# Confined subgroups and high transitivity

Adrien Le Boudec  CNRS, UMPA - ENS Lyon, 46 allée d’Italie, 69364 Lyon, France
<EMAIL_ADDRESS>and  Nicolás Matte Bon  CNRS, Institut Camille
Jordan (ICJ, UMR CNRS 5208), Université de Lyon, 43 blvd. du 11 novembre 1918,
69622 Villeurbanne, France<EMAIL_ADDRESS>
(Date: November 19, 2021)

###### Abstract.

An action of a group $G$ is highly transitive if $G$ acts transitively on
$k$-tuples of distinct points for all $k\geq 1$. Many examples of groups with
a rich geometric or dynamical action admit highly transitive actions. We prove
that if a group $G$ admits a highly transitive action such that $G$ does not
contain the subgroup of finitary alternating permutations, and if $H$ is a
confined subgroup of $G$, then the action of $H$ remains highly transitive,
possibly after discarding finitely many points.

This result provides a tool to rule out the existence of highly transitive
actions, and to classify highly transitive actions of a given group. We give
concrete illustrations of these applications in the realm of groups of
dynamical origin. In particular we obtain the first non-trivial classification
of highly transitive actions of a finitely generated group.

ALB supported by ANR-19-CE40-0008-01 AODynG

##  1\. Introduction

Given a group $G$, we let $\operatorname{Sub}(G)$ be the space of subgroups of
$G$, endowed with the induced topology from the set $2^{G}$ of all subsets of
$G$. The group $G$ acts on $\operatorname{Sub}(G)$ by conjugation. A subgroup
$H$ of $G$ is called a confined subgroup if the closure of the $G$-orbit of
$H$ in $\operatorname{Sub}(G)$ does not contain the trivial subgroup.
Equivalently, $H$ is a confined subgroup if there exists a finite subset $P$
of non-trivial elements of $G$ such that $gHg^{-1}\cap P$ is not empty for
every $g\in G$. A subgroup $H$ of $G$ is a uniformly recurrent subgroup (URS)
if the orbit closure of $H$ is minimal [GW15], i.e. does not contain any
proper closed $G$-invariant subspace. These two notions are closely related,
as every non-trivial subgroup $H$ that is a URS is confined; and conversely if
$H$ is confined, then the orbit closure of $H$ contains a non-trivial URS.

An early appearance of the notion of confined subgroups can be traced back to
the study of ideals in group algebras of locally finite groups, notably in
[SZ93, Zal95, HZ97]. In particular the terminology was introduced in [HZ97].
Confined subgroups and URSs are intimately connected to stabilizers of group
actions on compact spaces. In this context they have been implicitly
considered in the literature under different names, see [GNS00, Gri11].
Recently these notions turned out to be useful tools to understand reduced
group $C^{*}$-algebras, thanks to a connection with topological boundary
actions [KK17, Ken20, BKKO17]. The systematic study of confined subgroups and
URSs of various countable groups in [LBMB18, LB21, MB18, LBMB20] has led to
several applications and rigidity results.

In this article we establish a connection between confined subgroups and
highly transitive actions. Throughout the paper $\Omega$ will always be an
infinite set, and we denote by $\operatorname{Sym}(\Omega)$ the group of all
permutations of $\Omega$. An action of a group $G$ on $\Omega$ is
$k$-transitive if $G$ acts transitively on the set of $k$-tuples of distinct
elements of $\Omega$. An action is highly transitive if it is $k$-transitive
for all $k$. Equivalently, the action of $G$ on $\Omega$ is highly transitive
if the induced group homomorphism from $G$ to $\operatorname{Sym}(\Omega)$ has
a dense image, where $\operatorname{Sym}(\Omega)$ is endowed with the
pointwise convergence topology.

An example of a highly transitive action is for instance given by the action
of the group $\operatorname{Sym}_{f}(\Omega)$ of finitary permutations of
$\Omega$, or even its subgroup $\operatorname{Alt}_{f}(\Omega)$ of alternating
permutations. Here a permutation is termed finitary if it moves only finitely
many points. Recall that $\operatorname{Alt}_{f}(\Omega)$ and
$\operatorname{Sym}_{f}(\Omega)$ are normal subgroups of
$\operatorname{Sym}(\Omega)$, and every non-trivial normal subgroup of
$\operatorname{Sym}(\Omega)$ contains $\operatorname{Alt}_{f}(\Omega)$ [Ono29,
SU33]. For a group $G$ acting faithfully and highly transitively on $\Omega$,
one easily verifies that the image of $G$ in $\operatorname{Sym}(\Omega)$
contains a non-trivial finitary permutation if and only if it contains
$\operatorname{Alt}_{f}(\Omega)$. In this situation we will say that $G$ is
partially finitary. This can be reformulated as an intrinsic property of the
group, see §2.3. These are somehow considered as trivial examples, and we are
mostly interested in highly transitive actions of groups that are not
partially finitary.

The first goal of this article is to prove the following result, which asserts
that high transitivity is essentially inherited by confined subgroups.
Whenever $H$ is a subgroup of $\operatorname{Sym}(\Omega)$, we denote by
$\Omega_{H,f}$ the union of the finite $H$-orbits in $\Omega$.

######  Theorem 1.

Suppose that a group $G$ admits a faithful and highly transitive action on
$\Omega$ and that $G$ is not partially finitary. If $H$ is a confined subgroup
of $G$, then $\Omega_{H,f}$ is finite, and the action of $H$ on
$\Omega\setminus\Omega_{H,f}$ is highly transitive.

######  Remark 1.1.

In the setting of the theorem, the set $\Omega_{H,f}$ is not empty in general,
for instance because the stabilizer in $G$ of a point or of a finite subset of
$\Omega$ may very well be a confined subgroup of $G$. See e.g. Remark 3.13 or
Section 5.

######  Remark 1.2.

In the case where $G$ is partially finitary, a similar conclusion holds, up to
passing to a finite $H$-invariant partition of $\Omega\setminus\Omega_{H,f}$;
see Theorem 3.12.

Recently there has been a growing interest for a better understanding of which
(say countable) groups admit faithful highly transitive actions. Actually the
question is already rich for primitive actions, and was investigated in depth
by Gelander and Glasner [GG08], partly motivated by earlier results of
Margulis and Soifer [MS81]. See [GGS19] for a recent survey. A series of
consecutive works shows that various groups admitting a rich geometric or
dynamical action also admit a highly transitive action. This is for instance
the case for free groups [McD77, Dix90, Ols15, EG16, LM18], surface groups
[Kit12], hyperbolic groups [Cha12], outer automorphism groups of free groups
[GG13a], acylindrically hyperbolic groups [HO16], unbounded Zariski dense
subgroups of $\mathrm{SL}_{2}$ over a local field [GGS19], and many groups
acting on trees [FLMMS20]. Hence Theorem 1 applies in all these situations.

On the other hand the study of the existence of highly transitive actions
within a given class of groups still suffers from the lack of obstructions
that prevent a group from having such actions. In other words, given a group
$G$, we know very few restrictions on $G$ that are imposed by the existence of
a faithful highly transitive action of $G$. See §2.4 for a brief discussion
around this. The following consequence of Theorem 1 provides a new criterion
to rule out the existence of highly transitive actions.

######  Corollary 2.

Let $G$ be a group that is not partially finitary, and let $H$ be a confined
subgroup of $G$. If $H$ does not admit any faithful highly transitive action,
then neither does $G$.

We illustrate this criterion by providing an application to a class of groups
of dynamical origin, namely subgroups of the groups of piecewise linear
homeomorphisms of suspension flows introduced in [MBT20]. We briefly recall
this construction and refer to §4.2 for detailed definitions. Let $X$ be a
Stone space (i.e. a compact totally disconnected space), and $\varphi$ be a
homeomorphism of $X$. The pair $(X,\varphi)$ will be called a Stone system.
Let $Y^{\varphi}$ be the suspension space (or mapping torus) of $(X,\varphi)$,
and $\Phi\colon\mathbb{R}\times Y^{\varphi}\to Y^{\varphi}$ be the associated
suspension flow. The space $Y^{\varphi}$ is locally homeomorphic to
$X\times\mathbb{R}$, and its path-connected components are exactly the
$\Phi$-orbits. Following [MBT20], we let $\mathsf{PL}(\varphi)$ be the group
of all homeomorphisms of $Y^{\varphi}$ which preserve every $\Phi$-orbit and
act as piecewise linear homeomorphism of $\mathbb{R}$ on each of them, and
whose displacement along every $\Phi$-orbit is locally constant in the
transverse direction.

Recall that an action of a group $G$ on a compact space $Y$ by homeomorphisms
is topologically free if the set $\operatorname{Fix}(g)$ of fixed points of
$g$ has empty interior for every non-trivial element $g$ of $G$. Conversely
the action is topologically nowhere free if every point $y$ in $Y$ is
contained in the interior of $\operatorname{Fix}(g)$ for some non-trivial
element $g$ of $G$. This condition implies that the stabiliser $G_{y}$ of
every $y\in Y$ is a confined subgroup of $G$. Using Corollary 2, we show the
following result.

######  Theorem 3.

Let $(X,\varphi)$ be a minimal Stone system. Let $G$ be a subgroup of
$\mathsf{PL}(\varphi)$ whose action on $Y^{\varphi}$ is topologically nowhere
free. Then $G$ does not admit any faithful highly transitive action.

Examples of groups to which the criterion applies are the groups
$\mathsf{T}(\varphi)\leq\mathsf{PL}(\varphi)$ introduced and studied in
[MBT20], which are analogues in $\mathsf{PL}(\varphi)$ of Thompson’s group $T$
acting on the circle. It is shown in [MBT20] that the group
$\mathsf{T}(\varphi)$ is simple when the system $(X,\varphi)$ is minimal, and
finitely generated when $(X,\varphi)$ is conjugate to a subshift over a finite
alphabet. In §4.3 we describe a related family of examples to which Theorem 3
also applies, which retain similar properties, and for which we are
additionally able to check that none of the previously known obstructions to
high transitivity applies.

Beyond the problem of deciding which groups admit highly transitive actions,
it is natural to try to classify all highly transitive actions of a given
group. In the case of primitive actions, similar questions have been
considered by Gelander–Glasner in [GG08, GG13b] for countable linear groups.
It follows from an old result of Jordan-Wielandt that the group
$\operatorname{Alt}_{f}(\Omega)$ admits a unique highly transitive action up
to conjugacy, and more generally the same is true for every partially finitary
group (see §2.3). Apart from this observation, there was so far no example of
a finitely generated highly transitive group whose highly transitive actions
are completely understood. Theorem 1 provides a tool towards such
classification results. We illustrate this mechanism by considering the family
of Higman–Thompson’s groups $V_{d}$ [Hig74], whose definition is recalled in
§5.1. It is well-known that for the natural action of $V_{d}$ on the Cantor
set, the action on each orbit is highly transitive. Using Theorem 1, we prove
the following:

######  Theorem 4.

Every faithful highly transitive action of the Higman–Thompson group $V_{d}$
is conjugate to its natural action on an orbit in the Cantor set.

Here we focus on the family of groups $V_{d}$ instead of trying to reach the
most optimal level of generality, but the argument is likely to be adaptable
to other groups acting on compact spaces with suitable dynamical properties
(see Section 5, and in particular Remark 5.7).

To end this introduction, we point out that the analogue of Theorem 1 is also
true in the measured setting of invariant random subgroups (IRSs), i.e.
$G$-invariant Borel probability measures on $\operatorname{Sub}(G)$ [AGV14].
That is, whenever $G$ is not partially finitary, high transitivity is
inherited by all IRSs of $G$; see Proposition 6.1. This fact is a rather
direct application of the classical de Finetti theorem from probability
theory, whose importance to the study of IRSs was recognised by Vershik
[Ver12]. We point out that we are not aware of any application of this
statement, for instance to classify or to rule out the existence of highly
transitive actions of certain groups. In the groups considered in Theorem 3
and Theorem 4, the conjugacy class closure in $\operatorname{Sub}(G)$ of the
confined subgroups that are involved does not carry any $G$-invariant
probability measure. In other words, we deal with URSs that are not IRSs.
Hence the above result for IRSs does not provide any help here, and the
appropriate tool is indeed Theorem 1.

### Outline

The proof of Theorem 1 is given in Section 3. The crucial point of the proof
is Theorem 3.7, which provides a description of the confined subgroups of the
symmetric group $\operatorname{Sym}(\Omega)$. The reason why this result is
relevant regarding Theorem 1 comes from the observation that confined
subgroups behave well with respect to a dense embedding of a group $G$ into a
topological group $L$: the image closure in $L$ of every confined subgroup of
$G$ is a confined subgroup of $L$ (see Lemma 3.11). Since a faithful highly
transitive action of $G$ on $\Omega$ is the same as a dense embedding into
$\operatorname{Sym}(\Omega)$, every confined subgroup of $G$ thus gives rise
to a confined subgroup of $\operatorname{Sym}(\Omega)$. With Theorem 3.7 and
this observation in hand, the proof of Theorem 1 follows easily. In the course
of the proof of Theorem 3.7, we make use of the commutator lemma for confined
subgroups established in [LBMB20]. Roughly speaking, this result says that
when a group admits an action with non-trivial rigid stabilizers, then under
certain assumptions every confined subgroup must contain a large part of some
rigid stabilizer. We refer to Theorem 3.5 for a precise statement and for the
terminology. As a remark, we note that it seems reasonable to believe that our
mechanism of proof here with the help of Theorem 3.5 might be applicable to
study dense embeddings into other large topological groups.

Section 4 is dedicated to Theorem 3. An ingredient of the proof is an
auxiliary result about the subquotients of the group $\mathsf{PL}(I)$ of
piecewise linear homeomorphisms of an interval, which implies in particular
that every finite subquotient of the group $\mathsf{PL}(I)$ must be solvable;
see Proposition 4.1. That result is an improvement of the Brin-Squier result
[BS85] asserting that $\mathsf{PL}(I)$ does not contain any non-abelian free
subgroup, and provides an immediate obstruction to high transitivity for
subgroups of $\mathsf{PL}(I)$ (see §2.4). This conclusion does not generalize
to the group $\mathsf{PL}(\varphi)$ (which contains free subgroups). However,
it does generalize to the stabilizers of the natural action of
$\mathsf{PL}(\varphi)$ on $Y^{\varphi}$ (Lemma 4.8). Since the latter are
confined subgroups, we deduce Theorem 3 from Corollary 2.

Section 5 contains the proof of Theorem 4, and is independent of Section 4.
The proof relies on an appropriate combination of Theorem 1 together with an
independent result about proximality of the action of certain subgroups of
$V_{d}$ on the Cantor set (Proposition 5.2).

### Acknowledgements

We thank the referee for a careful reading of the article and for useful
comments.

##  2\. Preliminaries

###  2.1. Notation for group actions

Suppose that a group $G$ acts on a set $Y$. We denote by
$\operatorname{Fix}(g)\subset Y$ the set of fixed points of an element $g\in
G$, and define its support as the set
$\operatorname{Supp}(g)=Y\setminus\operatorname{Fix}(g)$. We use the notation
$G_{y}$ for the stabiliser of a point $y\in Y$, and $G_{y_{1},\ldots,y_{n}}$
for the pointwise fixator of an $n$-tuple of points. If $Y$ has a topology,
the germ-stabilizer of $y$ is the subgroup $G^{0}_{y}$ consisting of elements
that fix pointwise some neighbourhood of $y$. Similarly
$G^{0}_{y_{1},\cdots,y_{n}}$ is the subgroup of elements that fix pointwise a
neighbourhood of each of the points $y_{1},\ldots,y_{n}$.

###  2.2. Confined subgroups

Let $H,G$ be two subgroups of a group $L$. We say that $H$ is confined by $G$
if there is a finite subset $P$ of non-trivial elements of $L$ such that
$gHg^{-1}\cap P$ is not empty for every $g\in G$. Such a subset $P$ is called
a confining subset for the pair $(H,G)$. A confined subgroup of a group is a
subgroup which is confined by the whole ambient group. Note that if $H$
contains a subgroup confined by a group $G$, then $H$ is also confined by $G$.

