin in
[63, Proposition 4.13].

The stabilizers of vertices in $T$ are conjugates of $G$ by elements of
$\widetilde{G}$, with $G$ itself being the stabilizer of the vertex
$\Lambda=G\in V(T)$. Therefore, the action of $\widetilde{G}$ on $T$ is
faithful (so that $\widetilde{G}$ embeds into $\mathop{\rm
Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$) if and only if the core of $G$
in $\widetilde{G}$ (i.e., the maximal normal subgroup of $\widetilde{G}$ in
$G$) is trivial:

$\mathrm{Core}(G)=\bigcap_{x\in\widetilde{G}}xGx^{-1}=\\{1\\}.$ (6.1)

In practice it is easier to check an equivalent to (6.1) condition:
$\mathrm{Core}\left(\cap_{n=1}^{\infty}\sigma^{n}(G)\right)=\\{1\\}$ (it is
equivalent since $\sigma^{n}(G)$ for each $n$ is a conjugate of $G$ by a power
of $t$). By construction the stabilizer of the vertex $\Lambda$ corresponding
to the coset $1\cdot G=G$ is equal to $G$. The above facts are summarized in
the following theorem.

######  Theorem 6.1.

Let $G$ be a group with a subgroup $H$ isomorphic to $G$ and of finite index
$d$ and let $\sigma\colon G\rightarrow H$ be an isomorphism. Let
$\widetilde{G}=G*_{\sigma}$ be the ascending HNN-extension of $G$ with respect
to $\sigma$, whose presentation is given in equation (1.5). Then

  * (i)

$\widetilde{G}$ acts on its Bass-Serre tree $T$, which is isomorphic to
$\widetilde{T}_{d+1}$, by automorphisms fixing one of its ends $\omega$.

  * (ii)

The action of $\widetilde{G}$ on $T$ is vertex transitive with the stabilizer
of the vertex $\Lambda$ of $T$ corresponding to the coset $1\cdot G$ equal to
$G$.

  * (iii)

The action of $\widetilde{G}$ on $T$ is faithful if and only if
$\mathrm{Core}(G)=\\{1\\}$ (equivalently, if
$\mathrm{Core}\left(\cap_{n=1}^{\infty}\sigma^{n}(G)\right)=\\{1\\}$). In this
case $\widetilde{G}$ embeds into $\mathop{\rm
Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$ as a group acting vertex
transitively on $\widetilde{T}_{d+1}$ and its closure in $\mathop{\rm
Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$ is a scale group. Moreover, the
image of $G$ under this embedding coincides with the stabilizer of the
distinguished vertex $\Lambda=1\cdot G$ of $T$.

######  Remark 6.2.

The above theorem works also in the case $d=[G:\sigma(G)]=\infty$, but in this
case we get an action of $\widetilde{G}=G*_{\sigma}$ on a tree
$\widetilde{T}_{\infty}$ that is not locally finite.

To use the above theorem we need to deal with groups that are not finitely co-
Hopfian, i.e. that have finite index subgroups isomorphic to the whole group.
The obvious examples of such groups are $\mathbb{Z}^{d},d\geq 1$ (of course,
we are interested in finitely generated examples). Less obvious examples come
from scale-invariant groups that were defined at the beginning of Section 5.
To get embeddings of such groups into $\mathop{\rm
Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$ the last condition of Theorem 6.1
(on faithfulness of the action) needs to be satisfied.

######  Example 6.3.

Our first example illustrating the use of Theorem 6.1 is the lamplighter group
$\mathcal{L}=(\mathbb{Z}/2\mathbb{Z})\wr\mathbb{Z}$, studied in Example 5.4.
Recall that the standard presentation of $\mathcal{L}$ coming from the
“lamplighter” interpretation of the action, is given with respect to a
generating set $\\{x=a^{-1}b,s=b\\}$, where $\\{a,b\\}$ is the generating set
from Example 5.4:

