# Continuity of the stabilizer map on maximally highly proximal flows

Adrien Le Boudec  CNRS, UMPA–ENS Lyon
46 allée d’Italie
69364 Lyon
France<EMAIL_ADDRESS>and  Todor Tsankov  Institut Camille
Jordan
Université Claude Bernard Lyon 1
43, boulevard du 11 novembre 1918
69622 Villeurbanne cedex
France – and – Institut Universitaire de France<EMAIL_ADDRESS>
(Date: February 6, 2023)

###### Abstract.

Let $G$ be a locally compact group and let $G\curvearrowright X$ be a
$G$-flow. We prove that if the flow $X$ is maximally highly proximal, then the
stabilizer map $x\mapsto G_{x}$, from $X$ to the space of closed subgroups of
$G$, is continuous. This provides in particular a common generalization of a
theorem of Frolík (that the set of fixed points of a homeomorphism of an
extremally disconnected compact space is open) and a theorem of Veech (that
the action of a locally compact group on its greatest ambit is free).

###### Key words and phrases:

Locally compact groups, stabilizer map, URS, MHP flows.

###### 2020 Mathematics Subject Classification:

Primary: 37B05. Secondary: 22D12, 06E15, 54H15.

##  1\. Introduction

Let $G$ be a topological group. Recall that a _$G$ -flow_ is a continuous
action $G\curvearrowright X$ on a compact space $X$ (all our compact spaces
are Hausdorff). A $G$-flow is _minimal_ if every orbit is dense. A continuous,
$G$-equivariant map $\pi\colon Y\to X$ between $G$-flows is called a _$G$
-map_. If $\pi$ is surjective, we also say that $Y$ is an _extension_ of $X$,
or that $X$ is a _factor_ of $Y$.

A map $\pi\colon Y\to X$ between compact spaces is called _irreducible_ if
every non-empty open $U\subseteq Y$ contains the fiber $\pi^{-1}(\\{x\\})$ for
some $x\in X$, or, equivalently, if the image of any proper closed subset of
$Y$ is a proper subset of $X$. Irreducible maps were studied by Gleason
[Gleason1958], who proved that to every compact space $X$, one can associate
an extremally disconnected compact space $\hat{X}$, the Stone space of the
Boolean algebra $\operatorname{RO}(X)$ of regular open subsets of $X$, with an
irreducible map $\hat{X}\to X$ which is universal with respect to irreducible
maps $Y\to X$. Recall that a space is _extremally disconnected_ if the closure
of every open subset is clopen.

An extension $\pi\colon Y\to X$ between $G$-flows is called _highly proximal_
if $\pi$ is irreducible. This notion was studied by Auslander and Glasner
[Auslander1977]. For minimal flows, this is equivalent to asking that the
fibers of $\pi$ can be compressed to a point by a net of elements of $G$
[Auslander1977] (which justifies the name). Highly proximal extensions are
thought as being rather small extensions. They preserve many dynamical
properties such as minimality, proximality, strong proximality, and
disjointness. When the spaces $Y,X$ are metrizable, an extension $\pi\colon
Y\to X$ is highly proximal iff it is _almost one-to-one_ (the set
$\big{\\{}y\in Y:\pi^{-1}(\\{\pi(y)\\})=\\{y\\}\big{\\}}$ is dense in $Y$).
Almost one-to-one extensions are an important tool in topological dynamics
(used, for example, to construct symbolic representations of continuous
systems), and the notion of highly proximal extension is the appropriate
generalization that allows the existence of universal objects and the
development of a general theory.

For every $G$-flow $X$, there exists a $G$-flow $\hat{X}_{G}$ and a highly
proximal extension $\pi_{X}\colon\hat{X}_{G}\to X$ with the following
universal property: for every highly proximal extension $\pi\colon Y\to X$,
there exists a $G$-map $p\colon\hat{X}_{G}\to Y$ such that $\pi\circ
p=\pi_{X}$. Moreover, $\hat{X}_{G}$ is unique up to isomorphism. It is called
the _universal highly proximal extension_ of $X$. For minimal flows, the
existence and uniqueness of $\hat{X}_{G}$ were established in [Auslander1977]
and the general case is due to Zucker [Zucker2021]. A $G$-flow $X$ is called
_maximally highly proximal (MHP)_ if $\hat{X}_{G}=X$. Equivalently, $X$ is MHP
if $X$ admits no non-trivial highly proximal extension. The correspondence
$X\mapsto\hat{X}_{G}$ is idempotent and its image is the class of MHP
$G$-flows. Thus the class of $G$-flows is partitioned into equivalence
classes, where $X$ and $Y$ are equivalent if they admit a common highly
proximal extension; or equivalently if $\hat{X}_{G}$ and $\hat{Y}_{G}$ are
isomorphic. Each class contains a unique representative that is MHP.

For discrete groups, the construction of $\hat{X}_{G}$ reduces to the one by
Gleason, and we have that $\hat{X}_{G}=\hat{X}$ [Gleason1958, Th. 3.2]. In
that setting, a $G$-flow $X$ is MHP iff it is extremally disconnected. This
depends only on the topology of $X$, and not on $G$. This is no longer true
for non-discrete groups. Examples of MHP flows that arise in the non-discrete
setting are $X=G/H$, where $H$ is a closed, cocompact subgroup of $G$, and $G$
acts on $X$ by left-translations. MHP flows of Polish groups were extensively
studied by Zucker in [Zucker2021], where many more interesting examples can be
found. More general topological groups were considered by Basso and Zucker in
[Basso2021a].

The highly proximal equivalence relation and MHP flows are useful to express
certain rigidity properties among $G$-flows. An instance of this is a theorem
of Rubin that asserts that any two $G$-flows that are faithful and _micro-
supported_ are highly proximally equivalent [Rubin-loc-mov-book]. Combined
with [CLB-commens, Prop. 2.3], this implies that every group $G$ that admits a
faithful micro-supported $G$-flow admits exactly one faithful micro-supported
$G$-flow that is MHP. For certain non-discrete totally disconnected locally
compact groups, this flow is the Stone space of the centralizer lattice of
$G$, a Boolean algebra constructed from the local structure of the group [CRW-
part1]. See the references above for the definition of a “micro-supported”
action and more details.

### The main result

In certain contexts, MHP flows are better behaved than general flows. The main
result of this paper is an illustration of such a situation. For the remainder
of the introduction, we suppose that $G$ is a locally compact group, and we
denote by $\operatorname{Sub}(G)$ the space of closed subgroups of $G$.
Endowed with the Chabauty topology, the space $\operatorname{Sub}(G)$ is
compact, and the action of $G$ on $\operatorname{Sub}(G)$ by conjugation is
continuous. To every $G$-flow $X$, we can associate the stabilizer map
$X\to\operatorname{Sub}(G)$, $x\mapsto G_{x}$, which is $G$-equivariant. The
stabilizer map is always upper semi-continuous (see e.g. [Glasner2015]), but
fails to be continuous in general. This lack of continuity is not just a
technical issue, but is inherent to the study of $G$-flows. For instance it
witnesses the difference between free and topologically free actions (see
below). We show that for MHP flows, this defect disappears.

######  Theorem 1.1.

Let $G$ be a locally compact group and let $X$ be an MHP $G$-flow. Then the
stabilizer map $X\to\mathrm{Sub}(G)$, $x\mapsto G_{x}$, is continuous.

As mentioned above, when $G$ is a discrete group, $X$ is MHP if and only if
$X$ is extremally disconnected. In that case, Theorem 1.1 is equivalent to
saying that the set of fixed points in $X$ of every element $g\in G$ is an
open subset of $X$. This is a theorem of Frolík [Frolik1968].

Another special case of Theorem 1.1 is a well-known theorem of Veech that the
action of a locally compact group on its greatest ambit $\operatorname{Sa}(G)$
is free. One can apply Theorem 1.1 because the greatest ambit is an MHP flow
and the free left translation action $G\curvearrowright G$ embeds into it
densely (cf. Corollary 5.8). A relativized version of Veech’s theorem was
considered by Matte Bon and Tsankov in [MatteBon2020], where it was proved
that the stabilizer map for the flow $\operatorname{Sa}(G/H)$, where $H$ is a
closed subgroup of $G$, is continuous. This is again a special case of Theorem
1.1 because the flow $\operatorname{Sa}(G/H)$ is also MHP [Zucker2021].

As Theorem 1.1 is a common generalization of Frolík’s and Veech’s theorem, it
is perhaps not surprising that its proof mixes ideas from the proofs of both.
We also rely on the topometric structure on MHP flows introduced by Zucker
[Zucker2021] (extending a construction of [BenYaacov2017] for
$\operatorname{Sa}(G)$), which while being rather simple for locally compact
groups, is still useful for us.