Suppose that a group $G$ acts on a compact space $Y$. We say that the action
is topologically nowhere free if $G^{0}_{y}\neq\\{1\\}$ for every $y\in Y$.
The notions of confined subgroups and topologically nowhere free actions are
related through the following lemma.

######  Lemma 2.1.

Suppose that the action of $G$ on the compact space $Y$ is topologically
nowhere free. Then for every $y\in Y$, the germ stabilizer $G^{0}_{y}$ (and
hence the stabilizer $G_{y}$) is a confined subgroup of $G$.

###### Proof.

For every $y$ one can find a non-trivial element of $G$ acting trivially on a
neighbourhood of $y$. Hence by compactness one can find a finite subset $P$ of
non-trivial elements such that the interiors of the set of fixed points of
elements of $P$ cover $Y$, and it easily follows that $P$ is confining for the
pair $(G^{0}_{y},G)$ for every $y\in Y$. ∎

###  2.3. Partially finitary groups

Recall from the introduction that we call a group $G$ partially finitary if
$G$ admits an embedding into the symmetric group $\operatorname{Sym}(\Omega)$
over an infinite set such that the image of $G$ contains
$\operatorname{Alt}_{f}(\Omega)$.

######  Lemma 2.2.

A group $G$ is partially finitary if and only if there exists an infinite set
$\Omega$ and an injective group homomorphism
$\varphi:\operatorname{Alt}_{f}(\Omega)\to G$ such that the image of $\varphi$
is normal and has trivial centralizer in $G$.

###### Proof.

The direct implication is clear, and the converse follows from the fact that
the conjugation action of $G$ on the image of $\varphi$ provides an injective
homomorphism from $G$ into the automorphism group of
$\operatorname{Alt}_{f}(\Omega)$. Since the latter is
$\operatorname{Sym}(\Omega)$ [DM96, Th. 8.2.A], the claim follows. ∎

The following is a classical result from the theory of permutation groups due
to Jordan-Wielandt [DM96, Th. 3.3.D]. Recall that an action of a group $G$ on
a set $\Omega$ is primitive if it preserves no partition of $\Omega$ other
than the one block partition and the partition into singletons. For transitive
actions, this is equivalent to saying that point stabilizers are maximal
subgroups. For instance any $2$-transitive action is primitive.

######  Theorem 2.3.

If the action of a subgroup $G\leq\operatorname{Sym}(\Omega)$ on the infinite
set $\Omega$ is primitive and if $G$ contains a non-trivial finitary
permutation, then $G$ contains $\operatorname{Alt}_{f}(\Omega)$.

In particular this result has the following easy consequence:

######  Proposition 2.4.

If a group $G$ is partially finitary, then $G$ admits a unique faithful
2-transitive action, which is highly transitive.

###### Proof.

Suppose that $G$ admits an embedding into $\operatorname{Sym}(\Omega)$ such
that the image of $G$ contains $A=\operatorname{Alt}_{f}(\Omega)$. We claim
that actually the action of $G$ on $\Omega$ is its only $2$-transitive action.
Indeed suppose $H$ is a point stabilizer for a faithful $2$-transitive action
of $G$. Since $A$ is normal in $G$, the partition of $G/H$ into $A$-orbits
forms a $G$-invariant partition, which is not the partition into points as $A$
must act faithfully. Since a $2$-transitive action is primitive, it follows
that this partition must be trivial, and thus the action of $A$ on $G/H$ is
transitive. If $A\cap H$ is trivial, then $G=A\rtimes H$ and the action of $H$
on $G/H$ coincides with the conjugation action of $H$ on $A$. This is clearly
impossible since non-trivial elements of $A$ are not all conjugate. Therefore
$H$ must intersect $A$ non-trivially, thus contains elements with finite
support on $\Omega$. However $H$ cannot contain the whole group $A$, as we
already noted that $A$ acts transitively on $G/H$, and thus cannot fix $H$.
From this we deduce that the action of $H$ on $\Omega$ is not primitive by the
Jordan-Wielandt theorem. Hence by maximality of $H$ in $G$ we see that $H$ is
the stabilizer of a non-trivial partition $\mathcal{P}$ of $\Omega$, and the
2-transitive action we started with is the action on the $G$-orbit
$\mathcal{O}_{\mathcal{P}}$ of $\mathcal{P}$. Note that the action of $G$ on
pairs of distinct elements $(\mathcal{P}_{1},\mathcal{P}_{2})$ of
$\mathcal{O}_{\mathcal{P}}$ must preserve the number of blocks of
$\mathcal{P}_{1}$ which do not belong to $\mathcal{P}_{2}$. If $\mathcal{P}$
has at least 3 blocks, then by considering images of $\mathcal{P}$ under
elements of $A$ it is easy to find two pairs for which this number is finite
and different, contradicting 2-transitivity. We deduce that $\mathcal{P}$ must
be of the form $\left\\{\Omega_{1},\Omega_{2}\right\\}$ with
$\Omega_{1},\Omega_{2}$ non-empty. Equivalently, $H$ is just the stabilizer of
$\Omega_{1}$ in $G$, and the action of $G$ on $G/H$ is the action of $G$ on
the $G$-orbit of $\Omega_{1}$. But a very similar reasoning shows that this
action can be $2$-transitive only when $\Omega_{1}$ or the complement of
$\Omega_{1}$ is a singleton. So $H$ is a point stabilizer for the $G$-action
on $\Omega$, and the claim is proven. ∎

###  2.4. A brief discussion about high transitivity

In this paragraph we discuss some basic facts about highly transitive actions,
and collect some observations that allow to exclude the existence of highly
transitive actions of a given group. As far as we know, at the moment these
observations provide the only known obstructions to the existence of highly
transitive actions of a countable group.

Let $G$ be a group admitting a faithful highly transitive action on $\Omega$.
The first preliminary observation is that the action of every non-trivial
normal subgroup of $G$ remains highly transitive. This follows for instance
from the classical result that every non-trivial normal subgroup of
$\operatorname{Sym}(\Omega)$ contains the group
$\operatorname{Alt}_{f}(\Omega)$ [Ono29, SU33], since highly transitive
actions correspond to homomorphisms to $\operatorname{Sym}(\Omega)$ with dense
image. Therefore in particular every non-trivial normal subgroup of $G$ has
trivial centralizer in $G$.

If $\Sigma$ is a finite subset of $\Omega$ of cardinality $k$, then the
stabilizer of $\Sigma$ in $G$ admits a homomorphism to
$\operatorname{Sym}(\Sigma)$, that is surjective by the definition of high
transitivity. Hence the group $G$ admits all finite symmetric groups (hence
all finite groups) as subquotients. Recall that $Q$ is a subquotient of a
group $G$ if $G$ admits a subgroup $H$ such that $Q$ is isomorphic to a
quotient of $H$. This implies for instance that $G$ cannot be of bounded
exponent, or more generally that $G$ cannot be a torsion group whose element
orders are divisible by only finitely many primes. This also implies that $G$
cannot satisfy a non-trivial identity (or law), i.e. there does not exist a
non-trivial element $w$ in the free group $\mathbb{F}_{n}$ of rank $n$ such
that every homomorphism from $\mathbb{F}_{n}$ to $G$ is trivial on $w$.

This generalizes to the more general notion of mixed identities. If $w$ is an
element of the free product $G\ast\mathbb{F}_{n}$, we say that $G$ satisfies
the mixed-identity $w=1$ if every homomorphism from $G\ast\mathbb{F}_{n}$ to
$G$ that is the identity on $G$ is trivial on $w$. We say that $G$ satisfies a
(non-trivial) mixed-identity if one can find $n\geq 1$ and a non-trivial
element $w\in G\ast\mathbb{F}_{n}$ such that $G$ satisfies the mixed-identity
$w=1$. By [HO16, Theorem 5.9], if a group $G$ admits a faithful highly
transitive action and $G$ is not partially finitary, then $G$ does not satisfy
a mixed-identity. Basic examples of groups that satisfy a non-trivial mixed
identity are non-trivial direct products, and more generally any group that
admits two non-trivial commuting normal subgroups. Note that such groups may
very well not satisfy an identity, for example because they might contain free
subgroups. Slightly less obvious examples are groups admitting a fixed point
free action by homeomorphisms on the real line such that some non-trivial
elements have a relatively compact support, e.g. Thompson’s group $F$ [Zar10].
Similar but slightly more elaborated phenomena happen for certain groups
acting on the circle or acting on trees [LBMB19, Prop. 3.7-4.7].

##  3\. Confined subgroups of infinite symmetric groups and Theorem 1

###  3.1. Confined subgroups of infinite symmetric groups

Whenever $H$ is a subgroup of $\operatorname{Sym}(\Omega)$, we denote by
$\Omega_{H,f}$ the union of the finite $H$-orbits in $\Omega$, and by
$\Omega_{H,\infty}$ the union of the infinite $H$-orbits. Given a partition
$\mathcal{Q}=(\Omega_{i})_{i\in I}$ of $\Omega$, the subsets $\Omega_{i}$ are
called the blocks of $\mathcal{Q}$. We say that $\mathcal{Q}$ is a thick
partition if every block of $\mathcal{Q}$ has cardinality at least two.

######  Lemma 3.1.

Let $H\leq\operatorname{Sym}(\Omega)$ be a subgroup that is confined by a
subgroup $G\leq\operatorname{Sym}(\Omega)$ that is highly transitive on
$\Omega$, and let $P$ be a confining subset for $(H,G)$ of cardinality
$r=r_{1}+r_{2}$, where $r_{1}$ and $r_{2}$ are respectively the number of
elements of $P$ with finite support and with infinite support. Then the
following hold:

  1. (i)

$H$ has at most $r-1$ fixed points in $\Omega$.

  2. (ii)

$\Omega_{H,f}$ is finite.

  3. (iii)

Every $H$-invariant thick partition $\mathcal{Q}=(\Omega_{i})_{i\in I}$ of
$\Omega$ is finite; and:

    1. (a)

if $r_{1}=0$ then $\mathcal{Q}$ has exactly one infinite block
$\Omega_{i_{0}}$, and $|\Omega\setminus\Omega_{i_{0}}|\leq r-1$;

    2. (b)

if $r_{1}\geq 1$ then $\mathcal{Q}$ has at most $r_{1}$ infinite blocks.

###### Proof.

(i). Suppose for a contradiction that $H$ fixes $r$ distinct fixed points
$x_{1},\ldots,x_{r}$. We choose a set $\Sigma\subseteq\Omega$ of cardinality
at most $r$ such that every element of $P$ moves a point in $\Sigma$. Since
the action of $G$ is highly transitive, one can find $g\in G$ such that
$g(\Sigma)\subseteq\left\\{x_{1},\ldots,x_{r}\right\\}$. Then every element of
$gPg^{-1}$ moves at least one $x_{i}$, and hence $gPg^{-1}\cap H$ is empty, a
contradiction.

We show the statement (iii). Note that (ii) will follow in view of (i). Assume
that $\mathcal{Q}=(\Omega_{i})_{i\in I}$ is a thick $H$-invariant partition of
$\Omega$ . We decompose $P=P_{1}\cup P_{2}$, where $P_{1}$ is the set of
elements of $P$ having finite support. Let $S\subset\Omega$ be the union of
the supports of elements of $P_{1}$. We write
$P_{2}=\left\\{\sigma_{r_{1}+1},\ldots,\sigma_{r}\right\\}$. Note that for
every finite set $F\subset\Omega$ and every $\sigma_{i}\in P_{2}$, there
exists $x\in\Omega$ such that $x\neq\sigma_{i}(x)$ and $x,\sigma_{i}(x)\notin
F$ (just choose
$x\in\operatorname{Supp}(\sigma_{i})\setminus(F\cup\sigma_{i}^{-1}(F))$, which
is non-empty since the support $\operatorname{Supp}(\sigma_{i})$ is infinite).
Using this observation repeatedly, we can choose points $x_{i}$ and
$x_{i}^{\prime}$ such that

$x_{r_{1}+1},x_{r_{1}+1}^{\prime},\ldots,x_{r},x_{r}^{\prime},\sigma_{r_{1}+1}(x_{r_{1}+1}),\sigma_{r_{1}+1}(x_{r_{1}+1}^{\prime})\ldots,\sigma_{r}(x_{r}),\sigma_{r}(x_{r}^{\prime})$

are all distinct and all outside $S$.

Now we write

$T=S\cup\left\\{x_{r_{1}+1},x_{r_{1}+1}^{\prime},\ldots,x_{r},x_{r}^{\prime},\sigma_{r_{1}+1}(x_{r_{1}+1}),\sigma_{r_{1}+1}(x_{r_{1}+1}^{\prime})\ldots,\sigma_{r}(x_{r}),\sigma_{r}(x_{r}^{\prime})\right\\}.$

Assume for a contradiction that $\mathcal{Q}$ has infinitely many blocks.
Since the group $G$ is highly transitive and all blocks have cardinality at
least two, we can find $g\in G$ with the following properties:

  * •

for any two distinct $s,s^{\prime}\in S$, $g(s)$ and $g(s^{\prime})$ belong to
distinct blocks;

  * •

for all $i$ there exist ${j(i)},{k(i)},{l(i)}$ such that
$\Omega_{k(i)},\Omega_{l(i)}$ are distinct and
$g(x_{i}),g(x_{i}^{\prime})\in\Omega_{j(i)}$,
$g(\sigma_{i}(x_{i}))\in\Omega_{j(i)}$ and
$g(\sigma_{i}(x_{i}^{\prime}))\in\Omega_{l(i)}$.

For such an element $g$, we claim that $gPg^{-1}$ cannot intersect $H$.
Indeed, assume first that there is $h\in H\cap gP_{1}g^{-1}$. Then on the one
hand the element $h$ must preserve the partition, and on the other hand the
support of $h$ intersects every block of the partition along at most a
singleton. This is of course impossible, so $gP_{1}g^{-1}$ does not intersect
$H$. Now if $i$ is such that $g\sigma_{i}g^{-1}$ belongs to $H$, then the
block $\Omega_{j(i)}$ must be sent to both $\Omega_{k(i)}$ and
$\Omega_{l(i)}$, which is again impossible. So we have obtained a
contradiction, as desired. So $\mathcal{Q}$ is finite.

Assume now that $r_{1}=0$, i.e. every element of $P$ has infinite support. It
follows from the previous paragraph that $\mathcal{Q}$ has at least one
infinite block $\Omega_{i_{0}}$. We have to argue that the complement of
$\Omega_{i_{0}}$ has cardinality at most $r-1$. Suppose this is not the case.
Keeping the same notation as before, again by high transitivity of the group
$G$ we can find $g\in G$ such that
$g(x_{i}),g(x_{i}^{\prime}),g(\sigma_{i}(x_{i}))\in\Omega_{i_{0}}$ and
$g(\sigma_{i}(x_{i}^{\prime}))\notin\Omega_{i_{0}}$ for all $i$. This implies
that none of the elements $g\sigma_{i}g^{-1}$ sends $\Omega_{i_{0}}$ onto a
block of $\mathcal{Q}$. Hence $gPg^{-1}$ does not intersect $H$, and again we
have a contradiction.

Finally we assume that $r_{1}\geq 1$ and that $\mathcal{Q}$ has at least
$r_{1}+1\geq 2$ infinite blocks, say
$\Omega_{j_{1}},\ldots,\Omega_{j_{r_{1}+1}}$. We choose a finite subset
$Y=\\{y_{1},\ldots,y_{k}\\}\subset S$ of cardinality $k\leq r_{1}$ such that
every element of $P_{1}$ moves a point in $Y$. We choose $g\in G$ such that
$g(y_{i})\in\Omega_{j_{i}}$ for all $i=1,\ldots,k$ and $g(S\setminus
Y)\subset\Omega_{j_{k+1}}$.

