# Topological dynamics of kaleidoscopic groups

Gianluca Basso  Institut Camille Jordan
Université Claude Bernard Lyon 1
Université de Lyon
43, boulevard du 11 novembre 1918
69622 Villeurbanne cedex
France<EMAIL_ADDRESS>and  Todor Tsankov  Institut Camille Jordan
Université Claude Bernard Lyon 1
Université de Lyon
43, boulevard du 11 novembre 1918
69622 Villeurbanne cedex
France<EMAIL_ADDRESS>
###### Abstract.

Kaleidoscopic groups are a class of permutation groups recently introduced by
Duchesne, Monod, and Wesolek. Starting with a permutation group $\Gamma$, the
kaleidoscopic construction produces another permutation group
$\mathcal{K}(\Gamma)$ which acts on a Ważewski dendrite (a densely branching
tree-like compact space). In this paper, we study how the topological dynamics
of $\mathcal{K}(\Gamma)$ can be expressed in terms of the one of $\Gamma$,
when the group $\Gamma$ is transitive. By proving a Ramsey theorem for
decorated rooted trees, we show that the universal minimal flow (UMF) of
$\mathcal{K}(\Gamma)$ is metrizable iff $\Gamma$ is oligomorphic and the UMF
of $\Gamma$ is metrizable. More generally, we give concrete calculations, in
an appropriate model-theoretic framework, of the UMF of $\mathcal{K}(\Gamma)$
when the UMF of a point stabilizer $\Gamma_{c}$ has a comeager orbit. Our
results also give a large class of examples of non-metrizable UMFs with a
comeager orbit. These results extend previous work of Kwiatkowska and Duchesne
about the full homeomorphism groups.

###### Key words and phrases:

kaleidoscopic groups, topological dynamics, universal minimal flow, comeager
orbit

###### 2020 Mathematics Subject Classification:

37B05, 05D10, 22F50

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries on topological dynamics
  3. 3 The model theory of permutation groups
  4. 4 Kaleidoscopic and root-kaleidoscopic groups
  5. 5 Structures for kaleidoscopic groups and endpoint stabilizers
  6. 6 A Ramsey interlude
  7. 7 The universal highly proximal extension of the action on $W$
  8. 8 Universal minimal flows of kaleidoscopic groups
  9. 9 The case of the full homeomorphism group

##  1\. Introduction

Dendrites are one-dimensional, compact topological spaces that appear
naturally in dynamics (for example, as Julia sets of quadratic polynomials).
They are tree-like, but can have dense branching. This is the case for the
generic spaces in this class, the _Ważewski dendrites_ $W_{n}$ (where
$n=3,4,\ldots,\infty$ is the order of the branch points), which can also be
viewed as the Fraïssé limits of finite trees and whose rich homeomorphism
groups have been studied in topology, in dynamics, and in model theory
[Charatonik1980, Kwiatkowska2018, Duchesne2019, Duchesne2020]. Even though the
spaces $W_{n}$ are connected, their homeomorphism groups are _non-archimedean_
(i.e., admit a basis at the identity consisting of open subgroups) and are
best studied as permutation groups of the countable set of branch points of
the dendrite.

_Kaleidoscopic groups_ are subgroups of $\operatorname{Homeo}(W_{n})$ that
were recently introduced by Duchesne, Monod, and Wesolek in [Duchesne2019a].
For us a _permutation group $\Gamma\curvearrowright M$_ is simply a _closed_
subgroup of the full symmetric group $\operatorname{Sym}(M)$, endowed with the
topology of pointwise convergence. Starting with a permutation group
$\Gamma\curvearrowright M$, where $M$ is a countable set with $n\geq 3$
elements, the kaleidoscopic construction produces another group
$\mathcal{K}(\Gamma)\leq\operatorname{Homeo}(W_{n})$ whose _local action_ is
prescribed by $\Gamma$, that is, for any branch point $x\in W_{n}$, the action
of $\mathcal{K}(\Gamma)_{x}$ on the connected components of
$W_{n}\setminus\\{x\\}$ is isomorphic to the action $\Gamma\curvearrowright
M$. The groups $\mathcal{K}(\Gamma)$ are Polish, non-archimedean,
algebraically simple, and they have many other interesting properties, as
discussed in [Duchesne2019a]. They can also be represented as permutation
groups acting on the branch points of the dendrite. The construction was
inspired by a classical paper of Burger and Mozes [Burger2000] on groups
acting on trees but in contrast with [Burger2000], kaleidoscopic groups are
never locally compact because of the density of branch points.

