# Piecewise strongly proximal actions,
free boundaries and the Neretin groups

Pierre-Emmanuel Caprace F.R.S.-FNRS Senior Research Associate.
<EMAIL_ADDRESS>UCLouvain, 1348 Louvain-la-Neuve, Belgium  Adrien Le
Boudec CNRS Researcher<EMAIL_ADDRESS>UCLouvain, 1348 Louvain-
la-Neuve, Belgium  Nicolás Matte Bon CNRS Researcher<EMAIL_ADDRESS>lyon1.fr UCLouvain, 1348 Louvain-la-Neuve, Belgium

(December 8, 2022)

###### Abstract

A closed subgroup $H$ of a locally compact group $G$ is confined if the
closure of the conjugacy class of $H$ in the Chabauty space of $G$ does not
contain the trivial subgroup. We establish a dynamical criterion on the action
of a totally disconnected locally compact group $G$ on a compact space $X$
ensuring that no relatively amenable subgroup of $G$ can be confined. This
property is equivalent to the fact that the action of $G$ on its Furstenberg
boundary is free. Our criterion applies to the Neretin groups. We deduce that
each Neretin group has two inequivalent irreducible unitary representations
that are weakly equivalent. This implies that the Neretin groups are not of
type I, thereby answering a question of Y. Neretin.

##  1 Introduction

Let $G$ be a locally compact group. A compact $G$-space $X$ is a compact space
equipped with a continuous action of $G$. The action of $G$ on $X$ is strongly
proximal if for every $\mu\in\mathrm{Prob}(X)$, the closure of the $G$-orbit
of $\mu$ in $\mathrm{Prob}(X)$ contains a Dirac measure, where the space
$\mathrm{Prob}(X)$ of Borel probability measures on $X$ is endowed with the
weak∗ topology. The $G$-space $X$ is a (topological) $G$-boundary if the
$G$-action is minimal and strongly proximal [20]. If $G$ is an amenable group,
the only $G$-boundary is the one-point space; and this property actually
characterizes amenability among locally compact groups.

Every group $G$ admits a $G$-boundary, unique up to isomorphism, with the
property that every $G$-boundary is a factor of it. It is called the
Furstenberg boundary of $G$; we denote it by $\partial_{F}G$. The group $G$
acts faithfully on $\partial_{F}G$ if and only if the only amenable normal
subgroup of $G$ is the trivial subgroup $\langle e\rangle$ [19]. In this note
we are interested in the following:

######  Problem 1.1.

Determine when the action of $G$ on $\partial_{F}G$ is free.

A key motivation for that question is that the freeness of the $G$-action on
$\partial_{F}G$ is equivalent to various other properties of the group $G$. We
say that the action of $G$ on a minimal compact $G$-space $X$ is topologically
free if there is a dense set of points in $X$ that have a trivial stabilizer
in $G$. We say that $H$ is relatively amenable in $G$ if $H$ fixes a
probability measure on every compact $G$-space. Clearly, every amenable
subgroup is relatively amenable; when $G$ is discrete, the converse holds, see
[11]. A uniformly recurrent subgroup (or URS for short) of $G$ is a minimal
$G$-invariant closed subset of the Chabauty space $\mathrm{Sub}(G)$ of closed
subgroups of $G$ (we recall that $\mathrm{Sub}(G)$ is compact). A closed
subgroup $H\leq G$ is confined in $G$ if the closure of the conjugacy class of
$H$ in $\mathrm{Sub}(G)$ avoids the trivial subgroup $\langle e\rangle$.

######  Theorem 1.2 (See [25, 7, 27] in the case of discrete groups).

Let $G$ be a locally compact group. The following conditions are equivalent.

  1. (i)

The $G$-action on $\partial_{F}G$ is free.

  2. (ii)

There is a $G$-boundary on which the $G$-action is topologically free.

  3. (iii)

No relatively amenable closed subgroup of $G$ is confined.

  4. (iv)

The only relatively amenable URS of $G$ is the trivial subgroup.

If in addition $G$ is discrete, then those are also equivalent to:

  1. 5.

$G$ is $C^{\ast}$-simple, i.e. the reduced $C^{\ast}$-algebra of $G$ is a
simple $C^{\ast}$-algebra.

In the case where the group $G$ is discrete, it follows from the recent works
[25, 7, 27] that the five conditions in Theorem 1.2 are equivalent. For a
general (e.g. indiscrete) locally compact group $G$, the implications
$\textrm{(i)}\Rightarrow\textrm{(ii)}\Leftrightarrow\textrm{(iii)}\Leftrightarrow\textrm{(iv)}$
are rather straightforward, at least if $G$ is second countable (see
Proposition 2.2 and Remark 2.3 below), while the converse implication
$\textrm{(i)}\Leftarrow\textrm{(ii)}$ is a particular case of a result from
[32]. However, it is an important open problem to determine whether (v) is
also equivalent to (i)–(iv). It is worth noting that each of those conditions
implies that the group $G$ is totally disconnected (see [39] for (v), and the
discussion in the following paragraphs for (iv)).

Conditions (iii) and (iv) highlight an essential feature of Problem 1.1,
namely that it can be reformulated in terms of the $G$-action on the space of
its closed subgroups. Indeed, this allows one to avoid constructing explicitly
any $G$-boundary, and simply study the conjugation action of $G$ on its
relatively amenable subgroups. For example, if $G$ is a discrete hyperbolic
group, every amenable subgroup is virtually cyclic, hence finitely generated.
In particular $G$ has countably many amenable subgroups, and it is a general
fact that for a group $G$ with this property, the only amenable URS of $G$ is
trivial, provided $G$ has no amenable normal subgroup other than the trivial
subgroup. This situation also covers CAT(0) groups by [1, Cor. B]. Many other
discrete groups admitting an isometric action satisfying a combination of weak
forms of properness and of non-positive curvature can be proved to have a free
Furstenberg boundary in a similar (but more elaborate) way, see [15, 14, 7].
Note that the requirement of a certain form of properness cannot be dropped
according to [29]. The reformulation of Problem 1.1 in terms of confined
subgroups has also been exploited in the realm of discrete groups of dynamical
origin in [30].

For non-discrete groups, it turns out that Problem 1.1 has a very different
flavour. Indeed, contrary to the discrete case where numerous familiar groups
have no non-trivial amenable URS, many natural non-discrete locally compact
groups do admit a non-trivial relatively amenable URS. This is notably the
case for semisimple Lie groups and semisimple algebraic groups over local
fields: any such group $G$ indeed has a cocompact amenable subgroup $P$. By
cocompactness, the conjugacy class of $P$ is closed, so $P$ must be confined.
Beyond the classical case, in a locally compact group $G$ acting properly and
strongly transitively on a locally finite building of arbitrary (not
necessarily Euclidean) type, every maximal compact subgroup is confined (this
follows from [10, Th. 4.10]). Thus, many natural examples of non-discrete
locally compact groups fail to satisfy the condition of Problem 1.1. Note that
the above fact for semisimple Lie groups together with [8, Th. 3.3.3] imply
that every locally compact group $G$ such that the only relatively amenable
URS of $G$ is the trivial subgroup, must be totally disconnected.

The first goal of this note is to contribute to Problem 1.1 by establishing a
sufficient criterion for a locally compact group $G$ that ensures a positive
answer to Problem 1.1. In view of the preceding discussion, there is no loss
of generality in restricting to the case where $G$ is totally disconnected.
Given a compact $G$-space $X$ and a clopen subset $\alpha$ of $X$, the rigid
stabilizer of $\alpha$ in $G$ is the pointwise fixator of
$X\\!\setminus\\!\alpha$. It is denoted by
$R_{G}(\alpha)=\textrm{Fix}_{G}(X\\!\setminus\\!\alpha)$. We say that the
action of $G$ on $X$ is piecewise minimal-strongly-proximal if the action of
$R_{G}(\alpha)$ on $\alpha$ is minimal and strongly proximal for every non-
empty clopen subset $\alpha$ of $X$.

######  Theorem 1.3.

Let $G$ be a totally disconnected locally compact group. Suppose that there
exists a totally disconnected compact $G$-space $X$ such that the $G$-action
on $X$ is faithful and piecewise minimal-strongly-proximal. Then $G$ does not
have any relatively amenable confined subgroup. Equivalently, $G$ acts freely
on its Furstenberg boundary $\partial_{F}G$.

Note that the piecewise minimal-strongly-proximal property of the $G$-action
on $X$ implies in particular that $X$ is a $G$-boundary, and also that this
action is very far from being free. Hence the meaning of Theorem 1.3 is that
the existence of a $G$-boundary that is non-free and satisfies a certain
strong compressibility property, ensures the existence of another $G$-boundary
where the $G$-action is free.

