# On closure operations in the space of subgroups and applications

Dominik Francoeur  Newcastle University, Newcastle upon Tyne, NE1 7RU, United
Kingdom<EMAIL_ADDRESS>and  Adrien Le Boudec  CNRS, UMPA -
ENS Lyon, 46 allée d’Italie, 69364 Lyon, France<EMAIL_ADDRESS>
(Date: July 14, 2024)

###### Abstract.

We establish some interactions between uniformly recurrent subgroups (URSs) of
a group $G$ and cosets topologies $\tau_{\mathcal{N}}$ on $G$ associated to a
family $\mathcal{N}$ of normal subgroups of $G$. We show that when
$\mathcal{N}$ consists of finite index subgroups of $G$, there is a natural
closure operation $\mathcal{H}\mapsto\mathrm{cl}_{\mathcal{N}}(\mathcal{H})$
that associates to a URS $\mathcal{H}$ another URS
$\mathrm{cl}_{\mathcal{N}}(\mathcal{H})$, called the
$\tau_{\mathcal{N}}$-closure of $\mathcal{H}$. We give a characterization of
the URSs $\mathcal{H}$ that are $\tau_{\mathcal{N}}$-closed in terms of
stabilizer URSs. This has consequences on arbitrary URSs when $G$ belongs to
the class of groups for which every faithful minimal profinite action is
topologically free. We also consider the largest amenable URS
$\mathcal{A}_{G}$, and prove that for certain coset topologies on $G$, almost
all subgroups $H\in\mathcal{A}_{G}$ have the same closure. For groups in which
amenability is detected by a set of laws, we deduce a criterion for
$\mathcal{A}_{G}$ to be a singleton based on residual properties of $G$.

This work had been initiated within the framework of the Labex Milyon (ANR-10-
LABX-0070) of Universite de Lyon, within the program "Investissements
d’Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency
(ANR)

##  1\. Introduction

Let $G$ be a group. We denote by $\mathcal{N}_{G}$ the set of normal subgroups
of $G$. Let $\mathcal{N}\subseteq\mathcal{N}_{G}$ be a family of normal
subgroups of $G$ that is filtering: for every $N_{1},N_{2}\in\mathcal{N}$
there exists $N_{3}\in\mathcal{N}$ such that $N_{3}\leq N_{1}\cap N_{2}$.
There is a group topology $\tau_{\mathcal{N}}$ on $G$ associated to
$\mathcal{N}$, defined by declaring that the family of cosets $gN$, $g\in G$,
$N\in\mathcal{N}$, forms a basis for $\tau_{\mathcal{N}}$. When $\mathcal{N}$
is the family of all finite index normal subgroups of $G$,
$\tau_{\mathcal{N}}$ is the profinite topology on $G$. If $p$ is a prime and
$\mathcal{N}$ is the family of finite index normal subgroups $N$ of $G$ such
that $G/N$ is a $p$-group, $\tau_{\mathcal{N}}$ is the pro-$p$ topology.

If $H$ is a subgroup of $G$, the closure of $H$ with respect to
$\tau_{\mathcal{N}}$ is denoted by $\mathrm{cl}_{\mathcal{N}}(H)$. In the case
of the profinite topology, we use the shorter notation $\mathrm{cl}(H)$. The
closure operation defines a map

$\mathrm{cl}_{\mathcal{N}}:\operatorname{Sub}(G)\to\operatorname{Sub}(G),\,H\mapsto\mathrm{cl}_{\mathcal{N}}(H).$

Here $\operatorname{Sub}(G)$ is the set of subgroups of $G$. That set is
equipped with the topology inherited from the set $\left\\{0,1\right\\}^{G}$
of all subsets of $G$, equipped with the product topology. The space
$\operatorname{Sub}(G)$ is a compact space. The group $G$ acts on
$\operatorname{Sub}(G)$ by conjugation, and this action is by homeomorphisms.
The first object of study of this article is the behaviour of the map
$\mathrm{cl}_{\mathcal{N}}$ with respect to the dynamical system
$G\curvearrowright\operatorname{Sub}(G)$.

It follows from the definitions that the map $\mathrm{cl}_{\mathcal{N}}$ is
always increasing, idempotent, and $G$-equivariant. In general
$\mathrm{cl}_{\mathcal{N}}$ is far from being continuous. This failure of
continuity already happens in the most classical case where
$\tau_{\mathcal{N}}$ is the profinite topology. An elementary example
illustrating this is the group $G=\mathbb{Z}[1/p]$ of $p$-adic rational
numbers, for which the map $\mathrm{cl}$ is not upper semi-continuous on
$\operatorname{Sub}(G)$ (see Remark 3.6). Another example is $G=F_{k}$ (a
finitely generated non-abelian free group of rank $k$). M. Hall showed that
every finitely generated subgroup $H$ of $F_{k}$ verifies $\mathrm{cl}(H)=H$
[Hal49] (i.e. $F_{k}$ is a LERF group). Since finitely generated subgroups
always form a dense subset in the space of subgroups, it follows that
$\mathrm{cl}$ is the identity on a dense set of points. However $\mathrm{cl}$
is not the identity everywhere, for instance because $F_{k}$ admits infinite
index subgroups $H$ such that $\mathrm{cl}(H)=F_{k}$ (e.g. any infinite index
maximal subgroup). So $\mathrm{cl}$ is not lower semi-continuous on
$\operatorname{Sub}(F_{k})$.

The starting result of this article is that if we restrict to minimal
subsystems of $\operatorname{Sub}(G)$ (i.e. non-empty closed minimal
$G$-invariant subsets of $\operatorname{Sub}(G)$), the situation is better
behaved. Recall that a minimal subsystem
$\mathcal{H}\subset\operatorname{Sub}(G)$ is called a URS (Uniformly Recurrent
Subgroup) [GW15].

######  Proposition 1.

Let $\mathcal{N}\subseteq\mathcal{N}_{G}$ be a family of finite index normal
subgroups of $G$, and let $\mathcal{H}$ be a URS of $G$. Then the following
hold:

  1. (1)

The restriction
${\mathrm{cl}_{\mathcal{N}}}_{|\mathcal{H}}:\mathcal{H}\to\operatorname{Sub}(G)$
is upper semi-continuous.

  2. (2)

There exists a unique URS contained in
$\overline{\left\\{\mathrm{cl}_{\mathcal{N}}(H):H\in\mathcal{H}\right\\}}$,
denoted $\mathrm{cl}_{\mathcal{N}}(\mathcal{H})$, and called the
$\tau_{\mathcal{N}}$-closure of $\mathcal{H}$.

The proposition also holds in a more general situation not necessarily
requiring that $\mathcal{N}$ consists of finite index subgroups of $G$ (see
Proposition 3.4).

Statement (2) says that there is a natural closure operation

$\mathrm{URS}(G)\to\mathrm{URS}(G),\,\mathcal{H}\mapsto\mathrm{cl}_{\mathcal{N}}(\mathcal{H}),$

where $\mathrm{URS}(G)$ is the set of URSs of the group $G$. We say that a URS
$\mathcal{H}$ is closed for the topology $\tau_{\mathcal{N}}$ if
$\mathrm{cl}_{\mathcal{N}}(\mathcal{H})=\mathcal{H}$. When $G$ is a countable
group, this happens if and only if there is a dense $G_{\delta}$-set of points
$H\in\mathcal{H}$ such that $H$ is closed for the topology
$\tau_{\mathcal{N}}$.

Recently URSs were studied and appeared in a large amount of works, including
[LBMB18, BH21, FG23, LBMB22]. We refer notably to the introduction of [LBMB22]
for more references. A common theme is to establish rigidity results saying
that the set of URSs of certain groups is restricted, or to establish
connections between certain group theoretic properties of the ambient group
and properties of its URSs. We believe that in certain situations the above
process $\mathcal{H}\mapsto\mathrm{cl}_{\mathcal{N}}(\mathcal{H})$, and more
generally the consideration of coset topologies on the ambient group, can be
profitably used to study properties of URSs. In Sections 4 and 5 we exhibit
situations where it is indeed the case. In the remainder of this introduction
we shall describe these results.

When $\mathcal{N}$ consists of finite index subgroups, the property that a URS
$\mathcal{H}$ is closed for the topology $\tau_{\mathcal{N}}$ admits the
following natural characterization. Glasner–Weiss showed that to every minimal
action of $G$ on a compact space $X$, there is a naturally associated URS of
$G$, called the stabilizer URS of $X$, and denoted $S_{G}(X)$ [GW15]. We say
that the action of $G$ on a compact space $X$ is pro-$\mathcal{N}$ if $G\times
X\to X$ is continuous, where $G$ is equipped with the topology
$\tau_{\mathcal{N}}$ (see Proposition 4.2 for characterizations of this
property).

######  Proposition 2.

Suppose that $G$ is a countable group and that $\mathcal{N}$ consists of
finite index subgroups of $G$. For a URS $\mathcal{H}$ of $G$, the following
are equivalent:

  1. (1)

$\mathcal{H}$ is closed for the topology $\tau_{\mathcal{N}}$.

  2. (2)

There exists a pro-$\mathcal{N}$ compact minimal $G$-space $X$ such that
$S_{G}(X)=\mathcal{H}$.

In the case of the profinite topology, the notion of pro-$\mathcal{N}$
$G$-space coincides with the classical notion of profinite $G$-space. So in
that situation the above proposition says that a URS $\mathcal{H}$ is closed
for the profinite topology if and only if $\mathcal{H}$ is the stabilizer URS
associated to a minimal profinite action of $G$. Consequences on all URSs can
be drawn out of this when $G$ belongs to the class of groups for which, for a
faithful minimal compact $G$-space, profinite implies topologically free. See
Proposition 4.17. This class of groups includes non-abelian free groups, and
more generally any group $G$ admitting an isometric action on a hyperbolic
space with unbounded orbits such that the $G$-action on its limit set is
faithful. It also includes hereditarily just-infinite groups. Recall that a
group $G$ is just-infinite if $G$ is infinite and $G/N$ is finite for every
non-trivial normal subgroup $N$, and $G$ is hereditarily just-infinite if
every finite index subgroup of $G$ is just-infinite. We call a subgroup $H$ of
$G$ quasi-dense in $G$ for the profinite topology if the profinite closure of
$H$ has finite index in $G$. For hereditarily just-infinite groups we obtain:

######  Proposition 3.

Let $G$ be a hereditarily just-infinite group, and let $\mathcal{H}$ be a non-
trivial URS of $G$. Then for every $H\in\mathcal{H}$, $H$ is quasi-dense in
$G$ for the profinite topology.