In this paper, we are interested in the topological dynamics of kaleidoscopic
groups and, more precisely, in their continuous actions on compact spaces (or
_flows_). In particular, we study which dynamical properties are preserved
when passing from $\Gamma$ to $\mathcal{K}(\Gamma)$. We recall that a flow is
called _minimal_ if every orbit is dense, and that, for any topological group
$G$, there exists a _universal minimal flow (UMF)_ , denoted by
$\mathcal{M}(G)$, characterized by the property that it maps onto any other
minimal $G$-flow. $\mathcal{M}(G)$ captures a fair amount of the dynamical
information about $G$: the action on $\mathcal{M}(G)$ is free whenever $G$
admits a free flow, and $\mathcal{M}(G)$ is a singleton iff every $G$-flow has
a fixed point, for instance. For locally compact, non-compact groups $G$,
universal minimal flows are very large, non-metrizable objects that are
difficult to understand but, somewhat surprisingly, for large Polish groups,
the UMFs often admit a concrete description and carry interesting
combinatorial information. The landmark paper of Kechris, Pestov, and
Todorčević [Kechris2005] made explicit the connection between Ramsey theory
and UMFs of non-archimedean Polish groups and motivated a lot of the recent
work in structural Ramsey theory and in abstract topological dynamics. At the
heart of the theory is the correspondence between the Ramsey property for the
finite substructures a given ultrahomogeneous structure ${\mathbf{N}}$ and the
fact that the automorphism group $\operatorname{Aut}({\mathbf{N}})$ is
_extremely amenable_ , i.e., such that its UMF is a singleton.

In the recent investigations of minimal flows of Polish groups, two dividing
lines have emerged. The better established one is metrizability: if
$\mathcal{M}(G)$ is metrizable, it is always the completion of a homogeneous
space of the form $G/H$ for some closed, extremely amenable $H\leq G$ that is
unique up to conjugation [BenYaacov2017, Melleray2016]. In this case, one has
a good understanding of the topological dynamics of $G$: for example, the
minimal flows are all metrizable, they are concretely classifiable
[Melleray2016], and for any $G$-flow $X$, the collection of minimal points in
$X$ is closed [Zucker2018a]. Following [Basso2021a], we say that a Polish
group is _CAP_ if its UMF is metrizable. Notice that if $G$ is compact, then
$\mathcal{M}(G)\cong G$, so in particular it is metrizable. Another important
property is that if $G$ is CAP, then $\mathcal{M}(G)$, and therefore all
minimal $G$-flows, have a comeager orbit.

In fact, the existence of a comeager orbit in $\mathcal{M}(G)$ provides the
second interesting dividing line. It is a tameness condition, which still
implies a strong structure on the category of minimal $G$-flows, but is not
completely understood at the moment. Indeed, it was only recently that
Kwiatkowska [Kwiatkowska2018] found the first examples of groups $G$ such that
$\mathcal{M}(G)$ has a comeager orbit but is not metrizable, namely the
homeomorphism groups of _generalized Ważewski dendrites_. It is a consequence
of our results about UMFs of kaleidoscopic groups that many of them have
comeager orbits without being metrizable, suggesting that this phenomenon is
more prevalent than previously thought. Generalizing the metrizable case,
Zucker [Zucker2021] showed that if $\mathcal{M}(G)$ has a comeager orbit, then
it is of the form $\operatorname{Sa}(G/H)$, where $H\leq G$ is closed and
extremely amenable, and $\operatorname{Sa}$ denotes the _Samuel
compactification_. We refer to Section 2 for all relevant definitions and more
details.

The main results of this paper connect properties of the original permutation
group $\Gamma$ and its UMF with properties of the UMF of the kaleidoscopic
group $\mathcal{K}(\Gamma)$. Prior work about the full homeomorphism group
$G=\operatorname{Homeo}(W_{n})$ had been done by Kwiatkowska [Kwiatkowska2018]
(for all $n$) and Duchesne [Duchesne2020] (for $n=\infty$), who proved that
$\mathcal{M}(G)$ is metrizable and identified the appropriate subgroup $H\leq
G$ such that $\mathcal{M}(G)\cong\widehat{G/H}$. Duchesne also proposed (cf.
[Duchesne2020]*Theorem 1.12) a concrete description of
$\mathcal{M}(\operatorname{Homeo}(W_{\infty}))$ as a space of linear orders on
the branch points of the dendrite, which is unfortunately not correct. We
discuss this in detail in Section 9. In another paper [Duchesne2021p],
Duchesne proved that the group $\mathcal{K}(C_{3})$, where $C_{3}$ is the
cyclic group of order $3$ acting on itself by translations, also has a
metrizable UMF. All of these previous results can be deduced from our general
theorem below.

Our first result is a characterization of when
$\mathcal{M}(\mathcal{K}(\Gamma))$ is metrizable. Recall that a permutation
group $\Gamma\curvearrowright M$ is called _oligomorphic_ if the diagonal
action $\Gamma\curvearrowright M^{k}$ has finitely many orbits for all
$k\in\mathbb{N}$.