### Freeness vs topological freeness

Recall that $G\curvearrowright X$ is _free_ if $G_{x}$ is trivial for every
$x\in X$, and $G\curvearrowright X$ is called _topologically free_ if for
every compact $K\subseteq G$ with $1_{G}\notin K$, the closed set $\\{x\in
X:x\in K\cdot x\\}$ has empty interior. (When $G$ is second countable,
topological freeness is equivalent to saying that there is a dense set of
points $x\in X$ such that $G_{x}$ is trivial.) The difference between freeness
and topological freeness is detected by the failure of continuity of the
stabilizer map: a topologically free action is free if and only if the
stabilizer map is continuous. Also the property of being topologically free is
invariant under highly proximal equivalence. Hence the following is a
consequence of Theorem 1.1.

######  Corollary 1.2.

Let $G\curvearrowright X$ be a $G$-flow. Then the following are equivalent:

  1. (i)

$X$ is topologically free;

  2. (ii)

$\hat{X}_{G}$ is free.

In particular, an MHP flow if topologically free if and only if it is free.

This has the following application. Recall that a $G$-flow is called _strongly
proximal_ if the closure of the $G$-orbit of every Borel probability measure
on $X$ contains a Dirac measure. The flow $X$ is called a _boundary_ if $X$ is
minimal and strongly proximal. Every group $G$ admits a boundary
$\partial_{F}G$, unique up to isomorphism, such that every boundary is a
factor of $\partial_{F}G$ [Glasner1976, §III]. It is called the _Furstenberg
boundary_ of $G$. By [Glasner1975, Lemma 5.2] and [Glasner1976, Lemma 4.1] the
flow $\partial_{F}G$ is MHP.

######  Corollary 1.3.

For every locally compact group $G$, the stabilizer map is continuous on
$\partial_{F}G$. In particular the following are equivalent:

  1. (i)

$G$ admits a topologically free boundary;

  2. (ii)

$G$ acts freely on $\partial_{F}G$.

###### Proof.

The first assertion follows from the fact that $\partial_{F}G$ is MHP and
Theorem 1.1. For the second assertion, if $G$ admits a topologically free
boundary $G\curvearrowright X$, then the action of
$G\curvearrowright\partial_{F}G$ is also topologically free since there is a
factor map $\partial_{F}G\to X$. Since $\partial_{F}G$ is MHP, Corollary 1.2
implies that $G\curvearrowright\partial_{F}G$ is free. The other direction is
clear. ∎

When $G$ is a discrete group, the equivalence in Corollary 1.3 was already
known as it follows from [Frolik1968]. Whether this property holds true in a
given group $G$ was recently shown to be equivalent to the simplicity of the
reduced $\mathrm{C}^{\ast}$-algebra of $G$ [KK].

### Stabilizer flows

Theorem 1.1 has interest beyond the case of topologically free actions. Recall
that a _uniformly recurrent subgroup (URS)_ of a locally compact group $G$ is
a minimal closed, $G$-invariant subset of $\operatorname{Sub}(G)$
[Glasner2015]. Every minimal $G$-flow $X$ gives rise to a URS of $G$, called
the _stabilizer URS_ associated to $X$, defined as the unique minimal closed
$G$-invariant subset of the closure of the image of the stabilizer map in
$\mathrm{Sub}(G)$ (Glasner–Weiss [Glasner2015]).

Theorem 1.1 allows us to associate a _stabilizer flow_ to any $G$-flow $X$,
without a minimality assumption: we consider the MHP flow $\hat{X}_{G}$, and
simply take the image of $\hat{X}_{G}$ in $\operatorname{Sub}(G)$ by the
stabilizer map (cf. Definition 5.1). In Section 5, we prove some basic
properties of the stabilizer flow. We show in particular that when $X$ is
minimal, the stabilizer flow and the stabilizer URS are equal. Hence in that
situation the stabilizer flow is an alternative description of the stabilizer
URS.

######  Corollary 1.4.

Let $X$ be a minimal $G$-flow. Then the stabilizer URS of $X$ is equal to
$\\{G_{z}:z\in\hat{X}_{G}\\}$.

### Acknowledgments

We thank Nicolás Matte Bon for interesting discussions about this work.

##  2\. The universal highly proximal extension of a $G$-flow

In this section, we give a new construction of the universal highly proximal
extension of a $G$-flow $G\curvearrowright X$, where $G$ is an arbitrary
topological group. The existence of such an extension was proved by Auslander
and Glasner [Auslander1977] for minimal flows using an abstract argument and a
construction without a minimality assumption, in terms of near-ultrafilters,
was given by Zucker [Zucker2018] for Polish groups and Basso and Zucker
[Basso2021a] for arbitrary topological groups. Our construction is in some
sense dual to theirs: instead of constructing the points of $\hat{X}_{G}$
directly, we describe the lattice of continuous functions $C(\hat{X}_{G})$ and
use an appropriate duality theorem to recover the space.

###  2.1. The non-archimedean case

A Boolean algebra is called _complete_ if it admits suprema (and infima) of
arbitrary subsets. A Boolean algebra $\mathcal{B}$ is complete iff its Stone
space $\operatorname{S}(\mathcal{B})$ is _extremally disconnected_ , i.e., for
every open $U\subseteq\operatorname{S}(\mathcal{B})$, the set $\overline{U}$
is also open. If $\\{A_{i}\\}_{i\in I}$ is a family of clopen sets in
$\operatorname{S}(\mathcal{B})$, their supremum in $\mathcal{B}$ is the clopen
set $\overline{\bigcup_{i}A_{i}}$.

An open subset $U$ of a topological space $X$ is called _regular_ if
$U=\operatorname{Int}\overline{U}$. The collection $\operatorname{RO}(X)$ of
regular open subsets of $X$ forms a complete Boolean algebra with operations
$\cup$ given by the union, and complement given by $\neg
U=\operatorname{Int}(X\setminus U)$. If $X$ is Baire, $\operatorname{RO}(X)$
can also be viewed as the quotient of the Boolean algebra of Baire measurable
subsets of $X$ by the ideal of meager sets. See [Kechris1995]*Section 8. We
denote by $\hat{X}$ the Stone space of the algebra $\operatorname{RO}(X)$. If
$X$ is compact, there is a natural surjective, continuous map
$\ell_{X}\colon\hat{X}\to X$ given by

$\\{\ell_{X}(p)\\}=\bigcap_{U\in p}\overline{U},$

where $p$ is viewed as an ultrafilter on $\operatorname{RO}(X)$.

The construction $X\mapsto\hat{X}$ only depends on the topology of $X$, so if
$G$ is a group acting on $X$ by homeomorphisms, it also acts on $\hat{X}$. If
$G$ is a discrete group and $G\curvearrowright X$ is a $G$-flow, then
$G\curvearrowright\hat{X}$ is also a $G$-flow and it is the universal highly
proximal extension of $G\curvearrowright X$. This follows from the results of
Gleason [Gleason1958].

The problem when $G$ has non-trivial topology is that the action
$G\curvearrowright\hat{X}$ is not necessarily continuous even if the original
action of $G$ on $X$ is. In the case where $G$ is non-archimedean, this is
easy to fix. Recall that a topological group $G$ is called _non-archimedean_
if it admits a basis at $1_{G}$ consisting of open subgroups. For locally
compact groups, by a well known theorem of van Dantzig, being non-archimedean
is equivalent to being totally disconnected (or _tdlc_ , for short).

If $\mathcal{B}$ is a Boolean algebra on which $G$ acts and $V\leq G$, we will
denote by $\mathcal{B}_{V}$ the subalgebra of $\mathcal{B}$ of elements fixed
by $V$. Note that if $\mathcal{B}$ is complete, then $\mathcal{B}_{V}$ is
complete, too.

If $X$ is a $G$-flow, we let

$\operatorname{RO}(G,X)\coloneqq\bigcup\\{\operatorname{RO}_{V}(X):V\text{
open subgroup of }G\\}$

and note that, as a direct limit of Boolean algebras, $\operatorname{RO}(G,X)$
is also a Boolean algebra but that it is not necessarily complete. Note also
that $\operatorname{RO}(G,X)$ is invariant under the action of $G$ and that
the action $G\curvearrowright\operatorname{RO}(G,X)$ is continuous (where
$\operatorname{RO}(G,X)$ is taken to be discrete).

######  Lemma 2.1.

Let $G$ be a non-archimedean group and let $G\curvearrowright X$ be a
$G$-flow. Then the elements of $\operatorname{RO}(G,X)$ form a basis for the
topology of $X$.

###### Proof.

By regularity of $X$, it suffices to see that for every $x\in
U\in\operatorname{RO}(X)$ there exists $U^{\prime}\in\operatorname{RO}(G,X)$
such that $x\in U^{\prime}\subseteq U$. By continuity of the action, there
exists an open subgroup $V$ of $G$ and an open subset $U_{1}\subseteq U$ with
$x\in U_{1}$ such that $VU_{1}\subseteq U$. Then
$U^{\prime}=\operatorname{Int}\overline{VU_{1}}$ works, because $U^{\prime}$
is $V$-invariant and $U^{\prime}\subseteq\operatorname{Int}\overline{U}=U$,
since $U$ is regular. ∎

We denote by $X^{\ast}_{G}$ the Stone space of $\operatorname{RO}(G,X)$. The
action of $G$ on $X^{\ast}_{G}$ is continuous. Note that being the Stone space
of a Boolean algebra, $X^{\ast}_{G}$ is zero-dimensional.