In cases where we know a priori that the group $G$ has the property that the
only subgroups that are quasi-dense are the finite index subgroups, we deduce
that such a group $G$ admits no continuous URS (a URS is continuous if it is
not a finite set). See Corollary 4.20, and the surrounding discussion for
context and examples.

Another setting in which we show that the consideration of a coset topology
$\tau_{\mathcal{N}}$ is fruitful with respect to the study of URSs is the case
amenable URSs. A URS $\mathcal{H}$ is amenable if it consists of amenable
subgroups. Every group $G$ admits a largest amenable URS (with respect to a
natural partial order), which is the stabilizer URS associated to the action
of $G$ on its Furstenberg boundary (the largest minimal and strongly proximal
compact $G$-space). This URS is denoted $\mathcal{A}_{G}$ and is called the
Furstenberg URS of $G$. The action of $G$ on $\mathcal{A}_{G}$ is minimal and
strongly proximal. $\mathcal{A}_{G}$ is either a singleton, in which case we
have $\mathcal{A}_{G}=\left\\{\mathrm{Rad}(G)\right\\}$, where
$\mathrm{Rad}(G)$ is the amenable radical of $G$, or $\mathcal{A}_{G}$ is
continuous. We refer to [LBMB18] for a more detailed discussion.

Let $\mathcal{F}$ denote the class of groups $G$ such $\mathcal{A}_{G}$ is a
singleton. Equivalently, $G$ belongs to $\mathcal{F}$ if and only if every
amenable URS of $G$ lives inside the amenable radical of $G$. The class
$\mathcal{F}$ is known to be very large. It plainly contains amenable groups.
It also contains all linear groups, all groups with non-vanishing
$\ell^{2}$-Betti numbers, all hyperbolic groups, and more generally all
acylindrically hyperbolic groups. We refer to [BKKO17] for references and
details. Examples of groups outside the class $\mathcal{F}$ have been given in
[LB17].

The following result provides a criterion for a group to be in $\mathcal{F}$
that is based on residual properties of the group.

######  Theorem 1.

Let $G$ be a group such that every amenable subgroup of $G$ is virtually
solvable. If $G$ is residually-$\mathcal{F}$, then $G$ is in $\mathcal{F}$.

We point out that this theorem is applicable without necessarily relying on
other methods related to $\mathcal{F}$ to verify the assumption that the group
is residually-$\mathcal{F}$. The point is that the statement applies provided
that $G$ is residually-$\mathcal{C}$ for some subclass $\mathcal{C}$ of
$\mathcal{F}$ that is potentially much smaller. For instance the theorem
applies and is already interesting if $G$ is residually finite.

One interest of such a statement is that it is based on intrinsic algebraic
properties of the group. It does not require the group $G$ to admit a rich
action of geometric flavour, or to have an explicit minimal and strongly
proximal compact $G$-space at our disposal. The residual properties are used
as a tool in Theorem 1, but the confrontation of residual properties and the
class $\mathcal{F}$ is also motivated by the fact that it is not known whether
there exist residually finite groups $G$ with trivial amenable radical such
that $G$ does not belong to $\mathcal{F}$. The groups from [LB17] are never
residually finite (and some of them are virtually simple).

As an application, Theorem 1 allows to recover the following result from
[BKKO17]:

######  Corollary 1 (Breuillard–Kalantar–Kennedy–Ozawa).

If $G$ is a linear group, then $G$ is in $\mathcal{F}$.

The proof from [BKKO17] relies on linear groups technology. Here the argument
to deduce Corollary 1 from Theorem 1 uses a reduction to the case of finitely
generated groups, and then only appeals to Malcev’s theorem that finitely
generated linear groups are residually finite, and the Tits alternative.

The consideration of the class $\mathcal{F}$ is also motivated by the result
of Kalantar–Kennedy that a group $G$ belongs to $\mathcal{F}$ if and only if
the quotient of $G$ by its amenable radical is a $C^{\ast}$-simple group (that
is, its reduced $C^{\ast}$-algebra is simple) [KK17]. We refer to the survey
of de la Harpe [dlH07] for an introduction and historical developments on
$C^{\ast}$-simple groups, and to the Bourbaki seminar of Raum for recent
developments [Rau20]. Hence using the result of Kalantar–Kennedy, Theorem 1
can be reinterpreted as a criterion to obtain $C^{\ast}$-simplicity (under the
assumption on amenable subgroups) based on residual properties of the group.
See Corollary 5.14. We are not aware of other results of this kind.

The proof of Theorem 1 is based on the following proposition, of independent
interest. Given a group $G$, we denote by $\mathcal{N}_{G}(\mathcal{F})$ the
set of normal subgroups of $G$ such that $G/N\in\mathcal{F}$. The set
$\mathcal{N}_{G}(\mathcal{F})$ is stable under taking finite intersections
(Lemma 5.7), and we can consider the coset topology on $G$ associated to
$\mathcal{N}_{G}(\mathcal{F})$ (and more generally to a subset
$\mathcal{N}\subseteq\mathcal{N}_{G}(\mathcal{F})$). The following result says
that within the Furstenberg URS $\mathcal{A}_{G}$, almost all points have the
same closure for such a topology (for technical reasons we are led to make
some countability assumptions).

######  Proposition 4.

Let $G$ be a countable group, and let $\mathcal{N}$ be a countable subset of
$\mathcal{N}_{G}(\mathcal{F})$. Then there exists a normal subgroup $M$ of $G$
and a comeager subset $\mathcal{H}_{0}\subseteq\mathcal{A}_{G}$ such that
$\mathrm{cl}_{\mathcal{N}}(H)=M$ for every $H\in\mathcal{H}_{0}$.

The proof of the proposition makes crucial use of the strong proximality of
the action of $G$ on $\mathcal{A}_{G}$. The proof of Theorem 1 is easily
deduced from the proposition. The additional point is to ensure that the
closed normal subgroup $M$ appearing in the conclusion of the proposition
remains amenable, and this is where the two assumptions in the theorem are
used. We refer to Section 5 for details. Here we only mention that the actual
setting in which we prove Theorem 1 does not necessarily require amenable
subgroups to be virtually solvable. The assumption that we need is that
amenability within subgroups of $G$ can be detected by a set of laws
(Definition 5.10), a property that can be thought of as a version of the Tits
alternative. See Theorem 5.12 for the more general formulation of the theorem.

Acknowledgements. Thanks are due to Uri Bader and Pierre-Emmanuel Caprace. We
can trace back that the possibility of using Proposition 2.1 specifically in
the space of subgroups to build a URS starting from another one and a semi-
continuous map had been originally brought to our attention by them several
years ago.

##  2\. Preliminaries

A space $X$ is a $G$-space if $G$ admits a continuous action $G\times
X\rightarrow X$. Throughout the paper we make the standing assumption that
$G$-spaces are non-empty. The action (or the $G$-space $X$) is minimal if all
orbits are dense. For $x\in X$ we write $G_{x}$ for the stabilizer of $x$ in
$G$, and $G_{x}^{0}$ for the set of $g\in G$ such that $g$ acts trivially on a
neighbourhood of $x$. The action of $G$ on $X$ is free if
$G_{x}=\left\\{1\right\\}$ for every $x\in X$, and topologically free if
$G_{x}^{0}=\left\\{1\right\\}$ for every $x\in X$.

Let $X,Y$ be compact spaces. A continuous surjective map $\pi:Y\to X$ is
called irreducible if every proper closed subset of $Y$ has a proper image in
$X$. If $X,Y$ are compact $G$-spaces and $\pi:Y\to X$ is a continuous
surjective $G$-equivariant map, we say that $X$ is a factor of $Y$, and that
$Y$ is an extension of $X$. When $\pi:Y\to X$ is irreducible, we also say that
$Y$ is an irreducible extension of $X$. If $\pi:Y\to X$ is irreducible, then
$X$ is minimal if and only if $Y$ is minimal. Also for $X,Y$ minimal,
$\pi:Y\to X$ is irreducible if and only if it is highly proximal: for every
$x\in X$ the fiber $\pi^{-1}(x)$ is compressible [AG77].

###  2.1. Semi-continuous maps

If $Y$ is a locally compact space, we denote by $2^{Y}$ the space of closed
subsets of $Y$, endowed with the Chabauty topology. The space $2^{Y}$ is
compact.

Let $X$ be a compact $G$-space. A map $\varphi\colon X\to 2^{Y}$ is upper
semi-continuous if for every compact subset $K$ of $Y$, $\left\\{x\in
X:\varphi(x)\cap K=\emptyset\right\\}$ is open in $X$. It is lower semi-
continuous if for every open subset $U$ of $Y$, $\left\\{x\in X:\varphi(x)\cap
U\neq\emptyset\right\\}$ is open in $X$. We say that $\varphi$ is semi-
continuous if it is either upper or lower semi-continuous.