_Below and everywhere in the paper, when we consider a permutation group
$\Gamma\curvearrowright M$, the set $M$ is tacitly assumed to be countable
with $n\geq 3$ elements._ This is so that the corresponding Ważewski dendrite
and kaleidoscopic group $\mathcal{K}(\Gamma)$ are well-defined.

######  Theorem 1.1 (cf. Theorem 8.6).

Let $\Gamma\curvearrowright M$ be a transitive permutation group. Then the
following are equivalent:

  1. (i)

$\mathcal{M}(\Gamma)$ is metrizable and the action $\Gamma\curvearrowright M$
is oligomorphic;

  2. (ii)

$\mathcal{M}(\mathcal{K}(\Gamma))$ is metrizable.

Of course, item (i) above is trivially satisfied when $M$ is finite, so we
obtain the following.

######  Corollary 1.2.

Let $\Gamma\curvearrowright M$ be a transitive permutation group with $M$
finite. Then $\mathcal{M}(\mathcal{K}(\Gamma))$ is metrizable.

As the action $\mathcal{K}(\Gamma)\curvearrowright W_{n}$ is minimal, it is
clear that if we want to search for an extremely amenable subgroup of
$\mathcal{K}(\Gamma)$, we have to look at subgroups of the stabilizers of the
points in $W_{n}$. As the action $\mathcal{K}(\Gamma)\curvearrowright W_{n}$
has a comeager orbit, namely that of the _endpoints_ (for the definition of
the various types of points in $W_{n}$ and their properties, see Section 4),
it is natural to study the stabilizer $\mathcal{K}(\Gamma)_{\xi}$ for some
fixed endpoint $\xi\in W_{n}$.

To that end, if $\Delta\curvearrowright M$ is a permutation group that
stabilizes some distinguished point $c\in M$, we construct a _root-
kaleidoscopic group_ ${\mathcal{K}}^{*}(\Delta)$ that is a subgroup of the
stabilizer $\operatorname{Homeo}(W_{n})_{\xi}$. The root-kaleidoscopic
construction ${\mathcal{K}}^{*}(\Delta)$ has a similar behavior to
$\mathcal{K}(\Gamma)$: for any branch point $x\in W_{n}$, the local action of
${\mathcal{K}}^{*}(\Delta)_{x}$ on the connected components of
$W_{n}\setminus\\{x\\}$ is isomorphic to $\Delta\curvearrowright M$ _and the
isomorphism is such that the component containing the root $\xi$ is always
labeled by $c$_.

This construction has the crucial property that if $\Gamma\curvearrowright M$
is transitive, then ${\mathcal{K}}^{*}(\Gamma_{c})=\mathcal{K}(\Gamma)_{\xi}$.
An important insight of our work is that, therefore, in order to describe
$\mathcal{M}(\mathcal{K}(\Gamma))$, for transitive $\Gamma$, one needs to
understand the dynamics of the point stabilizers of the action
$\Gamma\curvearrowright M$ rather than that of $\Gamma$ itself.

Even though all results above are stated in the language of permutation
groups, our combinatorial work is done in the model-theoretic setting of
ultrahomogeneous structures (which we recall in Section 3). We associate
ultrahomogeneous structures ${\mathbf{N}}$ and
${\mathcal{K}}^{*}({\mathbf{N}})$ to $\Delta$ and ${\mathcal{K}}^{*}(\Delta)$,
respectively. The finite substructures of ${\mathcal{K}}^{*}({\mathbf{N}})$
are rooted trees decorated by substructures of ${\mathbf{N}}$.

We then prove a transfer Ramsey theorem (Theorem 6.8), which states that trees
decorated with Ramsey structures have the Ramsey property. This generalizes a
result of Sokić [Sokic2015], in which the decorations are linear orders. The
proof goes by induction on the height of such trees, and uses a group-
theoretic argument about permutational wreath products for the induction step
(see 6.2). Theorem 6.8 translates to the following statement about root-
kaleidoscopic groups.

######  Theorem 1.3 (cf. Corollary 6.9).

Let $\Delta\curvearrowright M$ be a permutation group, let $c\in M$, and
suppose that $\Delta$ stabilizes $c$. If $\Delta$ is extremely amenable, then
so is ${\mathcal{K}}^{*}(\Delta)$.

Using this theorem, together with the techniques from [Kechris2005] and
[Zucker2021], we obtain the following, from which Theorem 1.1 can be deduced
as a consequence.