Since in addition $r_{1}+1\geq 2$, we can also arrange $g$ to satisfy
$g(x_{i}),g(x_{i}^{\prime}),g(\sigma_{i}(x_{i}))\in\Omega_{j_{1}}$ and
$g(\sigma_{i}(x_{i}^{\prime}))\in\Omega_{j_{2}}$ for all $i=r_{1}+1,\ldots,r$.
We claim that $H$ cannot intersect $gPg^{-1}$. Indeed assume by contradiction
that $H$ contains $g\sigma g^{-1}$ with $\sigma\in P_{1}$. Let $i$ be such
that $\sigma(y_{i})\neq y_{i}$. Then $g\sigma g^{-1}$ moves
$g(y_{i})\in\Omega_{j_{i}}$. However the support of $g\sigma g$ intersects
$\Omega_{j_{1}}$ only along $\\{g(y_{i})\\}$, as
$g(S)\cap\Omega_{j_{i}}=\\{g(y_{i})\\}$ by construction. Hence $g\sigma
g^{-1}$ must necessarily preserve the block $\Omega_{j_{i}}$ and hence fix
$g(y_{i})$, which is absurd. Finally all elements of $gP_{2}g^{-1}$ must send
one of the points $g(x_{i})\in\Omega_{j_{1}}$ inside $\Omega_{j_{1}}$ and the
corresponding point $g(x_{i})^{\prime}\in\Omega_{j_{1}}$ inside
$\Omega_{j_{2}}$, so that none of them can preserve $\mathcal{Q}$. Thus $H$
does not intersect $gP_{2}g^{-1}$. ∎

In the sequel we borrow terminology and notation from [LBMB20, §3].

######  Definition 3.2.

Let $P$ be a finite set of non-trivial elements of
$\operatorname{Sym}(\Omega)$, and let
$\left\\{\Omega_{\sigma}\right\\}_{\sigma\in P}$ be a family of non-empty
subsets of $\Omega$ indexed by $P$. We say that
$\left\\{\Omega_{\sigma}\right\\}_{\sigma\in P}$ is a displacement
configuration for $P$ if the following hold:

  1. i)

for all $\sigma,\rho\in P$, either $\Omega_{\sigma}=\Omega_{\rho}$, or
$\Omega_{\sigma}$ and $\Omega_{\rho}$ are disjoint;

  2. ii)

for all $\sigma,\rho\in P$, either $\sigma$ fixes $\Omega_{\rho}$ pointwise,
or $\sigma(\Omega_{\rho})$ is disjoint from $\bigcup_{\alpha\in
P}\Omega_{\alpha}$;

  3. iii)

for all $\sigma\in P$, $\sigma(\Omega_{\sigma})$ is disjoint from
$\bigcup_{\alpha\in P}\Omega_{\alpha}$ and also from $\bigcup_{\alpha\in
P}\sigma^{-1}(\Omega_{\alpha})$.

Note that by the last condition, the subsets
$\sigma^{-1}(\Omega_{\sigma}),\Omega_{\sigma},\sigma(\Omega_{\sigma})$ are
disjoint for all $\sigma\in P$. In particular this prevents $\sigma$ from
being of order two. We will make use of the following easy lemma, which is a
partial converse of the previous observation.

######  Lemma 3.3.

Let $\sigma_{1},\ldots,\sigma_{q}\in\operatorname{Sym}(\Omega)$ such that
$\sigma_{i}^{2}$ has infinite support for all $i$. Then there exist infinite
subsets $\Sigma_{1},\ldots,\Sigma_{q}\subset\Omega$ that form a displacement
configuration for $\left\\{\sigma_{1},\ldots,\sigma_{q}\right\\}$.

###### Proof.

Lemma 4.1 in [LBMB20] states that if $X$ is a Hausdorff space without isolated
points, and if $\sigma_{1},\ldots,\sigma_{q}$ are homeomorphisms of $X$ such
that $\sigma_{i}^{2}$ is non-trivial, then there is a displacement
configuration for $\\{\sigma_{1},\ldots,\sigma_{q}\\}$ consisting of open
subsets. The present statement can be deduced from that one as follows.
Consider the action of $\operatorname{Sym}(\Omega)$ on the Stone-Čech
compactification $\beta\Omega$. Recall that there is a one-to-one
correspondence between subsets of $\Omega$ and clopen subsets of
$\beta\Omega$, given by $\Lambda\mapsto U_{\Lambda}$, where $U_{\Lambda}$ is
the set of ultrafilters $\omega\in\beta\Omega$ such that $\omega(\Lambda)=1$.
The fact that $\sigma_{i}^{2}$ moves infinitely many points in $\Omega$ is
equivalent to saying that $\sigma_{i}^{2}$ does not act trivially on
$\beta^{*}\Omega:=\beta\Omega\setminus\Omega$. Hence we are in position to
apply [LBMB20, Lemma 4.1] to $\left\\{\sigma_{1},\ldots,\sigma_{q}\right\\}$.
By this lemma, there exist open subsets $U_{1},\ldots,U_{q}$ of
$\beta^{*}\Omega$ that form a displacement configuration for
$\left\\{\sigma_{1},\ldots,\sigma_{q}\right\\}$ in $\beta^{*}\Omega$, and the
existence of infinite subsets $\Sigma_{1},\ldots,\Sigma_{q}$ that form a
displacement configuration for $\left\\{\sigma_{1},\ldots,\sigma_{q}\right\\}$
in $\Omega$ easily follows. ∎

######  Remark 3.4.

Alternatively, the lemma can be easily proven directly, without using the
Stone-Čech compactification, by adapting the argument of [LBMB20, Lemma 4.1].

The following is Theorem 3.17 from [LBMB20]. Whenever $\Sigma$ is a subset of
$\Omega$ and $G$ is a group acting on $\Omega$, we denote by
$\operatorname{R}_{G}(\Sigma)$ the pointwise fixator of the complement of
$\Sigma$, and call it the rigid stabilizer of $\Sigma$. In the sequel by the
trivial conjugacy class we mean the singleton consisting of the trivial
element of the group.

######  Theorem 3.5 (Commutator lemma for confined subgroups).

Let $H,G\leq\operatorname{Sym}(\Omega)$ be two subgroups such that $H$ is
confined by $G$, and let $P$ be a confining subset for $(H,G)$, and $r=|P|$.
Assume that $\left\\{\Omega_{\sigma}\right\\}_{\sigma\in P}$ is a displacement
configuration for $P$ such that for all $\sigma\in P$ the group
$\operatorname{R}_{G}(\Omega_{\sigma})$ is non-trivial, and every non-trivial
conjugacy class in $\operatorname{R}_{G}(\Omega_{\sigma})$ has cardinality
$>r$.

Then there exists $\rho\in P$ such that $H$ contains a non-trivial subgroup
$N\leq\operatorname{R}_{G}(\Omega_{\rho})$ whose normalizer in
$\operatorname{R}_{G}(\Omega_{\rho})$ has index at most $r$.

We will also use the following simplified version of [LBMB20, Lemma 3.2].

######  Lemma 3.6.

Let $G,H$ be subgroups of a group $L$ such that $H$ is confined by $G$.
Suppose that every confining subset $P\subset L$ for $(H,G)$ contains an
element of order 2. Then there exists an abelian subgroup $K\leq L$ which is
confined by $G$.

The following result provides a description of the confined subgroups of
$\operatorname{Sym}(\Omega)$. If we think of confined subgroups as
generalizations of normal subgroups, it might be compared to the classical
result that every non-trivial normal subgroup of $\operatorname{Sym}(\Omega)$
contains $\operatorname{Alt}_{f}(\Omega)$ [Ono29, SU33], or to the description
of all normal subgroups of $\operatorname{Sym}(\Omega)$ [DM96, Th. 8.1.A]. For
a subset $\Sigma$ of $\Omega$, the rigid stabilizer of $\Sigma$ in
$\operatorname{Alt}_{f}(\Omega)$ is naturally isomorphic to
$\operatorname{Alt}_{f}(\Sigma)$. By abuse of notation we will denote by
$\operatorname{Alt}_{f}(\Sigma)$ this rigid stabilizer.

######  Theorem 3.7.

For a subgroup $H\leq\operatorname{Sym}(\Omega)$, the following are
equivalent:

  1. i)

$H$ is a confined subgroup of $\operatorname{Sym}(\Omega)$;

  2. ii)

$H$ is confined by $\operatorname{Alt}_{f}(\Omega)$;

  3. iii)

$\Omega_{H,f}$ is finite, and there exists an $H$-invariant partition
$\Omega_{H,\infty}=\Omega_{1}\cup\ldots\cup\Omega_{k}$ of $\Omega_{H,\infty}$
into infinite subsets $\Omega_{1},\ldots,\Omega_{k}$ such that $H$ contains
$\mathrm{Alt}_{f}(\Omega_{1})\times\ldots\times\mathrm{Alt}_{f}(\Omega_{k})$.

######  Remark 3.8.

$\ref{item-confbysym}\implies\ref{item-confbyalt}$ is trivial and $\ref{item-
nicesbgp}\Rightarrow\ref{item-confbysym}$ is an easy verification (see below).
The main content of the theorem is the implication $\ref{item-
confbyalt}\implies\ref{item-nicesbgp}$. Theorem 1 in [SZ93] provides a
characterization of the confined subgroups $H$ of
$\operatorname{Alt}_{f}(\Omega)$, which turns out to be the same as the one
obtained in Theorem 3.7. In other words, that theorem gives the implication
$\ref{item-confbyalt}\implies\ref{item-nicesbgp}$ within the group
$\operatorname{Alt}_{f}(\Omega)$. The argument there uses in an essential way
the fact that all elements of $H$ have finite support, and hence does not
apply to the situation of Theorem 3.7.

######  Proposition 3.9.

Let $H\leq\operatorname{Sym}(\Omega)$ be a subgroup that is confined by
$\operatorname{Alt}_{f}(\Omega)$. Let $P$ be a confining subset for
$(H,\operatorname{Alt}_{f}(\Omega))$, and $r$ the number of elements of $P$
and $r_{1}$ the number of elements of $P$ with finite support. Then
$\Omega_{H,f}$ is finite, and there exists an $H$-invariant partition
$\Omega_{H,\infty}=\Omega_{1}\cup\ldots\cup\Omega_{k}$ of $\Omega_{H,\infty}$
into infinite subsets $\Omega_{1},\ldots,\Omega_{k}$ with
$k\leq\max(1,r_{1})$, such that $H$ contains
$\mathrm{Alt}_{f}(\Omega_{1})\times\ldots\times\mathrm{Alt}_{f}(\Omega_{k})$.
If in addition $r_{1}=0$ then $k=1$ and
$|\Omega_{H,f}|=|\Omega\setminus\Omega_{1}|\leq r-1$.

###### Proof.

According to Lemma 3.1, the subgroup $H$ has finitely many orbits in $\Omega$.
Let $O_{1},\ldots,O_{\ell}$ be the infinite orbits of $H$. By Lemma 3.1 again,
every $H$-invariant partition of $\Omega_{H,\infty}$ that does not restrict to
the trivial partition into singletons to any $O_{i}$, has cardinality at most
$\max(1,r_{1})$. Let $\Omega_{H,\infty}=\cup_{i,j}O_{i,j}$ be such a partition
with maximal cardinality, where $O_{i}=\cup_{j}O_{i,j}$ for all $i$. Since the
partition $O_{i}=\cup_{j}O_{i,j}$ cannot be further refined, the stabilizer of
$O_{i,j}$ in $H$ acts primitively on $O_{i,j}$. Note that this implies in
particular that $H$ cannot be abelian (since a maximal subgroup in an abelian
group must have finite prime index). Note also that Lemma 3.1 also ensures
$|\Omega\setminus\Omega_{H,\infty}|\leq r-1$ when $r_{1}=0$.

The core of the proof consists in showing that for each block $O_{i,j}$ the
subgroup $H$ contains a non-trivial element whose support is finite and
contained in $O_{i,j}$. We proceed by contradiction, and assume that
$O_{i,j}=O$ is a block that does not satisfy the desired property. According
to Lemma 3.6 and the above remark that no subgroup that is confined by
$\operatorname{Alt}_{f}(\Omega)$ can be abelian, we may find a subset $Q$ that
is confining for $(H,\operatorname{Alt}_{f}(\Omega))$ such that $Q$ has no
element of order two. We write $Q=Q_{0}\sqcup Q_{1}$, where $Q_{0}$ is the set
of elements of $Q$ such that $\sigma^{2}$ has finite support, and write
$Q_{1}=\left\\{\sigma_{1},\ldots,\sigma_{q}\right\\}$.

According to Lemma 3.3, there exist infinite subsets
$\Sigma_{1},\ldots,\Sigma_{q}\subset\Omega$ that form a displacement
configuration for $\left\\{\sigma_{1},\ldots,\sigma_{q}\right\\}$. For each
$i$ we choose a finite subset $F_{i}\subset\Sigma_{i}$ of cardinality $q+5$.
Upon replacing $Q$ by a conjugate by an element of
$\operatorname{Alt}_{f}(\Omega)$ (which is also a confining subset for
$(H,\operatorname{Alt}_{f}(\Omega))$), we may assume that
$\mathrm{supp}(\sigma^{2})\subset O$ for all $\sigma\in Q_{0}$ and also that
$F_{i}\subset O$ for all $i$. Since $Q$ is confining for
$(H,\operatorname{Alt}_{f}(\Omega))$, clearly $Q$ is confining for
$(H,\operatorname{Alt}_{f}(O))$. We claim that actually $Q_{1}$ is confining
for $(H,\operatorname{Alt}_{f}(O))$. Indeed $gQ_{0}g^{-1}\cap H$ is empty for
all $g\in\operatorname{Alt}_{f}(O)$, because otherwise if $g\sigma g^{-1}$
belongs to $H$ with $\sigma\in Q_{0}$, then $g\sigma^{2}g^{-1}$ would be a
non-trivial element of $H$ whose support is finite and contained in $O$, which
contradicts our assumption on the block $O$. Hence $Q_{1}$ is confining for
$(H,\operatorname{Alt}_{f}(O))$.

Now by construction $F_{1},\ldots,F_{q}$ form a displacement configuration for
$Q_{1}$. Since $|F_{i}|=q+5\geq 5$, every non-trivial element of
$\operatorname{Alt}_{f}(F_{i})$ has a conjugacy class of size $>q$. Hence it
follows that all the assumptions of Theorem 3.5 are satisfied. By applying the
theorem (with $r=q$), we deduce that there is $i$ such that $H$ contains a
non-trivial subgroup of $\operatorname{Sym}(F_{i})$. In particular $H$
contains non-trivial elements with finite support in $O$ since $F_{i}\subset
O$. We have therefore reached a contradiction with the definition of $O$.

It now follows easily that $H$ contains $\mathrm{Alt}_{f}(O_{i,j})$ for all
$i,j$. Indeed, fix a non-trivial element $h_{i,j}\in H$ whose support is
finite and contained in $O_{i,j}$. Then the stabilizer of $O_{i,j}$ in $H$
acts primitively on $O_{i,j}$, and its image in $\operatorname{Sym}(O_{i,j})$
contains the element of finite support $h_{i,j}$, so it follows from Jordan-
Wielandt theorem that this image actually contains $\mathrm{Alt}_{f}(O_{i,j})$
(Theorem 2.3). Now the rigid stabilizer $\operatorname{R}_{H}(O_{i,j})$ is
non-trivial (it contains $h_{i,j}$) and is normalized by
$\mathrm{Alt}_{f}(O_{i,j})$, so it follows that $H$ contains
$\mathrm{Alt}_{f}(O_{i,j})$, as desired. ∎

###### Proof of Theorem 3.7.

$\ref{item-confbysym}\implies\ref{item-confbyalt}$ is trivial. $\ref{item-
confbyalt}\implies\ref{item-nicesbgp}$ follows from Proposition 3.9. For
$\ref{item-nicesbgp}\implies\ref{item-confbysym}$ assume that $H$ satisfies
the condition in iii). Consider a finite subset $F$ of $\Omega$ of cardinality
$2k+1+|\Omega_{H,f}|$, and let $P$ be the set of $3$-cycles with support in
$F$. We claim that $P$ is a confining subset for $H$. Indeed for every
$g\in\operatorname{Sym}(\Omega)$, the intersection of $g(F)$ with
$\Omega_{H,\infty}$ must have cardinality at least $2k+1$, and hence there
exists $i$ such that $|g(F)\cap\Omega_{i}|\geq 3$. It follows that
$gPg^{-1}\cap\operatorname{Alt}_{f}(\Omega_{i})$ is not empty, and hence
$gPg^{-1}\cap H$ is not empty either. ∎

######  Remark 3.10.

Theorem 3.7 provides a description of the confined subgroups for an arbitrary
partially finitary group $G$: a subgroup $H\leq G$ is confined if and only if
$H$ satisfies condition iii) of the theorem. A well-studied family of
partially finitary groups are Houghton’s groups $H_{n}$. Recall that if
$\Gamma_{n}$ is the graph obtained by gluing at one point $n$ infinite one-
sided rays $\Delta_{1},\cdots,\Delta_{n}$, then $H_{n}$ is the group of all
permutations of the vertex set of $\Gamma_{n}$ which, outside of a finite set,
coincide with a translation on each ray $\Delta_{i}$. By the above result we
have a description of all confined subgroups of the groups $H_{n}$.