Let $\varphi:X\to 2^{Y}$ be a semi-continuous map, and $X_{\varphi}\subseteq
X$ be the set of points where $\varphi$ is continuous. Let

$F_{\varphi}:=\overline{\left\\{\left(x,\varphi(x)\right)\,:\,x\in
X\right\\}}\subseteq X\times 2^{Y},$

$E_{\varphi}:=\overline{\left\\{\left(x,\varphi(x)\right)\,:\,x\in
X_{\varphi}\right\\}}\subseteq F_{\varphi}(X),$

$T_{\varphi}:=\overline{\left\\{\varphi(x)\,:\,x\in X\right\\}},$

$S_{\varphi}:=\overline{\left\\{\varphi(x)\,:\,x\in X_{\varphi}\right\\}}.$

We denote by $\eta:X\times 2^{Y}\to X$ and $p:X\times 2^{Y}\to 2^{Y}$ the
projections to the first and second coordinate. If $Y$ is second-countable,
semi-continuity of $\varphi$ implies that $X_{\varphi}$ is a comeager subset
of $X$ [Kur28, Theorem VII].

######  Proposition 2.1.

Suppose $X$ is a minimal compact $G$-space, $Y$ is a locally compact
$G$-space, and $\varphi:X\to 2^{Y}$ is $G$-equivariant and semi-continuous.
Then the following hold:

  1. (i)

$F_{\varphi}$ has a unique non-empty minimal closed $G$-invariant subset
$E_{\varphi}^{\prime}$ , and $T_{\varphi}$ has a unique minimal closed
$G$-invariant subset $S_{\varphi}^{\prime}$, and
$p(E_{\varphi}^{\prime})=S_{\varphi}^{\prime}$ .

  2. (ii)

The extension $\eta:E_{\varphi}^{\prime}\to X$ is highly proximal.

If moreover $Y$ is second-countable, then $E_{\varphi}^{\prime}=E_{\varphi}$
and $S_{\varphi}^{\prime}=S_{\varphi}$ .

###### Proof.

See Glasner [Gla75, Theorem 2.3] and Auslander–Glasner [AG77, Lemma I.1]. ∎

###  2.2. The space of subgroups and URSs

We denote by $\operatorname{Sub}(G)$ the space of subgroups of $G$, equipped
with the product topology from $\left\\{0,1\right\\}^{G}$. It is a compact
$G$-space, where $G$ acts by conjugation. If $H\in\operatorname{Sub}(G)$, we
denote by $H^{G}$ the $G$-conjugates of $H$, i.e. the $G$-orbit of $H$ in
$\operatorname{Sub}(G)$.

A URS of $G$ is a (non-empty) minimal closed $G$-invariant subset of
$\operatorname{Sub}(G)$. By Zorn’s lemma every (non-empty) closed
$G$-invariant subset of $\operatorname{Sub}(G)$ contains a URS. A URS is
finite if it is a finite $G$-orbit. A URS that is not finite is called
continuous. By minimality and compactness, a continuous URS has no isolated
points. The singleton $\left\\{\left\\{1\right\\}\right\\}$ is called the
trivial URS. If $\mathcal{P}$ is a property of groups, we say that a URS
$\mathcal{H}$ has $\mathcal{P}$ if $H$ has $\mathcal{P}$ for every
$H\in\mathcal{H}$.

######  Definition 2.2.

If $\mathcal{H}$ is a URS of $G$, we denote by $\mathrm{Env}(\mathcal{H})$ the
subgroup generated by all subgroups $H$ in $\mathcal{H}$. The subgroup
$\mathrm{Env}(\mathcal{H})$ is normal in $G$, and it is the smallest normal
subgroup of $G$ containing some subgroup $H\in\mathcal{H}$.

Every minimal compact $G$-space naturally gives rise to a URS [GW15]:

######  Proposition 2.3.

If $X$ is a compact $G$-space, then the stabilizer map
$S:X\to\operatorname{Sub}(G)$, $x\mapsto G_{x}$, is $G$-equivariant and upper
semi-continuous. In particular if $X$ is minimal, then Proposition 2.1
applies.

######  Definition 2.4.

If $X$ is a minimal compact $G$-space, the unique URS contained in
$\overline{\left\\{G_{x}\,:\,x\in X\right\\}}$ is denoted $S_{G}(X)$, and is
called the stabilizer URS associated to the $G$-space $X$.

One verifies that the $G$-action on $X$ is topologically free if and only if
the URS $S_{G}(X)$ is trivial.

######  Lemma 2.5.

Let $H,K,L$ be subgroups of $G$ such that $H\leq L$. If $K$ belongs to the
closure of the $L$-orbit of $H$ in $\operatorname{Sub}(G)$, then $K\leq L$.

###### Proof.

The subset $\operatorname{Sub}(L)$ is a closed subset of
$\operatorname{Sub}(G)$, and contains the $L$-orbit of $H$ since $H\leq L$. ∎

######  Lemma 2.6.

Let $N$ be a normal subgroup of $G$. Then the map
$\operatorname{Sub}(G)\to\operatorname{Sub}(G)$, $H\mapsto HN$, is
$G$-equivariant and lower semi-continuous.

###### Proof.

It is $G$-equivariant because $N$ is normal in $G$. Since $G$ is discrete,
lower semi-continuity means that for every $g\in G$ and every
$H\in\operatorname{Sub}(G)$ such that $g\in HN$, there is a neighbourhood of
$H$ in which $g\in H^{\prime}N$ remains true. If $h\in H$ is such that $g\in
hN$, then the set of subgroups of $G$ containing $h$ is such a neighbourhood.
∎

###  2.3. Coset topologies on groups

Let $G$ be a group. We denote by $\mathcal{N}_{G}$ the set of normal subgroups
of $G$. We make the standing convention that when considering a family
$\mathcal{N}\subseteq\mathcal{N}_{G}$ of normal subgroups of $G$, we always
assume that $\mathcal{N}$ is non-empty.

######  Definition 2.7.

Let $\mathcal{N}\subseteq\mathcal{N}_{G}$. If $H$ is a subset of $G$, we
denote

$\mathrm{cl}_{\mathcal{N}}(H)=\bigcap_{N\in\mathcal{N}}HN.$

We say that $\mathcal{N}$ is filtering if for every
$N_{1},N_{2}\in\mathcal{N}$ there exists $N_{3}\in\mathcal{N}$ such that
$N_{3}\leq N_{1}\cap N_{2}$. We record the following [Bou71, Chap. III]:

######  Proposition 2.8.

Fix $\mathcal{N}\subseteq\mathcal{N}_{G}$ . Then:

  1. (1)

the family of cosets $gN$, $g\in G$, $N\in\mathcal{N}$, forms a subbasis for a
group topology $\tau_{\mathcal{N}}$ on $G$.

  2. (2)

The topology $\tau_{\mathcal{N}}$ is Hausdorff if and only if
$\bigcap_{\mathcal{N}}N=\left\\{1\right\\}$.

  3. (3)

Suppose that $\mathcal{N}$ is filtering. Then for every subset $H$ of $G$, the
closure of $H$ with respect to $\tau_{\mathcal{N}}$ is equal to
$\mathrm{cl}_{\mathcal{N}}(H)$.

If $\mathcal{C}$ is a class of groups, we denote by
$\mathcal{N}_{G}(\mathcal{C})$ the normal subgroups of $G$ such that
$G/N\in\mathcal{C}$. Note that a group $G$ is residually-$\mathcal{C}$ if and
only if $\bigcap_{\mathcal{N}_{G}(\mathcal{C})}N=\left\\{1\right\\}$.

When $\mathcal{C}$ is the class of all finite groups and
$\mathcal{N}=\mathcal{N}_{G}(\mathcal{C})$, $\tau_{\mathcal{N}}$ is the
profinite topology on $G$. For simplicity we write $\mathrm{cl}(H)$ for the
closure in the profinite topology. When $\mathcal{C}$ is the class of all
finite $p$-groups ($p$ is a prime number) and
$\mathcal{N}=\mathcal{N}_{G}(\mathcal{C})$, $\tau_{\mathcal{N}}$ is the
pro-$p$ topology. In that case we write $\mathrm{cl}_{p}(H)$ for the closure
in the pro-$p$ topology.

###  2.4. Laws

Let $w=w(x_{1},\ldots,x_{k})$ be a word in $k$ letters $x_{1},\ldots,x_{k}$,
meaning that $w$ is an element of the free group $F_{k}$ freely generated by
$x_{1},\ldots,x_{k}$. Given a group $G$, the word $w$ naturally defines a map
$G^{k}\to G$, a $k$-tuple $(g_{1},\ldots,g_{k})$ being mapped to the element
$w(g_{1},\ldots,g_{k})$ of $G$ that is obtained by replacing each $x_{i}$ by
$g_{i}$. We denote by $\Sigma_{w}(G)\subseteq G^{k}$ the set of
$(g_{1},\ldots,g_{k})$ such that $w(g_{1},\ldots,g_{k})=1$. We say that $G$
satisfies the law $w$ if $\Sigma_{w}(G)=G^{k}$.

######  Lemma 2.9.

Suppose $G$ is a Hausdorff topological group, and let $w\in F_{k}$. Then
$\Sigma_{w}(G)$ is a closed subset of $G^{k}$. In particular if a subgroup $H$
of $G$ satisfies the law $w$, then so does its closure.

###### Proof.

Since $G$ is Hausdorff, $\left\\{1\right\\}$ is closed in $G$. The map
$G^{k}\to G$ associated to $w$ being continuous, the preimage $\Sigma_{w}(G)$
of $\left\\{1\right\\}$ is a closed subset of $G^{k}$. ∎

##  3\. The $\tau_{\mathcal{N}}$-closure of a URS

Let $\mathcal{H}$ be a closed subset of $\operatorname{Sub}(G)$, and $L$ a
subgroup of $G$. For every $\Sigma\subseteq L^{G}$, we write

$\mathcal{H}_{\Sigma}=\left\\{H\in\mathcal{H}:\forall K\in L^{G},\,H\subset
K\Leftrightarrow K\in\Sigma\right\\}.$

######  Lemma 3.1.

If $\mathcal{H}_{\Sigma}\cap\mathcal{H}_{\Sigma^{\prime}}\neq\emptyset$ then
$\Sigma=\Sigma^{\prime}$, and $\mathcal{H}$ is the disjoint union of the
$\mathcal{H}_{\Sigma}$ when $\Sigma$ ranges over subsets of $L^{G}$.