Although the reformulation of Problem 1.1 in terms of confined subgroups is
helpful, we emphasize that the Chabauty space $\mathrm{Sub}(G)$ and its
$G$-invariant closed subspaces are typically delicate to describe. Moreover,
the general properties of the space $\mathrm{Sub}(G)$ are often more subtle in
the case of non-discrete groups (see for instance [12, §20.1]; see also [18,
§1.2] for a remarkable recent result ensuring that, in a simple Lie group of
rank $\geq 2$, every discrete confined subgroup is a lattice). In the case of
discrete groups, Theorem 1.3 is already known [30, Cor. 3.6]. It is actually
consequence of the following more general result: if $G$ is a discrete group
and $X$ a faithful $G$-space, then for every confined subgroup $H$ of $G$,
there exists a non-empty open subset $\alpha$ of $X$ such that $H$ contains
the commutator subgroup of $R_{G}(\alpha)$ [31, Th. 1.1]. Here the assumption
made in Theorem 1.3 implies that $R_{G}(\alpha)$ is non-amenable, so in this
situation it follows in particular that $H$ is not amenable either. However it
is worth noting that, as shown by classical examples, the stronger conclusion
of [31, Th. 1.1] completely fails for non-discrete groups; see Section 3.

Examples of groups to which Theorem 1.3 applies are the Neretin groups
$\mathcal{N}_{d,k}$ of almost automorphisms of quasi-regular tree
$\mathcal{T}_{d,k}$. The groups $\mathcal{N}_{d,k}$ are non-discrete,
compactly generated, simple, totally disconnected locally compact groups [9,
§6.3]. We refer to [21] for details. The groups $\mathcal{N}_{d,k}$ can be
defined as groups of homeomorphisms of the space of ends
$\partial\mathcal{T}_{d,k}$, and the action of $\mathcal{N}_{d,k}$ on
$\partial\mathcal{T}_{d,k}$ is piecewise minimal-strongly-proximal. The
following result is a direct consequence of Theorem 1.3.

######  Corollary 1.4.

For all integers $d,k\geq 2$, the Neretin group $\mathcal{N}_{d,k}$ does not
have any confined relatively amenable subgroup. In particular
$\mathcal{N}_{d,k}$ does not have any cocompact amenable subgroup.

To the best of our knowledge, the Neretin group $\mathcal{N}_{d,k}$ is the
first known example of a non-discrete compactly generated simple locally
compact group acting freely on its Furstenberg boundary. It is likely that
Theorem 1.3 will apply to many other simple groups.

Using that result, we establish the following representation theoretic
property of the Neretin group.

######  Theorem 1.5.

For all integers $d,k\geq 2$, the Neretin group $\mathcal{N}_{d,k}$ is not a
type I group.

This answers negatively Question 1.4(2) from [37]. We recall that a locally
compact group $G$ is of type I is for every unitary representation $\pi$, the
von Neumann algebra $\pi(G)^{\prime\prime}$ is of type I. By Glimm’s theorem
[23], a second countable group $G$ is of type I if and only if any two weakly
equivalent irreducible unitary representations of $G$ are unitarily
equivalent. We refer to [16] and [4] for detailed expositions.

Let us also mention that Y. Neretin has proved in [37, Th. 1.2] that the group
$\mathcal{N}_{d,d+1}$ has an open subgroup $A$ such that
$(\mathcal{N}_{d,d+1},A)$ forms a generalized Gelfand pair. Since
$\mathcal{N}_{d,d+1}$ admits no cocompact amenable subgroup by Corollary 1.4,
we deduce that N. Monod’s result on Gelfand pairs [35] cannot be extended to
generalized Gelfand pairs, even among simple groups (see Remark 4.3).

Motivated by the understanding of the confined subgroups of
$\mathcal{N}_{d,k}$, we also provide a complete classification of the closed
cocompact subgroups of $\mathcal{N}_{d,k}$, inspired by [3]. This
classification says that there are as few proper cocompact subgroups in
$\mathcal{N}_{d,k}$ as one might hope: any such subgroup is a finite index
open subgroup of the stabilizer of an end $\xi\in\partial\mathcal{T}_{d,k}$,
see Theorem 4.9. This description notably implies that $\mathcal{N}_{d,k}$ is
an isolated point of its Chabauty space (see Corollary 4.10). This last
phenomenon contrasts with the case of the automorphism group of a regular
tree; see Remark 4.11.

The proof of Theorem 1.3 is presented in Section 3. It is fairly elementary.
The argument actually establishes non-confinement for an appropriate class of
subgroups of $G$, which properly contains the class of relatively amenable
subgroups. It uses in an essential way that the fixator $G_{F}$ of a finite
subset $F$ of $X$ in $G$ admits a natural subgroup $J$ that is commensurated
in $G_{F}$, and which is an exhaustion of subgroups that are built up from a
compact open subgroup of $G_{F}$ and a certain rigid stabilizer in $G$. The
proof consists in combining general considerations on confined subgroups
(Lemma 3.5 and Proposition 3.7) together with an approximation argument at the
level of the above subgroup $J$. The application to the Neretin groups and the
description of the closed cocompact subgroups of the Neretin groups are given
in Section 4.

### Acknowledgement

We thank Sven Raum for his comments on an earlier version of this paper. This
work was supported by ANR-19-CE40-0008-01 AODynG.

##  2 Preliminaries

Let $G$ be a locally compact group. The Chabauty space of closed subgroups of
$G$, which is compact, is denoted by $\mathrm{Sub}(G)$. We refer to [6, Ch.
VIII, §5] for a description of some of its basic properties.

Recall that if $X$ is a compact $G$-space, the stabilizer map
$X\to\mathrm{Sub}(G)$, $x\mapsto G_{x}$, is upper semi-continuous, meaning
that for every net $(x_{i})$ in $X$ converging to $x$ and such that
$(G_{x_{i}})$ converges to $L$ in $\mathrm{Sub}(G)$, one has $L\leq G_{x}$.

Recall that a closed subgroup $H$ of a locally compact group $G$ is relatively
amenable in $G$ if $H$ fixes a probability measure on every compact $G$-space
[11]. The following is well-known (it is implicit in [11], and also appears in
[35]).

######  Proposition 2.1.

A subgroup $H$ of $G$ is relatively amenable in $G$ if and only if $H$ fixes a
probability measure on $\partial_{F}G$.

A closed subgroup $H\leq G$ is called confined if the closure of its conjugacy
class in $\mathrm{Sub}(G)$ avoids the trivial subgroup $\\{e\\}$.

######  Proposition 2.2.

Let $G$ be a second countable locally compact group. Then there exists a
$G$-boundary on which the action is topologically free if and only if no
relatively amenable subgroup of $G$ is confined.

###### Proof.

The ‘only if’ part is a fairly direct consequence of the definitions; it holds
regardless of the second countability assumption. Suppose conversely that the
$G$-action on each of the $G$-boundaries is not topologically free. In
particular, this is true for the Furstenberg boundary $\partial_{F}G$. Since
$G$ is second countable, the space $\mathrm{Sub}(G)$ is metrizable, and it
follows from semi-continuity that there exists a point $z\in\partial_{F}G$ at
which the stabilizer map $\partial_{F}G\to\mathbf{Sub}(G):x\mapsto G_{x}$, is
continuous [28, Th. VII]. By hypothesis, the stabilizer $G_{z}$ is relatively
amenable and non-trivial. By [22, Prop. 1.2], the closure of the conjugacy
class of $G_{z}$ in $\mathbf{Sub}(G)$ is a URS. In particular $G_{z}$ is
confined. ∎

######  Remark 2.3.

The proposition remains true regardless of the second countability assumption.
Indeed, in view of the main results of [32], for every locally compact group
$G$, the stabilizer map $\partial_{F}G\to\mathbf{Sub}(G)$ is continuous
everywhere.

##  3 The proof of Theorem 1.3

Let $X$ be a compact $G$-space. Given a subset $\alpha\subseteq X$, we define
the rigid stabilizer of $\alpha$ in $G$ as the pointwise fixator of the
complement $X\setminus\alpha$. It is denoted by

$\operatorname{Rist}_{G}(\alpha)=\textrm{Fix}_{G}(X\setminus\alpha).$

Let us now assume that $X$ is totally disconnected. We say that the $G$-action
on $X$ is piecewise minimal (resp. piecewise strongly proximal) if for every
non-empty clopen set $\alpha\subseteq X$, the action of the rigid stabilizer
$\operatorname{Rist}_{G}(\alpha)$ on $\alpha$ is minimal (resp. strongly
proximal). If the $G$-action has both properties, we say that the $G$-action
is piecewise minimal-strongly-proximal; we shall focus on that situation.

The criterion of non-confinement we shall establish in order to prove Theorem
1.3 is Theorem 3.3 below. It requires the following definition.

######  Definition 3.1.

Let $G$ be a locally compact and $X$ be a totally disconnected compact
$G$-space. We denote by $\mathcal{S}_{X}$ the subset of $\mathrm{Sub}(G)$
consisting of those closed subgroups $H\leq G$ such that for every non-empty
clopen subset $\alpha$ of $X$, the stabilizer of $\alpha$ in $H$ fixes a
probability measure on $\alpha$.

We observe that $\mathcal{S}_{X}$ indeed contains all relatively amenable
subgroups of $G$.