######  Theorem 1.4 (cf. Theorem 8.4).

Let $\Gamma\curvearrowright M$ be a transitive permutation group and let $c\in
M$. If $\mathcal{M}(\Gamma_{c})$ has a comeager orbit, then so does
$\mathcal{M}(\mathcal{K}(\Gamma))$. More precisely, if
$\mathcal{M}(\Gamma_{c})=\operatorname{Sa}(\Gamma_{c}/\Delta)$ for some closed
$\Delta\leq\Gamma_{c}$, then

$\mathcal{M}(\mathcal{K}(\Gamma))=\operatorname{Sa}(\mathcal{K}(\Gamma)/{\mathcal{K}}^{*}(\Delta)).$

In order to obtain a more concrete description of the Samuel compactification,
we develop model-theoretic framework to represent it as an appropriate type
space. It encompasses the relational expansions, which are used in
[Kechris2005] to describe metrizable UMFs of automorphism groups.

As opposed to Theorem 1.1, which is a characterization, we do not have a
converse of Theorem 1.4. This is because it is not known whether the property
of the UMF having a comeager orbit is closed under taking open subgroups (cf.
Question 2.9).

Theorem 1.4 is already interesting when $\Gamma_{c}$ is the trivial group (so
that one can take $\Delta=\Gamma_{c}$ above). Recall that a permutation action
$\Gamma\curvearrowright M$ is called _simply transitive_ if it is transitive
and free (so that it is isomorphic to the left translation action of the group
on itself).

######  Corollary 1.5.

Let $\Gamma\curvearrowright M$ be a simply transitive action. Then

$\mathcal{M}(\mathcal{K}(\Gamma))=\operatorname{Sa}(\mathcal{K}(\Gamma)/\mathcal{K}(\Gamma)_{\xi})$

and $\mathcal{M}(\mathcal{K}(\Gamma))$ is metrizable iff $M$ is finite.

As a consequence of the results of [Zucker2021], another way to view the space
$\operatorname{Sa}(\mathcal{K}(\Gamma)/\mathcal{K}(\Gamma)_{\xi})$ that
appears in the above corollary is as the _universal highly proximal extension_
of the action $\mathcal{K}(\Gamma)\curvearrowright W_{n}$. In Section 7, we
give an explicit description of this flow as a type space in an appropriate
model-theoretic structure.

Corollary 1.5 gives a large supply of non-metrizable UMFs with a comeager
orbit and a fairly concrete description of them, given by Theorem 7.1.
Continuum many such flows had been exhibited by Kwiatkowska [Kwiatkowska2018],
as mentioned before. Her examples are parametrized by elements of the Cantor
set, while Corollary 1.5 gives examples parametrized by countable groups,
which, from descriptive-set theoretic point of view, are much more
complicated.

In some cases, we can also calculate the _Furstenberg boundary_ (or the
_universal strongly proximal flow_) of $\mathcal{K}(\Gamma)$. This result
generalizes one of Duchesne [Duchesne2020] about the full homeomorphism group.

######  Theorem 1.6 (cf. Theorem 8.11).

Let $\Gamma\curvearrowright M$ be a transitive permutation group and let $c\in
M$. If $\Gamma_{c}$ is amenable and $\mathcal{M}(\Gamma_{c})$ is metrizable,
then the Furstenberg boundary of $\mathcal{K}(\Gamma)$ is isomorphic to
$\operatorname{Sa}(\mathcal{K}(\Gamma)/\mathcal{K}(\Gamma)_{\xi})$.

Finally, we say a few words about our methods. It is well-known that model
theory provides useful tools for studying oligomorphic permutation groups,
because of Ryll-Nardzewski’s theorem. To aid with the study of a non-
oligomorphic permutation group $\Gamma\curvearrowright M$, we develop the
notion of a _full structure_ for $\Gamma$ which associates to it an
$\aleph_{0}$-categorical structure in a possibly uncountable language (cf.
Theorem 3.3) and which coincides with the classical construction in the
oligomorphic case. While for model-theorists, $\aleph_{0}$-categorical
structures in an uncountable language are probably just a curiosity, this
viewpoint seems useful in dynamics: in particular, we obtain an interesting
model-theoretic description of the universal highly proximal extension of the
action $\mathcal{K}(\Gamma)\curvearrowright W_{n}$ even when it is non-
metrizable (cf. Theorem 7.1).

After some preliminary results in dynamics and model theory, most of the
technical work in the paper is about constructing the root-kaleidoscopic
structures, establishing the connection between them and point-stabilizers in
kaleidoscopic groups (in Section 5), and proving the appropriate Ramsey
theorem about them (Section 6). The results about UMFs are collected and
proved in Section 8.

In Section 9, we apply the techniques we have developed to the full
homeomorphism group $\operatorname{Homeo}(W_{n})$ and study the relation of
its UMF to the flow of _convex, converging linear orders (CCLO)_ on the branch
points introduced by Duchesne, which he proposed as the UMF of
$\operatorname{Homeo}(W_{\infty})$. More precisely, we prove the following.