###### Proof.

The first assertion is consequence of the definitions. The second assertion is
also clear since for every $H\in\mathcal{H}$, one has
$H\in\mathcal{H}_{\Sigma}$ with $\Sigma=\left\\{K\in L^{G}:H\subset
K\right\\}.$ ∎

######  Lemma 3.2.

Let $L$ be a subgroup of $G$ such that $L^{G}$ is finite. Suppose
$\mathcal{H}$ is a URS of $G$. Then $\mathcal{H}_{\Sigma}$ is a clopen subset
of $\mathcal{H}$ for every $\Sigma\subseteq L^{G}$.

###### Proof.

For $H$ in $\mathcal{H}$, we let $n(H)$ be the number of conjugates of $L$
containing $H$. By our assumption, the number $n(H)$ is finite. We claim that
$n(H)$ is constant on $\mathcal{H}$. In order to see this, take
$H\in\mathcal{H}$ such that $n(H)=r$ is minimal. Since not being contained in
a subgroup is an open condition, one can find a neighbourhood $V$ of $H$ such
that $n(H^{\prime})\leq r$ for every $H^{\prime}\in V$. Hence by minimality of
$r$ we have $n(H^{\prime})=r$ for every $H^{\prime}\in V$. Now for every
$K\in\mathcal{H}$, by minimality of the $G$-action on $\mathcal{H}$ the subset
$V$ contains a conjugate of $K$. Since $n(K)$ is invariant under conjugation,
we deduce $n(K)=r$.

Now fix $\Sigma\subseteq L^{G}$ such that $\mathcal{H}_{\Sigma}$ is non-empty,
and let $H\in\mathcal{H}_{\Sigma}$. Again there is a neighbourhood $V$ of $H$
in $\mathcal{H}$ such that for every $H^{\prime}$ in $V$, we have
$H^{\prime}\not\subset J$ for every $J\in L^{G}\setminus\Sigma$. Moreover by
the previous paragraph we have $n(H^{\prime})=n(H)$. Hence by the pigeonhole
principle we deduce that $H^{\prime}\subset J$ for every $J\in\Sigma$. This
shows that $\mathcal{H}_{\Sigma}$ is open. Since the family
$(\mathcal{H}_{\Sigma})$ forms a partition of $\mathcal{H}$ by Lemma 3.1, it
follows that $\mathcal{H}_{\Sigma}$ is also closed. ∎

######  Definition 3.3.

Let $\mathcal{N}\subseteq\mathcal{N}_{G}$. We say that a URS $\mathcal{H}$ of
$G$ is $\mathcal{N}$-finitary if $(HN)^{G}$ is finite for every
$H\in\mathcal{H}$, $N\in\mathcal{N}$.

######  Proposition 3.4.

Let $\mathcal{N}\subseteq\mathcal{N}_{G}$, and let $\mathcal{H}$ be a URS of
$G$ that is $\mathcal{N}$-finitary. Then the map
$\mathcal{H}\to\operatorname{Sub}(G)$, $H\mapsto\mathrm{cl}_{\mathcal{N}}(H)$,
is upper semi-continuous.

###### Proof.

Let $K$ be a finite subset of $G$, and let $H\in\mathcal{H}$ such that
$\mathrm{cl}_{\mathcal{N}}(H)\cap K=\emptyset$. One shall prove that
$\mathrm{cl}_{\mathcal{N}}(H^{\prime})\cap K=\emptyset$ remains true for every
$H^{\prime}$ inside a neighbourhood of $H$ in $\mathcal{H}$. Let $g\in K$. By
definition of $\mathrm{cl}_{\mathcal{N}}(H)$, there exists
$N_{g}\in\mathcal{N}$ such that $g\notin HN_{g}$. Since $L=HN_{g}$ verifies
that $(HN_{g})^{G}$ is finite, according to Lemma 3.2 one can find a
neighbourhood $V_{g}$ of $H$ in $\mathcal{H}$ such that $H^{\prime}\leq
HN_{g}$ for every $H^{\prime}\in V_{g}$. A fortiori we have
$H^{\prime}N_{g}\leq HN_{g}$ and hence
$\mathrm{cl}_{\mathcal{N}}(H^{\prime})\leq HN_{g}$. Since $K$ is finite,
taking the intersection over all $g\in K$ we obtain a neighbourhood $V$ of $H$
in $\mathcal{H}$ such that
$\mathrm{cl}_{\mathcal{N}}(H^{\prime})\leq\bigcap_{g\in K}HN_{g}$ for every
$H^{\prime}\in V$. Since $K$ does not intersect $\bigcap_{g\in K}HN_{g}$, the
neighbourhood $V^{\prime}$ satisfies the required property. ∎

######  Remark 3.5.

If $\mathcal{N}$ consists of finite index normal subgroups of $G$, then
trivially every URS of $G$ is $\mathcal{N}$-finitary. Hence the previous
proposition applies.

######  Remark 3.6.

Here we still consider the case where $\mathcal{N}$ consists of finite index
normal subgroups of $G$, and we point out that in general the map
$\operatorname{Sub}(G)\to\operatorname{Sub}(G)$,
$H\mapsto\mathrm{cl}_{\mathcal{N}}(H)$, is not upper semi-continuous.
Therefore it is necessary to restrict to a URS in Proposition 3.4 in order to
obtain upper semi-continuity. As an illustration, consider the group
$G=\mathbb{Z}[1/p]$ of $p$-adic rational numbers. For $n\geq 1$, let
$H_{n}=p^{n}\mathbb{Z}$. Then $G/H_{n}$ is a Prüfer $p$-group, and hence has
no proper finite index subgroup. So $G$ has no proper finite index subgroup
containing $H_{n}$, or equivalently $\mathrm{cl}(H_{n})=G$. On the other hand
$(H_{n})$ converges to the trivial subgroup in $\operatorname{Sub}(G)$, which
is closed for the profinite topology since $G$ is residually finite. Hence
$H\mapsto\mathrm{cl}(H)$ is not upper semi-continuous.

######  Corollary 3.7.

Let $\mathcal{N}\subseteq\mathcal{N}_{G}$, and $\mathcal{H}$ a URS of $G$ that
is $\mathcal{N}$-finitary. Then the set

$\overline{\left\\{\mathrm{cl}_{\mathcal{N}}(H):H\in\mathcal{H}\right\\}}$

contains a unique URS of $G$, that will be denoted
$\mathrm{cl}_{\mathcal{N}}(\mathcal{H})$. Moreover when $G$ is countable,
there is a dense $G_{\delta}$ subset $\mathcal{H}_{0}\subseteq\mathcal{H}$
such that

$\mathrm{cl}_{\mathcal{N}}(\mathcal{H})=\overline{\left\\{\mathrm{cl}_{\mathcal{N}}(H):H\in\mathcal{H}_{0}\right\\}}.$

###### Proof.

Proposition 3.4 asserts that $\mathcal{H}\to\operatorname{Sub}(G)$,
$H\mapsto\mathrm{cl}_{\mathcal{N}}(H)$, is upper semi-continuous. This allows
to invoke Proposition 2.1, from which the statement follows. ∎

######  Corollary 3.8.

Let $\mathcal{N}\subseteq\mathcal{N}_{G}$, and $\mathcal{H}$ a URS of $G$ that
is $\mathcal{N}$-finitary.

  1. (1)

If there exists $H\in\mathcal{H}$ such that $H=\mathrm{cl}_{\mathcal{N}}(H)$,
then $\mathcal{H}=\mathrm{cl}_{\mathcal{N}}(\mathcal{H})$.

  2. (2)

If $G$ is countable, then
$\mathrm{cl}_{\mathcal{N}}(\mathrm{cl}_{\mathcal{N}}(\mathcal{H}))=\mathrm{cl}_{\mathcal{N}}(\mathcal{H})$.

###### Proof.

The assumption in (1) implies that

$\mathcal{H}\cap\overline{\left\\{\mathrm{cl}_{\mathcal{N}}(H):H\in\mathcal{H}\right\\}}\neq\emptyset.$

So by minimality $\mathcal{H}$ is contained in
$\overline{\left\\{\mathrm{cl}_{\mathcal{N}}(H):H\in\mathcal{H}\right\\}}$.
Corollary 3.7 then implies
$\mathcal{H}=\mathrm{cl}_{\mathcal{N}}(\mathcal{H})$. (2) follows from the
second statement in Corollary 3.7 and (1). ∎

######  Definition 3.9.

Suppose that $\mathcal{N}$ is filtering, and let $\mathcal{H}$ be a URS of $G$
that is $\mathcal{N}$-finitary. We say that a URS $\mathcal{H}$ is closed for
the topology $\tau_{\mathcal{N}}$ if
$\mathrm{cl}_{\mathcal{N}}(\mathcal{H})=\mathcal{H}$.

##  4\. On profinite closures of a URS

###  4.1. Profinitely closed URSs and profinite actions

In all this section we assume that $\mathcal{N}\subseteq\mathcal{N}_{G}$ is
filtering, and that $\mathcal{N}$ consists of finite index subgroups of $G$.
Let $\widehat{G}^{\mathcal{N}}$ be the inverse limit of the inverse system of
finite groups $G/N$, $N\in\mathcal{N}$, and
$\psi:G\to\widehat{G}^{\mathcal{N}}$ the associated canonical group
homomorphism (for simplicity we omit $\mathcal{N}$ in the notation in $\psi$).
The group $\widehat{G}^{\mathcal{N}}$ is profinite, and
$\psi:G\to\widehat{G}^{\mathcal{N}}$ is continuous, where $G$ is equipped with
the topology $\tau_{\mathcal{N}}$. Recall that if $H$ is a subgroup of $G$,
one has $\psi^{-1}(\overline{\psi(H)})=\mathrm{cl}_{\mathcal{N}}(H)$.