######  Lemma 3.2.

Let $G$ be a locally compact and $X$ be a totally disconnected compact
$G$-space. Every relatively amenable closed subgroup of $G$ is contained in
$\mathcal{S}_{X}$.

###### Proof.

Let $H$ be subgroup of $G$ that is amenable relative to $G$. If $\alpha$ is a
non-empty clopen subset of $X$, then $\operatorname{Stab}_{G}(\alpha)$ is open
in $G$, and therefore $\operatorname{Stab}_{H}(\alpha)$ is amenable relative
to $\operatorname{Stab}_{G}(\alpha)$ [11, Lemma 11]. So
$\operatorname{Stab}_{H}(\alpha)$ fixes a probability measure on $\alpha$, and
$H\in\mathcal{S}_{X}$. ∎

In view Lemma 3.2, we see that Theorem 1.3 is an immediate consequence of the
following, which is the main result of this section. Here and in the rest of
this paper, we user the abbreviation tdlc for totally disconnected locally
compact.

######  Theorem 3.3.

Let $G$ be a tdlc group and $X$ be a totally disconnected compact $G$-space on
which the $G$-action is faithful and piecewise minimal-strongly-proximal. Then
no subgroup $H\in\mathcal{S}_{X}$ is confined.

The proof requires a few preparations. We recall the the commensurator of a
subgroup $H$ of a group $G$, denoted by $\mathrm{Comm}_{G}(H)$, consists of
those elements $g\in G$ such that the intersection $H\cap gHg^{-1}$ is of
finite index both in $H$ and in $gHg^{-1}$. We say that $H$ is a commensurated
subgroup of $G$ if $\mathrm{Comm}_{G}(H)=G$.

######  Lemma 3.4.

Let $G$ be a group acting on a set $X$ and let $\Lambda$ be a commensurated
subgroup of $G$. Then the union of the finite $\Lambda$-orbits in $X$ is a
$G$-invariant subset of $X$.

###### Proof.

Let $x\in X$ having a finite $\Lambda$-orbit and let $g\in G$. We want to see
that $\Lambda gx$ is finite. Since $g$ commensurates $\Lambda$, there exist
$g_{1},\ldots,g_{r}\in G$ such that $\Lambda g$ is covered by $\cup
g_{i}\Lambda$. So $\Lambda gx$ in contained in a finite union of finite sets,
and hence is finite. ∎

######  Lemma 3.5.

Let $G$ be a locally compact group and $X$ be a totally disconnected locally
compact $G$-space $X$ on which the $G$-action is faithful. Assume moreover
that for every non-empty clopen subset $\alpha$ of $X$, the action of
$\operatorname{Rist}_{G}(\alpha)$ on $\alpha$ is minimal and proximal. For
every closed subset $\mathcal{H}\subset\mathrm{Sub}(G)$ not containing the
trivial subgroup $\\{e\\}$, there exists $r\geq 1$ such that every element of
$\mathcal{H}$ has at most $r$ finite orbits in $X$.

###### Proof.

Since $\mathcal{H}$ is Chabauty-closed and does not contain the trivial
subgroup $\\{e\\}$, it follows that there exists a compact subset $P\subset G$
with $e\not\in P$ that intersects non-trivially every element of
$\mathcal{H}$.

Since all elements of $P$ act non-trivially on $X$, by compactness one can
find non-empty disjoint compact open subsets $\alpha_{0},\ldots,\alpha_{r}$
such that for every $g\in P$ there is $i$ such that $g(\alpha_{i})$ and
$\alpha_{i}$ are disjoint. Suppose that there exists $H\in\mathcal{H}$ with
$r+1$ distinct fixed points $x_{0},\ldots,x_{r}$. Upon replacing
$\alpha_{0},\ldots,\alpha_{r}$ with smaller subsets, we can assume that
$x_{i}$ does not belong to $\alpha_{j}$ for all $i,j$. Consider compact open
neighbourhoods $\beta_{i}$ of $x_{i}$ that are all disjoint and such that
$\beta_{i}$ does not intersect any $\alpha_{j}$. Since
$\operatorname{Rist}_{G}(\alpha_{i}\cup\beta_{i})$ acts minimally on
$\alpha_{i}\cup\beta_{i}$, there is
$g_{i}\in\operatorname{Rist}_{G}(\alpha_{i}\cup\beta_{i})$ such that
$g_{i}(x_{i})\in\alpha_{i}$. Then $g=g_{0}\cdots g_{r}$ verifies
$g(x_{i})\in\alpha_{i}$ for all $i$, so that $gHg^{-1}$ fixes a point inside
$\alpha_{i}$ for all $i$. So $gHg^{-1}$ does not intersect $P$, which is a
contradiction.

Suppose now that there exists $H\in\mathcal{H}$ with $r+1$ distinct finite
orbits $F_{0},\ldots,F_{r}$. Again we choose compact open neighbourhoods
$\beta_{i}$ of $F_{i}$ that are all disjoint. Using that the action of
$\operatorname{Rist}_{G}(\beta_{i})$ on $\beta_{i}$ is proximal, for each $i$,
one can find a net $g_{i,k}\in\operatorname{Rist}_{G}(\beta_{i})$ and a point
$x_{i}\in\beta_{i}$ such that $g_{i,k}(x)$ converges to $x_{i}$ for all $x\in
F_{i}$. Set $g_{k}=g_{0,k}\cdots g_{r,k}$. Then any accumulation point of
$g_{k}H{g_{k}}^{-1}$ in $\mathcal{H}$ fixes $x_{0},\ldots,x_{r}$, in
contradiction with the previous paragraph. ∎

Recall that a locally compact group is locally elliptic if every compact
subset is contained in a compact subgroup of $G$. The locally elliptic radical
$\mathrm{Rad_{LE}}(G)$ of $G$ is the union of all closed normal locally
elliptic subgroups of $G$; it is a closed normal subgroup of $G$ [38]. The
topological FC-center $B(G)$ of a locally compact group $G$ is the set of
elements with a relatively compact conjugacy class. It is a (not necessarily
closed) normal subgroup of $G$. More generally, if $H$ is a subgroup of $G$,
we let $B_{H}(G)$ be the set of elements of $G$ with a relatively compact
$H$-conjugacy class. Note that when $H$ is normal in $G$, $B_{H}(G)$ is also
normal in $G$. An elements $g\in G$ is periodic if the subgroup generated by
$g$ is relatively compact in $G$. The set of periodic elements of $G$ is
denoted $P(G)$.

The following result was proven by Wu–Yu in [42].

######  Proposition 3.6.

Let $G$ be a tdlc group such that $B(G)$ is dense in $G$. Then
$G/\mathrm{Rad_{LE}}(G)$ is a discrete torsion free abelian group.

In particular for every tdlc group $G$, the topological FC-center $B(G)$ is
contained in the amenable radical of $G$.

###### Proof.

Assume that $B(G)$ is dense in $G$. Under this assumption, Theorem 4 in [42]
asserts that $P(G)=\overline{P(G)\cap B(G)}$, that $P(G)$ is a characteristic
open subgroup of $G$, and that $G/P(G)$ is torsion free abelian. Hence to
conclude it is enough to see that $P(G)=\mathrm{Rad_{LE}}(G)$. By a result of
Usakov (see e.g. Theorem A in [42]), we have $P(G)\cap
B(G)\leq\mathrm{Rad_{LE}}(G)$. Since $\mathrm{Rad_{LE}}(G)$ is closed, we
infer that $P(G)\leq\mathrm{Rad_{LE}}(G)$. Now $G/P(G)$ has trivial locally
elliptic radical, as it is discrete and torsion free abelian, so we actually
have $P(G)=\mathrm{Rad_{LE}}(G)$, as desired.

Now for a general tdlc group $G$, the previous statement can be applied to
$H=\overline{B(G)}$. This subgroup is therefore an extension of a locally
elliptic group by an abelian group, and in particular is amenable. So $B(G)$
lies in the amenable radical of $G$, as desired. ∎

######  Proposition 3.7.

Let $G$ be tdlc group, $J$ an open subgroup of $G$ and $N$ a normal subgroup
of $G$ contained in $J$. Let $H_{0}$ be a closed subgroup of $G$ such that
$H_{0}\cap J=\\{e\\}$. Let $\mathcal{H}_{0,N}$ be the closure of the $N$-orbit
of $H_{0}$ in $\mathrm{Sub}(G)$, and let
$\mathcal{H}^{\prime}\subseteq\mathcal{H}_{0,N}$ be a closed minimal
$N$-invariant subset. Then $H\leq B_{N}(G)$ for every
$H\in\mathcal{H}^{\prime}$.

###### Proof.

We first observe that $H\cap J=\\{e\\}$ for all $H\in\mathcal{H}_{0,N}$.
Indeed, the map $\mathcal{H}_{0,N}\to\mathrm{Sub}(J)$, $H\mapsto H\cap J$, is
continuous since $J$ is open in $G$, and $N$-equivariant since $N$ is
contained in $J$. So the set of subgroup $H$ in $\mathcal{H}_{0,N}$ such that
$H\cap J=\\{e\\}$ is closed and $N$-invariant in $\mathcal{H}_{0,N}$, and
hence is equal to $\mathcal{H}_{0,N}$ since it contains $H_{0}$.