$\mathcal{L}\cong\langle x,s\mid x^{2}=[x,x^{s^{n}}]=1,n\geq 1\rangle.$

It was shown in [52] that the substitution $\alpha\colon x\mapsto
xx^{s},s\mapsto s$ extends to an injective endomorphism
$\alpha\colon\mathcal{L}\to\mathcal{L}$, whose image has index 2 in
$\mathcal{L}$. The homomorphism $\alpha$ written in terms of generating set
$\\{a,b\\}$ coincides with the lifting $\sigma_{1}$ defined in (5.1). Thus,
the image of $\alpha$ is equal to $\mathop{\rm St}\nolimits_{\mathcal{L}}(1)$
and has index $2$ in $\mathcal{L}$. It is also shown in [52] that HNN
extension of $\mathcal{L}$ with respect to $\alpha$ is a metabelian group
given by finite presentation

$\widetilde{\mathcal{L}}_{\alpha}=\langle x,s,t\mid
x^{2}=[t,s]=[x,x^{s}]=1,x^{t}=xx^{s}\rangle.$

By Corollary 5.2 this group embeds into $\mathop{\rm
Aut}\nolimits(\widetilde{T}_{3})_{\omega}$. However, Theorem 6.1 can also be
applied here by verifying that the core of $\mathcal{L}$ in
$\widetilde{\mathcal{L}}$ is trivial. We start with the following lemma.

######  Lemma 6.4.

For each $n\geq 0$ we have

$\alpha^{2^{n}}(x)=x\cdot x^{s^{2^{n}}},$ $\alpha^{2^{n}}(s)=s.$

###### Proof.

Since $\alpha(s)=s$, the second claim of the lemma is trivial. To prove the
first claim we use induction on $n$. The base of induction is trivial and to
prove the step we assume that the statement is correct for some $n\geq 0$.
Then

$\alpha^{2^{n+1}}(x)=\alpha^{2^{n}}\left(\alpha^{2^{n}}(x)\right)=\alpha^{2^{n}}\left(x\cdot
x^{s^{2^{n}}}\right)=x\cdot x^{s^{2^{n}}}\cdot\left(x\cdot
x^{s^{2^{n}}}\right)^{s^{2^{n}}}\\\ =x\cdot x^{s^{2^{n}}}\cdot
x^{s^{2^{n}}}\cdot x^{s^{2^{n+1}}}=x\cdot x^{s^{2^{n+1}}}.$

∎

######  Lemma 6.5.

$\bigcap_{n=1}^{\infty}\alpha^{n}(\mathcal{L})=\langle s\rangle$.

###### Proof.

It is enough to show that
$\bigcap_{n=1}^{\infty}\alpha^{2^{n}}(\mathcal{L})=\langle s\rangle$. Since
$s\in\alpha^{2^{n}}(\mathcal{L})$ for all $n\geq 1$, one inclusion is obvious.
We will show below that
$\bigcap_{n=1}^{\infty}\alpha^{2^{n}}(\mathcal{L})\subset\langle s\rangle$.

Every element $g$ of $\mathcal{L}$ can be uniquely written as

$g=\left(\prod_{i=1}^{k}x^{s^{n_{i}}}\right)\cdot s^{l},$ (6.2)

where $k\in\mathbb{Z}$ and $n_{1}<n_{2}<\cdots<n_{k}$ are distinct integers.
In the lamplighter interpretation such an element corresponds to turning on
lamps at positions $n_{i}$ and moving the pointer to position $l$.

By Lemma 6.4 $\alpha^{2^{n}}(\mathcal{L})$ is generated by $\\{x\cdot
x^{s^{2^{n}}},s\\}$. Define commuting elements

$x_{2^{n},i}=\left(x\cdot x^{s^{2^{n}}}\right)^{s^{i}}=x^{s^{i}}\cdot
x^{s^{2^{n}}+i},i\in\mathbb{Z}$

Then every element $h$ of $\alpha^{2^{n}}(\mathcal{L})$ can be written
uniquely as

$h=\left(\prod_{i=1}^{k}x_{2^{n},m_{i}}\right)\cdot s^{l},$

where $l\in\mathbb{Z}$ and $m_{1}<m_{2}<\cdots<m_{k}$ are distinct integers.
In the case $k\geq 1$ (i.e., when $h\notin\langle s\rangle$), the above
decomposition rewritten in the form (6.2), will satisfy $n_{1}=m_{1}$ and
$n_{k}=m_{k}+2^{n}\geq n_{1}+2^{n}$. Therefore, every element in
$\alpha^{2^{n}}(\mathcal{L})\setminus\langle s\rangle$ will turn on 2 lamps
that are at least $2^{n}$ units apart. This shows that there is no element in
$\mathcal{L}\setminus\langle s\rangle$ that belongs to
$\alpha^{2^{n}}(\mathcal{L})$ for all $n\geq 1$. ∎

######  Lemma 6.6.

$\mathrm{Core}\bigl{(}\bigcap_{n=1}^{\infty}\alpha^{n}(\mathcal{L})\bigr{)}=\\{1\\}$.