###  3.2. Application to high transitivity

The following lemma shows that confined subgroups behave well with respect to
a dense embedding of a group $G$ into a topological group $S$.

######  Lemma 3.11.

Let $\varphi:G\to S$ be an injective group homomorphism from a group $G$ into
a topological group $S$ such that the image of $G$ in $S$ is dense. If $H$ is
a confined subgroup of $G$, then $K=\overline{\varphi(H)}$ is a confined
subgroup of $S$, and if $P$ is a confining subset for $(H,G)$ then so is
$\varphi(P)$ for $(K,S)$.

###### Proof.

We have a $G$-equivariant map $\psi:G/H\to S/K$ with dense image. That $P$ is
confining for $(H,G)$ means that the union $\cup_{\sigma\in
P}\mathrm{Fix}_{G/H}(\sigma)$ is equal to the entire $G/H$. Here
$\mathrm{Fix}_{G/H}(\sigma)$ is the set of fixed points of $\sigma$ for the
left action of $G$ on $G/H$. The coset space $S/K$ being Hausdorff, the subset
$\cup_{\sigma\in P}\mathrm{Fix}_{S/K}(\psi(\sigma))$ is closed in $S/K$, and
it contains $\cup_{\sigma\in P}\psi(\mathrm{Fix}_{G/H}(\sigma))=\psi(G/H)$.
Therefore it is also dense, and hence it is equal to $S/K$. So $\varphi(P)$ is
confining for $(K,S)$. ∎

The following covers Theorem 1 from the introduction, which corresponds to the
case where $G$ is not partially finitary.

######  Theorem 3.12.

Suppose that a group $G$ admits a faithful and highly transitive action on a
set $\Omega$. Suppose that $H$ is a confined subgroup of $G$, that $P$ is
confining for $(H,G)$, and let $r$ be the number of elements of $P$ and
$r_{1}$ the number of elements of $P$ with finite support in $\Omega$. Then
the following hold:

  1. i)

$\Omega_{H,f}$ is finite. Moreover if $r_{1}=0$ then $\Omega_{H,f}$ has
cardinality at most $r-1$.

  2. ii)

There exist infinite subsets $\Omega_{1},\ldots,\Omega_{k}$ with with
$k\leq\max(1,r_{1})$ such that
$\Omega_{H,\infty}=\Omega_{1}\cup\ldots\cup\Omega_{k}$, and if $H_{i}$ is the
stabilizer of $\Omega_{i}$ in $H$, then the action of $H_{i}$ on $\Omega_{i}$
is highly transitive for all $i$.

So if $G$ is not partially finitary, then $\Omega_{H,f}$ has cardinality at
most $r-1$, and the action of $H$ on $\Omega\setminus\Omega_{H,f}$ is faithful
and highly transitive

###### Proof.

We write $S=\operatorname{Sym}(\Omega)$, $\varphi:G\to S$ the homomorphism
associated to the action of $G$ on $\Omega$, and $K=\overline{\varphi(H)}$.
Note that $H$ and $K$ have the same orbits in $\Omega$. By Lemma 3.11 the
subset $\varphi(P)$ is confining for $(K,S)$. Hence we are in position to
apply Proposition 3.9. It follows from the proposition that $\Omega_{K,f}$ is
finite, and there exist infinite subsets $\Omega_{1},\ldots,\Omega_{k}$ with
with $k\leq\max(1,r_{1})$ such that
$\Omega_{K,\infty}=\Omega_{1}\cup\ldots\cup\Omega_{k}$, and $K$ contains
$\operatorname{Sym}(\Omega_{i})$ for all $i$ (since $K$ is closed). Therefore
$\Omega_{H,f}=\Omega_{K,f}$ is indeed finite, and the action of $H_{i}$ on
$\Omega_{i}$ is highly transitive. Finally when $r_{1}=0$ we indeed have $k=1$
and $|\Omega_{H,f}|\leq r-1$ by the proposition. This shows i) and ii).

For the last statement, if $G$ is not partially finitary then we may very well
remove the elements of $P$ with finite support, and obtain a subset that is
still confining for $(H,G)$. By the previous paragraph the action of $H$ on
$\Omega_{H,\infty}$ is highly transitive, and it is also faithful since
otherwise $G$ would contain non-trivial elements with finite support, which is
not the case by assumption. ∎

######  Remark 3.13.

The bound $|\Omega_{H,f}|\leq r-1$ in Theorem 3.12 is optimal. In order to see
this, consider a group $\Gamma$ of homeomorphisms of the Cantor space $Y$
acting minimally on $Y$, and let $G$ be the topological full group of
$\Gamma$. We fix a point $y\in Y$, and denote by $\Omega$ the orbit of $y$.
Then one easily checks that the action of $G$ on $\Omega$ is faithful and
highly transitive. Moreover the stabilizer subgroup $H$ of $y$ is confined in
$G$, with a confining subset of cardinality $2$ (indeed if $P$ consists of two
non-trivial elements with disjoint supports, then for every $g$ at least one
element of $gPg^{-1}$ must fix $y$, and hence $H\cap
gPg^{-1}\neq\varnothing$). So in this example $H$ has exactly one fixed point
in $\Omega$. Similarly by considering the stabilizer $H$ of $r-1$ fixed points
in $G$, which admits a confining subset of cardinality $r$, we see that the
bound $|\Omega_{H,f}|\leq r-1$ is optimal.

######  Remark 3.14.

In the case where $G$ is partially finitary, the action of $H$ on
$\Omega_{H,\infty}$ need not be highly transitive, and it is indeed necessary
to pass to a partition of $\Omega_{H,\infty}$. This is illustrated by the
confined subgroups of the group $\operatorname{Alt}_{f}(\Omega)$ described in
Theorem 3.7.

##  4\. Application to groups of PL-homeomorphisms

In this section we give applications of Corollary 2 to groups of piecewise
linear homeomorphisms of suspension flows. First in §4.1 we prove an auxiliary
result on groups of piecewise linear homeomorphisms of an interval. In §4.2 we
recall the setting of [MBT20] and prove Theorem 3. Finally in §4.3 we discuss
a family of examples of groups covered by this theorem and prove that they do
not satisfy the obstructions to high transitivity described in §2.4.

###  4.1. Subquotients of $PL(I)$

In the sequel we let $I=[0,1]$ be the compact unit interval in the real line,
and denote by $\mathsf{PL}(I)$ the group of piecewise linear orientation
preserving homeomorphisms of $I$. These are the homeomorphisms $g$ of $I$
fixing the extremities of $I$ and such that there are $0=x_{0}<\ldots<x_{n}=1$
and real numbers $a_{0},\ldots,a_{n-1},b_{0},\ldots,b_{n-1}$ such that for all
$i$ and all $x\in[x_{i},x_{i+1}]$ one has $g(x)=a_{i}x+b_{i}$.

Recall that Brin and Squier have shown that the group $\mathsf{PL}(I)$ does
not contain any non-abelian free subgroup [BS85]. Following a similar
strategy, we prove a slight generalisation of this result, which places
constraints on the subquotients of $\mathsf{PL}(I)$. Recall also that two
subgroups $M,N$ of a group $G$ are commuting subgroups if every element of $M$
commutes with every element of $N$. Given a group $Q$ we denote by
$Q^{\prime}$ its commutator subgroup.

######  Proposition 4.1.

Let $Q$ be a non-trivial finitely generated subquotient of $\mathsf{PL}(I)$.
Suppose that $Q$ does not admit two non-trivial commuting normal subgroups.
Then $Q^{\prime}$ is not finitely generated.

Note that groups containing non-abelian free subgroups admit all countable
groups as subquotients, so this restriction on the subquotients of
$\mathsf{PL}(I)$ is indeed formally a generalisation of the Brin–Squier
result. Before going into the proof, we isolate the following consequence.

######  Corollary 4.2.

Every finite subquotient of $\mathsf{PL}(I)$ is solvable.

###### Proof of Corollary 4.2.

Assume by contradiction that $Q$ is a non-solvable finite subquotient of
$\mathsf{PL}(I)$ whose cardinality $|Q|$ is minimal. Let $N$ be a maximal
proper normal subgroup of $Q$ and $S=Q/N$ the associated simple quotient.
Suppose $S$ is not abelian. Then one can apply Proposition 4.1 to $S$, and we
deduce that $S^{\prime}=S$ is not finitely generated. Since $S$ is finite by
assumption, this is absurd. Hence $S$ must be abelian of prime order. Since we
chose $Q$ with minimal cardinality, $N$ must be solvable, and thus $Q$ is
solvable as well, which is a contradiction. ∎

Let us now turn to the proof of Proposition 4.1. Recall from §2.1 that we
define the support of an element $g\in\mathsf{PL}(I)$ as the set
$\operatorname{Supp}(g)=\\{x\in I\colon gx\neq x\\}$. Note that
$\operatorname{Supp}(g)$ is an open set; and moreover the definition of
$\mathsf{PL}(I)$ implies that $\operatorname{Supp}(g)$ has only finitely many
connected components. Given $H\leq\mathsf{PL}(I)$, we define its support as
$\operatorname{Supp}(H):=\bigcup_{h\in H}\operatorname{Supp}(h)$. When $H$ is
finitely generated it is enough to take the union for $h$ in a finite
generating set, so in this case $\operatorname{Supp}(H)$ has only finitely
many connected components.

######  Lemma 4.3.

Let $Q$ be a finitely generated subquotient of $\mathsf{PL}(I)$. Suppose that
$Q$ does not admit two non-trivial commuting normal subgroups. Then there
exists a subgroup $L$ of $\mathsf{PL}(I)$ that admits $Q$ as a quotient and
such that the support of $L$ is an open interval.

###### Proof.

Let $H$ be a subgroup of $\mathsf{PL}(I)$ that admits $Q$ as a quotient and
such that the number $n$ of connected components of $\operatorname{Supp}(H)$
is minimal. Observe that $Q$ is the quotient of a finitely generated subgroup
of $\mathsf{PL}(I)$, so $n<\infty$. Suppose for a contradiction that the
statement is not true, that is to say $n\geq 2$. Let $J$ be a connected
component of $\operatorname{Supp}(H)$, and denote by $H_{1}$ the subgroup of
$\mathsf{PL}(I)$ induced by the action of $H$ on $J$, and by $N_{1}$ the
subgroup of $H$ consisting of elements that are supported in the complement of
$J$. We also denote by $H_{2}$ the subgroup of $\mathsf{PL}(I)$ induced by the
action of $H$ on the complement of $J$, and by $N_{2}$ the subgroup of $H$
consisting of elements that are supported in $J$. Note that $H/N_{i}$ is
isomorphic to $H_{i}$ for $i=1,2$.

Let $N$ be a normal subgroup of $H$ such that $H/N$ is isomorphic to $Q$. By
the choice of $H$, neither $H_{1}$ nor $H_{2}$ can admit $Q$ as a quotient
since the number connected components of their support is $<n$. That means
that neither $N_{1}$ nor $N_{2}$ is contained in $N$. Since $N_{1}$ and
$N_{2}$ commute, it follows that their images in $Q$ are commuting non-trivial
normal subgroups, which is impossible by our assumption. Hence we have reached
a contradiction, and hence $n$ must equal to $1$. ∎

The following lemma is based on the same argument as in [BS85].

######  Lemma 4.4.

Let $H$ be a subgroup of $\mathsf{PL}(I)$ such that the support of $H$ is an
open interval. Suppose that $H$ admits the group $Q$ as a quotient. Then for
every finitely generated subgroup $K\leq Q^{\prime}$, there exists $g\in Q$
such that $gKg^{-1}$ and $K$ commute. In particular if $Q^{\prime}$ is
finitely generated, then the group $Q$ is metabelian.

###### Proof.

Assume that $\operatorname{Supp}(H)=(c,d)$, with $c<d$. Let $\pi:H\to Q$ be a
surjective group homomorphism. Suppose $q_{1},\ldots,q_{n}$ are generators of
$K$. Since $K$ lies in $Q^{\prime}$, one can find $h_{1},\ldots,h_{n}$ in
$H^{\prime}$ such that $\pi(h_{i})=q_{i}$ for all $i$. Since $h_{i}$ is in
$H^{\prime}$, the derivative of $h_{i}$ at the extreme points $c$ and $d$ must
be equal to 1. Thus the closure of $\operatorname{Supp}(h_{i})$ is contained
in $\operatorname{Supp}(H)$, so that we can find $c<x<y<d$ such that for all
$i$ one has $h_{i}(t)=t$ for all $t\leq x$ and all $t\geq y$. Since $H$ acts
without fixed points in $(c,d)$, one can find $g\in H$ such that $g(x)\geq y$,
and it follows that $g(\operatorname{Supp}(h_{i}))$ is disjoint from
$\operatorname{Supp}(h_{j})$ for all $i,j$. In particular $gh_{i}g^{-1}$
commutes with $h_{i}$, and hence $\pi(g)K\pi(g)^{-1}$ and $K$ commute. This
shows the first statement, and the second statement follows by taking
$K=Q^{\prime}$. ∎

###### Proof of Proposition 4.1.

Since $Q$ is not trivial and $Q$ does not admit a non-trivial abelian normal
subgroup, the group $Q$ is not metabelian, and the statement then follows from
Lemmas 4.3 and 4.4. ∎

###  4.2. Groups of $PL$-homeomorphisms of suspension flows

We start by recalling the setting of [MBT20]. Throughout this section we let
$X$ be a Stone space, i.e. a compact totally disconnected space, and
$\varphi\colon X\to X$ be a homeomorphism. The dynamical system $(X,\varphi)$
will be called a Stone system. We denote $Y^{\varphi}$ the suspension (or
mapping torus) of $(X,\varphi)$, which is defined as the quotient

$Y^{\varphi}:=(X\times\mathbb{R})/\mathbb{Z}$

with respect to the action of $\mathbb{Z}$ on $X\times\mathbb{R}$ given by
$n\cdot(x,t)=(\varphi^{n}(x),t-n)$. We denote by $\pi\colon
X\times\mathbb{R}\to Y^{\varphi}$ the projection, and by $[x,t]:=\pi(x,t)$.
The suspension is endowed with a flow

$\Phi\colon\mathbb{R}\times Y^{\varphi}\to
Y^{\varphi},\quad\Phi^{t}([x,s])=[x,s+t].$

The group of self-flow-equivalences of $(X,\varphi)$ is the group
$\mathsf{H}(\varphi)$ of all homeomorphisms of $Y^{\varphi}$ which send every
$\Phi$-orbit to a $\Phi$-orbit in an orientation preserving way. This is a
well-studied object in symbolic dynamics, see [BCE17, BC18, APP16]. We let
$\mathsf{H}_{0}(\varphi)$ be its subgroup consisting of homeomorphisms
isotopic to the identity.