Now let $\mathcal{H}^{\prime}\subseteq\mathcal{H}_{0,N}$ be a closed minimal
$N$-invariant subset. Fix a compact open subgroup $U$ of $G$ contained in $J$,
a subgroup $H$ in $\mathcal{H}^{\prime}$ and an element $h_{0}$ in $H$. The
set

$\mathcal{O}=\left\\{K\in\mathcal{H}^{\prime}|K\cap
h_{0}U\neq\emptyset\right\\}$

is an open subset of $\mathcal{H}^{\prime}$, which is non-empty since
$H\in\mathcal{O}$. So by minimality and compactness one can find
$g_{1},\ldots,g_{r}$ in $N$ such that
$\mathcal{H}^{\prime}=\bigcup_{i}g_{i}\mathcal{O}g_{i}^{-1}$. Set
$P=\bigcup_{i}g_{i}h_{0}Ug_{i}^{-1}$. We infer that $K\cap P\neq\emptyset$ for
all $K$ in $\mathcal{H}^{\prime}$.

Consider the canonical projection $\pi\colon G\to G/N$. We have
$\pi(P)\subset\pi(h_{0})\pi(U)$, so that $P\subset h_{0}J$ since $UN\leq J$.
In particular, for any two elements $p_{1},p_{2}\in P$, we have
$p_{1}^{-1}p_{2}\in J$. Since $H\cap J$ is trivial, this implies that $H\cap
P=\left\\{h_{0}\right\\}$. Moreover, since $K\cap J=\\{e\\}$ for all
$K\in\mathcal{H}^{\prime}$, we see that $gHg^{-1}\cap h_{0}J$ contains at most
one element for any $g\in N$. Since $gHg^{-1}\cap P$ is non-empty and since
$P\subset h_{0}J$, we deduce that $gHg^{-1}\cap h_{0}J=gHg^{-1}\cap P$. On the
other hand, we have $gh_{0}g^{-1}\in gHg^{-1}\cap h_{0}J$ since $g\in N\leq
J$. Therefore we have $gh_{0}g^{-1}\in P$. This shows that the $N$-conjugacy
class of $h_{0}$ is entirely contained in the compact set $P$, and hence
$h_{0}\in B_{N}(G)$. This is valid for all $h_{0}\in H$ and all
$H\in\mathcal{H}^{\prime}$, so the statement is proved. ∎

###### Proof of Theorem 3.3.

Before going into the proof, let us first observe that if the space $X$ is
finite, then $X$ is a singleton by strong proximality of the $G$-action on
$X$. Since $G$ acts faithfully, it is the trivial group, and the required
conclusion is trivially true. We assume henceforth that $X$ is infinite.

We first observe that if a closed subgroup $H$ of $G$ fixes probability
measure $\mu$ on $X$, then by strong proximality of $G$ we can find a net
$(g_{i})$ in $G$ such that $g_{i}\mu$ converges to a Dirac measure
$\delta_{x}$. By upper semi-continuity, every accumulation point of
$(H^{g_{i}})$ in $\mathrm{Sub}(G)$ fixes the point $x$.

Suppose for a contradiction that there exists $H_{0}\in\mathcal{S}_{X}$ that
is confined, and let $\mathcal{H}$ denote the orbit closure of $H_{0}$ in
$\mathrm{Sub}(G)$. Hence $\mathcal{H}$ does not contain $\\{e\\}$. For
$K\in\mathcal{H}$, let $n_{f}(K)$ be the number of finite $K$-orbits in $X$.
According to Lemma 3.5, the supremum $r=\sup\\{n_{f}(K)\mid
K\in\mathcal{H}\\}$ is finite. Let $L$ in $\mathcal{H}$ such that
$n_{f}(L)=r$. Note that $n_{f}(L)\geq 1$ by the initial observation in the
second paragraph of this proof. Let $F\subset X$ be the set of points with a
finite $L$-orbit. We denote by $G_{F}$ the pointwise fixator of $F$ and by
$G_{(F)}$ the setwise stabilizer of $F$ in $G$. We also let $G_{F}^{0}$ be the
subgroup of $G_{F}$ consisting of those elements fixing pointwise a
neighbourhood of $F$. In other words, the group $G_{F}^{0}$ is the union,
taken over all clopen subsets $\alpha\subset X$ with $\alpha\cap
F=\varnothing$, of the rigid stabilizer $\operatorname{Rist}_{G}(\alpha)$.
Note that $G_{F}$ is a finite index open subgroup of $G_{(F)}$, and that
$G_{F}^{0}$ is a normal subgroup of $G_{(F)}$. We fix a compact open subgroup
$U$ of $G_{(F)}$ contained in $G_{F}$, and we denote by $J$ the subgroup of
$G_{(F)}$ generated by $U$ and $G_{F}^{0}$. Since $G_{F}^{0}$ is normal in
$G_{(F)}$ and $U$ is commensurated in $G_{(F)}$, the subgroup $J$ is
commensurated in $G_{(F)}$.

Let $\mathcal{K}$ be the subset of $\mathcal{H}$ consisting of those elements
$H$ such that $H$ is contained in $G_{(F)}$. Note that $\mathcal{K}$ is closed
and $G_{(F)}$-invariant. By the definition of $F$, we have $L\in\mathcal{K}$,
so that $\mathcal{K}$ is non-empty.

######  Lemma 3.8.

We have $H\cap J\neq\\{e\\}$ for all $H\in\mathcal{K}$.

###### Proof of Lemma 3.8.

Suppose for a contradiction that this is not the case, i.e. there exists
$K_{0}\in\mathcal{K}$ such that $K_{0}\cap J=\\{e\\}$. By invoking Proposition
3.7 for the group $G_{(F)}$, the open subgroup $J$ and the normal subgroup
$G_{F}^{0}$, we infer that there exists $K\in\mathcal{K}$ such that $K$ is
contained in $B_{G_{F}^{0}}(G_{(F)})$.

Now, we claim that the subgroup $B_{G_{F}^{0}}(G_{(F)})$ is actually trivial.
Otherwise, since $B_{G_{F}^{0}}(G_{(F)})$ is a normal subgroup of $G_{(F)}$,
the classical commutator lemma for normal subgroups (see e.g. [36, Lem. 4.1]
for a modern reference) asserts that there exists a compact open subset
$\alpha$ of $X\setminus F$ such that the derived subgroup
$\operatorname{Rist}_{G}(\alpha)^{\prime}$ is contained in
$B_{G_{F}^{0}}(G_{(F)})$. Since $\operatorname{Rist}_{G}(\alpha)$ is contained
in $G_{F}^{0}$, this actually implies that
$\operatorname{Rist}_{G}(\alpha)^{\prime}$ is contained in
$B(\operatorname{Rist}_{G}(\alpha))$, and hence also in the amenable radical
of $\operatorname{Rist}_{G}(\alpha)$ according to Proposition 3.6. On the
other hand, by assumption the action of $\operatorname{Rist}_{G}(\alpha)$ on
$\alpha$ is faithful, minimal and strongly proximal, so the group
$\operatorname{Rist}_{G}(\alpha)$ has a trivial amenable radical (indeed
$\alpha$ is not a singleton since $X$ is infinite by hypothesis). In
particular $\operatorname{Rist}_{G}(\alpha)$ has trivial topological FC-
center, and $\operatorname{Rist}_{G}(\alpha)^{\prime}$ is trivial. This
implies that $\operatorname{Rist}_{G}(\alpha)$ would be abelian, hence
amenable, which again is impossible. It follows that $B_{G_{F}^{0}}(G_{(F)})$
is indeed trivial, and hence so is the subgroup $K$. We deduce that
$\mathcal{H}$ contains the trivial subgroup, which is a contradiction. ∎

We next record the following.

######  Lemma 3.9.

For every $H\in\mathcal{K}$, the subgroup $H\cap J$ does not have any finite
orbit in $X\setminus F$.

###### Proof of Lemma 3.9.

The set

$\left\\{H\cap J|H\in\mathcal{K}\right\\}$

is a closed $J$-invariant subset of $\mathrm{Sub}(J)$, which does not contain
the trivial subgroup according to Lemma 3.8. Let us fix some
$H\in\mathcal{K}$. Since $J$ contains $G_{F}^{0}$, we deduce that the
$J$-action on $X\setminus F$ satisfies the hypotheses of Lemma 3.5. Hence, the
number of finite $H\cap J$-orbits is finite. Since $J$ is a commensurated
subgroup of $G_{(F)}$, the subgroup $H\cap J$ is commensurated in $H$, and
hence the union of the finite orbits of $H\cap J$ is an $H$-invariant set (see
Lemma 3.4). By the definition of $\mathcal{K}$, every finite $H$-orbit is
contained in $F$. This confirms that every finite $H\cap J$-orbit is contained
in $F$, as desired. ∎

We shall now finish the proof of Theorem 3.3. Let $A$ be the collection of
$U$-invariant clopen neighbourhoods of $F$. For $\alpha\in A$, the subgroup
$\operatorname{Rist}_{G}(X\\!\setminus\\!\alpha)$ is normalized by $U$ since
$U$ stabilizes $\alpha$, and
$J_{\alpha}:=\operatorname{Rist}_{G}(X\\!\setminus\\!\alpha)U$ is a subgroup
of $J$. We have $J_{\alpha}\subset J_{\alpha^{\prime}}$ for all
$\alpha^{\prime}\subseteq\alpha$. Since $A$ forms a basis of neighbourhoods of
$F$, we have
$\bigcup_{A}\operatorname{Rist}_{G}(X\\!\setminus\\!\alpha)=G_{F}^{0}$, so
that $\bigcup_{A}J_{\alpha}=J$.