######  Theorem 1.7 (cf. Theorem 9.3).

Let $n=3,4,\ldots,\infty$, let $G=\operatorname{Homeo}(W_{n})$, and let $X$
denote the set of branch points of $W_{n}$. Then the following hold:

  1. (i)

If $n$ is finite, then the flow $G\curvearrowright\mathrm{CCLO}(X)$ is
isomorphic to $\mathcal{M}(G)$;

  2. (ii)

If $n=\infty$, then the flow $G\curvearrowright\mathrm{CCLO}(X)$ is a proper
factor of $\mathcal{M}(G)$. In particular, $\mathrm{CCLO}(X)$ is not
isomorphic to $\mathcal{M}(G)$.

### Acknowledgments

We are grateful to Andy Zucker for explaining to us the actual generality of
Effros’s theorem and its proof (cf. Fact 2.5).

##  2\. Preliminaries on topological dynamics

Let $G$ be a topological group. A _$G$ -flow_ is a continuous action
$G\curvearrowright X$ on a compact Hausdorff space $X$. A $G$-flow is called
_minimal_ if every orbit is dense. A _factor map_ between flows is a
continuous, surjective, $G$-equivariant map. It is a classical fact, due to
Ellis, that among all minimal $G$-flows there is a universal one that admits a
factor map to any minimal $G$-flow and this property characterizes it uniquely
up to isomorphism. It is called the _universal minimal flow_ (_UMF_ for short)
of $G$ and we will denote it by $\mathcal{M}(G)$. A group $G$ is _extremely
amenable_ if $\mathcal{M}(G)$ is a singleton. Following [Basso2021a], we say
that a Polish group $G$ is _CAP_ if $\mathcal{M}(G)$ is metrizable. (The name
comes from a characterization of CAP groups: a group $G$ is CAP iff in every
$G$-flow, the set of almost periodic points is closed. It turns out that this
a more robust property that also makes sense for general topological groups.)

If $Y$ is a uniform space, we will denote by $\operatorname{Sa}(Y)$ the
_Samuel compactification_ of $Y$, i.e., the Gelfand space of the algebra
$\operatorname{UCB}(Y)$ of all uniformly continuous, bounded functions on $Y$.
It is well-known and easy to see that for a metrizable uniform space $Y$,
$\operatorname{Sa}(Y)$ is metrizable iff $Y$ is precompact and in that case,
$\operatorname{Sa}(Y)$ is simply the completion of $Y$. The Samuel
compactification has the following universal property: any uniformly
continuous function $Y\to X$, with $X$ a compact Hausdorff space, extends
(uniquely) to a continuous function $\operatorname{Sa}(Y)\to X$.

Let $H\leq G$ be a closed subgroup of $G$. The homogeneous space $G/H$ carries
a natural uniformity inherited from the right uniformity of $G$: a basis of
entourages is given by sets of the form

$\\{(gH,vgH):v\in V,g\in G\\},$

where $V$ varies over (a basis of) symmetric open neighborhoods of $1_{G}$. In
the case where $G$ is non-archimedean, it is convenient to restrict to open
subgroups $V\leq G$ because then the sets above are equivalence relations
whose classes are the $(V,H)$-double cosets. We will say that $H$ is _co-
precompact_ in $G$ if the space $G/H$ is precompact in this uniformity or,
equivalently, if for every open $V\ni 1_{G}$, there exists a finite set
$F\subseteq G$ such that $VFH=G$; if $G$ is non-archimedean, this just means
that the set $\\{VgH:g\in G\\}$ of double $(V,H)$-cosets is finite for every
open $V\leq G$. We say that $H$ is _presyndetic_ if for every open $V\ni
1_{G}$, there exists a finite set $F\subseteq G$ such that $FVH=G$.

The Samuel compactification $\operatorname{Sa}(G/H)$ is always a $G$-flow.
When $G$ is Polish, $\operatorname{Sa}(G/H)$ is metrizable iff $H$ is co-
precompact and it is minimal iff $H$ is presyndetic ([Zucker2021]*Proposition
6.6). Flows of the type $\operatorname{Sa}(G/H)$ appear naturally when one
studies metrizable UMFs and UMFs with a comeager orbit. In the following fact,
we collect the results that we will need.

######  Fact 2.1.

Let $G$ be a Polish group. Then the following statements hold:

  1. (i)

For any closed $H\leq G$, $\mathcal{M}(G)=\operatorname{Sa}(G/H)$ iff $H$ is
extremely amenable and presyndetic.

  2. (ii)

$\mathcal{M}(G)$ has a comeager orbit iff there exists $H\leq G$ extremely
amenable and presyndetic. In that case,
$\mathcal{M}(G)=\operatorname{Sa}(G/H)$.