######  Remark 4.5.

Note that elements of $\mathsf{H}_{0}(\varphi)$ preserve all $\Phi$-orbits,
since these are precisely the path-components of $Y^{\varphi}$. The converse
is false, namely a homeomorphism of $Y^{\varphi}$ which preserves every
$\Phi$-orbit is not necessarily in $\mathsf{H}_{0}(\varphi)$ [BCE17]; however,
this holds true in the important case where $(X,\varphi)$ is minimal (i.e. all
its orbits are dense), see Aliste-Prieto and Petite [APP16]. We also mention
the following equivalent characterisation of $\mathsf{H}_{0}(\varphi)$, proven
in [BCE17]: a homeomorphism $g$ of $Y^{\varphi}$ belongs to
$\mathsf{H}_{0}(\varphi)$ if and only if there exists a continuous function
$\alpha\colon Y^{\varphi}\to\mathbb{R}$ such that $g(y)=\Phi^{\alpha(y)}(y)$
for all $y\in Y^{\varphi}$.

Given a clopen set $C\subset X$, its smallest return time is :

$\tau_{C}:=\min\\{n\geq 1\colon\varphi^{n}(C)\cap
C\neq\varnothing\\}\in\mathbb{N}\cup\\{\infty\\}.$

Assume that $I\leq\mathbb{R}$ is an interval with length $|I|<\tau_{C}$. In
this case we will say that the pair $(C,I)$ is $\varphi$-admissible; note that
this is automatically the case if $|I|<1$. If $(C,I)$ is $\varphi$-admissible,
then the restriction $\pi_{C,I}:=\pi|_{C\times I}$ is injective, and is called
a chart. We denote by $U_{C,I}\subset Y^{\varphi}$ its image, which by abuse
of terminology will also be called a chart.

A homeomorphism $f\colon I\to J$ between two bounded intervals
$I,J\subset\mathbb{R}$ is said to be PL if it is piecewise linear, with
finitely many discontinuity points for the derivative. Moreover, we say that
$f$ is PL-dyadic if all the discontinuity points of its derivative are dyadic
rational, and in restriction to each interval of continuity of the derivative
it is of the form $x\mapsto ax+b$, with $a=2^{n},n\in\mathbb{Z}$, and
$b\in\mathbb{Z}[\frac{1}{2}]$. Recall that Thompson’s group $F$ is the group
of all PL-dyadic homeomorphisms of the interval $(0,1)$, and Thompson’s group
$T$ is the group of all PL-dyadic homeomorphisms of the circle
$\mathbb{R}/\mathbb{Z}$.

######  Definition 4.6 ([MBT20]).

We let $\mathsf{PL}(\varphi)$ be the subgroup of $\mathsf{H}_{0}(\varphi)$
consisting of all elements $g$ with the following property: for every $y\in
Y^{\varphi}$, there exist a clopen set $C\subset X$, two dyadic intervals
$I,J$ such that $(C,I)$ and $(C,J)$ are $\varphi$-admissible, and a PL-
homeomorphism $f\colon I\to J$ such that $y\in U_{C,I},g(U_{C,I})=U_{C,J}$ and
the restriction of $g$ to $U_{C,I}$ is given in coordinates by

$\pi_{C,J}^{-1}\circ g\circ\pi_{C,I}=\operatorname{Id}\times f\colon C\times
I\to C\times J.$

We further let $\mathsf{T}(\varphi)$ be the subgroup of $\mathsf{PL}(\varphi)$
defined in a similar way but with the additional requirement that $f$ is PL-
dyadic.

A proof of the following result can be found in [MBT20].

######  Theorem 4.7.

If $\varphi$ is a minimal homeomorphism of $X$, then $\mathsf{T}(\varphi)$ is
a simple group. If $(X,\varphi)$ is conjugate to a subshift, then
$\mathsf{T}(\varphi)$ is finitely generated.

For every $\varphi$-admissible pair $(C,I)$, we can define a subgroup
$\mathsf{PL}_{C,I}\leq\mathsf{PL}(\varphi)$ isomorphic to $\mathsf{PL}(I)$ and
supported in the chart $U_{C,I}$. The group $\mathsf{PL}_{C,I}$ is defined as
the group of all homeomorphisms that act as the identity on the complement of
$U_{C,I}$ and that are given in coordinates by $\operatorname{Id}\times f$ on
$U_{C,I}$ for some $f\in\mathsf{PL}(I)$. We also let $F_{C,I}$ be the subgroup
of $\mathsf{PL}_{C,I}$ defied by the additional requirement that $f$ is PL-
dyadic. Note that $F_{C,I}$ is always isomorphic to the group $F_{I}$ of PL-
dyadic homeomorphisms of $I$; in particular if the endpoints of $I$ are dyadic
rationals then $F_{I}$ is isomorphic to Thompson’s group $F:=F_{(0,1)}$. It is
shown in [MBT20] that the groups $F_{C,I}$ generate $\mathsf{T}(\varphi)$ when
$(C,I)$ varies over all admissible pairs.

The following lemma is essentially Lemma 7.2 in [MBT20]. It analyses the
structure of the germ stabilizers $\mathsf{PL}(\varphi)_{y}^{0}$ for $y\in
Y^{\varphi}$ when $\varphi$ is minimal.

######  Lemma 4.8.

Assume that $\varphi$ is a minimal homeomorphism of $X$, and fix $y\in
Y^{\varphi}$. Let $S\subset\mathsf{PL}(\varphi)_{y}^{0}$ be a finite subset.
Then there exist disjoint charts $U_{C_{1},I_{1}},\ldots,U_{C_{n},I_{n}}$
whose closure does not contain $y$ such that $S$ is contained in the subgroup
$\langle\mathsf{PL}_{C_{1},I_{1}},\ldots,\mathsf{PL}_{C_{n},I_{n}}\rangle\simeq\mathsf{PL}({I_{1}})\times\cdots\times\mathsf{PL}({I_{n}})$.

###### Proof.

The analogous statement for the group $\mathsf{T}(\varphi)$ is [MBT20, Lemma
7.2], and the same proof works for the whole group $\mathsf{PL}(\varphi)$. For
completeness we recall a sketch of the argument. Since $S$ is a finite subset
of $\mathsf{PL}(\varphi)_{y}^{0}$, its elements must act trivially on a chart
$U_{C,J}\ni y$ for some interval $J=(a,b)$ with $|J|=b-a<1$. Consider the
first return time function $T_{C}\colon C\to\mathbb{N}_{>0}$, namely
$T_{C}(x):=\min\\{n\geq 1\colon\varphi^{n}(x)\in C\\}$. By minimality, this
function is bounded and locally constant on $C$, so we can find a clopen
partition $C=C_{1}\sqcup\cdots\sqcup C_{n}$ such that $T_{C}\equiv t_{i}$ on
$C_{i}$, with $t_{1},\cdots,t_{n}\in\mathbb{N}_{>0}$. For each $i$ consider
the interval $I_{i}=(b,t_{i}-a)$, and note that $|I_{i}|=t_{i}-|J|$. By
construction we have
$Y^{\varphi}\setminus\overline{U}_{C,J}=\bigsqcup_{i=1}^{n}U_{C_{i},I_{i}}$,
and the group generated by $S$ preserves each chart $U_{C_{i},I_{i}}$. After
taking a refinement of the partition $C=C_{1}\sqcup\cdots\sqcup C_{n}$ if
necessary, we reach the conclusion of the lemma. ∎

We are now in position to prove Theorem 3 from the introduction, which is
recalled here.

######  Theorem 4.9.

Let $(X,\varphi)$ be a minimal Stone system, and let
$G\leq\mathsf{PL}(\varphi)$ be any subgroup whose action on $Y^{\varphi}$ is
topologically nowhere free. Then $G$ does not admit any faithful highly
transitive action. In particular this holds true for the group
$G=\mathsf{T}(\varphi)$.

###### Proof.

Note that the group $\mathsf{PL}(\varphi)$ is torsion-free, and hence does not
contain any finitary alternating group. The germ stabilizers $G_{y}^{0}$ are
confined in $G$ since the action is topologically nowhere free (Lemma 2.1).
Moreover every finitely generated subgroup of $G_{y}^{0}$ is isomorphic to a
subgroup of $\mathsf{PL}(I)$ by Lemma 4.8, and hence has all its finite
subquotients solvable by Corollary 4.2. Therefore $G_{y}^{0}$ cannot admit any
faithful highly transitive action (see §2.4). The conclusion then follows from
Corollary 2. ∎

###  4.3. A variant on a non-orientable quotient of $Y^{\varphi}$

It is natural to compare Theorem 4.9 with the criteria to exclude high
transitivity of a group $G$ described in §2.4. For instance the groups
$\mathsf{T}(\varphi)$ always contain non-abelian free subgroups [MBT20,
Corollary 8.2], and thus do not satisfy any restriction on their subquotients.
Moreover their simplicity rules out the existence of a similar restriction in
a non-trivial normal subgroup. It is not clear whether $\mathsf{T}(\varphi)$
can satisfy a non-trivial mixed identity. In this subsection, we describe a
related family of subgroups of $\mathsf{PL}(\varphi)$ covered by Theorem 4.9,
which retain the same main properties, and for which we can exclude this as
well. This family is inspired by Hyde and Lodha’s examples of finitely
generated simple groups acting on the real line [HL19], which can be embedded
in $\mathsf{PL}(\varphi)$ for a suitable $(X,\varphi)$. In fact, relying on
subsequent work of the same authors and Navas and Rivas [HLNR21], one can show
that the Hyde–Lodha examples arise as special cases of the construction
described here (see Remark 4.19 below).

The main idea is to consider a 2-to-1 quotient of the space $Y^{\varphi}$ in
such a way that the natural partition of $Y^{\varphi}$ into $\Phi$-orbits
passes to the quotient, but no longer admits an orientation coming from a
globally defined flow. To this end we let again $X$ be a Stone space, and
consider an action $D_{\infty}\curvearrowright X$ of the infinite dihedral
group $D_{\infty}:=\mathbb{Z}\rtimes\mathbb{Z}/2\mathbb{Z}$. Such an action is
given by a pair of homeomorphisms $(\varphi,\sigma)$ of $X$ such that
$\sigma^{2}=\operatorname{Id}$ and $\sigma\varphi\sigma=\varphi^{-1}$.
Explicitly an element $(n,j)\in
D_{\infty}=\mathbb{Z}\rtimes\mathbb{Z}/2\mathbb{Z}$ acts on $x$ by $(n,j)\cdot
x=\varphi^{n}\sigma^{j}(x)$. We call the triple $(X,\varphi,\sigma)$ a Stone
$D_{\infty}$-system.

_Throughout this section, we make the standing assumption that the action of
$D_{\infty}$ defined by $(\varphi,\sigma)$ is free._

Consider the diagonal action of $D_{\infty}$ on $X\times\mathbb{R}$, where the
action on $\mathbb{R}$ is the natural action by isometries given by by
$(n,j)\cdot t:=(-1)^{j}t-n$ for $(n,j)\in D_{\infty},t\in\mathbb{R}$. We
define a space $Y^{\varphi,\sigma}$ as the quotient

$Y^{\varphi,\sigma}=(X\times\mathbb{R})/D_{\infty}.$

We let again $\pi\colon X\times\mathbb{R}\to Y^{\varphi,\sigma}$ be the
quotient map, and set $[x,t]=\pi(x,t)$.

Note that given $(X,\varphi,\sigma)$ we can also consider the Stone system
$(X,\varphi)$ obtained by dropping $\sigma$. Then $\sigma$ naturally induces a
homeomorphism $\hat{\sigma}$ of order 2 of $Y^{\varphi}$ given by:

(1) $\hat{\sigma}\colon Y^{\varphi}\to
Y^{\varphi},\quad\hat{\sigma}([x,t])=[\sigma(x),-t],$

and we have $Y^{\varphi,\sigma}=Y^{\varphi}/\langle\hat{\sigma}\rangle$.

For $x\in X$ the restriction of $\pi$ to $\\{x\\}\times\mathbb{R}$ is
injective, and we denote $\ell_{x}$ its image, and let
$\gamma_{x}\colon\mathbb{R}\to\ell_{x}$ be the natural parametrisation

(2) $\gamma_{x}\colon\mathbb{R}\to\ell_{x},\quad\gamma_{x}(t)=\pi(x,t).$

Subsets $\ell\subset Y^{\varphi,\sigma}$ of the form $\ell=\ell_{x}$ for some
$x$ will be called leaves. Note that $Y^{\varphi,\sigma}$ is a disjoint union
of leaves, which coincide with the path-connected components of
$Y^{\varphi,\sigma}$. However note that the parametrisation (2) depends on the
choice of a point $x\in X$ such that $\ell=\ell_{x}$; a different choice of
$x$ amounts to precomposing $\gamma_{x}$ by an element of $D_{\infty}$. Since
the action of $D_{\infty}$ on $\mathbb{R}$ contains reflections, leaves _do
not_ have a well-defined preferred orientation.

Given a clopen subset $C\subset X$ and an interval $I$, we say that the pair
$(C,I)$ is $(\varphi,\sigma)$-admissible if the restriction of $\pi$ to
$C\times I$ is injective; equivalently if $C\times I$ is disjoint from its
image under all non-trivial elements of $D_{\infty}$. If $(C,I)$ is an
admissible pair, we denote by $\pi_{C,I}$ the restriction of the quotient map
$\pi$ to $C\times I$, and by $U_{C,I}\subset Y^{\varphi,\sigma}$ its image. In
this context, the map $\pi_{C,I}$, or simply its image $U_{C,I}$, will be
called a chart.

The following lemma guarantees that charts form indeed a basis for the
topology of $Y^{\varphi,\sigma}$, and moreover a bounded segment on a leaf can
always be “thickened” to a chart.

######  Lemma 4.10.

For every $(x,t)\in X\times\mathbb{R}$, and for every bounded interval
$I\subset\mathbb{R}$ containing $t$, there exists a clopen neighbourhood
$C\subset X$ of $x$ such that $(C,I)$ is $(\varphi,\sigma)$-admissible.

###### Proof.

Since the isometric action of $D_{\infty}$ on $\mathbb{R}$ is proper, the set
$F:=\\{g\in D_{\infty}\setminus\\{1\\}\colon g\cdot I\cap I\neq\varnothing\\}$
is finite. Since moreover the action of $D_{\infty}$ on $x$ is free, we have
$g(x)\neq x$ for every $g\in F$. Thus there exists a clopen neighbourhood $C$
of $x$ such that $g(C)\cap C=\varnothing$ for every $g\in F$. Now for $g\in
D_{\infty}$ the set $g\cdot(C\times I)=(g\cdot C)\times(g\cdot I)$, is
disjoint from $C\times I$: if $g\notin F$ this follows looking at the second
factor, and for $g\in F$ it follows by looking at the first factor. Thus the
pair $(C,I)$ is $(\varphi,\sigma)$-admissible. ∎

The following definition is analogous to Definition 4.6.

######  Definition 4.11.

We let $\mathsf{PL}(\varphi,\sigma)$ be the subgroup of
$\mathsf{H}_{0}(\varphi,\sigma)$ consisting of elements $g$ such that for
every $y\in Y^{\varphi,\sigma}$, there exist $(\varphi,\sigma)$-admissible
pairs $(C,I)$ and $(C,J)$ such that $y\in U_{C,I}$, $g(U_{C,I})=U_{C,J}$ and
there exists a PL homeomorphism $f\colon I\to J$ such that the restriction of
$g$ to $U_{C,I}$ is given in coordinates by

$\pi_{C,J}\circ g\circ\pi_{C,I}^{-1}=\operatorname{Id}\times f\colon C\times
I\to C\times J.$

We let $\mathsf{T}(\varphi,\sigma)$ be the subgroup of
$\mathsf{PL}(\varphi,\sigma)$ defined by the additional requirement that $f$
is PL-dyadic.