######  Proposition 4.1.

Let $\mathcal{N}$ be as above. Then the following hold:

  1. (1)

Let $H$ be a subgroup of $G$ such that $H=\mathrm{cl}_{\mathcal{N}}(H)$. Then
the closure of the conjugacy class of $H$ contains a unique URS.

  2. (2)

Let $\mathcal{H}$ be a URS of $G$. Let $H\in\mathcal{H}$, and
$L=\overline{\psi(H)}$. Then the stabilizer URS associated to the left
translation action of $G$ on $\widehat{G}^{\mathcal{N}}/L$ s equal to
$\mathrm{cl}_{\mathcal{N}}(\mathcal{H})$.

###### Proof.

Write $L=\overline{\psi(H)}$ and $X=\widehat{G}^{\mathcal{N}}/L$, which is a
minimal compact $G$-space since $G$ has dense image in $\widehat{G}$. The
stabilizer of the coset $L\in X$ in $G$ is
$\psi^{-1}(L)=\mathrm{cl}_{\mathcal{N}}(H)$. So in case
$H=\mathrm{cl}_{\mathcal{N}}(H)$, one has

$\overline{H^{G}}\subseteq\overline{\left\\{G_{x}\,:\,x\in X\right\\}}.$

By Zorn’s lemma $\overline{H^{G}}$ contains at least one URS, and it follows
that is contains exactly one because $\overline{\left\\{G_{x}\,:\,x\in
X\right\\}}$ has this property by Proposition 2.3. Hence (1) holds.

For (2), we have

$\overline{\left\\{G_{x}\,:\,x\in
X\right\\}}\cap\overline{\left\\{\mathrm{cl}_{\mathcal{N}}(K)\,:\,K\in\mathcal{H}\right\\}}\neq\emptyset.$

Each one of these two sets contains a unique URS, namely $S_{G}(X)$ and
$\mathrm{cl}_{\mathcal{N}}(\mathcal{H})$. Hence equality
$S_{G}(X)=\mathrm{cl}_{\mathcal{N}}(\mathcal{H})$ follows. ∎

######  Proposition 4.2.

Let $\mathcal{N}$ be as above, and let $X$ be a compact totally disconnected
$G$-space. The following are equivalent:

  1. (1)

$G\times X\to X$ is continuous, where $G$ is equipped with the topology
$\tau_{\mathcal{N}}$;

  2. (2)

for every clopen subset $U$ of $X$, the stabilizer of $U$ in $G$ is open for
the topology $\tau_{\mathcal{N}}$;

  3. (3)

$G\times X\to X$ extends to a continuous action of $\widehat{G}^{\mathcal{N}}$
on $X$.

If $X$ is a minimal $G$-space, these are also equivalent to:

  1. (4)

there exists a closed subgroup $L$ of $\widehat{G}^{\mathcal{N}}$ such that
$X$ is isomorphic to $\widehat{G}^{\mathcal{N}}/L$ as a $G$-space (where $G$
acts on $\widehat{G}^{\mathcal{N}}/L$ by left translations).

###### Proof.

Since $X$ is totally disconnected, clopen subsets form a basis of the topology
on $X$. The equivalence between (1) and (2) is therefore a consequence of the
definitions. Suppose these conditions hold, and let $L$ be the closure of the
image of $G$ in the group $\mathrm{Homeo}(X)$. Since it follows in particular
from (2) that every clopen subset of $X$ has a finite $G$-orbit, the group $L$
is a profinite group. Since $G\to L$ is continuous, by the universal property
of $\widehat{G}^{\mathcal{N}}$ [Wil98, Prop. 1.4.1–1.4.2], $G\to L$ extends to
a continuous homomorphism $\widehat{G}^{\mathcal{N}}\to L$. So (3) holds.
Finally (3) implies (1) because $\psi:G\to\widehat{G}^{\mathcal{N}}$ is
continuous.

The last statement is clear since a minimal continuous action of a compact
group on a compact space is necessarily transitive. ∎

######  Definition 4.3.

The $G$-action on $X$ is called pro-$\mathcal{N}$ if it satisfies the
equivalent conditions (1)-(2)-(3). We also say that the $G$-space $X$ is
pro-$\mathcal{N}$.

In case where $\mathcal{N}$ consists of all finite index normal subgroups of
$G$, this corresponds to the common notion of profinite $G$-space.

######  Proposition 4.4.

Let $\mathcal{H}$ be a URS of a countable group $G$. Then the following are
equivalent:

  1. (1)

$\mathcal{H}$ is closed for the topology $\tau_{\mathcal{N}}$.

  2. (2)

For every $H\in\mathcal{H}$, the stabilizer URS associated to the left
translation action of $G$ on $\widehat{G}^{\mathcal{N}}/\overline{\psi(H)}$ is
equal to $\mathcal{H}$.

  3. (3)

There exists a minimal $G$-space $X$ that is pro-$\mathcal{N}$ such that
$S_{G}(X)=\mathcal{H}$.

###### Proof.

Proposition 4.1 implies that (1) and (2) are equivalent. (2) clearly implies
(3), so we only have to see that (3) implies (1). Let $X$ be a minimal
$G$-space that is pro-$\mathcal{N}$ that admits $\mathcal{H}$ as a stabilizer
URS. By Proposition 4.2 there exists a closed subgroup $L$ of
$\widehat{G}^{\mathcal{N}}$ such that $X$ is isomorphic to
$\widehat{G}^{\mathcal{N}}/L$ as a $G$-space. The stabilizers in $G$ for the
action on $\widehat{G}^{\mathcal{N}}/L$ are closed for the topology
$\tau_{\mathcal{N}}$ on $G$. Since $G$ is countable, there is a dense set of
points $x$ in $\widehat{G}^{\mathcal{N}}/L$ such that $G_{x}\in
S_{G}(\widehat{G}^{\mathcal{N}}/L)$ (Proposition 2.3). Since
$S_{G}(\widehat{G}^{\mathcal{N}}/L)$ is equal to $\mathcal{H}$ by assumption,
it follows that $\mathcal{H}$ contains some elements that are closed for the
topology $\tau_{\mathcal{N}}$. By Corollary 3.8 this implies
$\mathcal{H}=\mathrm{cl}(\mathcal{H})$. ∎

######  Remark 4.5.

Matte Bon–Tsankov and Elek showed that every URS $\mathcal{H}$ is equal to the
stabilizer URS associated to some compact $G$-space [MBT20, Ele18], and among
the compact $G$-spaces associated to $\mathcal{H}$ there is a unique one that
is universal in a certain sense [MBT20]. We point out that this $G$-space is
very different from the $G$-space
$\widehat{G}^{\mathcal{N}}/\overline{\psi(H)}$ associated to the specific
setting considered in Proposition 4.4.

######  Remark 4.6.

In the case of the profinite topology, Proposition 4.4 says that a URS
$\mathcal{H}$ is closed for the profinite topology if and only if
$\mathcal{H}$ is the stabilizer URS associated to a minimal profinite
$G$-space. It is worth noting that if $\mathcal{H}$ is such a URS, then the
$G$-action on $\mathcal{H}$ need not be profinite. Such a phenomenon has been
exploited by Matte Bon [MB17] and Nekrashevych [Nek20].

We end this section by showing that in general the restriction of
$\mathrm{cl}$ to a URS is not continuous. Recall that a closed subset $F$ of a
space $X$ is regular if $F$ equals the closure of its interior.

######  Lemma 4.7.

Let $X$ be a compact minimal $G$-space such that $\mathrm{Fix}_{X}(g)$ is a
regular closed set of $X$ for every $g\in G$. Then for every $x\in X$, we have
$G_{x}\leq\mathrm{cl}(G_{x}^{0})$.

###### Proof.

Fix $x\in X$, $g\in G_{x}$, and a finite index normal subgroup $N$ of $G$. We
want to see that $g\in G_{x}^{0}N$. Since $N$ has finite index in $G$ and $G$
acts minimally on $X$, each minimal closed $N$-invariant subset of $X$ is
clopen, and the minimal closed $N$-invariant subsets form a finite partition
$\left\\{U_{1},\ldots,U_{n}\right\\}$ of $X$. Let $U_{i}$ be the one
containing $x$. Since $U_{i}$ is a neighbourhood of $x$ and $g\in G_{x}$, the
assumption that $\mathrm{Fix}_{X}(g)$ is regular implies that there exists a
non-empty open subset $V\subseteq U_{i}$ on which $g$ acts trivially. Since
$N$ acts minimally on $U_{i}$, one can find $h\in N$ such that $y=hx\in V$. It
follows that $g\in G_{y}^{0}=hG_{x}^{0}h^{-1}$, and since $h\in N$ we deduce
that $g\in G_{x}^{0}N$. ∎

######  Proposition 4.8.

Suppose that $G$ is countable. Let $X$ be a minimal profinite compact
$G$-space, and $\mathcal{H}=S_{G}(X)$. Suppose that $\mathrm{Fix}_{X}(g)$ is a
regular closed set of $X$ for every $g\in G$, and the stabilizer map
$X\to\operatorname{Sub}(G)$ is not continuous on $X$. Then
$\mathrm{cl}:\mathcal{H}\to\operatorname{Sub}(G)$ is the identity on a dense
set of points, but is not the identity everywhere on $\mathcal{H}$. In
particular it is not continuous.

###### Proof.

By Proposition 4.4 the URS $\mathcal{H}$ is closed for the profinite topology.
Since $G$ is countable, this means that there is a dense set of
$H\in\mathcal{H}$ such that $\mathrm{cl}(H)=H$. Since $x\mapsto G_{x}$ is not
continuous on $X$, one easily verifies that one can find $x\in X$ and
$H\in\mathcal{H}$ such that $H\lneq G_{x}$ and $G_{x}^{0}\leq H$ (see [LBMB18,
Lemma 2.8]). It follows from Lemma 4.7 that
$G_{x}\leq\mathrm{cl}(G_{x}^{0})\leq\mathrm{cl}(H)$, and hence $H$ is properly
contained in $\mathrm{cl}(H)$ since it is properly contained in $G_{x}$. ∎

######  Remark 4.9.

An example of the above situation is provided by $G$ the Grigorchuk group and
$X$ the boundary of the defining rooted tree of $G$ [Gri11, Sec. 7].