We fix a point $\xi$ in $X\setminus F$, and $\alpha\in A$ such that
$\xi\notin\alpha$. We also fix an element $H\in\mathcal{K}$. The subgroup
$H\cap J_{\alpha}$ stabilizes the clopen subset $X\\!\setminus\\!\alpha$, and
hence fixes a probability measure $\nu_{\alpha}$ on $X\\!\setminus\\!\alpha$
by the assumption that $H$ belongs to $\mathcal{S}_{X}$. Since the action of
$\operatorname{Rist}_{G}(X\\!\setminus\\!\alpha)$ on $X\\!\setminus\\!\alpha$
is minimal and strongly proximal, one can find a net $(g_{k})$ in
$\operatorname{Rist}_{G}(X\\!\setminus\\!\alpha)$ such that
$g_{k}(\nu_{\alpha})$ converges to the Dirac measure at $\xi$. By compactness
we may assume that $H^{g_{k}}$ converges to a point $H^{(\alpha)}$ in
$\mathcal{K}$. Since $J_{\alpha}$ is open, it follows that $H^{g_{k}}\cap
J_{\alpha}$ converges to $H^{(\alpha)}\cap J_{\alpha}$, which must therefore
fix $\xi$ by the choice of the net $(g_{k})$.

Now by compactness again, upon passing to a subnet we may assume that
$H^{(\alpha)}$ converges in $\mathcal{K}$. Let $K$ be its limit. We claim that
$K\cap J$ fixes $\xi$. Since $J=\bigcup_{A}J_{\beta}$, it suffices to check
that $K\cap J_{\beta}$ fixes $\xi$ for all $\beta\in A$. We fix $\beta\in A$.
Using again that $J_{\beta}$ is open, the subgroup $K\cap J_{\beta}$ is the
limit of $H^{(\alpha)}\cap J_{\beta}$. Now eventually we have $J_{\beta}\leq
J_{\alpha}$. Therefore $H^{(\alpha)}\cap J_{\beta}$ is a subgroup of
$H^{(\alpha)}\cap J_{\alpha}$, and hence $H^{(\alpha)}\cap J_{\beta}$ fixes
$\xi$. By upper semi-continuity, so does $K\cap J_{\beta}$. This proves the
claim. So $K$ is an element of $\mathcal{K}$ such that $K\cap J$ fixes $\xi$.
Since Lemma 3.9 above says that no element of $\mathcal{K}$ can have this
property, we have obtained a contradiction, thereby finishing the proof of the
theorem. ∎

As mentioned in the introduction, the conclusion of Theorem 3.3 was already
known when $G$ is a discrete group, as it follows from Theorem 1.1 of [31].
This result asserts that when $G$ is discrete and $X$ is a $G$-space on which
$G$ acts faithfully, then every confined subgroup of $G$ contains the
commutator subgroup of the rigid stabilizer of some non-empty open subset of
$X$. Here we note that this much stronger conclusion no longer holds outside
the realm of discrete groups. For instance let $G=\mathrm{Aut}(T_{d})$ be the
automorphism group of a $d$-regular tree, $d\geq 3$. The group $G$ acts
faithfully on $\partial T$. If $\Gamma$ is a discrete and cocompact subgroups
for $G$ (take for instance $\Gamma$ to be the free product of $d$ copies of
the cyclic group $C_{2}$), then $\Gamma$ has a closed conjugacy class in
$\mathrm{Sub}(G)$ because $\Gamma$ is cocompact. So in particular $\Gamma$ is
a confined subgroup of $G$. But on the other hand the pointwise fixator in
$\Gamma$ of an infinite subtree is trivial; and hence $\Gamma$ obviously
cannot contain the commutator subgroup the rigid stabilizer of a non-empty
open subset of $\partial T$.

##  4 Neretin groups

### 4.1 Definitions

We briefly review basic notions concerning Neretin groups (see [9, §6.3] and
[21] for more details). Let $d,k\geq 2$ be integers, and let
$\mathcal{T}_{d,k}$ be a rooted tree such that the root has $k$ descendants,
and every vertex distinct from the root has $d$ descendants. Notice that
$\mathcal{T}_{d,d}$ is the regular rooted tree of degree $d$. Moreover, the
graph $\mathcal{T}_{d,d+1}$ is isomorphic to the regular non-rooted tree of
degree $d+1$. An almost automorphism of $\mathcal{T}_{d,k}$ is a triple of the
form $(A,B,\varphi)$, where $A$ and $B$ are finite subtrees of
$\mathcal{T}_{d,k}$ containing the root such that $|\partial A|=|\partial B|$,
and $\varphi$ is an isomorphism of forests $\mathcal{T}_{d,k}\setminus
A\to\mathcal{T}_{d,k}\setminus B$. The group of almost automorphisms of
$\mathcal{T}_{d,k}$, denoted by $\mathcal{N}_{d,k}$, is the quotient of the
set of all almost automorphisms by the relation which identifies two almost
automorphisms $(A,B,\varphi)$ and $(A^{\prime},B^{\prime},\varphi^{\prime})$
if there exists some finite subtree $A^{\prime\prime}$ containing $A\cup
A^{\prime}$ and such that $\varphi$ and $\varphi^{\prime}$ coincide on
$\mathcal{T}_{d,k}\setminus A^{\prime\prime}$. It is easy to verify that
$\mathcal{N}_{d,k}$ is indeed a group. It carries a unique locally compact
group topology such that the natural inclusion of the profinite group
$\operatorname{Aut}(\mathcal{T}_{d,k})$ in $\mathcal{N}_{d,k}$ is continuous
and open. In particular $\mathcal{N}_{d,k}$ is a totally disconnected locally
compact group. Moreover, as observed in [9, §6.3], the group
$\mathcal{N}_{d,k}$ has a finitely generated dense subgroup, and is thus
compactly generated.

Observe that for all integers $d,k$, the topological group $\mathcal{N}_{d,k}$
is naturally isomorphic to $\mathcal{N}_{d,k+d-1}$, so that the values of $k$
in the set $\\{2,3,\dots,d\\}$ suffice to account for all isomorphism classes
of the Neretin groups.

The collection of those elements of $\mathcal{N}_{d,k}$ that can be
represented by an almost automorphism $(A,B,\varphi)$ such that $A=B$ is a
ball around the root of $\mathcal{T}_{d,k}$ is a subgroup of
$\mathcal{N}_{d,k}$ that we denote by $\mathcal{O}_{d,k}$. Clearly, the full
automorphism group $\operatorname{Aut}(\mathcal{T}_{d,k})$ of the rooted tree
is a compact open subgroup of $\mathcal{N}_{d,k}$ contained in
$\mathcal{O}_{d,k}$. The group $\mathcal{O}_{d,k}$ is the directed union of
its compact open subgroups and is thus amenable. Moreover $\mathcal{O}_{d,k}$
is topologically simple (see [9, Rem. 6.8 and Lem. 6.9]).

The set of ends $\partial\mathcal{T}_{d,k}$ is a compact
$\mathcal{N}_{d,k}$-space on which $\mathcal{N}_{d,k}$ acts faithfully. Given
a vertex $v$ of $\mathcal{T}_{d,k}$, we denote by $\mathcal{T}_{v}$ the
subtree spanned by all vertices separated from the root by $v$. In particular,
if $v_{0}$ denotes the root, we have $\mathcal{T}_{v_{0}}=\mathcal{T}_{d,k}$.
For each vertex $v$, the set $\partial\mathcal{T}_{v}$ is a basic clopen
subset of $\partial\mathcal{T}_{d,k}$; the collection of those clopen subsets
forms a basis of the topology on $\partial\mathcal{T}_{d,k}$.

### 4.2 Application of Theorem 1.3

The following important feature follows from the self-replicating properties
of the Neretin groups.

######  Lemma 4.1.

The action of $\mathcal{N}_{d,k}$ on $\partial\mathcal{T}_{d,k}$ is piecewise
minimal-strongly-proximal.

###### Proof.

Set $G=\mathcal{N}_{d,k}$. Let us first show that, for every $d$ and $k$, the
$G$-action on $\partial\mathcal{T}_{d,k}$ is minimal and strongly proximal.