For every $(\varphi,\sigma)$-admissible pair, we have subgroups
$\mathsf{PL}_{C,I}\leq\mathsf{PL}(\varphi,\sigma)$ and
$F_{C,I}\leq\mathsf{T}(\varphi,\sigma)$ defined analogously as in the previous
subsection.

By viewing $Y^{\varphi,\sigma}=Y^{\varphi}/\langle\hat{\sigma}\rangle$ as in
(1), the group $\mathsf{PL}(\varphi,\sigma)$ can alternatively be seen as a
subgroup of $\mathsf{PL}(\varphi)$.

######  Proposition 4.12.

The group $\mathsf{PL}(\varphi,\sigma)$ lifts to a group of homeomorphisms of
$Y^{\varphi}$, namely the subgroup of $\mathsf{PL}(\varphi)$ consisting of
elements that commute with $\hat{\sigma}$.

###### Proof.

It is clear that every element of $\mathsf{PL}(\varphi)$ which commutes with
$\hat{\sigma}$ descends to a homeomorphism of $Y^{\varphi,\sigma}$ that
belongs to $\mathsf{PL}(\varphi,\sigma)$. Conversely, given
$g\in\mathsf{PL}(\varphi,\sigma)$ we check that $g$ lifts to an element
$\tilde{g}\in\mathsf{PL}(\varphi)$ which commutes with $\hat{\sigma}$. By
definition we can find two covers
$Y^{\varphi,\sigma}=\bigcup_{i=1}^{n}U_{C_{i},I_{i}}=\bigcup_{i=1}^{n}U_{C_{i},J_{i}}$
by charts, and PL-homeomorphisms $f_{i}\colon I_{i}\to J_{i},i=1,\ldots,n$,
such that $g(U_{C_{i},I_{i}})=U_{C_{i},J_{i}}$ as in the definition of
elements of $\mathsf{PL}(\varphi,\sigma)$. Each pair $(C_{i},I_{i})$ defines
also a chart $U^{\varphi}_{C_{i},I_{i}}\subset Y^{\varphi}$ (here we use the
notation $U^{\varphi}_{C,I}$ for charts in $Y^{\varphi}$ to distinguish them
from charts in $Y^{\varphi,\sigma}$). Now the fact that $(C_{i},I_{i})$ is a
$(\varphi,\sigma)$-admissible pair for every $i$ is equivalent to the fact
that $\hat{\sigma}(U^{\varphi}_{C_{i},I_{i}})\cap
U^{\varphi}_{C_{i},I_{i}}=\varnothing$. Noting that
$\hat{\sigma}(U_{C_{i},I_{i}})=U_{\sigma(C_{i}),-I_{i}}$, and using that the
charts $U_{C_{i},I_{i}}$ and $U_{C_{i},J_{i}}$ form covers of
$Y^{\varphi,\sigma}$, we have two covers
$Y^{\varphi}=\bigcup_{i=1}^{n}\left(U^{\varphi}_{C_{i},I_{i}}\cup
U^{\varphi}_{\hat{\sigma}(C_{i}),-I_{i}}\right)=\bigcup_{i=1}^{n}\left(U^{\varphi}_{C_{i},J_{i}}\cup
U^{\varphi}_{\hat{\sigma}(C_{i}),-J_{i}}\right)$. Then we can consider the
element $\tilde{g}\in\mathsf{PL}(\varphi)$ which in restriction to each chart
$U^{\varphi}_{C_{i},I_{i}}$ is given in coordinates by the homeomorphism
$f_{i}\colon I_{i}\to J_{i}$, and in restriction to each chart
$U^{\varphi}_{\sigma(C_{i}),-I_{i}}$ is given by $\bar{f_{i}}\colon-I_{i}\to
J_{i}$, where $\bar{f}(t):=-f(-t)$. It is routine to check that $\tilde{g}$ is
a well-defined lift of $g$, and this gives the desired conclusion. ∎

The goal of the remaining part of this section is to prove the following.

######  Theorem 4.13.

Let $(X,\varphi,\sigma)$ be a free Stone $D_{\infty}$-system such that
$(X,\varphi)$ is minimal. Then the group $\mathsf{T}(\varphi,\sigma)$
satisfies the following:

  1. (i)

it does not admit any faithful highly transitive action;

  2. (ii)

it does not satisfy any non-trivial mixed identity;

  3. (iii)

it contains non-abelian free subgroups.

We will prove each property separately. The first part is a direct application
of Theorem 4.9, thanks to Proposition 4.12.

###### Proof of Theorem 4.13 (i).

Choose $\tilde{y}\in Y^{\varphi}$ which projects to $y$. By Proposition 4.12,
the group $G=\mathsf{T}(\varphi,\sigma)$ lifts to an isomorphic subgroup
$\tilde{G}\leq\mathsf{PL}(\varphi)$. Moreover for every $y\in
Y^{\varphi,\sigma}$, and every $\tilde{y}\in Y^{\varphi}$ which projects to
$y$ we have $G_{y}^{0}=\tilde{G}^{0}_{\tilde{y}}$. Hence $\tilde{G}$ has non-
trivial germ-stabilizers. The conclusion then follows from Theorem 4.9. ∎

We now turn to the absence of non-trivial mixed identity. To this end, we
first rule out the existence of a specific type of non-trivial mixed
identities within the group $\operatorname{Homeo}_{0}(\mathbb{R})$ of
orientation-preserving homeomorphisms of $\mathbb{R}$. For
$f\in\operatorname{Homeo}_{0}(\mathbb{R})$, we say that $f$ is unboundedly
positive if the set $\\{t\in\mathbb{R}\colon f(t)>t\\}$ accumulates to both
$+\infty$ and $-\infty$. We say that $f$ is unboundedly negative if $f^{-1}$
is unboundedly positive.

Assume that $G$ is a group and $w$ is an element of the free product $\langle
z\rangle*G$, where $z$ is a generator of an infinite cyclic group. For $h\in
G$ we denote $w(h)\in G$ the image of $w$ under the homomorphism from $\langle
z\rangle*G$ to $G$ which is the identity on $G$ and maps $z$ to $h$. By
definition the group $G$ satisfies the mixed identity $w=1$ if $w(h)=1$ for
every $h\in G$.

######  Lemma 4.14.

Let $w\in\langle z\rangle*\operatorname{Homeo}_{0}(\mathbb{R})$ be a non-
trivial element given by the reduced word $w=z^{n_{k}}g_{k}\cdots
z^{n_{1}}g_{1}$, where $n_{1},\cdots,n_{k}\neq 0$ and
$g_{1},\cdots,g_{k}\in\operatorname{Homeo}_{0}(\mathbb{R})$ are all
unboundedly positive. Then there exists
$h\in\operatorname{Homeo}_{0}(\mathbb{R})$ such that $w(h)\neq 1$.

Note that the lemma is also true if we assume instead that
$g_{1},\cdots,g_{k}$ are unboundedly negative.

###### Proof.

If $k=1$ the conclusion is obvious, so we can assume that $k\geq 2$. Using the
assumption, we can find points $t_{1},\cdots,t_{k}\in\mathbb{R}$ such that

$t_{1}<g_{1}(t_{1})<t_{2}<g_{2}(t_{2})<\cdots<t_{k}<g_{k}(t_{k}).$

Choose points $p_{i}\in(t_{i},g_{i}(t_{i}))$ for $i=1,\cdots,k$. Choose an
element $h\in\operatorname{Homeo}_{0}(\mathbb{R})$ which fixes $p_{i}$ for
$i=1,\cdots,k$, acts trivially on the interval $(-\infty,p_{1})$ and on
$(p_{k},+\infty)$, and such that for $i=1,\cdots,k$ it satisfies
$h(g_{i}(t_{i}))>t_{i+1}$ if $n_{i}>0$, and $h(t_{i+1})<g_{i}(t_{i})$ if
$n_{i}<0$. Note that in both cases, we have $h^{n_{i}}(g_{i}(t_{i}))>t_{i+1}$
for $i=1,\cdots,k-1$. Then one readily checks that $w(h)(t_{1})\geq
t_{k}>t_{1}$. In particular $w(h)\neq 1$ as desired. ∎

######  Remark 4.15.

If we drop the assumption on the parameters $g_{1},\cdots,g_{k}$, the
conclusion is false. Indeed the group $\operatorname{Homeo}_{0}(\mathbb{R})$
does satisfy non-trivial mixed identities [Zar10].

For every $x\in X$ the group $\mathsf{T}(\varphi,\sigma)$ preserves the leaf
$\ell_{x}\subset Y^{\varphi,\sigma}$. Its action on $\ell_{x}$ provides a
representation
$\rho_{x}\colon\mathsf{T}(\varphi,\sigma)\to\operatorname{Homeo}_{0}(\mathbb{R})$
given by

$\rho_{x}(g)=\gamma_{x}^{-1}\circ g|_{\ell_{x}}\circ\gamma_{x},$

where $\gamma_{x}$ is the parametrisation (2). The representations $\rho_{x}$
have the following properties.

######  Lemma 4.16.

Let $(X,\varphi,\sigma)$ be a free Stone $D_{\infty}$-system. Fix $x\in X$,
and consider the representation
$\rho_{x}\colon\mathsf{T}(\varphi,\sigma)\to\operatorname{Homeo}_{0}(\mathbb{R})$.
Then:

  1. (i)

The image of $\rho_{x}$ is dense in $\operatorname{Homeo}_{0}(\mathbb{R})$
endowed with the compact-open topology.

  2. (ii)

If $(X,\varphi)$ is minimal, then $\rho_{x}$ is faithful and the image of
every non-trivial element is both unboundedly positive and unboundedly
negative.

###### Proof.

(i) Let $h\in\operatorname{Homeo}_{0}(\mathbb{R})$. By definition of the
compact-open topology, we need to show that for every compact interval
$K\subset\mathbb{R}$ and every $\varepsilon>0$ there exists
$g\in\mathsf{T}(\varphi,\sigma)$ such that
$|\rho_{x}(g)(t)-h(t)|\leq\varepsilon$ for every $t\in K$. Let
$I\subset\mathbb{R}$ be a large enough bounded open interval such that $0\in
I$ and $K\cup h(K)\subset I$. By Lemma 4.10 we can find a clopen neighbourhood
$C$ of $x$ such that the pair $(C,I)$ is $(\varphi,\sigma)$-admissible. Then
the image under $\rho_{x}$ of the group
$F_{C,I}\leq\mathsf{T}(\varphi,\sigma)$ preserves $I$ and its action induces
the standard action of $F_{I}$. Then, using that $F_{I}$ is dense in
$\operatorname{Homeo}_{0}(I)$, we can find $g\in F_{C,I}$ such that
$|\rho_{x}(g)(t)-h(t)|\leq\varepsilon$ for every $t\in K$, as desired.

(ii) If $(X,\varphi)$ is minimal, then every leaf $\ell_{x}$ is dense in
$Y^{\varphi,\sigma}$, and thus the action of $\mathsf{T}(\varphi,\sigma)$ on
it is faithful. Choose $g\in\mathsf{T}(\varphi,\sigma)$ non-trivial and
$y\in\ell_{x}$ such that $g(y)\neq y$. Write $y=\gamma_{x}(t_{0})$ for some
$t_{0}\in\mathbb{R}$. Choose also a chart $U_{C,I}$ containing $y$, with
$(x,t_{0})\in C\times I$, such $g$ is given in coordinates by
$\operatorname{Id}\times f$ for some PL-dyadic homeomorphism $f\colon I\to J$.
Note that $g(y)\neq y$ implies that $f(t_{0})\neq t_{0}$, so that upon
restricting $I$ we can assume without loss of generality that $f$ has no fixed
points in $I$. Moreover, after replacing $g$ by its inverse if necessary, we
can assume that $f(t)>t$ for every $t\in I$. By construction, the restriction
of $\rho_{x}(g)$ to $I$ coincides with $f$.

Consider now the following two sets of integers:

$E_{+}=\\{n\in\mathbb{Z}\colon\varphi^{n}(x)\in C\\},\quad\quad
E_{-}=\\{n\in\mathbb{Z}\colon\varphi^{n}\sigma(x)\in C\\}.$

By minimality of $(X,\varphi)$ both $E_{+}$ and $E_{-}$ contain sequences
accumulating to both $+\infty$ and $-\infty$. For $n\in E_{+}$ let $I_{n}$ be
the image of $I$ under the translation $t\mapsto t+n$ (that is which
corresponds to the element $(n,0)^{-1}\in D_{\infty}$). For $n\in E_{-}$ let
${I_{n}}$ be the image of $I$ under the map $t\mapsto-t-n$ (that is the
element $(n,1)^{-1}=(n,1)\in D_{\infty}$). Note that the set $\bigcup_{n\in
E_{+}\cup E_{-}}I_{n}$ is precisely the sets of times $t\in\mathbb{R}$ such
that $\gamma_{x}(t)\in U_{C\times I}$. More precisely for $t\in I_{n}$ with
$n\in E_{+}$ we have $\gamma_{x}(t)=\pi(\varphi^{n}(x),t-n)$, so that
$\gamma_{x}(t)$ runs through the arc $\pi(\\{\varphi^{n}(x)\\}\times I)$
according to the orientation given by the orientation of $I$. Instead if $t\in
I_{n}$ with $n\in E_{-}$ we have
$\gamma_{x}(t)=\pi(\varphi^{n}\sigma(x),-t-n)$, so that $\gamma_{x}(t)$ runs
through the arc $\pi(\\{\varphi^{n}\sigma(x)\\}\times I)$ in the opposite
orientation. Now note that for $n\in E_{+}$ the restriction of $\rho_{x}(g)$
to the interval $I_{n}$ coincides with the conjugate of $f$ by the translation
$t\mapsto t+n$, and thus satisfies $\rho_{x}(g)(t)>t$ for every $t\in I_{n}$.
In contrast, when $n\in E_{-}$ the restriction of $\rho_{x}(g)$ to ${I}_{n}$
coincides with the conjugate of $f$ under the orientation reversing map
$t\mapsto-t-n$, and therefore satisfies $\rho_{x}(g)(t)<t$ for $t\in I_{n}$.
Since both sets $E_{+}$ and $E_{-}$ are unbounded from above and from below,
we deduce that $\rho_{x}(g)$ is unboundedly positive and unboundedly negative.
∎

###### Proof of Theorem 4.13 (ii).

We write $G=\mathsf{T}(\varphi,\sigma)$. According to [HO16, Remark 5.1], it
is enough to prove that there does not exist a non-trivial element
$w\in\mathbb{Z}\ast G$ such that $G$ satisfies the mixed-identity $w=1$.
Assume that $w\in\langle z\rangle*G$ is a non-trivial reduced word of the form
$w(z)=z^{n_{k}}g_{k}\cdots z^{n_{1}}g_{1}$, with $g_{1},\cdots,g_{k}\in G$
such that $w(h)=1$ for every $h\in G$. By Lemma 4.16 we can embed $G$ as a
dense subgroup of $\operatorname{Homeo}_{0}(\mathbb{R})$ such that all non-
trivial elements are unboundedly positive (and unboundedly negative). By
density this implies that $w(h)=1$ for every
$h\in\operatorname{Homeo}_{0}(\mathbb{R})$, which is in contradiction with
Lemma 4.14. ∎

We now wish to prove that the group $\mathsf{T}(\varphi,\sigma)$ contains free
subgroups. To this end, recall that an action of a group $G$ on a compact
space $Y$ is said to be extremely proximal if for every proper closed subset
$Z\subsetneq Y$, there exists a point $y\in Y$ such that for every
neighbourhood $V$ of $y$ there exists $g\in G$ such that $g(Z)\subset V$. If
an action of a group $G$ on a compact space is minimal and extremely proximal,
a well-known ping-pong argument implies that $G$ admits non-abelian free
subgroups [Gla74, Theorem 3.4]. Thus, it is enough to show the following.

######  Proposition 4.17.

Let $(X,\varphi,\sigma)$ be a minimal Stone $D_{\infty}$-system. Then the
action of $\mathsf{T}(\varphi,\sigma)$ on $Y^{\varphi,\sigma}$ is minimal and
extremely proximal.