  3. (iii)

$G$ is CAP iff there exists $H\leq G$ which is extremely amenable, co-
precompact (and presyndetic).

###### Proof.

(i), (ii) This follows from [Zucker2021]*Theorem 7.5.

(iii) ($\Leftarrow$) is due to Pestov [Pestov2002a] but the argument is simple
enough to include here. The flow $\operatorname{Sa}(G/H)$ has the universal
property that it maps to any $G$-flow with an $H$-fixed point. As $H$ is
extremely amenable, every $G$-flow has an $H$-fixed point, so any minimal
subflow of $\operatorname{Sa}(G/H)$ is isomorphic to $\mathcal{M}(G)$. As $H$
is co-precompact in $G$, $\operatorname{Sa}(G/H)$ is metrizable and so is any
subflow thereof.

($\Rightarrow$) is Theorem 1.1 in [BenYaacov2017]. ∎

######  Proposition 2.2.

Let $G$ be a topological group and let $K\leq H\leq G$ be closed subgroups. If
$H$ is presyndetic in $G$ and $K$ is presyndetic in $H$, then $K$ is
presyndetic in $G$. Similarly, if we substitute co-precompact for presyndetic.

###### Proof.

We prove the statement for presyndeticity; the proof for co-precompactness
follows the same lines.

Let a neighborhood $V\ni 1_{G}$ be given and let $U\ni 1_{G}$ be open such
that $U^{2}\subseteq V$. Let $F_{1}\subseteq H$ be finite such that
$F_{1}(U\cap H)K=H$. Let $U^{\prime}=\bigcap_{f\in F_{1}}fUf^{-1}$ and let
$F_{2}\subseteq G$ be finite such that $F_{2}U^{\prime}H=G$. We claim that
$(F_{2}F_{1})VK=G$. Indeed, let $g\in G$. There exist $f_{2}\in F_{2},u_{2}\in
U^{\prime}$, and $h\in H$ such that $g=f_{2}u_{2}h$. There are also $f_{1}\in
F_{1},u_{1}\in U$, and $k\in K$ such that $h=f_{1}u_{1}k$. Finally, we have
that

$g=f_{2}u_{2}h=f_{2}u_{2}f_{1}u_{1}k=f_{2}f_{1}(f_{1}^{-1}u_{2}f_{1})u_{1}k\in
F_{2}F_{1}UUK\subseteq(F_{2}F_{1})VK,$

as desired. ∎

######  Corollary 2.3.

Let $G$ be a Polish group. Then the following hold:

  1. (i)

If $H\leq G$ is a closed, presyndetic subgroup and
$\mathcal{M}(H)=\operatorname{Sa}(H/K)$ for some closed $K\leq H$, then
$\mathcal{M}(G)=\operatorname{Sa}(G/K)$.

  2. (ii)

If $G$ has a co-precompact CAP subgroup, then $G$ is CAP.

Spaces of the form $\operatorname{Sa}(G/H)$, where $G$ is non-archimedean, are
not difficult to describe. We have the following.

######  Fact 2.4.

Let $G$ be a non-archimedean topological group and let $H\leq G$ be a closed
subgroup. Then

$\operatorname{Sa}(G/H)=\varprojlim\beta(V\mkern 2.5mu\reflectbox{/}\mkern
1.0muG/H),$

where the inverse limit is taken over the collection of open subgroups of $G$
ordered by reverse inclusion and $\beta$ denotes the Stone–Čech
compactification.

###### Proof.

This follows from Gelfand duality and the fact that one can view the algebra
$\ell^{\infty}(V\mkern 2.5mu\reflectbox{/}\mkern 1.0muG/H)$ as a subalgebra of
$\operatorname{UCB}(G/H)$ by identifying it with the $V$-invariant bounded
functions and the union of all of these is dense in $\operatorname{UCB}(G/H)$.
∎

Effros’s classical theorem from [Effros1965] about non-meager orbits of Polish
group actions is usually stated about actions on Polish spaces but it holds in
considerably greater generality. We are grateful to Andy Zucker for explaining
this to us. It is also used in the proof of [Zucker2021]*Theorem 5.5.