######  Proposition 2.2.

Let $G$ be a non-archimedean group and let $G\curvearrowright X$ be a
$G$-flow. Then $G\curvearrowright X^{\ast}_{G}$ is the universal highly
proximal extension of $X$.

###### Proof.

We denote by $\pi\colon\hat{X}\to X^{\ast}_{G}$ the dual map of the inclusion
$\operatorname{RO}(G,X)\subseteq\operatorname{RO}(X)$ and note that $\pi$ is
continuous and $G$-equivariant. By Lemma 2.1, if two elements of $\hat{X}$
have the same image by $\pi$, then they have the same image under the map
$\ell\colon\hat{X}\to X$. Hence there is a continuous $G$-equivariant map
$\ell_{G}\colon X^{\ast}_{G}\to X$ such that $\ell_{G}\circ\pi=\ell$. The map
$\ell_{G}\colon X^{\ast}_{G}\to X$ is irreducible because $\ell$ is. If $Y\to
X$ is a highly proximal extension of $X$, then $\hat{Y}=\hat{X}$. Thus
$\operatorname{RO}(X)=\operatorname{RO}(Y)$ and
$\operatorname{RO}(G,X)=\operatorname{RO}(G,Y)$. In particular, $Y$ is a
factor of $X^{\ast}_{G}=Y^{\ast}_{G}$. ∎

By continuity of the $G$-action on $X$, we have
$\operatorname{Clopen}(X)\subseteq\operatorname{RO}(G,X)$, where
$\operatorname{Clopen}(X)$ is the subalgebra of $\operatorname{RO}(X)$
consisting of clopen subsets of $X$. That this inclusion is an equality
actually characterizes MHP flows for non-archimedean groups.

######  Corollary 2.3.

Let $G$ be a non-archimedean group and let $G\curvearrowright X$ be a
$G$-flow. Then the following are equivalent:

  1. (i)

$X$ is MHP;

  2. (ii)

$\operatorname{RO}(G,X)=\operatorname{Clopen}(X)$.

###### Proof.

(i) $\Rightarrow$ (ii) follows from Proposition 2.2. Note that (ii) implies
that $X$ is zero-dimensional in view of Lemma 2.1, so the implication (ii)
$\Rightarrow$ (i) also follows from Proposition 2.2. ∎

###  2.2. The general case

When $G$ is a general topological group, one cannot hope to construct the
universal highly proximal extension as the Stone space of a Boolean algebra:
for example, if $G$ is connected, then all of its minimal flows are connected
and have no non-trivial clopen sets. So for the general case, we employ Riesz
spaces instead of Boolean algebras.

Recall that a _Riesz space_ is an ordered real vector space, which is a
_lattice_ for the ordering, i.e., all pairs of elements $a,b$ have a least
upper bound $a\vee b$ and a greatest lower bound $a\wedge b$. A Riesz space
$\mathcal{L}$ is called _archimedean_ if there exists a _unit_
${\mathbf{1}}\in\mathcal{L}$ such that for every $a\in\mathcal{L}$, there
exists $n\in{\mathbf{N}}$ with $a\leq n{\mathbf{1}}$. A unit also naturally
defines the _uniform norm_ :

$\lVert a\rVert\coloneqq\inf\\{r\in{\mathbf{R}}:|a|\leq r{\mathbf{1}}\\},$

where, as usual, $|a|=a\vee(-a)$.

A natural example of an archimedean Riesz space is the collection of real-
valued continuous functions $C(X)$ on a compact space $X$ with the usual
lattice operations and unit the constant function ${\mathbf{1}}$. Then the
uniform norm coincides with the $\sup$ norm. The Yosida representation
theorem, which we recall below, states that in fact every archimedean Riesz
space complete in the uniform norm is of this form.

For every archimedean Riesz space $\mathcal{L}$ with a unit ${\mathbf{1}}$
(and equipped with the uniform norm), we can consider its _spectrum_ :

$\operatorname{S}(\mathcal{L})=\\{x\in\mathcal{L}^{*}:x(a\vee b)=x(a)\vee
x(b)\text{ for all }a,b\in\mathcal{L}\text{ and }x({\mathbf{1}})=1\\}.$

$\operatorname{S}(\mathcal{L})$ is a compact space if equipped with the weak∗
topology and we have a map $\Gamma\colon\mathcal{L}\to
C(\operatorname{S}(\mathcal{L}))$ defined by

$\Gamma(a)(x)=x(a).$

$\Gamma$ is clearly a contractive homomorphism and in fact it is an isometric
isomorphism (see [Jonge1977]*Section 13). The algebra $B(X)$ has also been
considered before in a dynamical context by Keynes and Robertson in
[Keynes1968].

Let $G$ be a topological group, let $E$ be a Banach space and let
$G\curvearrowright E$ be an action by isometric isomorphisms. We will say that
an element $\phi\in E$ is _$G$ -continuous_ if the map $G\to E,g\mapsto
g\cdot\phi$ is norm-continuous.

######  Lemma 2.4.

Let $G$ be a topological group, let $X$ be a compact space and let
$G\curvearrowright X$ be an action by homeomorphisms. Then the following are
equivalent:

  1. (i)

$G\curvearrowright X$ is a $G$-flow (that is, the action is jointly
continuous);

  2. (ii)

Every function $\phi\in C(X)$ is $G$-continuous for the induced action
$G\curvearrowright C(X)$.

###### Proof.

  * (i) $\Rightarrow$ (ii).

This is obvious.

  * (iii) $\Rightarrow$ (i).

Let $U\subseteq X$ be open and let $x_{0}\in U$. Our goal is to find an open
$V\ni 1_{G}$ and an open $W\ni x_{0}$ such that $V\cdot W\subseteq U$. Let
$W\ni x_{0}$ be open such that $\overline{W}\subseteq U$. By Urysohn’s lemma,
there exists $\phi\in C(X)$ with $\phi|_{\overline{W}}=1$ and
$\phi|_{X\setminus U}=0$. As $\phi$ is $G$-continuous, there exists $V\ni
1_{G}$ such that for every $v\in V$, $\lVert v^{-1}\cdot\phi-\phi\rVert<1/2$.
This implies that $V\cdot W\subseteq U$. ∎

Next we will describe the universal highly proximal extension of a $G$-flow
$G\curvearrowright X$. Let $\mathcal{B}(X)$ denote the Riesz space of bounded
Borel functions on $X$ with unit the constant function ${\mathbf{1}}$ and let
$\mathcal{M}$ be the ideal given by:

$\mathcal{M}=\big{\\{}\phi\in\mathcal{B}(X):\\{x\in X:\phi(x)\neq 0\\}\text{
is meager}\big{\\}}.$

Denote $B(X)\coloneqq\mathcal{B}(X)/\mathcal{M}$ and let
$\lVert\cdot\rVert_{\mathcal{M}}$ be the seminorm on $\mathcal{B}(X)$, which
is the pullback of the uniform norm on $B(X)$. Equivalently,
$\lVert\cdot\rVert_{\mathcal{M}}$ is the essential supremum seminorm defined
by

$\lVert\phi\rVert_{\mathcal{M}}=\inf\big{\\{}r\in{\mathbf{R}}:\\{x\in
X:|\phi(x)|>r\\}\text{ is meager}\big{\\}}.$

Let $\hat{X}$ be the spectrum of $B(X)$. It can naturally be identified with
the Stone space of the Boolean algebra $\operatorname{RO}(X)$ (see
[Jonge1977]*Section 14).

We let $B_{G}(X)$ denote the set of $G$-continuous elements of $B(X)$. We note
that $B_{G}(X)$ is a closed subspace of $B(X)$ which is also closed under the
lattice operations, so we can define
$\hat{X}_{G}\coloneqq\operatorname{S}(B_{G}(X))$. It follows from Lemma 2.4
that $G\curvearrowright\hat{X}_{G}$ is a $G$-flow. We have the following.

######  Proposition 2.5.

Let $G$ be a topological group and let $G\curvearrowright X$ be a $G$-flow.
Then the flow $G\curvearrowright\hat{X}_{G}$ is the universal highly proximal
extension of $G\curvearrowright X$. In particular $X$ is MHP if and only if
the natural injection $C(X)\to B_{G}(X)$ is a bijection.