Clearly the action is transitive, hence minimal. It is easy to see that the
action is $n$-transitive (i.e. transitive on ordered $n$-tuples of distinct
points) for all $n$. In particular it is proximal. To show strong proximality,
it suffices by [34, Prop. VI.1.6] to show the existence of a compressible non-
empty open set. The existence of such an open set is clear since $G$ acts
transitively on the basic clopen sets arising as the set of ends of the sub-
rooted trees of the form $\mathcal{T}_{v}$, where $v$ is a vertex different
from the root.

Let now $\alpha$ be a non-empty clopen subset of $\partial\mathcal{T}_{d,k}$.
Then there is a finite set of vertices $\\{v_{1},\dots,v_{n}\\}$ distinct from
the root, such that $\alpha$ decomposes as the disjoint union
$\bigcup_{i=1}^{n}\partial\mathcal{T}_{v_{i}}$. Moreover, it follows from the
definitions that $\operatorname{Rist}_{G}(\alpha)$ is isormorphic to
$\mathcal{N}_{d,n}$, and that $\alpha$ is equivariantly homeomorphic to
$\partial T_{d,n}$. Therefore, it follows from the first part of the proof
that the $\operatorname{Rist}_{G}(\alpha)$-action on $\alpha$ is minimal and
strongly proximal. ∎

As announced in the introduction, the Neretin groups constitute an important
family of examples to which Theorem 1.3 applies.

###### Proof of Corollary 1.4.

The $\mathcal{N}_{d,k}$-action on the compact space
$X=\partial\mathcal{T}_{d,k}$ is continuous and faithful. Moreover it is
piecewise minimal-strongly proximal by Lemma 4.1. Thus the conclusion follows
from Theorem 1.3. ∎

We underline that for $X=\partial\mathcal{T}_{d,k}$, the Neretin group
$\mathcal{N}_{d,k}$ has a discrete subgroup $F_{d,k}$, also called Thompson’s
group $F_{d,k}$, which belongs to $\mathcal{S}_{X}$, while it is an open
problem to determine whether $F_{d,k}$ is relatively amenable in
$\mathcal{N}_{d,k}$ (not to mention the problem whether or not $F_{d,k}$ is
amenable). We refer to [9, §6.3] for the definition.

######  Corollary 4.2.

For all integers $d,k\geq 2$, Thompson’s group $F_{d,k}$ is not confined in
the Neretin group $\mathcal{N}_{d,k}$.

###### Proof.

Let $X=\partial\mathcal{T}_{d,k}$ and $\Gamma=F_{d,k}$. It follows from the
definition of $F_{d,k}$ that for any non-empty clopen subset $\alpha\subseteq
X$, the stabilizer $\operatorname{Stab}_{\Gamma}(\alpha)$ fixes a point in
$\alpha$. In particular $\operatorname{Stab}_{\Gamma}(\alpha)$ fixes a Dirac
probability measure on $\alpha$. Thus $\Gamma\in\mathcal{S}_{X}$, and the
conclusion follows from Theorem 3.3. ∎

######  Remark 4.3.

N. Monod [35] has proved that a locally compact group $G$ having a compact
subgroup $K$ such that $(G,K)$ forms a Gelfand pair, must have a cocompact
amenable closed subgroup. Y. Neretin [37, Th. 1.2] has proved that the group
$\mathcal{N}_{d,2}\cong\mathcal{N}_{d,d+1}$ has an open subgroup $A$ such that
$(\mathcal{N}_{d,2},A)$ forms a generalized Gelfand pair. In view of Corollary
1.4, we see that Monod’s result cannot be extended from Gelfand pairs to
generalized Gelfand pairs, even among simple groups. Without the condition of
simplicity, this can be observed by a much more straightforward argument.
Indeed, given a locally compact group $G$ and a subgroup $A$ containing a
closed normal subgroup $N$ of $G$, every irreducible unitary representation of
$G$ with a nonzero $A$-invariant vector factorizes through a representation of
$G/N$. In particular $(G,A)$ is a generalized Gelfand pair if and only if
$(G/N,A/N)$ is one. Taking $N$ to a discrete free group of countable rank,
$G=N\rtimes\mathrm{SL}_{2}(\mathbf{Q}_{p})$ (where
$\mathrm{SL}_{2}(\mathbf{Q}_{p})$ acts by permuting continuously the elements
of a free basis of $N$) and $A=N\rtimes\mathrm{SL}_{2}(\mathbf{Z}_{p})$, we
deduce that $(G,A)$ is a generalized Gelfand pair (because
$(\mathrm{SL}_{2}(\mathbf{Q}_{p}),\mathrm{SL}_{2}(\mathbf{Z}_{p}))$ is a
Gelfand pair) such that $G$ does not have any cocompact amenable subgroup.

### 4.3 Quasi-regular representations

We refer to [5, Appendix F] for general background about the Fell topology and
weak containment of unitary representations.

######  Proposition 4.4.

Let $G$ be a second countable locally compact group and $H\leq G$ be a closed
subgroup. If the closure of the conjugacy class of $H$ in $\mathrm{Sub}(G)$
contains the trivial subgroup $\\{e\\}$, then the quasi-regular representation
$\lambda_{G/H}$ weakly contains the regular representation $\lambda_{G}$.

###### Proof.

By a result of Fell [17], if a sequence $(H_{n})$ of closed subgroups
converges to $J$ in $\mathrm{Sub}(G)$, the corresponding sequence of quasi-
regular representations $(\lambda_{G/H_{n}})$ converges to $\lambda_{G/J}$ in
the Fell topology. By hypothesis, there exists a sequence $(H_{n})$ of
conjugates of $H$ in $G$ such that $\lim_{n}H_{n}=\\{e\\}$. Since $H_{n}$ is
conjugate to $H$, the quasi-regular representation $\lambda_{G/H_{n}}$ is
equivalent to $\lambda_{G/H}$. Hence, we deduce from Fell’s result that the
constant sequence $(\lambda_{G/H})$ converges to the regular representation
$\lambda_{G}$ in the Fell topology. This is a reformulation of the fact that
$\lambda_{G/H}$ weakly contains $\lambda_{G}$. ∎

Combining this with Corollary 1.4, we obtain the following.

######  Corollary 4.5.

Let $d,k\geq 2$ be integers. For any relatively amenable subgroup $A$ of
$G=\mathcal{N}_{d,k}$, the quasi-regular representation $\lambda_{G/A}$ is
weakly equivalent to the regular representation $\lambda_{G}$.

###### Proof.

Since $A$ is relatively amenable, the representation $\lambda_{G/A}$ is weakly
contained in $\lambda_{G}$. By Corollary 1.4, the closure of the conjugacy
class of $A$ in $\mathrm{Sub}(G)$ contains the trivial subgroup $\\{e\\}$. By
Proposition 4.4, this implies that $\lambda_{G}$ is weakly contained in
$\lambda_{G/A}$. ∎

The following proposition records a special case of well known results due to
G. Mackey [33].

######  Proposition 4.6.

Let $G$ be a second countable locally compact group and $H\leq G$ be an open
subgroup.

  1. (i)

If $\mathrm{Comm}_{G}(H)=H$, then the unitary representation $\lambda_{G/H}$
is irreducible.

  2. (ii)

If $J\leq G$ is an open subgroup such that every $H$-orbit on $G/J$ is
infinite, then the unitary representations $\lambda_{G/H}$ and $\lambda_{G/J}$
are not equivalent.

###### Proof.

The first assertion follows from [33, Theorem $6^{\prime}$]. To prove the
second assertion, it suffices to observe that $H$ has a non-zero invariant
vector in $\ell^{2}(G/H)$, while the hypothesis implies that the only
$H$-invariant vector in $\ell^{2}(G/J)$ is the zero function. ∎

We can now prove the following.

######  Proposition 4.7.

For all integers $d,k\geq 2$, the Neretin group $G=\mathcal{N}_{d,k}$ has two
open amenable subgroups $O,O^{\prime}$ such that the unitary representations
$\lambda_{G/O}$ and $\lambda_{G/O^{\prime}}$ are irreducible and inequivalent.

###### Proof.

Let $K$ be the full automorphism group of the rooted tree $\mathcal{T}_{d,k}$.
Thus $K$ is a compact open subgroup of $\mathcal{N}_{d,k}$ acting transitively
on the set of ends $X=\partial\mathcal{T}_{d,k}$. Thus $K$ fixes a unique
probability measure $\nu$ on $X$.