###### Proof.

According to Lemma 6.5 we only need to show that $\langle s\rangle$ does not
contain a non-trivial normal subgroup in $\widetilde{\mathcal{L}}_{\alpha}$.
This is the case because the conjugate $(s^{i})^{x}$ of any non-trivial
element $s^{i}$ of $\langle s\rangle$ is not a power of a shift $s$ as it
changes the states of 2 lamps, so $(s^{i})^{x}\notin\langle s\rangle$ and thus
$\langle s\rangle$ contains no normal subgroup of
$\widetilde{\mathcal{L}}_{\alpha}$. ∎

Now, since
$\mathrm{Core}(\mathcal{L})<\mathrm{Core}\bigl{(}\bigcap_{n=1}^{\infty}\alpha^{n}(\mathcal{L})\bigr{)}$,
the group $\widetilde{\mathcal{L}}_{\alpha}$ embeds into $\mathop{\rm
Aut}\nolimits(\widetilde{T}_{3})_{\omega}$ by Theorem 6.1.

The above example is interesting in particular because the group
$\widetilde{\mathcal{L}}_{\alpha}$ is a quotient by the relation $x^{2}=1$ of
Baumslag’s finitely presented metabelian group with the commutator isomorphic
to a free abelian group of infinite rank considered in [14]. Hence,
$\widetilde{\mathcal{L}}_{\alpha}$ is a finitely presented metabelian group
with the commutator subgroup isomorphic to an elementary 2-group of infinite
rank. Moreover, $\widetilde{\mathcal{L}}_{\alpha}$ is isomorphic to the
fundamental group of a closed Riemannian manifold with a non-integer
$L^{2}$-Betti number, constructed in [52] (as the first example answering the
original question of M. Atiyah and a counterexample to the Strong Atiyah
conjecture on $L^{2}$-Betti numbers).

Formally, the two constructions of the actions of HNN extensions on
$\widetilde{T}_{d+1}$ given by Theorem B and Theorem 6.1 are quite different
and it is quite unlikely that they lead to conjugate action of the same group
(although in some cases perhaps conjugacy of action is possible). Observe that
in the construction used in the proof of Theorem 6.1 we do not have a
requirement that the index $[G:\sigma(G)]$ is related to the degree of the
tree as we have in the case of Theorem 6.1. Moreover, this index in most
examples presented in Section 5 is infinite. On the other hand, for each
liftable self-similar group $G$ that acts transitively on the first level and
is defined by a lifting $\sigma$, the core of $G$ in its HNN extension
$\widetilde{G}$ from Theorem B is trivial. Indeed, since $G$ stabilizes the
vertex $\Lambda$ in $\widetilde{T}_{d+1}$, its conjugates by elements of
$\widetilde{G}$ are subgroups of the stabilizers of vertices of
$\widetilde{T}_{d+1}$ in $\widetilde{G}$. Moreover, since $\widetilde{G}$ acts
transitively on $\widetilde{T}_{d+1}$, one can conjugate $G$ into a stabilizer
of any vertex. Now, as the action of $\widetilde{G}$ on $\widetilde{T}_{d+1}$
is faithful, the intersection of all these stabilizers, which includes
$\mathrm{Core}(G)$ in $\widetilde{G}$, is trivial. Therefore, by Theorem 6.1,
$\widetilde{G}$ acts on the corresponding Bass-Serre tree
$\widetilde{T}_{d^{\prime}+1}$, where $d^{\prime}=[G:\sigma(G)]$ and in the
case $d^{\prime}<\infty$ this leads to the constructions of scale groups via
Theorem 6.1 (like in the last example).