###  4.2. Hereditarily minimal actions

######  Definition 4.10.

A compact $G$-space $X$ is hereditarily minimal if every finite index subgroup
of $G$ acts minimally on $X$.

######  Proposition 4.11.

Let $\mathcal{H}$ be a URS of $G$ that is hereditarily minimal. Then for every
$H,K\in\mathcal{H}$ we have
$\mathrm{cl}(H)=\mathrm{cl}(K)=\mathrm{cl}(\mathrm{Env}(\mathcal{H}))$.

This holds in particular if $\mathcal{H}=S_{G}(X)$ with $X$ a hereditarily
minimal compact $G$-space.

###### Proof.

Let $L$ be a finite index subgroup of $G$ such that $H\leq L$. Since $L$ acts
minimally on $\mathcal{H}$, Lemma 2.5 says that $K\leq L$. Consequently
$\mathrm{cl}(H)=\mathrm{cl}(K)$. Since $\mathcal{H}$ is $G$-invariant, it
follows that this common subgroup is normal in $G$. Call it $N$. We shall see
that $N=\mathrm{cl}(\mathrm{Env}(\mathcal{H}))$. Since
$H\leq\mathrm{Env}(\mathcal{H})$ for every $H\in\mathcal{H}$, the inclusion
$N\leq\mathrm{cl}(\mathrm{Env}(\mathcal{H}))$ is clear. On the other hand $N$
contains $\mathrm{Env}(\mathcal{H})$ since $N$ contains all elements of
$\mathcal{H}$. Since $N$ is closed in the profinite topology, $N$ contains
$\mathrm{cl}(\mathrm{Env}(\mathcal{H}))$. Hence equality holds.

As for the last claim, it follows from the fact that Propositions 2.1 and 2.3
ensure that the stabilizer URS associated to a hereditarily minimal compact
$G$-space is itself hereditarily minimal. ∎

######  Proposition 4.12.

Suppose $G$ is a residually finite group. Let $\mathcal{H}$ be a URS of $G$
that is hereditarily minimal, and suppose that $H\in\mathcal{H}$ satisfies the
law $w$. Then $\mathrm{Env}(\mathcal{H})$ also satisfies the law $w$.

###### Proof.

$G$ is residually finite, so the profinite topology on $G$ is Hausdorff. Hence
Lemma 2.9 says that $\mathrm{cl}(H)$ still satisfies $w$. Since
$\mathrm{cl}(H)$ contains $\mathrm{Env}(\mathcal{H})$ by Proposition 4.11,
$\mathrm{Env}(\mathcal{H})$ also satisfies $w$. ∎

Without the hereditarily minimal assumption, it does not hold in general that
a URS satisfying a law $w$ lives inside a normal subgroup of $G$ satisfying
$w$, as the following example shows:

######  Example 4.13.

Let $(F_{n})$ be a sequence of non-abelian finite groups. Suppose that for
every $n$ there is an abelian subgroup $A_{n}$ of $F_{n}$ such that the only
normal subgroup $N$ of $F_{n}$ containing $A_{n}$ is $N=F_{n}$. Let
$\mathbb{G}=\prod_{n}F_{n}$, and let $G$ be a countable dense subgroup of
$\mathbb{G}$ containing $\bigoplus_{n}F_{n}$. Consider the $G$-action on
$X=\prod_{n}F_{n}/A_{n}$. This action is minimal and $G_{x}$ is abelian for
every $x\in X$. In particular every $H\in S_{G}(X)$ is abelian. On the other
hand $\mathrm{Env}(\mathcal{H})$ contains the normal closure in $G$ of
$\bigoplus_{n}A_{n}$. In particular $\mathrm{Env}(\mathcal{H})$ contains
$\bigoplus_{n}F_{n}$, and hence $\mathrm{Env}(\mathcal{H})$ is not abelian.

######  Remark 4.14.

Continuing the previous example, we note that by taking a sequence $(F_{n})$
such that no law $w$ is satisfied by all $F_{n}$ (for instance
$F_{n}=\mathrm{Sym}(n)$ for all $n$), we actually obtain an example where
$\mathrm{Env}(\mathcal{H})$ satisfies no law at all.

We note that examples of groups as above can be found among finitely generated
groups. For instance the groups constructed by B.H. Neumann in [Neu37, Ch.
III] satisfy this properties.

###  4.3. PIF groups

In this paragraph we focus on the class of groups for which, for a faithful
minimal compact $G$-space, profinite implies topologically free.

######  Definition 4.15.

We say that a group $G$ is PIF if for every faithful minimal compact $G$-space
$X$, if the $G$-action on $X$ is profinite then it is topologically free.

This notion was studied notably by Grigorchuk. We will use the following
proposition from [Gri11]. Recall that a group $G$ is just-infinite (JI) if $G$
is infinite and $G/N$ is finite for every non-trivial normal subgroup $N$.
Also $G$ is hereditarily just-infinite (HJI) if every finite index subgroup of
$G$ is JI.

######  Proposition 4.16.

Each one of the following conditions implies that $G$ is PIF:

  1. (1)

for every non-trivial subgroups $H_{1},H_{2}\leq G$ such that the normalizer
$N_{G}(H_{i})$ of $H_{i}$ has finite index in $G$ for $i=1,2$, we have that
$H_{1}\cap H_{2}$ is non-trivial.

  2. (2)

$G$ is hereditarily just-infinite.

###### Proof.

The first assertion is [Gri11, Proposition 4.11] (the formulation there is not
quite the same, but the argument is the same). The second assertion follows
from the first one because every HJI group satisfies (1). ∎

The first condition of the proposition is satisfied for instance by non-
abelian free groups, and also by all Gromov-hyperbolic groups with no non-
trivial finite normal subgroup, and more generally by any group $G$ admitting
an isometric action on a hyperbolic space $X$ with unbounded orbits such that
the $G$-action on its limit set $\partial_{X}G$ is faithful.

######  Proposition 4.17.

Suppose $G$ is PIF, and let $\mathcal{H}$ be a non-trivial URS of $G$. Then
there exists a non-trivial normal subgroup $N$ of $G$ such that
$N\leq\mathrm{cl}(H)$ for every $H\in\mathcal{H}$.

###### Proof.

Note that since $\mathcal{H}$ is not the trivial URS,
$\mathrm{cl}(\mathcal{H})$ is not the trivial URS either. Let
$H\in\mathcal{H}$, and $L=\overline{\psi(H)}$, where $\psi:G\to\widehat{G}$ is
the canonical map from $G$ to its profinite completion. By Proposition 4.1,
the stabilizer URS associated to the left translation action of $G$ on
$\widehat{G}/L$ is equal to $\mathrm{cl}(\mathcal{H})$, and hence is not
trivial. This means that the action of $G$ on $\widehat{G}/L$ is not
topologically free. Since this action is profinite and $G$ is PIF, the action
cannot be faithful. So there is a non-trivial normal subgroup $N$ of $G$ that
is contained in the stabilizer in $G$ of the coset $L$, which is
$\psi^{-1}(L)=\mathrm{cl}(H)$. Upper semi-continuity of $\mathrm{cl}$ on
$\mathcal{H}$ then implies that $N\leq\mathrm{cl}(H^{\prime})$ for every
$H^{\prime}\in\mathcal{H}$. ∎

######  Definition 4.18.

We say that a subgroup $H$ of a group $G$ is quasi-dense for the profinite
topology if $\mathrm{cl}(H)$ is a finite index subgroup of $G$.

######  Corollary 4.19.

Suppose $G$ is HJI, and let $\mathcal{H}$ be a non-trivial URS of $G$. Then
for every $H\in\mathcal{H}$, $H$ is quasi-dense in $G$ for the profinite
topology.

###### Proof.

Proposition 4.16 says that $G$ is PIF. So Proposition 4.17 applies, and gives
a non-trivial normal subgroup $N$ such that $N\leq\mathrm{cl}(H)$ for every
$H\in\mathcal{H}$. By the assumption $N$ must have finite index, and the
conclusion follows. ∎

Recall that every HJI-group is either virtually simple or residually finite.
Corollary 4.19 is void for virtually simple groups, so the focus here is on
residually finite HJI-groups. By Margulis normal subgroup theorem, every
irreducible lattice $\Gamma$ in a connected semisimple Lie group $\mathbf{G}$
(with trivial center and no compact factor) of rank $\geq 2$ is HJI. Under the
assumption that every simple factor of the ambient Lie group $\mathbf{G}$ has
rank $\geq 2$, it is known that every non-trivial URS of $\Gamma$ is just the
conjugacy class of a finite index subgroup [BH21, Cor. F]. The normal subgroup
theorem of Bader–Shalom asserts that any irreducible cocompact lattice
$\Gamma$ in a product $\mathbf{G_{1}}\times\mathbf{G_{2}}$, where
$\mathbf{G_{1}},\mathbf{G_{2}}$ are compactly generated topologically simple
locally compact groups, is HJI [BS06]. In this setting the URSs of $\Gamma$
are not understood.

Following [Cor06], we shall say that a group $G$ has property (PF) if for
every subgroup $H$ of $G$, $H$ is quasi-dense in $G$ for the profinite
topology only if $H$ has finite index. Let $p$ be a prime number. Following
[EJZ13], we say that a group $G$ is weakly $p$-LERF if for every subgroup $H$
of $G$, the closure $\mathrm{cl}_{p}(H)$ of $H$ for the pro-$p$ topology has
finite index in $G$ only if $H$ has finite index. Note that applying the
definition of weakly $p$-LERF to the trivial subgroup, we see that a just-
infinite group that is weakly $p$-LERF is necessarily residually-$p$. Every
group that is weakly $p$-LERF has property (PF). Among their striking
properties, the finitely generated groups obtained as the outputs of the
process carried out in [EJZ13] are HJI and weakly $p$-LERF.