###### Proof.

First, as $\hat{X}_{G}$ is a factor of $\hat{X}$, it is clear that the
extension $\hat{X}_{G}\to X$ is highly proximal. If $G\curvearrowright Y$ is a
highly proximal extension of $G\curvearrowright X$, then by the universal
property of $\hat{X}$, there exists an embedding $\iota\colon C(Y)\to
C(\hat{X})=B(X)$ (see above). It follows from Lemma 2.4 that every $\phi\in
C(Y)$ is $G$-continuous, so $\iota(\phi)$ is also $G$-continuous. Therefore
$\iota(C(Y))\subseteq B_{G}(X)$ and this gives a factor map $\hat{X}_{G}\to
Y$. ∎

It is proved in [Auslander1977] that for minimal flows $X$, the correspondence
$X\mapsto\hat{X}_{G}$ is functorial. Our description of $\hat{X}_{G}$ suggests
the correct formulation of this result for general flows. Recall that a
continuous map $\phi\colon X\to Y$ is called _category-preserving_ if
$\phi^{-1}(A)$ is nowhere dense for any nowhere dense $A\subseteq Y$. Every
homomorphism between minimal flows is category-preserving.

######  Proposition 2.6.

The correspondence $X\mapsto\hat{X}_{G}$ is a functor from the category of
$G$-flows with morphisms category-preserving $G$-maps to the category of MHP
flows.

###### Proof.

Let $\phi\colon X\to Y$ be a category-preserving homomorphism of $G$-flows.
The map $\phi$ preserves the ideal of meager sets, so we obtain a dual
homomorphism of Riesz spaces $\phi^{*}\colon B(Y)\to B(X)$ given by
$\phi^{*}([f])=[f\circ\phi]$, where $f\in\mathcal{B}(Y)$ and $[f]$ denotes its
equivalence class in $B(Y)$. The image of $B_{G}(Y)$ is contained in
$B_{G}(X)$, so by the duality theorem, this gives us a map
$\hat{X}_{G}\to\hat{Y}_{G}$. ∎

##  3\. Characterizations of MHP flows

Starting from this section, $G$ will denote a locally compact group. A
_pseudo-norm_ on $G$ is a continuous function $\lVert\cdot\rVert\colon
G\to{\mathbf{R}}_{+}$ satisfying:

  * •

$\lVert 1_{G}\rVert=0$;

  * •

$\lVert gh\rVert\leq\lVert g\rVert+\lVert h\rVert$ for all $g,h\in G$.

$\lVert\cdot\rVert$ is a _norm_ if it moreover satisfies that the only element
$g$ with $\lVert g\rVert=0$ is $1_{G}$. A norm is called _compatible_ if it
induces the topology of $G$.

Every pseudo-norm induces a right-invariant pseudo-metric $d_{r}$ on $G$
defined by

(3.1) $d_{r}(g,h)=\lVert gh^{-1}\rVert.$

Let $\lVert\cdot\rVert$ be some fixed pseudo-norm on $G$. We denote by $B_{r}$
the set of elements $g\in G$ such that $\lVert g\rVert<r$. If
$G\curvearrowright X$ is a $G$-flow, we can define a pseudo-metric $\partial$
on $X$ by

(3.2) $\partial(x,y)=\inf\\{\lVert g\rVert:g\in G,g\cdot x=y\\}.$

If $x$ and $y$ are not in the same orbit, then $\partial(x,y)=\infty$.

Note that $\partial$ is always lower semi-continuous for the compact topology
$\tau$ on $X$. Recall that a real-valued function $f$ is called _lower semi-
continuous (lsc)_ if for every real number $r$ the set $\left\\{f>r\right\\}$
is open. It is _upper semi-continuous (usc)_ if $\left\\{f<r\right\\}$ is
open.

When $G$ is metrizable, we can work throughout with a fixed compatible norm on
$G$, and $\partial$ is a metric on $X$ that refines the topology $\tau$, i.e.,
$(X,\tau,\partial)$ is a _compact topometric space_ in the sense of
[BenYaacov2013b]. In general, one can work with a _topouniform spaces_ as is
done in [Basso2021a], but we will not need this here. In the case where $G$ is
Polish, locally compact, the topometric space above is the same as the one
considered by Zucker [Zucker2021].

The following characterization of MHP flows is the main theorem of this
section.

######  Theorem 3.1.

Let $G$ be locally compact and let $G\curvearrowright X$ be a $G$-flow. Then
the following are equivalent:

  1. (i)

$X$ is MHP;

  2. (ii)

$V\overline{U}$ is open for every open neighborhood $V$ of $1_{G}$ and open
subset $U$ of $X$.

  3. (iii)

for every continuous pseudo-norm on $G$, and every open subset $U$ of $X$, the
function $X\to{\mathbf{R}}\cup\left\\{\infty\right\\}$,
$x\mapsto\partial(x,\overline{U})$, is continuous.

When $G$ is a tdlc group, these are also equivalent to:

  1. (iv)

$\operatorname{RO}(G,X)=\operatorname{Clopen}(X)$;

  2. (v)

$X$ is zero-dimensional and for every compact open subgroup $V$ of $G$, the
Boolean algebra $\operatorname{Clopen}_{V}(X)$ is complete.

Before going further, we make a few comments. The equivalence between (i) and
(ii) is already contained in [Zucker2021] (up to the observation that when $G$
is locally compact, Definition 3.1 from [Zucker2021] can be restated as in
(ii)). Here we provide an alternative proof of that equivalence. The proof of
(ii) $\Rightarrow$ (i) follows arguments close to [Gleason1958], while the
proof of the converse (which goes through (iii)) uses the characterization of
MHP flows given in Proposition 2.5.

We need some preliminaries.

######  Lemma 3.2.

Let $G\curvearrowright X$ be a $G$-flow and let $\partial$ be defined as
above. Then the following hold:

  1. (i)

If $F\subseteq X$ is closed, the function $x\mapsto\partial(x,F)$ is lsc.

  2. (ii)

If $U\subseteq X$ is open, the function $x\mapsto\partial(x,U)$ is usc.

###### Proof.

(i) Let $A=\\{x:\partial(x,F)\leq r\\}$. Let $x$ be a limit point of $A$ and
let $(x_{i},\epsilon_{i})_{i}$ be a net in $A\times{\mathbf{R}}^{+}$
converging to $(x,0)$. Let $y_{i}\in F$ be such that
$\partial(x_{i},y_{i})<r+\epsilon_{i}$. By passing to a subnet, we may assume
that $y_{i}\to y\in F$. Then taking limits and using the fact that $\partial$
is lsc, we obtain that $\partial(x,y)\leq r$.

(ii) Let $r>0$, and let $V$ be the open ball around $1_{G}$ of radius $r$.
Then

$\partial(x,U)<r\iff V\cdot x\cap U\neq\emptyset,$

which is an open condition. ∎

######  Remark 3.3.

In fact, Lemma 3.2 does not need $G$ to be locally compact (with the
appropriate definition of $\partial$ in the general case, see [Zucker2021]).
The proof of (i) works as above and (ii) is [Zucker2021]*Theorem 4.8 and it is
harder.

######  Lemma 3.4.

Let $X$ be a $G$-flow. Then $X$ satisfies condition (ii) of Theorem 3.1 if and
only if for every open neighborhood $V\ni 1_{G}$ and open subset $U\subseteq
X$, there exists an open neighborhood $V^{\prime}\ni 1_{G}$ with
$V^{\prime}\subseteq V$ such that $V^{\prime}\overline{U}$ is open.

###### Proof.

We only have to prove the implication from right to left. Suppose that the
property in the statement holds, and let $U$ be an open subset of $X$ and $V$
an open neighborhood of $1_{G}$. For every $g$ in $V$ one can find
$V^{\prime}_{g}$ an open neighborhood of $1_{G}$ such that $gV^{\prime}_{g}$
is contained in $V$ and $V^{\prime}_{g}\overline{U}$ is open. Writing
$V=\bigcup_{g\in V}V^{\prime}_{g}$, we then have $V\overline{U}=\bigcup_{g\in
V}V^{\prime}_{g}\overline{U}$, which is this thus open. ∎

######  Lemma 3.5.

Let $X$ be a $G$-flow that satisfies condition (ii) of Theorem 3.1. Then for
all open subsets $U_{1},U_{2}\subseteq X$, we have
$\overline{U_{1}}\cap\overline{U_{2}}\neq\emptyset$ if and only if $VU_{1}\cap
U_{2}\neq\emptyset$ for every open neighborhood $V\ni 1_{G}$.

###### Proof.

Suppose $\overline{U_{1}}\cap\overline{U_{2}}\neq\emptyset$, and let $V$ be an
open neighborhood of $1_{G}$. Then clearly
$V\overline{U_{1}}\cap\overline{U_{2}}\neq\emptyset$. Since
$V\overline{U_{1}}$ is open, this implies that $V\overline{U_{1}}\cap
U_{2}\neq\emptyset$. That condition is equivalent to $\overline{U_{1}}\cap
V^{-1}U_{2}\neq\emptyset$ and hence implies that $U_{1}\cap
V^{-1}U_{2}\neq\emptyset$. So $VU_{1}\cap U_{2}\neq\emptyset$, as desired. The
reverse implication is a general fact that follows from continuity of the
$G$-action. ∎

Recall that a subalgebra $A$ of a Boolean algebra $B$ is _dense_ if for every
non-zero element in $B$ there is a non-zero element in $A$ that is smaller. We
recall the following (see [Koppelberg1989]*Theorem 4.19).