Let $O$ denote the stabilizer of $\nu$ in $G$, namely $O=G_{\nu}$. Given
vertices $v,w$ of $\mathcal{T}_{d,k}$, the set of ends
$\partial\mathcal{T}_{v}$ and $\partial\mathcal{T}_{w}$ are basic clopen
subsets of $X$. Moreover, we have
$\nu(\partial\mathcal{T}_{v})=\nu(\partial\mathcal{T}_{w})$ if and only if $v$
and $w$ have the same level (i.e. their distance from the root are equal).
This implies that any element of $O$ can be represented by an almost
automorphism $(A,B,\varphi)$ such that $A=B$ is a ball around the root. It
follows that $O=\mathcal{O}_{d,k}$ (§4.1). Hence $O$ is the directed union of
its compact open subgroups; in particular $O$ is amenable. Using [9, Rem. 6.8
and Lem. 6.9], we see that $O$ is topologically simple. In particular, it does
not have any proper open subgroup of finite index. It follows that
$\mathrm{Comm}_{G}(O)=N_{G}(O)$. Since $\nu$ is the unique $K$-invariant
probability measure on $X$, it is also the unique $O$-invariant probability
measure. This implies that $N_{G}(O)\leq G_{\nu}=O$. Thus $\lambda_{G/O}$ is
irreducible by Proposition 4.6(i).

Let now $v_{1},\dots,v_{k}$ denote the $k$ descendants of the root of
$\mathcal{T}_{d,k}$. Set $\alpha_{i}=\partial\mathcal{T}_{v_{i}}$. Set
$O_{i}=\operatorname{Rist}_{O}(\alpha_{i})$. The group

$P=\langle O_{1}\cup\dots\cup O_{k}\rangle$

is isomorphic to the direct product $O_{1}\times\dots\times O_{k}$. We set
$O^{\prime}=N_{O}(P)$.

Notice that $O^{\prime}$ contains $K$, hence it is open. Moreover, we have
$O^{\prime}\leq O$ so that $O^{\prime}$ is amenable. For each $i$, the sub-
tree $\mathcal{T}_{v_{i}}$ is naturally isomorphic to $\mathcal{T}_{d,d}$ and
that map defines an isomorphism of $O_{i}$ to the group $\mathcal{O}_{d,d}$
defined above. Therefore, the group $O_{i}$ is topologically simple, so that
$P$ does not have any proper open subgroup of finite index. It follows that
$\mathrm{Comm}_{G}(O^{\prime})$ normalizes $P$, and thus acts by permutation
on the $k$ direct factors $O_{1},\dots,O_{k}$ by the Krull–Remak–Schmidt
theorem (see [40, 3.3.8 and 3.3.11]). Since $O_{i}$ fixes a unique probability
measure $\nu_{i}$ supported on $\alpha_{i}$, we deduce that
$\mathrm{Comm}_{G}(O^{\prime})$ permutes the measures $\nu_{i}$, hence it
fixes $\nu$, since we have $\nu=\frac{1}{k}\sum_{i=1}^{k}\nu_{i}$. Therefore
we have $\mathrm{Comm}_{G}(O^{\prime})\leq G_{\nu}=O$. This finally shows that
$\mathrm{Comm}_{G}(O^{\prime})=N_{O}(P)=O^{\prime}$, so that
$\lambda_{G/O^{\prime}}$ is irreducible by Proposition 4.6(i).

In order to show that $\lambda_{G/O}$ and $\lambda_{G/O^{\prime}}$ are not
equivalent, it suffices to show by Proposition 4.6(ii) that every $O$-orbit on
$G/O^{\prime}$ is infinite. Since $O$ is topologically simple, the existence
of a finite $O$-orbit on $G/O^{\prime}$ implies the existence of a fixed
point, which implies in turn that $O$ is contained in some conjugate
$gO^{\prime}g^{-1}$. Since $O$ and $O^{\prime}$ are both amenable and since
$\nu$ is the unique $O$-invariant measure on $\partial T$, this implies that
$O=gO^{\prime}g^{-1}$. This is impossible since $O$ is topologically simple
but $O^{\prime}$ is not. ∎

We can now complete the proof of Theorem 1.5.

###### Proof of Theorem 1.5.

By Proposition 4.7, the group $G=\mathcal{N}_{d,k}$ has two inequivalent
irreducible unitary representations of the form $\lambda_{G/O}$ and
$\lambda_{G/O^{\prime}}$, with $O,O^{\prime}$ open amenable subgroups of $G$.
In view of Corollary 4.5, those representations are both weakly equivalent to
the regular representation. By Glimm’s theorem (see [23, Theorem 1]), in a
second countable locally compact group of type I, any two weakly equivalent
irreducible unitary representations are equivalent. It follows that $G$ is not
a type I group. ∎

######  Remark 4.8.

We believe that the amenable group $\mathcal{O}_{d,k}$, which is an open
subgroup of $\mathcal{N}_{d,k}$, is not a type I group either. This
conjectural statement is formally stronger than Theorem 1.5 since the type I
property is inherited by open subgroups***This conjecture has been confirmed
by R. Arimoto [2] after the first version of the present paper was
circulated.. That conjecture, and the more ambitious problem of classifying
the irreducible unitary CCR representations of $\mathcal{O}_{d,k}$, raises
interesting questions on the asymptotic representation theory of the finite
symmetric groups.

### 4.4 Cocompact subgroups and Chabauty-isolation

In view of the result from [3] and its far-reaching generalization obtained in
[43], as well as Corollary 1.4, it is a natural problem to try to classify the
confined subgroups of the group $\mathcal{N}_{d,k}$. Since closed cocompact
subgroups are in a sense the most basic examples of confined subgroups,
classifying these is a natural first step towards this problem. The aim of
this section, which is independent of the previous ones, is to establish the
following:

######  Theorem 4.9.

Let $d,k\geq 2$ be integers. Any proper closed cocompact subgroup $H$ of
$G=\mathcal{N}_{d,k}$ fixes a point in $\xi\in\partial\mathcal{T}_{d,k}$.
Moreover $H$ is a finite index subgroup of $G_{\xi}$.

In particular, the only closed cocompact unimodular subgroup of $G$ is $G$
itself.

It should be noted that the stabilizer $G_{\xi}$ of a point
$\xi\in\partial\mathcal{T}_{d,k}$ does indeed have proper open subgroups of
finite index. Indeed $G_{\xi}$ admits an infinite cyclic quotient. This can be
seen for example by observing that $G_{\xi}$ is compactly generated (because
it is cocompact in the compactly generated group $G$) but not unimodular
(because $\partial\mathcal{T}_{d,k}$ does not carry any $G$-invariant
probability measure by Lemma 4.1).

We also record the following consequence of independent interest.

######  Corollary 4.10.

For all integers $d,k\geq 2$, the Neretin group $\mathcal{N}_{d,k}$ is an
isolated point of the Chabauty space $\mathrm{Sub}(\mathcal{N}_{d,k})$.

###### Proof.

A sequence $(H_{n})$ of proper closed subgroups converging to
$G=\mathcal{N}_{d,k}$ is eventually cocompact by [6, VIII.5.3, Proposition 6]
since $G$ is compactly generated. Hence upon extracting we may assume that
$H_{n}$ fixes a point $\xi_{n}\in\partial\mathcal{T}_{d,k}$ for all $n$ by
Theorem 4.9. The limit $G=\lim_{n}H_{n}$ must then fix any accumulation point
of $(\xi_{n})$ in $\partial\mathcal{T}_{d,k}$, which is absurd since $G$ is
transitive on $\partial\mathcal{T}_{d,k}$. ∎

######  Remark 4.11.

Corollary 4.10 lies in sharp contrast with the case of the full automorphism
group $\operatorname{Aut}(T)$ of a non-rooted regular tree of degree $n\geq
4$, which is not Chabauty-isolated. Indeed the group $\operatorname{Aut}(T)$
can be approximated by a sequence of closed boundary-$2$-transitive subgroups
of $\operatorname{Aut}(T)$ (see the Appendix to [13]).

The proof of Theorem 4.9 follows a similar strategy as in [3]. The key step is
provided by the following.

######  Proposition 4.12.

Let $d,k$ be integers. The only proper cocompact closed subgroups of
$O=\mathcal{O}_{d,k}$ are the stabilisers $O_{\xi}$ of points
$\xi\in\partial\mathcal{T}_{d,k}$.

The proof requires subsidiary facts on finite permutation groups.

#### 4.4.1 Factorizations of finite symmetric groups

The following result is based on a refinement of the arguments from §3 of [3].

######  Proposition 4.13.

Let $\mathbf{n}=\\{1,\dots,n\\}$. Assume that $n$ is large enough so that the
interval $[n/2,n]$ contains at least $3$ primes (we remark that the Prime
Number Theorem implies that the number of primes between $n/2$ and $n$ tends
to infinity with $n$).

Let $A,B$ be subgroups of the symmetric group $\operatorname{Sym}(\mathbf{n})$
such that

$\operatorname{Sym}(\mathbf{n})=AB,$

where $AB=\\{ab\mid a\in A,b\in B\\}$. If $A$ is transitive and imprimitive on
$\mathbf{n}$, then either $B$ fixes a point $x\in\mathbf{n}$ and contains the
full alternating group $\operatorname{Alt}(\mathbf{n}\setminus\\{x\\})$, or
$B$ contains $\operatorname{Alt}(\mathbf{n})$.

We first collect a couple of preliminaries. The following one is a theorem of
C. Jordan [24].

######  Proposition 4.14.

Let $\mathbf{k}=\\{1,\dots,k\\}$. A primitive (hence transitive) subgroup of
$\operatorname{Sym}(\mathbf{k})$ containing a prime cycle of order $p\leq k-3$
contains the full alternating group $\operatorname{Alt}(\mathbf{k})$.