######  Corollary 4.20.

Suppose $G$ is HJI and has property (PF). Then $G$ has no continuous URS.

###### Proof.

Let $\mathcal{H}$ be a URS of $G$. If $\mathcal{H}$ is trivial, then there is
nothing to show. Otherwise, for every $H\in\mathcal{H}$, $\mathrm{cl}(H)$ has
finite index in $G$ by Corollary 4.19. Hence so does $H$ since $G$ has (PF).
In particular $H$ has only finitely many conjugates, i.e. $\mathcal{H}$ is
finite. ∎

##  5\. The Furstenberg URS

######  Definition 5.1.

Given two closed subsets
$\mathcal{X}_{1},\mathcal{X}_{2}\subset\operatorname{Sub}(G)$, we write
$\mathcal{X}_{1}\preccurlyeq\mathcal{X}_{2}$ if there exist
$H_{1}\in\mathcal{X}_{1}$ and $H_{2}\in\mathcal{X}_{2}$ such that $H_{1}\leq
H_{2}$.

One verifies that, when restricted to the set $\mathrm{URS}(G)$, the relation
$\preccurlyeq$ is a partial order [LBMB18, Cor. 2.15].

Recall that a compact $G$-space $X$ is strongly proximal if the orbit closure
of every probability measure on $X$ in the space $\mathrm{Prob}(X)$ contains a
Dirac measure. The Furstenberg boundary $\partial_{F}G$ of $G$ is the
universal minimal and strongly proximal $G$-space [Gla76]. We denote by
$\mathrm{Rad}(G)$ the amenable radical of the group $G$. It coincides with the
kernel of the action of $G$ on $\partial_{F}G$.

######  Definition 5.2.

The stabilizer URS associated to the $G$-action on $\partial_{F}G$ is denoted
$\mathcal{A}_{G}$, and is called the Furstenberg URS of $G$.

A result of Frolík implies that the map $x\mapsto G_{x}$ is continuous on
$\partial_{F}G$, so that $\mathcal{A}_{G}$ is exactly the collection of point
stabilizers for the action of $G$ on $\partial_{F}G$ (see [Ken20] and
references there).

######  Proposition 5.3.

The following hold:

  1. (1)

$\mathcal{A}_{G}$ is amenable, and $\mathcal{X}\preccurlyeq\mathcal{A}_{G}$
for every non-empty closed $G$-invariant subset $\mathcal{X}$ of
$\operatorname{Sub}(G)$ consisting of amenable subgroups.

  2. (2)

$\mathcal{A}_{G}$ is invariant under the action of $\mathrm{Aut}(G)$ on
$\operatorname{Sub}(G)$.

  3. (3)

$\mathrm{Rad}(G)\leq H$ for every $H\in\mathcal{A}_{G}$.

  4. (4)

If $N\in\mathcal{N}_{G}$ is amenable and if $\operatorname{Sub}_{\geq N}(G)$
is the set of subgroups of $G$ containing $N$, then the natural map
$\varphi:\operatorname{Sub}(G/N)\rightarrow\operatorname{Sub}_{\geq N}(G)$
induces a $G$-equivariant homeomorphism between $\mathcal{A}_{G/N}$ and
$\mathcal{A}_{G}$.

  5. (5)

$\mathcal{A}_{G}$ is a singleton if and only if
$\mathcal{A}_{G}=\left\\{\mathrm{Rad}(G)\right\\}$. When this does not hold,
$\mathcal{A}_{G}$ is continuous.

  6. (6)

$\mathcal{A}_{G}=\left\\{\mathrm{Rad}(G)\right\\}$ if and only if every
amenable URS of $G$ lives inside the amenable radical: $H\leq\mathrm{Rad}(G)$
for every amenable URS $\mathcal{H}$ and every $H\in\mathcal{H}$.

###### Proof.

See [LBMB18] and references there. ∎

######  Lemma 5.4.

Let $N$ be a normal subgroup of $G$ such that $H\leq N$ for every
$H\in\mathcal{A}_{G}$. Then $\mathcal{A}_{N}=\mathcal{A}_{G}$. In particular
$N$ acts minimally on $\mathcal{A}_{G}$.

###### Proof.

$\mathcal{A}_{N}$ being $\mathrm{Aut}(N)$-invariant, it is $G$-invariant.
Hence $\mathcal{A}_{N}$ is an amenable URS of $G$. So
$\mathcal{A}_{N}\preccurlyeq\mathcal{A}_{G}$. On the other hand
$\mathcal{A}_{G}$ is a closed $N$-invariant subset of $\operatorname{Sub}(N)$
consisting of amenable subgroups, so
$\mathcal{A}_{G}\preccurlyeq\mathcal{A}_{N}$. Since $\preccurlyeq$ is an order
in restriction to URSs, $\mathcal{A}_{G}=\mathcal{A}_{N}$. ∎

######  Definition 5.5.

We denote by $\mathcal{F}$ the class of groups whose Furstenberg URS is a
singleton.

######  Lemma 5.6.

Suppose $G=\bigcup_{I}G_{i}$ is the directed union of subgroups $G_{i}$ such
that eventually $G_{i}$ is in $\mathcal{F}$ (resp. $\mathcal{A}_{G_{i}}$ is
trivial). Then $G$ is in $\mathcal{F}$ (resp. $\mathcal{A}_{G}$ is trivial).

###### Proof.

Write $R_{i}$ for the amenable radical of $G_{i}$, so that eventually
$\mathcal{A}_{G_{i}}=\left\\{R_{i}\right\\}$. Consider
$\varphi_{i}:\operatorname{Sub}(G)\to\operatorname{Sub}(G_{i})$, $H\mapsto
H\cap G_{i}$. This map is continuous and $G_{i}$-equivariant. Take
$H\in\mathcal{A}_{G}$. The subset $\mathcal{X}_{i}:=\overline{(H\cap
G_{i})^{G_{i}}}$ is a closed $G_{i}$-invariant subset of
$\operatorname{Sub}(G_{i})$ consisting of amenable subgroups, so
$\mathcal{X}_{i}\preccurlyeq\mathcal{A}_{G_{i}}=\left\\{R_{i}\right\\}$ by
Proposition 5.3. Since $\varphi_{i}(\mathcal{A}_{G})$ is closed and
$G_{i}$-invariant, $\mathcal{X}_{i}\subseteq\varphi_{i}(\mathcal{A}_{G})$. We
infer that there exists $K_{i}\in\mathcal{A}_{G}$ such that $K_{i}\cap
G_{i}\leq R_{i}$. Upon passing to a subnet we may assume that $(K_{i})$
converges to some $K\in\mathcal{A}_{G}$ and $(R_{i})$ converges to some $R$.
The subgroup $R$ is normal and amenable, so $R\leq\mathrm{Rad}(G)$. Since
$G=\bigcup_{I}G_{i}$, $(K_{i}\cap G_{i})$ also converges to $K$, and the
inclusion $K_{i}\cap G_{i}\leq R_{i}$ then implies $K\leq R$. So
$\mathcal{A}_{G}\preccurlyeq\left\\{\mathrm{Rad}(G)\right\\}$, which means
that $\mathcal{A}_{G}=\left\\{\mathrm{Rad}(G)\right\\}$ by Proposition 5.3. We
also immediately obtain that in case $R_{i}$ is trivial eventually, then
$\mathcal{A}_{G}$ is trivial. ∎

###  5.1. Proofs of the results

Recall that If $\mathcal{C}$ is a class of groups, we denote by
$\mathcal{N}_{G}(\mathcal{C})$ the normal subgroups of $G$ such that
$G/N\in\mathcal{C}$. In the sequel we mainly use this notation with
$\mathcal{C}=\mathcal{F}$. We have the following lemma:

######  Lemma 5.7.

$\mathcal{N}_{G}(\mathcal{F})$ is stable under taking finite intersections.

###### Proof.

Let $N_{1},N_{2}\in\mathcal{N}_{G}(\mathcal{F})$. Let $Q_{i}=G/N_{i}$, and let
$R_{i}=\mathrm{Rad}(Q_{i})$ be the amenable radical of $Q_{i}$. Let
$\pi_{i}:G\to Q_{i}$ be the canonical projection, and
$M_{i}:=\pi_{i}^{-1}(R_{i})$. Let also $X_{i}=\partial_{F}Q_{i}$. The subgroup
$R_{i}$ acts trivially on $X_{i}$, and the assumption that $Q_{i}$ belongs to
$\mathcal{F}$ means that the $Q_{i}/R_{i}$-action on $X_{i}$ is free.

We consider the $G$-action on the product $X_{1}\times X_{2}$. This action
remains strongly proximal [Gla76, III.1]. It follows that there exists a
unique minimal closed $G$-invariant subset $X\subseteq X_{1}\times X_{2}$, and
the $G$-action on $X$ is strongly proximal [Gla76, III.1]. The subgroup
$M_{1}\cap M_{2}$ of $G$ acts trivially on $X_{1}\times X_{2}$, and hence on
$X$. Moreover since the $Q_{i}/R_{i}$-action on $X_{i}$ is free, it follows
that for the $G$-action on $X$, every point stabilizer is equal to $M_{1}\cap
M_{2}$. Equivalently, the $G$-action on $X$ factors through a free action of
$G/M_{1}\cap M_{2}$. In particular $G/M_{1}\cap M_{2}$ is in $\mathcal{F}$.
Since the group $M_{1}\cap M_{2}/N_{1}\cap N_{2}$ is amenable (as it embeds in
the amenable group $R_{1}\times R_{2}$), and since being in $\mathcal{F}$ is
invariant under forming an extension with amenable normal subgroup, it follows
that $G/N_{1}\cap N_{2}$ is in $\mathcal{F}$. ∎

As a consequence of the lemma, it follows that the family of cosets $gN$,
$g\in G$, $N\in\mathcal{N}_{G}(\mathcal{F})$, forms a basis for a group
topology $\tau_{\mathcal{N}_{G}(\mathcal{F})}$ on $G$, and that the closure of
$H$ with respect to this topology is equal to
$\mathrm{cl}_{\mathcal{N}_{G}(\mathcal{F})}(H)$ (Proposition 2.8).