##  7 Scale-invariant groups and Haagerup property

We are going to explore scale-invariant groups to get more examples of scale
groups using Theorem 6.1. The main result of Nekrashevych and Pete in [70] and
its corollary state:

######  Theorem 7.1 (Theorem 1.1 in [70]).

Let $H$ be a countable scale-invariant group, with a descending sequence
$H=H_{0}>H_{1}>\cdots$ of finite index subgroups, each isomorphic to $H$, with
trivial intersection. Let $A$ be a countable automorphism group of $H$ leaving
all subgroups $H_{n}$ invariant. Assume that the action of $A$ is faithful on
each $H_{n}$, and that the semidirect products $H_{n}\rtimes A$ are isomorphic
to $H\rtimes A$. Then $G:=H\rtimes A$ is scale-invariant; in fact, there is a
required subgroup chain with $\cap_{n\geq 0}G_{n}=\\{1\\}$.

######  Corollary 7.2 (Corollary 1.2 in [70]).

The following groups are scale-invariant.

  * (1)

The lamplighter groups $G=F\wr\mathbb{Z}$, where $F$ is any finite Abelian
group.

  * (2)

The solvable Baumslag–Solitar groups $BS(1,m)=\langle a,b\mid
bab^{-1}=a^{m}\rangle$ with $m>1$.

  * (3)

The affine group $\mathbb{Z}^{d}\rtimes\mathrm{GL}(\mathbb{Z},d)$, and its
subgroups $\mathbb{Z}^{d}\rtimes A$ for any
$A\leq\mathrm{GL}(\mathbb{Z},d),d>1$.

The proof of the theorem uses (rooted) coset tree $\mathcal{T}$ associated
with $\\{H_{n}\\}_{n\geq 1}$ and the natural level transitive action of
$H\rtimes A$ on it by affine automorphisms. The sequence $\\{G_{n}\\}_{n\geq
1}$ is obtained as the sequence of stabilizers of vertices belonging to the
path in $\mathcal{T}$ joining the root vertex with infinity (i.e., with an end
of the tree). It is proved that the stabilizers are isomorphic to $G$. The
fact that they have finite index in $G$ is obvious from the construction, and
the main nontrivial point is to show that there are points in the boundary of
$\mathcal{T}$ with trivial stabilizers. The arguments provided in [70] show
that in fact the set of such points is a dense $G_{\delta}$ set, so is non-
empty (and comeager).

For our purposes we will need Proposition 2.2(i) from [70] and its proof. It
states that if, under conditions of Theorem 7.1, the sequence
$\\{H_{n}\\}_{n\geq 1}$ satisfies $H_{n}=\varphi^{n}(H)$ for some injective
endomorphism $\varphi\colon H\to H$, then there is an injective endomorphism
$\tilde{\varphi}\colon G\to G$ defined by (see equation (2.3) in [70])

$\tilde{\varphi}(h,\alpha)=\bigl{(}\varphi(h),\alpha\bigr{)}$ (7.1)

for $(h,\alpha)\in G$ such that
$[G:\tilde{\varphi}(G)]=[H:\varphi(H)]<\infty$, and such that
$\tilde{\varphi}^{n}(G)$ is the stabilizer in $G$ of a vertex of $\mathcal{T}$
of level $n$. Let

$\widetilde{G}=G*_{\tilde{\varphi}}=\langle G,t\mid\text{relations in}\
G,tgt^{-1}=\tilde{\varphi}(g)\ \text{for all}\ g\in G\rangle$

be the ascending HNN extension of $G$ with respect to $\tilde{\varphi}$.

######  Proposition 7.3.

The core of $G$ in $\widetilde{G}$ is trivial.

###### Proof.

By the proof of Proposition 2.2(i) in [70],
$t^{n}Gt^{-n}=\tilde{\varphi}^{n}(G)$ is the stabilizer in $G$ of a vertex of
$\mathcal{T}$ of level $n$. Since the action of $G$ on $\mathcal{T}$ is level
transitive, one can use conjugate $\tilde{\varphi}^{n}(G)$ by elements of $G$
into a stabilizer of any vertex of that (arbitrary) level. Since the action of
$G$ on $\mathcal{T}$ is faithful, the intersection of all vertex stabilizers
is trivial, which implies that $\mathrm{Core}(G)$, that is the intersection of
all conjugates of $G$ in $\widetilde{G}$ is also trivial. ∎

As a direct corollary of the above proposition and Theorem 6.1 we obtain:

######  Corollary 7.4.

The group $\widetilde{G}$ embeds into $\mathop{\rm
Aut}\nolimits(\widetilde{T}_{d+1})_{\omega}$ for $d=[G:\tilde{\varphi}(G)]$.