######  Lemma 3.6.

Let $A$ be a dense subalgebra of a Boolean algebra $B$. If $A$ is complete,
then $A=B$.

Before starting the proof, we also introduce some notation. If $\pi\colon Y\to
X$ is a continuous map between topological spaces and $U\subseteq Y$ is open,
we denote by $\pi_{*}(U)$ the _fiber image_ of $U$:

$\pi_{*}(U)\coloneqq\\{x\in X:\pi^{-1}(x)\subseteq U\\}.$

The set $\pi_{*}(U)$ is always open and if $\pi$ is irreducible, it is non-
empty for any non-empty $U$.

###### Proof of Theorem 3.1.

(ii) $\Rightarrow$ (i). Let $\pi\colon Y\to X$ be a highly proximal extension.
We shall prove that $\pi$ is injective. Suppose for a contradiction that there
exist distinct points $y_{1},y_{2}$ in $Y$ with the same image $x$ in $X$.
Then one can find an open $V\ni 1_{G}$ and open subsets $O_{1},O_{2}\subseteq
Y$ such that $y_{1}\in O_{1}$, $y_{2}\in O_{2}$ and $VO_{1}\cap
O_{2}=\emptyset$. The irreducibility of $\pi$ implies that
$\pi(O)\subseteq\overline{\pi_{*}(O)}$ for any open $O\subseteq Y$. Indeed, if
not, there is $y\in O$ and an open $W\ni\pi(y)$ disjoint from $\pi_{*}(O)$. By
irreducibility, $\pi^{-1}(W)\cap O$ contains a fiber, whose image must be in
$\pi_{*}(O)$, contradiction. Thus the sets $\overline{\pi_{*}(O_{1})}$ and
$\overline{\pi_{*}(O_{2})}$ both contain $x$. Hence by the assumption (ii) and
Lemma 3.5, we have $V\pi_{*}(O_{1})\cap\pi_{*}(O_{2})\neq\emptyset$. Since
$V\pi_{*}(O_{1})=\pi_{*}(VO_{1})$, we deduce that $VO_{1}$ and $O_{2}$
intersect each other, which is a contradiction.

(iii) $\Rightarrow$ (ii). Let $V$ be an open neighborhood of $1_{G}$. We can
always find a continuous pseudo-norm on $G$ such that $B_{1/2}$ is contained
in $V$ [Hewitt1979, Theorem 8.2]. If $\partial$ is the pseudo-metric on $X$
associated to this pseudo-norm, by assumption, the function
$f(x)\coloneqq\partial(x,\overline{U})$ is continuous. So
$B_{1/2}\overline{U}=\left\\{f<1/2\right\\}$ is open. Since $B_{1/2}\subseteq
V$ and $V$ was arbitrary, Lemma 3.4 ensures that (ii) holds.

(i) $\Rightarrow$ (iii). Let $\phi_{0}(x)=\partial(x,\overline{U})$ and
$\phi_{1}(x)=\partial(x,U)$. We have that $\phi_{0}$ is lsc and $\phi_{1}$ is
usc by Lemma 3.2. Moreover $\phi_{0}\leq\phi_{1}$, and both $\phi_{0}$ and
$\phi_{1}$ are $\partial$-contractive. First we show that the set
$\\{\phi_{0}<\phi_{1}\\}$ is meager. Note that

$\\{\phi_{0}<\phi_{1}\\}=\bigcup_{q_{1}<q_{2}\in{\mathbf{Q}}}\\{\phi_{0}\leq
q_{1}<q_{2}\leq\phi_{1}\\}$

and each set in the union is closed. So if $\\{\phi_{0}<\phi_{1}\\}$ is non-
meager, there exist $q_{1}<q_{2}$ such that $\\{\phi_{0}\leq
q_{1}<q_{2}\leq\phi_{1}\\}$ has non-empty interior $W$. The set
$\\{x:\partial(x,W)<q_{2}\\}$ is open and intersects $\overline{U}$, so it
must intersect $U$. So there exist $x\in U$, $y\in W$ with
$\partial(x,y)<q_{2}$, which contradicts the definition of $W$.

Now for $r>0$, set $\phi_{0,r}=\min(\phi_{0},r)$ and
$\phi_{1,r}=\min(\phi_{1},r)$. The functions $\phi_{0,r},\phi_{1,r}$ are
bounded and remain $\partial$-contractive (hence $G$-continuous). As $X$ is
MHP, by Proposition 2.5, there exists a continuous function $\theta$ on $X$
such that $\phi_{0,r}=\phi_{1,r}=\theta$ on a comeager set. As the sets
$\\{\theta<\phi_{0,r}\\}$ and $\\{\theta>\phi_{1,r}\\}$ are open, they must be
empty, and we must have that $\phi_{0,r}\leq\theta\leq\phi_{1,r}$. We claim
that $\theta$ is $\partial$-contractive. If not, there exist $x\in X$ and
$g\in G$ such that $|\theta(x)-\theta(g\cdot x)|>\lVert g\rVert$. However, the
set $\\{x:|\theta(x)-\theta(g\cdot x)|>\lVert g\rVert\\}$ is open, so as
$\theta=\phi_{0,r}$ on a comeager set, there exists $x$ such that
$|\phi_{0,r}(x)-\phi_{0,r}(g\cdot x)|>\lVert g\rVert$, contradiction. Note
that $\theta^{-1}(0)\supseteq U$, so by continuity,
$\theta^{-1}(0)\supseteq\overline{U}$. As $\theta$ is $\partial$-contractive
and $\theta=0$ on $\overline{U}$, for every $x\in X$, we have

$\theta(x)\leq\inf_{y\in\overline{U}}\partial(x,y)=\partial(x,\overline{U})=\phi_{0}(x).$

Since $\theta\leq r$, this shows that $\theta\leq\phi_{0,r}$, and hence
$\theta=\phi_{0,r}$. So $\phi_{0,r}$ is continuous. Since $r$ is arbitrary, it
follows that $\phi_{0}$ is continuous.

We now assume $G$ is a tdlc group. The equivalence between (i) and (iv)
follows from Corollary 2.3. Recall in particular that these imply that $X$ is
zero-dimensional. Hence the fact that (iv) implies (v) is clear since
$\operatorname{RO}_{V}(X)$ is always complete. It remains to see that (v)
implies (iv). To that end, let $V$ be a compact open subgroup of $G$. We want
to see that $\operatorname{RO}_{V}(X)=\operatorname{Clopen}_{V}(X)$. We claim
that $\operatorname{Clopen}_{V}(X)$ is a dense subalgebra of
$\operatorname{RO}_{V}(X)$. Indeed, if $U$ is a non-empty element of
$\operatorname{RO}_{V}(X)$, then we can find a non-empty clopen subset $U_{1}$
inside $U$ since $X$ is zero-dimensional. Since $V$ is compact and open, the
stabilizer of $U_{1}$ has finite index in $V$, so that $VU_{1}$ is a union of
finitely many clopen subsets, and hence is clopen. Moreover $VU_{1}\subseteq
U$ since $U$ is $V$-invariant. Hence $\operatorname{Clopen}_{V}(X)$ is dense
in $\operatorname{RO}_{V}(G,X)$. Since we make the assumption that
$\operatorname{Clopen}_{V}(X)$ is complete, Lemma 3.6 implies that
$\operatorname{RO}_{V}(X)=\operatorname{Clopen}_{V}(X)$, as desired. ∎

Compare the next corollary with [BenYaacov2017]*Lemma 2.4.

######  Corollary 3.7.

Let $G\curvearrowright X$ be an MHP flow. Then for $U_{1},U_{2}\subseteq X$
open,

$\partial(\overline{U_{1}},\overline{U_{2}})=\partial(U_{1},U_{2}).$

###### Proof.

Suppose that $\partial(\overline{U_{1}},\overline{U_{2}})<r$. Consider the set
$\\{x:\partial(x,\overline{U_{2}})<r\\}$. By Theorem 3.1, it is open and it
intersects $\overline{U_{1}}$, so it intersects $U_{1}$. Let
$W\subseteq\\{x:\partial(x,\overline{U_{2}})<r\\}$ be open, non-empty with
$\overline{W}\subseteq U_{1}$. Then by continuity of the function
$\partial(\cdot,\overline{W})$, there exists $x\in U_{2}$ with
$\partial(x,\overline{W})<r$. So $\partial(U_{1},U_{2})<r$. ∎

##  4\. Continuity of the stabilizer map

Let $Y$ be locally compact space and let $2^{Y}$ denote the space of closed
subsets of $Y$. The _Chabauty topology_ on $2^{Y}$ is given by the subbasis of
sets of the form

$O_{K}=\\{F\in 2^{Y}:F\cap K=\emptyset\\}\quad\text{ and }\quad O^{U}=\\{F\in
2^{Y}:F\cap U\neq\emptyset\\}$

with $K\subseteq Y$ compact and $U\subseteq Y$ open. The space $2^{Y}$
equipped with this topology is compact. A map $\phi\colon X\to 2^{Y}$ is
_upper semi-continuous_ if $\phi^{-1}(O_{K})$ is open for every compact subset
$K$ of $Y$ and it is _lower semi-continuous_ if $\phi^{-1}(O^{U})$ is open for
every open subset $U$ of $Y$.