######  Fact 2.5 (Effros).

Let $X$ be a Hausdorff space. Let $G$ be a Polish group, let
$G\curvearrowright X$ be a continuous action, and let $x_{0}\in X$ be a point
with a dense orbit. Then the following are equivalent:

  1. (i)

The orbit $G\cdot x_{0}$ is non-meager;

  2. (ii)

The map $G/G_{x_{0}}\to G\cdot x_{0},gG_{x_{0}}\mapsto g\cdot x_{0}$ is a
homeomorphism;

###### Proof.

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

This is Lemma 2.5 in [Effros1965]. Effros’s proof goes through in the
generality of our statement, except that one needs to replace Lemma 2.4 with
the following: for any open set $V\subseteq G$, the set $V\cdot x_{0}$ has the
Baire property in $X$. This is a consequence of the following two facts.
First, $V\cdot x_{0}$ is _analytic_ (i.e., a continuous image of the space
$\mathbb{N}^{\mathbb{N}}$), and second, every analytic set in a Hausdorff
space has the Baire property. The latter follows from the classical theorem of
Nikodým that the collection of sets with the Baire property is closed under
the Souslin operation [Kechris1995]*29.14 and that in a Hausdorff space,
analytic sets can be obtained via the Souslin operation from closed sets (see
[Bogachev2007]*6.6.8).

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

The hypothesis implies that $Y\coloneqq G\cdot x_{0}$ is Polish. Suppose that
it is meager. Then there exist closed, nowhere dense subsets $F_{n}\subseteq
X$, for $n\in\mathbb{N}$, with $Y\subseteq\bigcup_{n}F_{n}$. As $Y$ is non-
meager in itself, there exist a non-empty open $U\subseteq X$ and
$n\in\mathbb{N}$ such that $U\cap Y\subseteq F_{n}\cap Y$. But then
$U\subseteq\overline{U\cap Y}\subseteq F_{n}$, contradicting the fact that
$F_{n}$ is nowhere dense. ∎

If $X$ and $Y$ are topological spaces and $\pi\colon X\to Y$ is a surjective,
continuous map, we will say that $\pi$ is _irreducible_ if for every non-
empty, open $U\subseteq X$, there exists a point $y\in Y$ with
$\pi^{-1}(\\{y\\})\subseteq U$. We say that $\pi$ is _almost 1-to-1_ whenever
the set $\\{x\in X:\pi^{-1}(\pi(x))=\\{x\\}\\}$ is comeager. If $\pi$ is onto
and almost 1-to-1, then it is irreducible, and the converse holds if $X$ is
metrizable and Baire. See [deVries2014]*A.9 for more details on irreducible
maps.

If $\pi\colon X\to Y$ is a factor map between $G$-flows which is irreducible,
we will say that $X$ is a _highly proximal extension of $Y$_. It is a result
of Auslander and Glasner [Auslander1977], for minimal flows, and of Zucker
[Zucker2018] in general, that among all highly proximal extensions of a given
flow $X$, there exists a universal one, which we will denote by
$\operatorname{\mathcal{S}}_{G}(X)\to X$. It has the property that for any
other highly proximal extension $X^{\prime}\to X$, there is a (highly
proximal) map $\operatorname{\mathcal{S}}_{G}(X)\to X^{\prime}$ such that the
appropriate diagram commutes. An important feature of highly proximal
extensions is that they preserve many dynamical properties.

######  Fact 2.6 ([Auslander1977]).

Let $\pi\colon X\to Y$ be a highly proximal factor map between $G$-flows with
$Y$ minimal. Then $X$ is minimal and if $Y$ is proximal (strongly proximal),
then so is $X$.

A flow $X$ such that $\operatorname{\mathcal{S}}_{G}(X)=X$ is called
_maximally highly proximal_ , or _MHP_. Maximally highly proximal flows were
extensively studied by Zucker [Zucker2021]. If $H\leq G$ is a closed subgroup
of $G$, then the flow $\operatorname{Sa}(G/H)$ is always MHP (see
[Zucker2021]*Subsection 3.2.1). There is an important connection between
Samuel compactifications of homogeneous spaces and universal highly proximal
extensions of flows with a comeager orbit.

######  Proposition 2.7.

Let $G$ be a Polish group and let $G\curvearrowright X$ be a $G$-flow with a
comeager orbit $G\cdot x_{0}$. Then:

  1. (i)

$\operatorname{\mathcal{S}}_{G}(X)\cong\operatorname{Sa}(G/G_{x_{0}})$;

  2. (ii)

Any highly proximal extension of $X$ is almost 1-to-1;

  3. (iii)

If $X$ is moreover minimal, then for any two factor maps
$\pi,\pi^{\prime}\colon\operatorname{\mathcal{S}}_{G}(X)\to X$, there is an
automorphism $f$ of $\operatorname{\mathcal{S}}_{G}(X)$ such that $\pi\circ
f=\pi^{\prime}$. In particular, any factor map
$\operatorname{\mathcal{S}}_{G}(X)\to X$ is almost 1-to-1.