###### Proof.

The minimality of $(X,\varphi,\sigma)$ implies that all leaves of
$Y^{\varphi,\sigma}$ are dense. Since it is not difficult to see (for instance
from part (i) in Lemma 4.16) that the $\mathsf{T}(\varphi,\sigma)$-orbit of
every $y\in Y^{\varphi,\sigma}$ is dense in its leaf, it follows that the
action of $\mathsf{T}(\varphi,\sigma)$ on $Y^{\varphi,\sigma}$ is minimal. To
show that it is extremely proximal, we will use an argument similar to the
proof of Lemma 4.8. Let $Z\subsetneq Y^{\varphi,\sigma}$ be a closed subset,
and fix $y\in Y^{\varphi,\sigma}\setminus Z$. Let $V$ be a neighbourhood of
$y$ such that $V\cap Z=\varnothing$. Let $U_{C,I}$ be an open chart containing
$y$ and such that $\overline{U_{C,I}}\subset V$. Then, using that every leaf
that exits $U_{C,I}$ in both directions must return to it, and reasoning in a
similar way as in the proof of Lemma 4.8, we can find a decomposition of the
complement $Y^{\varphi,\sigma}\setminus\overline{U_{C,I}}$ as a disjoint union
of open charts

$Y^{\varphi,\sigma}\setminus\overline{U_{C,I}}=\bigsqcup_{i=1}^{n}U_{C_{i},I_{i}},$

such that the boundary of $\partial U_{C_{i},I_{i}}$ is contained in the
boundary of $\partial U_{C,I}$ (the only difference here from Lemma 4.8 is
that it may happen that the two components of the boundary of
$U_{C_{i},I_{i}}$ corresponding to the two extreme points of $I_{i}$ are both
contained in a same component of $\partial U_{C,I}$ corresponding to one of
the endpoints of $I$). Set $Z_{i}=Z\cap U_{C_{i},I_{i}}$, and $V_{i}=V\cap
U_{C_{i},I_{i}}$. Note that since $\partial U_{C_{i},I_{i}}\subset V$, if we
set $I_{i}=(a_{i},b_{i})$, then there exists $\varepsilon>0$ such that the set
$V_{i}$ contains $U_{C_{i},(a_{i},a_{i}+\varepsilon)}$ and
$U_{C_{i},(b_{i},b_{i}-\varepsilon)}$ for some $\varepsilon>0$, while the set
$Z_{i}$ is contained in $U_{C_{i},(a_{i}+\varepsilon,b_{i}-\varepsilon)}$.
Using that the group $F_{I_{i}}$ acts without fixed points on $I_{i}$, we can
find for every $i=1,\ldots,n$ an element $g_{i}\in F_{C_{i},I_{I}}$ such that
$g_{i}(Z_{i})\subset V_{i}$. Then the element $g=g_{1}\cdots
g_{n}\in\mathsf{T}(\varphi,\sigma)$ satisfies $g(Z)\subset Z$. Since $V$ was
an arbitrary neighbourhood of $y$, the conclusion follows. ∎

This concludes the proof of Theorem 4.13.

######  Remark 4.18.

Most other results on the groups $\mathsf{T}(\varphi)$ from [MBT20] can be
adapted to $\mathsf{T}(\varphi,\sigma)$ without much effort. In particular if
the Stone $D_{\infty}$-system $(X,\varphi,\sigma)$ is minimal then the group
$\mathsf{T}(\varphi,\sigma)$ is simple, and if the underlying Stone system
$(X,\varphi)$ is conjugate to a subshift then the group
$\mathsf{T}(\varphi,\sigma)$ is finitely generated. We do not explain the
details of the proofs of these results, since they can be proven by a routine
adaptation of the arguments in [MBT20], by replacing charts in $Y^{\varphi}$
and the subgroups $F_{C,I}$ by their analogues for
$\mathsf{T}(\varphi,\sigma)$, modulo some mild modifications.

######  Remark 4.19.

Here we elaborate on the connection between the family of groups
$\mathsf{T}(\varphi,\sigma)$ and the groups defined by Hyde and Lodha in
[HL19]. The starting point of their constructions is a bi-infinite sequence
$\rho=(u_{n})_{n\in\mathbb{Z}}$, where each element $u_{n}$ belongs to the
finite alphabet $A=\\{a,a^{-1},b,b^{-1}\\}$. The sequence $\rho$ is required
to satisfy suitable conditions (see [HL19]), among which is the requirement
that for every finite word $w=w_{0}\cdots w_{n}$ of consecutive elements of
$\rho$, the formal inverse $w^{-1}=w_{n}^{-1}\cdots w_{1}^{-1}$ should also
appear in $\rho$. To every such sequence $\rho$, they associate a group
$G_{\rho}\leq\operatorname{Homeo}_{0}(\mathbb{R})$ generated by a finite set
of explicit piecewise linear homeomorphisms of $\mathbb{R}$ with an infinite
discrete set of discontinuity points for the derivative. Roughly speaking,
each letter $u_{n}$ of $\rho$ prescribes how each generator acts in
restriction to the interval $[n,n+2]$. A more intrinsic description of which
homeomorphisms of $\mathbb{R}$ belong to the group $G_{\rho}$ was later
obtained by Hyde-Lodha-Navas-Rivas [HLNR21, Theorem 0.8]. The condition that
$\rho$ is closed under inverses ensures that elements of $G_{\rho}$ are both
unboundedly positive and negative, which plays a role in the combinatorial
analysis in [HL19, HLNR21].

Now, given a sequence $\rho$ as above, let $X\subset A^{\mathbb{Z}}$ be the
subshift given by the orbit-closure of $\rho$ under the shift map, and let
$\varphi$ be the restriction of the shift to $X$. The assumption that $\rho$
is closed under formal inverses implies that we can define an involution

$\sigma\colon X\to X,\quad\sigma((v_{n})_{n})=(v_{-n-1}^{-1})_{n}.$

One readily checks that the triple $(X,\varphi,\sigma)$ is a Stone
$D_{\infty}$-system. The description of the group $G_{\rho}$ provided by
[HLNR21, Theorem 0.8] essentially implies that $G_{\rho}$ coincides with the
group $\mathsf{T}(\varphi,\sigma)$ associated to this system.

##  5\. A classification result of highly transitive actions

In this section we explain how Theorem 1 can be used to classify all highly
transitive actions of certain groups. Below for simplicity we focus of the
case of the Higman–Thompson groups [Hig74], but the argument generalizes to
other classes of groups, see Remark 5.7.

###  5.1. Preliminaries

Throughout this section we let $X_{d}=\\{0,\cdots,d-1\\}^{\mathbb{N}}$ be the
Cantor set of one sided sequences over a $d$-letter alphabet. Given a finite
word $w\in\\{0,\cdots,d-1\\}^{*}$ we let $C_{w}\subset X_{d}$ be the cylinder
subsets of sequences that admit $w$ as a prefix. Recall that the
Higman–Thompson group $V_{d}$ is the group of all homeomorphisms of $X_{d}$
such that there exist two collections of finite words
$w_{1},\cdots,w_{k},u_{1},\cdots,u_{k}\in\\{0,\ldots,d-1\\}^{*}$ with
$X_{d}=C_{w_{1}}\sqcup\cdots\sqcup C_{w_{k}}=C_{u_{1}}\sqcup\cdots\sqcup
C_{u_{k}}$ and such that $g(w_{i}\xi)=u_{i}\xi$ for all $i=1\cdots k$ and all
$\xi\in X_{d}$. It follows easily from the definition that the $V_{d}$-orbits
in $X_{d}$ coincide with the cofinality classes of sequences, meaning that two
sequences $\xi=x_{0}x_{1}\cdots,\eta=y_{0}y_{1}\cdots$ are in the same
$V_{d}$-orbit if and only if there exists $m,n\geq 0$ such that
$x_{m+i}=y_{n+i}$ for all $i\geq 0$. Moreover the action of $V_{d}$ on each
orbit is highly transitive. Note also that the action of $V_{d}$ on $X_{d}$ is
topologically nowhere free, and hence the stabilizer in $V_{d}$ of every point
of $X_{d}$ is a confined subgroup of $V_{d}$.

The following description of the dynamics of individual elements of $V_{d}$ is
due to Brin [Bri04, §10.7].

######  Proposition 5.1.

For every $g\in V_{d}$, there exists a decomposition $X_{d}=Y_{g}\sqcup Z_{g}$
into two $g$-invariant clopen sets such that the following hold:

  1. (i)

The restriction of $g$ to $Y_{g}$ has finite order;

  2. (ii)

The subset $\operatorname{Per}_{\operatorname{hyp}}(g)\subset Z_{g}$ of points
with a finite $g$-orbit is finite, and there exists a partition
$\operatorname{Per}_{\operatorname{hyp}}(g)=\operatorname{Att}(g)\sqcup\operatorname{Rep}(g)$
such that for every neighbourhood $U$ of $\operatorname{Att}(g)$ and $V$ of
$\operatorname{Rep}(g)$, there exists $n>0$ such that $g^{n}(Z_{g}\setminus
V)\subset U$ and $g^{-n}(Z_{g}\setminus V)\subset U$.

Moreover $Z_{g}\neq\varnothing$ if and only if $g$ has infinite order, and in
this case $\operatorname{Att}(g)$ and $\operatorname{Rep}(g)$ are both non-
empty.

Points in $\operatorname{Per}_{\operatorname{hyp}}(g)$ are called the
hyperbolic periodic points of $g$, and $\operatorname{Att}(g)$ and
$\operatorname{Rep}(g)$ are respectively the attractive and repelling ones.

Given an action of a group $G$ on a compact space $X$, we say that a closed
subset $C\subset X$ is compressible if there exists a point $x\in X$ such that
for every neighbourhood $U$ of $x$ there exists $g\in G$ such that
$g(C)\subset U$. Similarly a Borel probability measure
$\mu\in\operatorname{Prob}(X)$ is compressible if the closure of $G\cdot\mu$
in the weak-* topology of $\operatorname{Prob}(X)$ contains a Dirac mass
$\delta_{x}$. Recall that the action of $G$ on $X$ is proximal if every pair
of points is compressible. When this holds, every finite subset is
compressible [Mar91, Ch. VI Cor. 1.4]. The action is strongly proximal if
every $\mu\in\operatorname{Prob}(X)$ is compressible. Recall also from Section
4 that the action is extremely proximal if every proper closed subset
$C\subsetneq X$ is compressible. Extreme proximality implies strong
proximality, which implies proximality, and the reverse implications are not
true in general [Gla74].

In our present setting we deduce from Proposition 5.1 the following
equivalence for subgroups of $V_{d}$, which was inspired by a reading of
[HM19, §5].

######  Proposition 5.2.

Let $H\leq V_{d}$ be a subgroup which acts minimally on $X_{d}$. Suppose that
$H$ contains an element of infinite order. Then the following are equivalent:

  1. (i)

the action of $H$ on $X_{d}$ is proximal;

  2. (ii)

the action of $H$ on $X_{d}$ is strongly proximal;

  3. (iii)

the action of $H$ on $X_{d}$ is extremely proximal.

###### Proof.

We only have to prove the implication (i)$\Rightarrow$(iii). First of all note
that $\bigcap_{g\in H}Y_{g}$ is a closed $H$-invariant subset of $X_{d}$.
Moreover since $H$ contains an element of infinite order, we have
$\bigcap_{g\in H}Y_{g}\subsetneq X_{d}$. Thus by minimality of the $H$-action
we have $\bigcap_{g\in H}Y_{g}=\varnothing$. By the finite intersection
property for compact sets, we can find $g_{1},\cdots,g_{n}\in H$ such that
$\bigcap_{i=1}^{n}Y_{g_{i}}=\varnothing$. Without loss of generality we may
assume that all $g_{i}$’s have infinite order (indeed if $g_{i}$ has finite
order then we have $Y_{g_{i}}=X_{d}$ and we can simply remove $g_{i}$ without
changing the conclusion). Moreover, after replacing each $g_{i}$ by one of its
powers, we can assume that $g_{i}$ is the identity on $Y_{g_{i}}$.

Now let $C\subsetneq X_{d}$ be a closed subset of $X_{d}$, and denote by
$\mathcal{O}_{C}$ the closure of the $H$-orbit of $C$ in the space of all
closed subset of $X_{d}$ (with respect to the Vietoris topology). We aim to
show that $\mathcal{O}_{C}$ contains a singleton. Since finite subsets are
compressible by $H$ by the assumption that the $H$-action is proximal, it is
enough to show that $\mathcal{O}_{C}$ contains a subset of the finite set
$\Sigma:=\bigcup_{i=1}^{n}\operatorname{Per}_{\operatorname{hyp}}(g_{i})$.
First observe that by proximality and minimality of the $H$-action, we can
find $f\in H$ such that $f(\Sigma)\subset X_{d}\setminus C$, so that
$C_{0}:=f^{-1}(C)\in\mathcal{O}_{C}$ satisfies $C_{0}\cap\Sigma=\varnothing$.
Now by Proposition 5.1 we can find a sequence $n_{i}$ of positive integers
tending to $\infty$ such that $g_{1}^{n_{i}}(C_{0})$ converges in
$\mathcal{O}_{C}$ to a closed subset $C_{1}\in\mathcal{O}_{C}$ such that
$C_{1}\subset(C_{0}\cap
Y_{g_{1}})\cup\operatorname{Per}_{\operatorname{hyp}}(g_{1})$. Note that since
$C_{0}$ does not intersect $\operatorname{Rep}(g_{2})$, every point in
$C_{1}\cap\operatorname{Rep}(g_{2})$ must be isolated in $C_{1}$. So again we
can find a sequence $(n_{i})$ such that $g_{2}^{n_{i}}(C_{1})$ converges in
$\mathcal{O}_{C}$ to a closed subset $C_{2}$ such that $C_{2}\subset(C_{0}\cap
Y_{g_{1}}\cap
Y_{g_{2}})\cup\operatorname{Per}_{\operatorname{hyp}}(g_{1})\cup\operatorname{Per}_{\operatorname{hyp}}(g_{2})$.
Proceeding in this way we can find for each $i=1\cdots,n$ a subset
$C_{i}\in\mathcal{O}_{C}$ such that $C_{i}\subset(C_{0}\cap
Y_{g_{1}}\cap\cdots\cap Y_{g_{i}})\cup\Sigma$. In particular
$C_{n}\subset\Sigma$ is finite. As observed above, this proves the statement.
∎

######  Remark 5.3.

The assumption that $H$ has elements of infinite order is necessary. Indeed
consider the subgroup $H\leq V_{d}$ consisting of elements defined by
partitions $X_{d}=C_{w_{1}}\sqcup\cdots\sqcup
C_{w_{k}}=C_{v_{i}}\sqcup\cdots\sqcup C_{v_{k}}$ such the words $w_{i}$ and
$v_{i}$ have the same length for every $i$. Then $H$ is an infinite locally
finite group, isomorphic to a block diagonal limit of symmetric groups
$\operatorname{Sym}(d^{n})$, and its action on $X_{d}$ is minimal, proximal,
and preserves the natural uniform Bernoulli measure on $X_{d}$. Hence this
action is not extremely proximal. We also note that by a result of Röver
[Rov99], every subgroup of $V_{d}$ without elements of infinite order is
locally finite.

We will use the following consequence of Proposition 5.2. Recall that the germ
of an element $g\in V_{d}$ at a point $\xi\in X_{d}$ is the equivalence class
$[g,\xi]$ of the pair $(g,\xi)$ under the equivalence relation defined by
$(g_{1},\xi_{1})\sim(g_{2},\xi_{2})$ if $\xi_{1}=\xi_{2}$ and there exists a
neighbourhood $U$ of $\xi_{1}$ such that $g_{1}|_{U}=g_{2}|_{U}$. We say that
a subgroup $H\leq V_{d}$ covers pairs of germs in $V_{d}$ if for every $g\in
V_{d}$ and every pair of points $\xi,\eta\in X_{d}$ there exists $h\in H$ such
that $[g,\xi]=[h,\xi]$ and $[g,\eta]=[h,\eta]$. Equivalently, $H$ covers pairs
of germs if $V_{d}=(V_{d})_{\xi,\eta}^{0}H$ for every $\xi,\eta\in X_{d}$,
where $(V_{d})_{\xi,\eta}^{0}$ is the subgroup of $V_{d}$ consisting of
elements acting trivially on an open subset containing $\xi,\eta$.