We apply Corollary 7.4 to get examples of scale groups obtained as a closures
of groups without Haagerup property. Recall that a (discrete) group $G$ has a
_Haagerup property_ (or is _a-T-menable_) if it admits a proper affine action
on a Hilbert space. There are several equivalent definitions of this property
(see, for instance, Section 1.1.1 and Theorem 2.1.1 in [26]). Also, there are
many definitions of _Kazhdan’s property (T)_ and its relative version for a
pair consisting of a group and its subgroup (see, for example, [26, Definition
1.4.1]). Property (T) groups also have Serre’s property (FA) meaning that
every action on a tree has a fixed vertex. The class of Haagerup groups
includes amenable groups, free groups, groups acting properly on trees,
$\mathbb{R}$-trees, and $\mathrm{CAT}(0)$ cube complexes. Examples of groups
with property (T) mostly come from lattices in higher rank Lie groups. A
nontrivial example of a pair with relative property (T) is
$(\mathbb{Z}^{2}\rtimes\mathrm{SL}(2,\mathbb{Z}),\mathbb{Z}^{2})$ (see, for
example, [16, Theorem 4.2.2]), which, in particular, implies that
$\mathbb{Z}^{2}\rtimes\mathrm{SL}(2,\mathbb{Z})$ does not have Haagerup
property.

######  Theorem 7.5.

  * (a)

There is a subgroup of $\mathop{\rm Aut}\nolimits(\widetilde{T}_{9})_{\omega}$
without the Haagerup property that acts vertex transitively on
$\widetilde{T}_{9}$ and whose closure is a scale group.

  * (b)

There is a subgroup of $\mathop{\rm Aut}\nolimits(\widetilde{T}_{5})_{\omega}$
without the Haagerup property that acts vertex transitively on
$\widetilde{T}_{5}$ and whose closure is a scale group.

###### Proof.

To prove item (a), consider a scale-invariant group $H=\mathbb{Z}^{3}$ with an
injective endomorphism $\varphi\colon H\to H$ defined by $\varphi(v)=2v$. This
defines a nested sequence $H_{n}=\varphi^{n}(H)$ of subgroups of $H$ with
trivial intersection such that $[H_{n}:H_{n+1}]=8$ for all $n\geq 0$. Then
according to the proof of Theorem 7.1 the group
$\Gamma=\mathbb{Z}^{3}\rtimes\mathrm{SL}(\mathbb{Z},3)$ consisting of affine
transformations of $\mathbb{Z}^{3}$, which is a group without Haagerup
property (as the class of groups with Haagerup property is closed under taking
subgroups), acts faithfully by automorphisms on the coset tree $\mathcal{T}$,
that is a regular rooted 8-ary tree. By Corollary 7.4 its HNN extension
$\widetilde{\Gamma}*_{\tilde{\varphi}}$, where $\tilde{\varphi}$ is defined by
equation (7.1), embeds into $\mathop{\rm
Aut}\nolimits(\widetilde{T}_{9})_{\omega}$. Its closure in $\mathop{\rm
Aut}\nolimits(\widetilde{T}_{9})_{\omega}$ is a scale group by Theorem
6.1(iii).

The proof of item (b) is very similar with the difference that we take
$H=\mathbb{Z}^{2}$ and $\Gamma=\mathbb{Z}^{2}\rtimes\mathrm{SL}(\mathbb{Z},2)$
as the corresponding scale-invariant group. Since now $[H:\varphi(H)]=4$, the
HNN extension $\widetilde{\Gamma}*_{\tilde{\varphi}}$, where $\tilde{\varphi}$
is defined by equation (7.1), embeds into $\mathop{\rm
Aut}\nolimits(\widetilde{T}_{5})_{\omega}$. Its closure in $\mathop{\rm
Aut}\nolimits(\widetilde{T}_{5})_{\omega}$ is again a scale group by Theorem
6.1(iii).

∎

## References

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