The proof of the following is the technical part of this section.

######  Proposition 5.8.

Let $G$ be a countable group. Then for every
$N\in\mathcal{N}_{G}(\mathcal{F})$, there exists a normal subgroup $M$ of $G$
such that $N\leq M$ and $M/N\leq\mathrm{Rad}(G/N)$, and a comeager subset
$\mathcal{H}_{0}\subseteq\mathcal{A}_{G}$ such that $NH=M$ for every
$H\in\mathcal{H}_{0}$.

###### Proof.

The map $\varphi_{N}:\mathcal{A}_{G}\to\operatorname{Sub}(G)$, $H\mapsto NH$,
is lower semi-continuous by Lemma 2.6. Hence the set
$\mathcal{H}_{0}\subseteq\mathcal{A}_{G}$ of points where $\varphi_{N}$ is
continuous is a comeager subset of $\mathcal{A}_{G}$, and

$E_{\varphi_{N}}=\overline{\left\\{\left(H,NH\right)\,:\,H\in\mathcal{H}_{0}\right\\}}\,\text{
and }\,S_{\varphi_{N}}=\overline{\left\\{NH\,:\,H\in\mathcal{H}_{0}\right\\}}$

satisfy the conclusion of Proposition 2.1. Note that $S_{\varphi_{N}}$ is
contained in the closed subset $\operatorname{Sub}_{\geq N}(G)$ of
$\operatorname{Sub}(G)$ consisting of subgroups of $G$ containing $N$.

Since the URS $\mathcal{A}_{G}$ is strongly proximal and strong proximality
passes to highly proximal extensions and factors [Gla75, Lemma 5.2], we deduce
that the $G$-action on $S_{\varphi_{N}}$ is minimal and strongly proximal. The
map $\pi_{N}:S_{\varphi_{N}}\to\operatorname{Sub}(G/N)$, $K\to K/N$, is a
$G$-equivariant homeomorphism onto its image (indeed, one easily verifies that
modding out by $N$ defines a homeomorphism from $\operatorname{Sub}_{\geq
N}(G)$ onto $\operatorname{Sub}(G/N)$). Hence $\pi_{N}(S_{\varphi_{N}})$ is a
strongly proximal URS of $G/N$. Moreover $\pi_{N}(S_{\varphi_{N}})$ consists
of amenable subgroups. If we let $R$ be the amenable radical of $G/N$, it
follows from the assumption that $G/N$ belongs to $\mathcal{F}$ that
$\pi_{n}(S_{\varphi_{N}})$ is contained in $\operatorname{Sub}(R)$. On the
other hand $R$ must act trivially on $\pi_{N}(S_{\varphi_{N}})$ by minimality
and strong proximality.

Consider the envelope $E=\mathrm{Env}(S_{\varphi_{N}})$. Since
$\pi_{N}(S_{\varphi_{N}})\subseteq\operatorname{Sub}(R)$, we have $E/N\leq R$.
So by the previous paragraph and the fact that
$\pi_{N}:S_{\varphi_{N}}\to\pi_{N}(S_{\varphi_{N}})$ is a $G$-equivariant
homeomorphism, it follows that $E$ acts trivially on $S_{\varphi_{N}}$. On the
other hand $E$ contains $H$ for every $H\in\mathcal{H}_{0}$, and since
$\mathcal{H}_{0}$ is dense in $\mathcal{A}_{G}$ this easily implies that $E$
contains $H$ for every $H\in\mathcal{A}_{G}$. Hence Lemma 5.4 can be applied
to $E$, and we infer that $E$ acts minimally on $\mathcal{A}_{G}$. Using
Proposition 2.1 and the fact that minimality passes to irreducible extensions,
we deduce that $E$ acts minimally on $S_{\varphi_{N}}$. All together, this
shows that $S_{\varphi_{N}}$ is a one-point space. The corresponding normal
subgroup $M$ of $G$ verifies the conclusion. ∎

######  Proposition 5.9.

Let $G$ be a countable group, and let $\mathcal{N}$ be a countable subset of
$\mathcal{N}_{G}(\mathcal{F})$. Then there exists a normal subgroup $M$ of $G$
and a comeager subset $\mathcal{H}_{0}\subseteq\mathcal{A}_{G}$ such that
$\mathrm{cl}_{\mathcal{N}}(H)=M$ for every $H\in\mathcal{H}_{0}$.

###### Proof.

We apply Proposition 5.8 for every $N\in\mathcal{N}$. We obtain a normal
subgroup $M_{N}$ of $G$ and a comeager subset $\mathcal{H}_{N}$ of
$\mathcal{A}_{G}$. Set $\mathcal{H}_{0}=\bigcap_{\mathcal{N}}\mathcal{H}_{N}$
and $M=\bigcap_{\mathcal{N}}M_{N}$. Since $\mathcal{N}$ is countable,
$\mathcal{H}_{0}$ is a comeager subset of $\mathcal{H}$. By construction for
every $H\in H_{0}$,

$cl_{\mathcal{N}}(H)=\bigcap_{\mathcal{N}}NH=\bigcap_{\mathcal{N}}M_{N}=M.$

∎

######  Definition 5.10.

We say that a set of laws $\mathbb{W}$ detects amenability in a group $G$ if
for every subgroup $H$ of $G$, one has that $H$ is amenable if and only if
there exists $w\in\mathbb{W}$ such that $H$ virtually satisfies $w$.

Note that if a set of laws detects amenability in a group $G$, it also detects
amenability in any subgroup of $G$. The following is an immediate consequence
of the definition and Lemma 2.9.

######  Lemma 5.11.

Let $G$ be a group such that there is a set of laws that detects amenability
in $G$, and let $(G,\tau)$ be a group topology on $G$ that is Hausdorff. Then
for every amenable subgroup $H$ of $G$, the $\tau$-closure of $H$ remains
amenable.

######  Theorem 5.12.

Let $G$ be a group such that there is a set of laws that detects amenability
in $G$. Then $G$ is in $\mathcal{F}$ if and only if $G$ is
residually-$\mathcal{F}$.

###### Proof.

Only one direction is non-trivial. Writing $G$ as the directed limit of its
countable subgroups and invoking Lemma 5.6, one sees that it suffices to prove
the result when $G$ is countable. Under this assumption, since $G$ is
residually-$\mathcal{F}$, one can find
$\mathcal{N}\subseteq\mathcal{N}_{G}(\mathcal{F})$ such that $\mathcal{N}$ is
countable and $\bigcap_{\mathcal{N}}N=\left\\{1\right\\}$. Since
$\mathcal{N}_{G}(\mathcal{F})$ is stable under taking finite intersections by
Lemma 5.7, we can replace $\mathcal{N}$ by the collection of finite
intersections of elements of $\mathcal{N}$, so that we may assume that
$\mathcal{N}$ is filtering. So for a subgroup $H$ of $G$,
$\mathrm{cl}_{\mathcal{N}}(H)$ equals the closure of $H$ in the topology
$\tau_{\mathcal{N}}$ (Proposition 2.8).

Proposition 5.9 provides a normal subgroup $M$ of $G$ such that
$\mathrm{cl}_{\mathcal{N}}(H)=M$ for every $H$ in a comeager subset of
$\mathcal{A}_{G}$. The topology $\tau_{\mathcal{N}}$ is Hausdorff since
$\bigcap_{\mathcal{N}}N=\left\\{1\right\\}$, so it follows from Lemma 5.11
that the closure of an amenable subgroup of $G$ remains amenable. This shows
$M$ is amenable, and it follows that $M\leq\mathrm{Rad}(G)$. By Proposition
5.3 this means that $\mathcal{A}_{G}=\left\\{\mathrm{Rad}(G)\right\\}$. ∎

######  Remark 5.13.

When the group $G$ is residually finite, there is a shorter way to obtain the
conclusion of Theorem 5.12. Indeed, since the $G$-space $\partial_{F}G$ is
proximal, it is hereditarily minimal [Gla76]. Moreover it follows from the
conclusion of Proposition 2.1 that being hereditarily minimal is inherited
from a $G$-space to its stabilizer URS. Hence $\mathcal{A}_{G}$ is a
hereditarily minimal URS. Hence Propositions 4.11 and 4.12 apply, and the
conclusion follows as above.

######  Corollary 5.14.

Let $G$ be a group such that there is a set of laws that detects amenability
in $G$, and suppose $\mathrm{Rad}(G)$ is trivial. If $G$ is
residually-$\mathcal{F}$, then $G$ is $C^{\ast}$-simple.

###### Proof.

The result follows from Theorem 5.12 and the main result of [KK17], which
asserts that $G$ is in $\mathcal{F}$ if and only if $G/\mathrm{Rad}(G)$ is
$C^{\ast}$-simple. ∎

###  5.2. Linear groups

We deduce Corollary 1 from the introduction, which asserts that linear groups
belong to $\mathcal{F}$.

###### Proof of Corollary 1.

Writing $G$ as the directed limit of its finitely generated subgroups and
invoking Lemma 5.6, one sees that without loss of generality we can assume
that $G$ is a finitely generated linear group. By Malcev’s theorem, the group
$G$ is residually finite. Also by the Tits alternative [Tit72], every amenable
subgroup of $G$ is virtually solvable (we are using again that $G$ is finitely
generated to have this version of the Tits alternative). Hence all the
assumptions of Theorem 5.12 are verified. The conclusion follows. ∎

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