If $G$ is a locally compact group, the set $\operatorname{Sub}(G)$ of closed
subgroups of $G$ is closed in $2^{G}$, and hence, a compact space. Moreover,
the conjugation action of $G$ on $\operatorname{Sub}(G)$ is continuous.

######  Definition 4.1.

Let $X$ be a $G$-flow. For $x\in X$, let $G_{x}$ denote the stabilizer of $x$.
The map $\operatorname{Stab}\colon X\to\operatorname{Sub}(G)$ defined by
$\operatorname{Stab}(x)=G_{x}$ is called the _stabilizer map_ associated to
the flow $X$.

It is easy to see that for every $G$-flow, the stabilizer map is
$G$-equivariant and upper semi-continuous (see, e.g., [Glasner2015]). It is
also well-known that in general it is not continuous. The main theorem of the
paper is the following.

######  Theorem 4.2.

Let $G$ be a locally compact group and let $X$ be an MHP $G$-flow. Then the
stabilizer map $X\to\mathrm{Sub}(G)$, $x\mapsto G_{x}$ is continuous.

The remainder of this section is devoted to the proof of the theorem. Let
$\lVert\cdot\rVert\colon G\to{\mathbf{R}}_{+}$ be a pseudo-norm on $G$. We
recall that $B_{r}$ denotes the set of elements $g\in G$ such that $\lVert
g\rVert<r$; we also let $\bar{B}_{r}$ be the set of elements $g\in G$ such
that $\lVert g\rVert\leq r$. We say that $\lVert\cdot\rVert$ is _proper_ if
$\bar{B}_{r}$ is compact for all $r$.

Let $X$ be a $G$-flow. Recall that to every pseudo-norm on $G$, we associated,
by the equation (3.2), a pseudo-metric $\partial$ on $X$. We will say that a
pseudo-norm $\lVert\cdot\rVert\colon G\to{\mathbf{R}}_{+}$ is _normal_ if for
every $g\in G$, the conjugation by $g$ is a uniformly continuous map of the
pseudo-metric space $(G,d_{r})$, where $d_{r}$ is the right-invariant metric
on $G$ associated to $\lVert\cdot\rVert$ given by (3.1). In particular, the
kernel $\\{g\in G:\lVert g\rVert=0\\}$ of a normal pseudo-norm is a normal
subgroup of $G$.

######  Proposition 4.3.

Let $G$ be a $\sigma$-compact locally compact group, and let $V$ be an open
neighborhood of $1_{G}$. Then there exists a continuous, proper, normal
pseudo-norm on $G$ and $r>0$ such that $B_{r}\subseteq V$.

###### Proof.

Choose and open neighborhood $W$ of $1_{G}$ such that $W^{2}\subseteq V$.
Since $G$ is $\sigma$-compact, theorems of Kakutani–Kodaira ([Hewitt1979,
Theorem 8.7]) and Struble [Struble1974] ensure that there exists a compact
normal subgroup $K$ of $G$ with $K\subseteq W$ such that $G/K$ admits a
compatible and proper norm $\lVert\cdot\rVert_{G/K}$. If we let $\lVert
g\rVert=\lVert gK\rVert_{G/K}$, then $\lVert\cdot\rVert$ is a pseudo-norm on
$G$ that is continuous and proper. Moreover since the image of $W$ in $G/K$ is
an open neighborhood of the identity in $G/K$ and $\lVert\cdot\rVert_{G/K}$ is
compatible, there is $r>0$ such that $\lVert gK\rVert_{G/K}<r$ implies $gK\in
WK$. Hence $\lVert g\rVert<r$ implies $g\in V$. Normality is clear since
$\lVert\cdot\rVert_{G/K}$ induces the topology on $G/K$. ∎

The following is the main lemma.

######  Lemma 4.4.

Let $G$ be a locally compact and let $\lVert\cdot\rVert$ be a continuous,
proper, normal pseudo-norm on $G$. Let $X$ be an MHP $G$-flow, let $g\in G$
and $r>0$. Then there exist $n\geq 1$ and a continuous function $\phi\colon
X\to{\mathbf{R}}^{n}$ such that for all $x\in X$

$\partial(g\cdot x,x)>r\implies\lVert\phi(g\cdot x)-\phi(x)\rVert_{\infty}\geq
r/3.$

###### Proof.

Since $\lVert\cdot\rVert$ is normal, $g$ and $g^{-1}$ are uniformly continuous
as self-maps of $(X,\partial)$. So let $\delta<r/3$ and $\delta^{\prime}$ be
such that

$\displaystyle\forall x,y\in X\quad\partial(x,y)<\delta\implies\partial(g\cdot
x,g\cdot y)<r/3$ $\displaystyle\forall x,y\in X\quad\partial(g\cdot x,g\cdot
y)<\delta^{\prime}\implies\partial(x,y)<\delta.$

Since $\lVert\cdot\rVert$ is continuous and proper, one can find
$g_{1},\ldots,g_{\ell}$ such that $\bar{B}_{2r}$ is contained in
$\bigcup_{i=1}^{\ell}g_{i}B_{\delta/2}$. By the pigeonhole principle, this
implies that a ball of radius $2r$ in $(X,\partial)$ cannot contain more than
$\ell$ points which are pairwise at least $\delta$ apart. That is, for every
$x,x_{1},\ldots,x_{\ell+1}\in X$ such that $\partial(x,x_{i})\leq 2r$ for all
$i$, there are $i\neq j$ such that $\partial(x_{i},x_{j})<\delta$. Similarly,
there is $k\in{\mathbf{N}}$ such that for all $x,x_{1},\ldots,x_{k+1}\in X$
such that $\partial(x,x_{i})\leq 2r$ for all $i$ there are $i\neq j$ such that
$\partial(x_{i},x_{j})<\delta^{\prime}$. Set $n=k+\ell+1$.

Set $M_{r}=\\{x:\partial(g\cdot x,x)>r\\}$. We will construct open sets
$U_{1},\ldots,U_{n}\subseteq X$ with the following properties:

  1. (i)

the closure of $\bigcup_{i}B_{\delta}U_{i}$ contains $M_{r}$;

  2. (ii)

$\partial(U_{i},U_{j})\geq\delta$ for $i\neq j$;

  3. (iii)

$\partial(g\cdot U_{i},U_{i})\geq r$ for all $i$.

Once the construction is completed, we finish the proof as follows. We set

$\phi_{i}(x)=\min(\partial(x,\overline{U_{i}}),r)$

and $\phi=(\phi_{i})_{i}$. By Theorem 3.1, $\phi$ is continuous. To see that
$\phi$ satisfies the conclusion, in view of (i) it is enough to see that
$||\phi(g\cdot x)-\phi(x)||_{\infty}\geq r/3$ for every $x$ in
$\bigcup_{i}B_{\delta}U_{i}$. So let $x\in B_{\delta}U_{i}$ and let $y\in
U_{i}$ be such that $\partial(x,y)<\delta$. Then $\partial(g\cdot x,g\cdot
y)<r/3$ and using Corollary 3.7, we obtain

$\begin{split}\partial(g\cdot x,\overline{U_{i}})&\geq\partial(g\cdot
y,\overline{U_{i}})-\partial(g\cdot y,g\cdot x)\\\
&\geq\partial(g\cdot\overline{U_{i}},\overline{U_{i}})-r/3\\\
&=\partial(g\cdot U_{i},U_{i})-r/3\geq 2r/3.\end{split}$

So

$||\phi(g\cdot x)-\phi(x)||_{\infty}\geq\phi_{i}(g\cdot x)-\phi_{i}(x)\geq
2r/3-\delta\geq r/3$

and we are done.

Now we proceed with the construction. Using Zorn’s lemma, we find a maximal
(under inclusion) tuple of open sets $(U_{i})$ satisfying (ii) and (iii)
above. We will show that it must also satisfy (i). If not, there exists
$x_{0}\in M_{r}$ and an open neighborhood $W_{0}$ of $x_{0}$ such that
$\partial(W_{0},U_{i})\geq\delta$ for all $i$. By lower semi-continuity of
$\partial$, there is an open neighborhood $W_{1}$ of $x_{0}$ such that
$\partial(W_{1},g\cdot W_{1})\geq r$. Suppose that there exists $j\leq n$ such
that

  * •

$\partial(g\cdot x_{0},U_{j})>r$;

  * •

$\partial(x_{0},g\cdot U_{j})>r$.