From Proposition 5.2 we deduce the following.

######  Proposition 5.4.

Let $H\leq V_{d}$ be a subgroup which covers pairs of germs in $V_{d}$. Then
the action of $H$ on $X_{d}$ is minimal and extremely proximal.

###### Proof.

The assumption implies in particular that $H$ has the same orbits as $V_{d}$
on pairs of distinct points. Since the action of $V_{d}$ on $X_{d}$ is minimal
and proximal, so is the $H$-action. Moreover if $g\in V_{d}$ is an element of
infinite order that admits $\xi\in Z_{g}$ as a hyperbolic attractive fixed
point, then any element $h\in H$ such that $[g,\xi]=[h,\xi]$ also has infinite
order. Thus Proposition 5.2 implies that the $H$-action is extremely proximal.
∎

###  5.2. Highly transitive actions of $V_{d}$

We now give the proof of the main result of this section.

######  Theorem 5.5.

Every faithful and highly transitive action of $V_{d}$ on a set is conjugate
to its action on an orbit in $X_{d}$.

In the proof we will invoke the following lemma, which is an immediate
consequence of well-known properties of the action of $V_{d}$ on $X_{d}$.
Given $G=V_{d}$ and points $\xi_{1},\cdots,\xi_{n}\in X_{d}$, we use the
notation $G_{\xi_{1},\ldots,\xi_{n}}$ for the pointwise stabilizer of
$\xi_{1},\ldots,\xi_{n}$.

######  Lemma 5.6.

Let $G=V_{d}$. For every $\xi_{1},\xi_{2}\in X_{d}$, the proper subgroups of
$G$ that contain $G_{\xi_{1},\xi_{2}}$ are
$G_{\xi_{1},\xi_{2}},G_{\xi_{1}},G_{\xi_{2}}$, and the stabilizer in $G$ of
$\\{\xi_{1},\xi_{2}\\}$.

###### Proof of Theorem 5.5.

In the sequel we denote $G=V_{d}$. Assume that $G$ acts highly transitively on
$\Omega$. Given $\xi\in X_{d}$, the subgroup $G_{\xi}$ is a confined subgroup
of $G$, and every pair of non-trivial elements with disjoint supports in
$X_{d}$ forms a confining subset for $(G_{\xi},G)$. Hence by Theorem 3.12
(applied with $r=2$; and note that $G$ is not partially finitary), either
$G_{\xi}$ fixes a unique point $\omega_{\xi}$ in $\Omega$ and acts highly
transitively on the complement, or the action of $G_{\xi}$ on $\Omega$ is
highly transitive.

Suppose there exists $\xi$ such that the first possibility holds. Then
$G_{\xi}$ being a maximal subgroup of $G$, we must have equality
$G_{\xi}=G_{\omega_{\xi}}$, and it follows that the $G$-action on $\Omega$ is
conjugate to the $G$-action on the orbit of $\xi$. Hence in that case the
conclusion holds. In the sequel we assume that for all $\xi$ the action of
$G_{\xi}$ on $\Omega$ is highly transitive, and we want to reach a
contradiction.

We fix a point $\xi\in X_{d}$. Given $\eta\in X_{d}$, the subgroup
$G_{\xi,\eta}$ is confined in $G_{\xi}$, and for the same reason as before
there exists a confining subset with two elements. Hence we can apply Theorem
3.12 again, but this time to the action of $G_{\xi}$ on $\Omega$. Suppose that
there exists $\eta$ such that $G_{\xi,\eta}$ fixes a point $\omega$ of
$\Omega$, i.e. $G_{\xi,\eta}\leq G_{\omega}$. Then by Lemma 5.6 we deduce that
$G_{\omega}$ fixes $\xi$, or $G_{\omega}$ fixes $\eta$, or $G_{\omega}$
stabilizes $\\{\xi,\eta\\}$. If $G_{\omega}$ fixes $\xi$ then by maximality of
$G_{\omega}$ in $G$ we must have $G_{\xi}=G_{\omega}$, which is impossible
because we make the assumption that the action of $G_{\xi}$ on $\Omega$ is
highly transitive. By the same argument $G_{\omega}$ cannot fix $\eta$. And
similarly $G_{\omega}$ cannot stabilize the pair $\\{\xi,\eta\\}$, because
otherwise the action of $G$ on $\Omega$ would be conjugate to the action of
$G$ on the orbit of $\\{\xi,\eta\\}$, which is impossible as the latter is not
even 2-transitive. Hence all possibilities lead to a contradiction, and hence
there is no $\eta$ such that $G_{\xi,\eta}$ fixes a point in $\Omega$.
Therefore by Theorem 3.12 we deduce that the action of $G_{\xi,\eta}$ on
$\Omega$ is highly transitive for all $\xi,\eta\in X_{d}$.

Since $G_{\xi,\eta}^{0}$ is normal in $G_{\xi,\eta}$, the action of
$G_{\xi,\eta}^{0}$ on $\Omega$ is also highly transitive. This implies that
whenever $\Sigma$ is a finite subset of $\Omega$, we have
$G=G_{\xi,\eta}^{0}G_{\Sigma}$, where $G_{\Sigma}$ is the pointwise fixator of
$\Sigma$; indeed given $g\in G$ there exists $h\in G_{\xi,\eta}^{0}$ which
coincides with $g$ on $\Sigma$, so that $h^{-1}g\in G_{\Sigma}$. Since the
equality $G=G_{\xi,\eta}^{0}G_{\Sigma}$ holds for all distinct points
$\xi,\eta$, we infer that the subgroup $G_{\Sigma}$ covers pairs of germs in
$G$. Therefore we deduce from Proposition 5.4 that the action of $G_{\Sigma}$
on $X_{d}$ is minimal and extremely proximal.

In order to derive a contradiction from this point, we adapt an easy argument
used in the proof of [LBMB19, Lemma 3.8]. Let $U\subset X_{d}$ be a non-empty
proper clopen subset, and choose an element $k\in\operatorname{R}_{G}(U)$ such
that $k^{2}\neq 1$, and a point $\omega_{1}\in\Omega$ whose $k$-orbit has
order $\geq 3$. Set $\omega_{2}=k(\omega_{1}),\omega_{3}=k^{2}(\omega_{1})$.
Choose $g\in G$ such that $g(\omega_{1})=\omega_{1},g(\omega_{2})=\omega_{2}$
and $\omega_{3}^{\prime}:=g(\omega_{3})\neq\omega_{3}$, and set
$\Sigma=\\{\omega_{1},\omega_{2},\omega_{3}^{\prime}\\}$. Since $G_{\Sigma}$
acts minimally and extremely proximally, we can choose $h\in G_{\Sigma}$ such
that $hg(U)\cap U=\varnothing$. Set $f=hg$ and $k^{\prime}=fkf^{-1}$. Note
that on the one hand we have $k^{\prime}(\omega_{1})=\omega_{2}$ and
$k^{\prime}(\omega_{2})=\omega_{3}^{\prime}$ because
$f(\omega_{i})=\omega_{i}$ for $1,2$ and $f(\omega_{3})=\omega_{3}^{\prime}$.
On the other hand $k^{\prime}$ commutes with $k$ since $k^{\prime}$ is
supported in $f(U)$ and $U$ and $f(U)$ are disjoint. Therefore
$k^{\prime}(\omega_{2})=k^{\prime}k(\omega_{1})=kk^{\prime}(\omega_{1})=k(\omega_{2})=\omega_{3}$.
So $\omega_{3}=\omega_{3}^{\prime}$, and we have reached a contradiction. By
the first paragraph above, this terminates the proof. ∎

######  Remark 5.7.

The attentive reader will have noticed that the only important point of the
proof of Theorem 5.5 where properties of the group $V_{d}$ are used is
Proposition 5.4 (which, in turns, relies on Proposition 5.2). Actually the
proof of Theorem 5.5 works for every group of homeomorphisms of a compact
space that fulfils the conclusions of Lemma 5.6 and Proposition 5.4.

##  6\. Invariant random subgroups and highly transitive actions

The purpose of this section is to prove the following. Recall that an IRS of a
group $G$ is a $G$-invariant Borel probability measure on
$\operatorname{Sub}(G)$ .

######  Proposition 6.1.

Let $G$ be a countable group that is not partially finitary, and let $\mu$ be
an IRS of $G$ such that $\mu(\\{1\\})=0$. Suppose that $G$ admits a faithful
and highly transitive action on a set $\Omega$. Then for $\mu$-almost every
$H\in\operatorname{Sub}(G)$ the action of $H$ on $\Omega$ is highly
transitive.

###### Proof.

We begin by showing that $\mu$-almost every $H$ acts transitively on $\Omega$.
To show this, it is enough to show the following claim: for every
$\omega\in\Omega$ and every probability measure $\lambda$ on
$\operatorname{Sub}(G)$ that is invariant by $G_{\omega}$, $\lambda$-almost
every $H$ must either fix $\omega$ or act transitively on $\Omega$. Indeed
assume that the claim is proven, and consider the events
$E_{\omega}:=\\{H\colon H\text{ fixes }\omega\\}$ and $E_{t}:=\\{H\colon
H\text{ acts transitively on }\Omega\\}$. Clearly $E_{\omega}\cap
E_{t}=\varnothing$ and the previous claim applied to $\lambda=\mu$ shows that
$\mu(E_{\omega}\sqcup E_{t})=1$ for every $\omega$. Since $\Omega$ is
countable, this implies that the event $\bigcap_{\omega}(E_{\omega}\sqcup
E_{t})=(\bigcap_{\omega}E_{\omega})\sqcup E_{t}$ has measure one. However
$\bigcap_{\omega}E_{\omega}$ is nothing but the event that $H$ acts trivially
on $\Omega$, i.e. that $H$ is the trivial subgroup. Since we are assuming that
$\mu(\\{1\\})=0$, it follows that $\mu(E_{t})=1$, as desired.

To prove the claim, fix $\omega\in\Omega$ and a probability measure $\lambda$
on $\operatorname{Sub}(G)$ that is invariant by $G_{\omega}$. By Choquet’s
theorem [Cho69, Theorem 27.6], $\lambda$ can be written as an integral of
$G_{\omega}$-invariant probability measures that are ergodic for the action of
$G_{\omega}$. Thus to prove the claim there is no loss of generaility in
assuming that $\lambda$ is $G_{\omega}$-ergodic. We write
$\Omega^{\ast}=\Omega\setminus\left\\{\omega\right\\}$, and we consider the
map
$\varphi\colon\operatorname{Sub}(G)\to\left\\{0,1\right\\}^{\Omega^{\ast}}$
which associates to a subgroup $H$ the $H$-orbit of the point $\omega$, from
which we remove $\omega$. The map $\varphi$ is equivariant for the actions of
$G_{\omega}$, and hence the push-forward $\nu$ of $\lambda$ is an invariant
ergodic probability measure on $\left\\{0,1\right\\}^{\Omega^{\ast}}$. Now
since the image of $G_{\omega}$ inside $\operatorname{Sym}(\Omega^{\ast})$ is
dense and the action of $\operatorname{Sym}(\Omega^{\ast})$ on
$\left\\{0,1\right\\}^{\Omega^{\ast}}$ is continuous, the measure $\nu$ is
also invariant under $\operatorname{Sym}(\Omega^{\ast})$. By de Finetti’s
theorem the ergodic probability measures on
$\left\\{0,1\right\\}^{\Omega^{\ast}}$ that are invariant under
$\operatorname{Sym}(\Omega^{\ast})$ are Bernoulli, so it follows that
$\nu=(p\delta_{0}+(1-p)\delta_{1})^{\Omega^{\ast}}$ for some $p\in[0,1]$. Now
given an infinite subset $\Sigma\subset\Omega\times\Omega$ that does not
intersect the diagonal, denote by $S_{\Sigma}$ the set of
$(x_{\alpha})\in\left\\{0,1\right\\}^{\Omega^{\ast}}$ such that
$x_{\alpha}=x_{\alpha^{\prime}}$ for all $(\alpha,\alpha^{\prime})\in\Sigma$.
Note that if $p\in]0,1[$ then $S_{\Sigma}$ has measure $0$. Now take a non-
trivial element $g$ of $G$ such that the set $\mathcal{H}_{g}$ of subgroups
$H$ of $G$ that contain $g$ has positive measure. Since $g$ moves infinitely
many points in $\Omega$, one can find a sequence $(\omega_{n})$ such that all
the points $\omega_{n}$ and $g(\omega_{n})$ are pairwise distinct. If we let
$\Sigma$ be the set of pairs $(\omega_{n},g(\omega_{n}))$, then by
construction one has $\varphi(\mathcal{H}_{g})\subset S_{\Sigma}$. Hence by
the choice of $g$ it follows that $S_{\Sigma}$ has positive $\nu$-measure, and
hence $p=0$ or $p=1$ by the above observation. This means that either $H$
fixes $\omega$ almost surely or $H$ acts transitively almost surely, and the
claim is proved. As explained at the beginning of the proof, it follows that
$\mu$-almost every $H$ acts transitively on $\Omega$.

We shall now argue that almost surely $H$ does not act freely on $\Omega$. Let
$S_{F}$ denote the set of subgroups $H$ of $G$ such that $H$ acts freely on
$\Omega$. Suppose for a contradiction that $\mu(S_{F})>0$. Then one can find a
non-trivial element $g$ such that $\mu(S_{F}\cap\mathcal{H}_{g})>0$. Choose
$\omega_{1},\omega_{2}$ that are distinct and such that
$g(\omega_{1})=\omega_{2}$. Then for every $\gamma\in G$ such that
$\gamma(\omega_{1})=\omega_{2}$ and $\gamma\neq g$, then one has that
$S_{F}\cap\mathcal{H}_{g}\cap\mathcal{H}_{\gamma}$ is empty. In particular
whenever $\gamma$ is a conjugate of $g$ by an element of
$G_{\omega_{1},\omega_{2}}$ and $\gamma$ is distinct from $g$, then the sets
$S_{F}\cap\mathcal{H}_{g}$ and $S_{F}\cap\mathcal{H}_{\gamma}$ have the same
(positive) measure and do not intersect each other. This implies that the set
of conjugates of $g$ by $G_{\omega_{1},\omega_{2}}$ is finite, i.e. $g$
centralizes a finite index subgroup of $G_{\omega_{1},\omega_{2}}$. Since this
subgroup acts highly transitively on the complement of
$\left\\{\omega_{1},\omega_{2}\right\\}$ in $\Omega$, this is possible only if
$g$ is the transposition that exchanges $\omega_{1},\omega_{2}$. This is
absurd since we assume that all non-trivial elements of $G$ have infinite
support. So almost surely $H$ does not act freely on $\Omega$.

Now given $k\geq 1$, we shall argue by induction that almost surely the action
of $H$ on $\Omega$ is $k$-transitive and the fixator in $H$ of $k$ distinct
elements of $\Omega$ is non-trivial. We have already treated the case $k=1$.
Suppose the result holds for $k$, and choose a subset $\Sigma$ in $\Omega$ of
cardinality $k$. The IRS induced on the fixator $G_{\Sigma}$ by the map
$H\mapsto H\cap G_{\Sigma}$ does not charge the identity by assumption, and
hence we can apply the case $k=1$ to the action of $G_{\Sigma}$ on the
complement of $\Sigma$ in $\Omega$. Combined with the induction hypothesis
this implies that almost surely the action of $H$ on $\Omega$ is
$k+1$-transitive and the fixator in $H$ of $k+1$ distinct elements of $\Omega$
is non-trivial. This terminates the induction, and it immediately follows that
almost surely the action of $H$ on $\Omega$ is highly transitive. ∎

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