###### Proof.

See Theorem 13.9 in [41]. ∎

The following result appears implicitly in §3 of [3].

######  Lemma 4.15.

Let $\mathbf{n}=\\{1,\dots,n\\}$ and $A,B$ be subgroups of the symmetric group
$\operatorname{Sym}(\mathbf{n})$ such that
$\operatorname{Sym}(\mathbf{n})=AB$. We assume that $A$ is transitive and
imprimitive on $\mathbf{n}$. If the interval $[n/2,n]$ contains at least $3$
primes, then there exists a subset $\Omega\subset\mathbf{n}$ of size $k\geq
n/2+4$ such that $B$ contains the full alternating group
$\operatorname{Alt}(\Omega)$.

For the sake of completeness, we reproduce the proof borrowed from [3].

###### Proof of Lemma 4.15.

Since $A$ is transitive but imprimitive, it is contained in a subgroup of
$\operatorname{Sym}(\mathbf{n})$ of the form
$\operatorname{Sym}(d)\wr\operatorname{Sym}(n/d)$ for some proper divisor $d$
of $n$. In particular no prime in the interval $[\frac{n+1}{2},n]$ divides the
order of $A$. Therefore, the transitivity of $B$ on
$\operatorname{Sym}(\mathbf{n})/A$ implies that any such prime divides $|B|$.
The hypothesis made on $n$ implies that $[\frac{n+1}{2},n]$ contains two
distinct primes, say $p<q$. Let $\alpha,\beta\in B$ be two elements of order
$p$ and $q$. Then $\alpha$ and $\beta$ act as $p$\- and $q$-cycles on
$\mathbf{n}$, whose supports intersect non-trivially. Let $\Omega$ be the
union of their support. We have $|\Omega|\geq q+1\geq p+3\geq n/2+4$. The
subgroup of $B$ generated by $\alpha$ and $\beta$ is transitive on $\Omega$.
Therefore it is also primitive, since it contains a $p$-cycle with
$p>|\Omega|/2$. We infer from Proposition 4.14 that it contains
$\operatorname{Alt}(\Omega)$. ∎

###### Proof of Proposition 4.13.

Throughout, we set $G=\operatorname{Sym}(\mathbf{n})$.

We first claim that $B$ has at most two orbits on $\mathbf{n}$, one of which
is of size $\geq n-1$. If this were not the case, then $B$ would stabilise a
proper subset of $\mathbf{n}$ for size $m$, with $2\leq m\leq n/2$. It then
follows that $B$ is contained in a subgroup $M\leq G$ of the form
$M\cong\operatorname{Sym}(m)\times\operatorname{Sym}(n-m)$. Since $A$ is
transitive on $G/B$ by hypothesis, it follows that $A$ is also transitive on
$G/M$, which is naturally in one-to-one correspondence with the set of subsets
of size $m$ in $\mathbf{n}$. By hypothesis $A$ is imprimitive on $\mathbf{n}$,
and it is easy to see that it can therefore not act transitively on the set of
unordered $m$-tuples in $\mathbf{n}$ (recalling that $2\leq m\leq n/2$). This
proves the claim.

Let now $\mathbf{o}\subseteq\mathbf{n}$ be the largest $B$-orbit. By the claim
we have either $\mathbf{o}=\mathbf{n}$ or
$\mathbf{o}=\mathbf{n}\setminus\\{x\\}$ for some $B$-fixed point
$x\in\mathbf{n}$. We then invoke Lemma 4.15, which has two consequences: the
$B$-action on $\mathbf{o}$ is primitive (since any invariant $B$-invariant
partition on $\mathbf{o}$ induces a $\operatorname{Stab}_{B}(\Omega)$
invariant partition on $\Omega$), and it contains a $3$-cycle. Therefore
Proposition 4.14 implies that $B$ contains $\operatorname{Alt}(\mathbf{o})$.
This finishes the proof. ∎

#### 4.4.2 Cocompact subgroups of $\mathcal{O}_{d,k}$ and $\mathcal{N}_{d,k}$

###### Proof of Proposition 4.12.

Set $O=\mathcal{O}_{d,k}$ and let $H<O$ be a cocompact closed subgroup. We
view $O$ as an ascending chain of the compact open subgroups $O_{n}$ defined
as in [3], namely $O_{n}$ consists of those elements represented by almost
automorphisms of the form $(A,A,\varphi)$ where $A$ is the $n$-ball around the
root of $\mathcal{T}_{d,k}$. By compactness, there exists $n_{0}$ such that
$O=O_{n}H$ for all $n\geq n_{0}$.

Let $H_{n}=O_{n}\cap H$. We have $O_{n}=O_{n_{0}}H_{n}$ for all $n\geq 0$. Let
now $U_{n}\leq O_{n}$ be the closed normal subgroup consisting of those
elements represented by almost automorphisms of the form $(A,A,\varphi)$ where
$A$ is the $n$-ball around the root of $\mathcal{T}_{d,k}$ and $\varphi$
stabilizes each connected component of the forest $\mathcal{T}_{d,k}\setminus
A$. The quotient $O_{n}/U_{n}$ is a symmetric group
$\operatorname{Sym}(k_{n})$ whose degree $k_{n}$ tends to infinity with $n$.
Let $A_{n}$ (resp. $B_{n}$) denote the image of $O_{n_{0}}$ (resp. $H_{n}$) in
the quotient $O_{n}/U_{n}$, which we identity with
$\operatorname{Sym}(k_{n})$. We have $\operatorname{Sym}(k_{n})=A_{n}B_{n}$.
Moreover $A_{n}$ is transitive but not primitive for any $n>n_{0}$.
Proposition 4.13 implies that $B_{n}$ contains $\operatorname{Alt}(k_{n})$ or
fixes a point and contains $\operatorname{Alt}(k_{n}-1)$.

The first case implies that the image of $H\cap O_{n-1}$ is the full quotient
$O_{n-1}/U_{n-1}\cong\operatorname{Sym}(k_{n-1})$. If that happens for
infinitely many values of $n$, then we have $(H\cap O_{n-1})U_{n-1}=O_{n-1}$
for infinitely many $n$’s, so that $HU_{n-1}=O$ for infinitely many $n$’s.
This means that $H$ is dense in $O$, hence $H=O$ since $H$ is closed.

Otherwise, we deduce that $H$ fixes a point $\xi\in\partial\mathcal{T}_{d,k}$
and a similar argument shows that $H$ contains a dense subgroup of the
stabilizer $O_{\xi}$. Thus $H=O_{\xi}$. ∎

###### Proof of Theorem 4.9.

Set $G=\mathcal{N}_{d,k}$ and let $H\leq G$ be a cocompact closed subgroup.
Since $O=\mathcal{O}_{d,k}$ is open in $G$, it has open orbits in $G/H$, hence
finitely many orbits, hence clopen orbits. In particular $OH/H\cong O/O\cap H$
is compact.

By Proposition 4.12, we have $O\cap H=O$ or $O\cap H=O_{\xi}$ for some
$\xi\in\partial\mathcal{T}_{d,k}$. The former case implies that $H$ is open,
hence of finite index, in $G$. Since $G$ is simple (see [26] for the case
where $k=d+1$; in the general case, a similar argument can be applied,
exploiting that $O$ is topologically simple by [9, Rem. 6.8 and Lem. 6.9]),
this implies $H=G$, and we are done in this case.

We assume henceforth that $O\cap H=O_{\xi}$.

We next claim $H$ fixes $\xi$. Indeed, there would otherwise exist $h\in H$
which does not fix $\xi$. We then see that $H$ contains $\langle O_{\xi}\cup
hO_{\xi}h^{-1}\rangle$. Since $O$ is $2$-transitive (in fact
$\infty$-transitive) on $\partial\mathcal{T}_{d,k}$, we infer that $H$ is
transitive on $\partial\mathcal{T}_{d,k}$. Therefore $G=G_{\xi}H$. Hence
$G_{\xi}/G_{\xi}\cap H$ is homeomorphic to $G/H$, and is thus compact. Since
$G_{\xi}\cap H$ contain $O_{\xi}=O\cap G_{\xi}$, it is relatively open in
$G_{\xi}$. Therefore $G/H\cong G_{\xi}/G_{\xi}\cap H$ is finite. As before,
this implies that $H=G$, contradicting that $O\cap H=O_{\xi}$.

This proves that $H$ fixes $\xi$. We have $O_{\xi}\leq H\leq G_{\xi}$, so that
$H$ is relatively open in $G_{\xi}$. Since $H$ is cocompact in $G$, it is
cocompact in $G_{\xi}$, hence of finite index. The result follows.

For the last claim, it remains to observe that the stabiliser $G_{\xi}$ is not
unimodular, because $G$ does not preserve any probability measure on
$\partial\mathcal{T}_{d,k}$ by Lemma 4.1. Therefore, if $H$ is unimodular, we
must have $H=G$. ∎

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