Since both conditions are open, there exists an open neighborhood $W_{2}$ of
$x_{0}$ such that $\partial(W_{2},g\cdot U_{j})\geq r$, and $\partial(g\cdot
W_{2},U_{j})\geq r$. This implies that if we set $W=W_{0}\cap W_{1}\cap
W_{2}$, we can add $W$ to $U_{j}$ without violating (ii) or (iii), thus
contradicting the maximality of $(U_{i})$. So our final task in order to
obtain a contradiction is to find $j$ satisfying the two conditions above.
First, note that

$|\\{i:\partial(g\cdot x_{0},U_{i})\leq r\\}|\leq\ell.$

Indeed, suppose to the contrary that there exist $y_{i_{0}}\in
U_{i_{0}},\ldots,y_{i_{\ell}}\in U_{i_{\ell}}$ with $\partial(y_{i_{s}},g\cdot
x_{0})<2r$ for all $s\leq\ell$. Then the $y_{i_{s}}$ are $\ell+1$ points in a
ball of radius $2r$ which are pairwise $\delta$ apart by (ii), which
contradicts the definition of $\ell$.

Similarly,

$|\\{i:\partial(x_{0},g\cdot U_{i})\leq r\\}|\leq k,$

because if there exist $y_{i_{0}}\in U_{i_{0}},\ldots,y_{i_{k}}\in U_{i_{k}}$
with $\partial(x_{0},g\cdot y_{i_{s}})<2r$ for all $s$, then by the choice of
$k$, there exist $s\neq t$ with $\partial(g\cdot y_{i_{s}},g\cdot
y_{i_{t}})<\delta^{\prime}$. Now the choice of $\delta^{\prime}$ implies that
$\partial(y_{i_{s}},y_{i_{t}})<\delta$, contradicting (ii). Now by the choice
of $n$, there exists $j$ as desired. ∎

###### Proof of Theorem 4.2.

It is a general fact that the stabilizer map is upper semi-continuous, so we
only have to prove lower semi-continuity. So for every open subset $O$ of $G$,
we have to prove that

$X_{G,O}\coloneqq\left\\{x\in X:G_{x}\cap O\neq\emptyset\right\\}$

is an open subset of $X$. Clearly it is enough to do this for every relatively
compact open subset $O$.

Let $L$ be the subgroup of $G$ generated by $O$. The subgroup $L$ is open, so
the $L$-flow $X$ is also MHP (by Theorem 3.1 (ii)). Moreover $L$ is compactly
generated, so in particular $L$ is $\sigma$-compact. Since $X_{G,O}=X_{L,O}$,
it follows that it is enough to prove the desired conclusion under the
assumption that the group is $\sigma$-compact. From now on, we make this
assumption.

We fix $x_{0}\in X_{G,O}$. Let $g\in O$ be such that $g\cdot x_{0}=x_{0}$ and
let $V\ni 1_{G}$ be open such that $Vg\subseteq O$. Now we find a neighborhood
$U_{0}$ of $x_{0}$ that is contained in $X_{G,Vg}\subseteq X_{G,O}$. Since $G$
is $\sigma$-compact, by Proposition 4.3, there are $r>0$ and a continuous,
proper, normal pseudo-norm $||\cdot||$ on $G$ such that $B_{2r}\subseteq V$.
If $\phi\colon X\to{\mathbf{R}}^{n}$ is a continuous function as given by
Lemma 4.4, then we have

$\displaystyle U_{0}:=\left\\{x\in X:||\phi(g\cdot
x)-\phi(x)||_{\infty}<r/3\right\\}$ $\displaystyle\subseteq\left\\{x\in
X:\partial(g\cdot x,x)\leq r\right\\}$ $\displaystyle\subseteq\left\\{x\in
X:x\in B_{2r}g\cdot x\right\\}$ $\displaystyle\subseteq X_{G,Vg}.$

So $U_{0}$ is an open neighborhood of $x_{0}$ that has the desired property. ∎

##  5\. Stabilizer flows

Throughout this section, let $G$ be a locally compact group. The continuity of
the stabilizer map allows us to associate to any MHP flow $X$ a subflow of
$\operatorname{Sub}(G)$, namely, the image of the stabilizer map. As every
flow has a unique universal highly proximal extension, this leads us to the
following definition.

######  Definition 5.1.

Let $G$ be locally compact and let $G\curvearrowright X$ be a $G$-flow. The
_stabilizer flow $\operatorname{S}_{G}(X)$_ of $X$ is the subflow of
$\operatorname{Sub}(G)$ given by

$\operatorname{S}_{G}(X)\coloneqq\operatorname{Stab}(\hat{X}_{G})=\\{G_{z}:z\in\hat{X}_{G}\\}.$

We have the following general facts about the stabilizer flow.

######  Proposition 5.2.

Let $G\curvearrowright X$ be a $G$-flow and let $\pi\colon\hat{X}_{G}\to X$ be
the universal highly proximal extension of $X$. Then the following hold:

  1. (i)

For any compact $K\subseteq G$, the set

$D_{K}\coloneqq\\{z\in\hat{X}_{G}:z\notin K\cdot z\text{ and }\pi(z)\in
K\cdot\pi(z)\\}$

is nowhere dense in $\hat{X}_{G}$.

  2. (ii)

For any dense subset $X^{\prime}\subseteq X$, we have that
$\operatorname{S}_{G}(X)\subseteq\overline{\operatorname{Stab}(X^{\prime})}$.

  3. (iii)

If $x\in X$ is a point of continuity of $\operatorname{Stab}$, then
$G_{x}\in\operatorname{S}_{G}(X)$.

  4. (iv)

If the set $X_{0}\subseteq X$ of continuity points of $\operatorname{Stab}$ is
dense in $X$, then
$\operatorname{S}_{G}(X)=\overline{\operatorname{Stab}(X_{0})}$.

###### Proof.

(i) Let $U\subseteq\hat{X}_{G}$ be non-empty, open. We will find a non-empty,
open subset of $U$ disjoint from $D_{K}$. Let $z_{0}\in U\cap D_{K}$ (if there
is no such $z_{0}$, we are done). The set
$\\{(z,z^{\prime})\in\hat{X}_{G}^{2}:z\notin K\cdot z^{\prime}\\}$ is open and
$z_{0}$ belongs to it, so there exists a neighborhood $U^{\prime}$ of $z_{0}$,
$U^{\prime}\subseteq U$ such that $K\cdot U^{\prime}\cap
U^{\prime}=\emptyset$. By irreducibility of $\pi$, the set
$\pi_{*}(U^{\prime})$ is non-empty and for any $x\in\pi_{*}(U^{\prime})$, we
have that $x\notin K\cdot x$. Thus the open set
$\pi^{-1}(\pi_{*}(U^{\prime}))\subseteq U^{\prime}$ is disjoint from $D_{K}$.

(ii) Let $z_{0}\in\hat{X}_{G}$ and let

$\mathcal{U}=\\{H\in\operatorname{Sub}(G):H\cap
O_{1}\neq\emptyset,\ldots,H\cap O_{n}\neq\emptyset,H\cap K=\emptyset\\},$

where $O_{1},\ldots,O_{n}\subseteq G$ are open and $K\subseteq G$ is compact,
be a neighborhood of $G_{z_{0}}$ in $\operatorname{Sub}(G)$. Our goal is to
find $x\in X^{\prime}$ with $G_{x}\in\mathcal{U}$. Let $U=\\{z\in
Z:G_{z}\in\mathcal{U}\\}$ and note that by continuity of the stabilizer map,
$U$ is open. By (i), the open set $U\setminus\overline{D_{K}}$ is non-empty.
We claim that any $x\in\pi_{*}(U\setminus\overline{D_{K}})\cap X^{\prime}$
works. Indeed, fix such an $x$ and let $z\in\hat{X}_{G}$ be such that
$\pi(z)=x$. As $z\notin D_{K}$, we have that $G_{x}\cap K=\emptyset$ and as
$G_{z}\leq G_{x}$, we also have that $G_{x}\cap O_{i}\neq\emptyset$ for all
$i$, so $G_{x}\in\mathcal{U}$.

(iii) Let $x$ be a point of continuity of $\operatorname{Stab}$ and let

$\mathcal{U}\coloneqq\\{H\in\operatorname{Sub}(G):H\cap
O_{1}\neq\emptyset,\ldots,H\cap O_{n}\neq\emptyset,H\cap K=\emptyset\\}$

be a neighborhood of $G_{x}$, where each $O_{i}\subseteq G$ is open and
$K\subseteq G$ is compact. Let $O_{i}^{\prime}\subseteq G$ be open, relatively
compact with $\overline{O_{i}^{\prime}}\subseteq O_{i}$ such that
$G_{x}\in\mathcal{U}^{\prime}$, where

$\mathcal{U}^{\prime}\coloneqq\\{H\in\operatorname{Sub}(G):H\cap
O_{1}^{\prime}\neq\emptyset,\ldots,H\cap O_{n}^{\prime}\neq\emptyset,H\cap
K=\emptyset\\}.$

By the continuity of $\operatorname{Stab}$ at $x$, there is an open $W\ni x$
with $\operatorname{Stab}(W)\subseteq\mathcal{U}^{\prime}$. By (i), the set
$\bigcup_{i}D_{\overline{O^{\prime}_{i}}}$ is nowhere dense, so there exists
$z\in\pi^{-1}(W)\setminus\bigcup_{i}D_{\overline{O^{\prime}_{i}}}$. Then
$z\in\overline{O_{i}^{\prime}}\cdot z\subseteq O_{i}\cdot z$ for every $i$ and
$G_{z}\cap K=\emptyset$ (because $\pi(z)\in W$ and $G_{z}\leq G_{\pi(z)}$).
Thus $G_{z}\in\mathcal{U}$. As $\mathcal{U}$ was arbitrary and
$\operatorname{S}_{G}(X)$ is closed, this implies that
$G_{x}\in\operatorname{S}_{G}(X)$.

(iv) follows from (ii) and (iii). ∎

The following is well-known and follows from [Kuratowski1928, Theorem VII]. We
include a short proof for completeness.

######  Lemma 5.3.

Let $X$ be a compact space, $Y$ a locally compact space, and let
$\varphi\colon X\to 2^{Y}$ be upper semi-continuous. Let $(U_{i})_{i\in I}$ be
a basis for the topology on $Y$ such that each $U_{i}$ is relatively compact.
For $i\in I$, we let

$X_{i}=\left\\{x\in X:\phi(x)\cap\overline{U_{i}}\neq\emptyset\right\\}.$

Then $\phi$ is continuous at each point of the set
$\bigcap_{i}(X\setminus\partial X_{i})$.

In particular, if $Y$ is second countable, then the set of continuity points
of $\phi$ is comeager.

###### Proof.

Let $x\in\bigcap_{i}(X\setminus\partial X_{i})$, and let $(x_{a})$ be a net in
$X$ converging to $x$ and such that $(\phi(x_{a}))$ converges to $F$. By upper
semi-continuity, we know that $F\subseteq\phi(x)$, and we want to prove
equality. Let $i$ such that $\phi(x)\cap\overline{U_{i}}\neq\emptyset$, i.e.,
$x\in X_{i}$. Since $x$ is in $X\setminus\partial X_{i}$ by assumption, $x$
must be in the interior of $X_{i}$. Since $(x_{a})$ converges to $x$,
eventually $x_{a}\in X_{i}$, that is,
$\phi(x_{a})\cap\overline{U_{i}}\neq\emptyset$. Since $\overline{U_{i}}$ is
compact, this implies $F\cap\overline{U_{i}}\neq\emptyset$. So whenever
$\phi(x)$ intersects $\overline{U_{i}}$, so does $F$. Since $(U_{i})_{i\in I}$
is a basis for the topology on $Y$, this shows that $\phi(x)\subseteq F$, as
desired.

Note that $X_{i}$ is always closed by upper semi-continuity, so
$X\setminus\partial X_{i}$ is a dense open subset. In case $Y$ is second
countable, $(U_{i})_{i\in I}$ can be chosen to be countable, and hence the
domain of continuity of $\phi$ is comeager. ∎

######  Corollary 5.4.

Let $G$ be second countable and let $G\curvearrowright X$ be a $G$-flow. Then
the set $X_{0}\subseteq X$ of continuity points of $\operatorname{Stab}$ is
dense $G_{\delta}$ in $X$ and we have

$\operatorname{S}_{G}(X)=\overline{\operatorname{Stab}(X_{0})}.$

###### Proof.

The first claim follows from the upper semi-continuity of
$\operatorname{Stab}$ and Lemma 5.3, and the second claim follows from (iv) of
Proposition 5.2. ∎

######  Remark 5.5.

When $G$ is not second countable, it is no longer true that there exists $x\in
X$ such that $G_{x}\in\operatorname{S}_{G}(X)$. Indeed, consider the group
$G=\mathrm{SO}(3,{\mathbf{R}})$, equipped with the discrete topology, acting
on the $2$-dimensional sphere $X=\mathbf{S}^{2}$. Then $G_{x}\neq\\{1_{G}\\}$
for all $x\in X$. On the other hand, every non-identity element has only two
fixed points in $X$, so the action is topologically free, which means that
$\operatorname{S}_{G}(X)=\big{\\{}\\{1_{G}\\}\big{\\}}$ (see Corollary 5.7).
Here the set of continuity points of $\operatorname{Stab}$ is empty.

In the case where $X$ is minimal, stabilizer flows have already been
considered in the literature under the name of stabilizer URSs. Recall that a
_uniformly recurrent subgroup (URS)_ of $G$ is a minimal subflow of
$\operatorname{Sub}(G)$. Glasner and Weiss [Glasner2015] associated to every
minimal $G$-flow its _stabilizer URS_ as follows. Upper semi-continuity of the
stabilizer map implies that $\overline{\operatorname{Stab}(X)}$ has a unique
minimal subflow (see [Auslander1977, Lemma 1.1] or [Glasner2015, Proposition
1.2]). Then the _stabilizer URS of $X$_ is simply defined to be this minimal
subflow. Proposition 5.2 implies that for minimal flows, our definition and
theirs coincide.

######  Corollary 5.6.

Let $X$ be a minimal $G$-flow. Then its stabilizer URS is equal to
$\operatorname{S}_{G}(X)$.

###### Proof.

Proposition 5.2 (ii) tells us that
$\operatorname{S}_{G}(X)\subseteq\overline{\operatorname{Stab}(X)}$. As $X$ is
minimal, $\hat{X}_{G}$ is also minimal and so is its factor
$\operatorname{S}_{G}(X)$. Now the conclusion follows from the fact that
$\overline{\operatorname{Stab}(X)}$ has a unique minimal subflow. ∎

Corollary 5.4 was also known for minimal $X$: see [Glasner2015]*Proposition
1.2.

Recall that a flow $G\curvearrowright X$ is called _topologically free_ if for
every compact $K\subseteq G$ that does not contain $1_{G}$, the closed set
$\\{x\in X:x\in K\cdot x\\}$ has empty interior. A point $x\in X$ is called
_free_ if the orbit map $G\to G\cdot x$, $g\mapsto g\cdot x$ is injective. A
flow is called _free_ if all points are free. It is clear that a flow for
which the free points are dense is topologically free, and a simple Baire
category argument shows that the converse is also true if $G$ is second
countable.

######  Corollary 5.7.

Let $G\curvearrowright X$ be a $G$-flow. Then the following are equivalent:

  1. (i)

$X$ is topologically free;

  2. (ii)

$\hat{X}_{G}$ is free;

  3. (iii)

$\operatorname{S}_{G}(X)=\big{\\{}\\{1_{G}\\}\big{\\}}$.

In particular, topologically free MHP flows are free.

###### Proof.

The equivalence of (ii) and (iii) follows from the definition of
$\operatorname{S}_{G}(X)$.

(i) $\Rightarrow$ (ii) Let $g\in G$, $g\neq 1_{G}$. Let $V\subseteq G$ be an
open, relatively compact subset with $g\in V$ and $1_{G}\notin\overline{V}$.
Then the set $\\{x:x\in V\cdot x\\}$ is open by Theorem 4.2 and has empty
interior by topological freeness, so it must be empty. So we conclude that
$g\cdot x\neq x$ for all $x$.

(ii) $\Rightarrow$ (i) Suppose, towards a contradiction, that there is a
compact $K\subseteq G$ with $1_{G}\notin K$ such that the set $\\{x\in X:x\in
K\cdot x\\}$ has non-empty interior $W$. By Proposition 5.2 (i), the set
$\pi^{-1}(W)\setminus D_{K}$ is non-empty and for any $z$ in this set, we have
that $z\in K\cdot z$, contradicting the freeness of $\hat{X}_{G}$. ∎

From this, it is not hard to deduce a well-known theorem of Veech.

######  Corollary 5.8 (Veech).

Every locally compact group admits a free flow.

###### Proof.

Let $G$ be a locally compact group and let $\operatorname{Sa}(G)$ denote its
_Samuel compactification_ , i.e., the spectrum of the Riesz space of right
uniformly continuous bounded functions on $G$. Then
$G\curvearrowright\operatorname{Sa}(G)$ is a $G$-flow and $G$ embeds densely
in $\operatorname{Sa}(G)$ as point evaluations. Also, the flow
$\operatorname{Sa}(G)$ is MHP by [Zucker2021]*3.2.1 (alternatively, it is not
difficult to verify condition (iii) of Theorem 3.1). As the left translation
action $G\curvearrowright G$ is free, Corollary 5.7 tells us that the flow
$\operatorname{Sa}(G)$ is also free. ∎

## References