###### Proof.

Consider the natural bijection $\rho\colon G/G_{x_{0}}\to G\cdot x_{0}$, which
by Effros’s theorem (Fact 2.5) is a homeomorphism. By the universal property
of $\operatorname{Sa}(G/G_{x_{0}})$, $\rho$ extends to a map
$\pi\colon\operatorname{Sa}(G/G_{x_{0}})\to X$. We prove that
$\pi^{-1}(x_{0})=\\{G_{x_{0}}\\}$. Indeed, suppose that
$p\in\operatorname{Sa}(G/G_{x_{0}})$ is such that $\pi(p)=x_{0}$. Since
$G/G_{x_{0}}$ embeds densely in $\operatorname{Sa}(G/G_{x_{0}})$, there is a
net $(g_{i}G_{x_{0}})$ of elements of $G/G_{x_{0}}$ which converges to $p$. By
continuity, $\pi(g_{i}G_{x_{0}})=\rho(g_{i}G_{x_{0}})=g_{i}\cdot x_{0}$
converges to $x_{0}=\rho(G_{x_{0}})$. But $\rho$ is a homeomorphism, so
$g_{i}G_{x_{0}}\to G_{x_{0}}$, that is, $p=G_{x_{0}}$.

By [Engelking1989]*Theorem 4.3.26, since $G/G_{x_{0}}$ is Polish, it is a
$G_{\delta}$ set in all of its compactifications, in particular in
$\operatorname{Sa}(G/G_{x_{0}})$. Therefore, $\pi$ is almost 1-to-1, as
$\pi^{-1}(\pi(gG_{x_{0}}))=g\pi^{-1}(x_{0})=\\{gG_{x_{0}}\\}$ for all
$gG_{x_{0}}\in G/G_{x_{0}}$.

Now (i) follows from the universal property of
$\operatorname{\mathcal{S}}_{G}(X)$: since $\pi$ is almost 1-to-1, it is
highly proximal, so $\operatorname{\mathcal{S}}_{G}(X)$ is a highly proximal
extension of $\operatorname{Sa}(G/G_{x_{0}})$. As
$\operatorname{Sa}(G/G_{x_{0}})$ is MHP, this extension must be an
isomorphism.

(ii) Since the universal highly proximal extension is almost 1-to-1, so is any
other highly proximal extension.

(iii) If $X$ is minimal, then
$\operatorname{\mathcal{S}}_{G}(X)\cong\operatorname{Sa}(G/G_{x_{0}})$ is
minimal, too. Let $\pi^{\prime}\colon\operatorname{\mathcal{S}}_{G}(X)\to X$
be any factor map. By Fact 2.5, $G\cdot x_{0}$ is Polish, so by
[Engelking1989]*Theorem 4.3.26, it is $G_{\delta}$ in $X$, so
$\pi^{\prime-1}(G\cdot x_{0})$ is $G_{\delta}$ and dense by minimality, so it
must intersect the dense $G_{\delta}$ set $G/G_{x_{0}}$ in
$\operatorname{Sa}(G/G_{x_{0}})$. In particular,
$\pi^{\prime}[G/G_{x_{0}}]=G\cdot x_{0}$.

Then $\phi\coloneqq\rho^{-1}\pi^{\prime}|_{G/G_{x_{0}}}$ is a $G$-equivariant
bijection of the homogeneous space $G/G_{x_{0}}$, so there exists an element
$h$ in the normalizer of $G_{x_{0}}$ in $G$ such that $\phi$ is left
multiplication by $h$. In particular, $\phi$ is a $G$-equivariant uniform
isomorphism of $G/G_{x_{0}}$, so it extends to an automorphism $f$ of the flow
$\operatorname{Sa}(G/G_{x_{0}})$. If we define $\pi$ as in (i), we get that
$\pi\circ f=\pi^{\prime}$ as desired. ∎

We finally record some of the closure properties of relevant classes of
topological groups. Recall that $G$ is called _amenable_ if every $G$-flow
carries an invariant probability measure.

######  Fact 2.8.

Let $G$ be a Polish group and let $V\leq G$ be an open subgroup. Let $K$ be a
topological group and let $\pi\colon G\to K$ be a continuous homomorphism with
dense image. If $G$ is amenable/extremely amenable/CAP, then so are $V$ and
$K$.

###### Proof.

All of these are standard. For $K$, it is a direct consequence of the
definitions. For $V$, it is possible to give a uniform proof using the co-
induction construction as follows. If we start with any flow
$V\curvearrowright X$, then one can define an action of $G$ on the space
$X^{G/V}$ and a $V$-equivariant map $X^{G/V}\to X$ (see, e.g.,
[Bodirsky2013]*Lemma 13). Now it is clear that if $X^{G/V}$ has an invariant
measure/fixed point/metrizable subflow, then so does $X$. ∎

######  Question 2.9.

If $G$ is Polish, $\mathcal{M}(G)$ has a comeager orbit and $V\leq G$ is an
open subgroup, is it true that $\mathcal{M}(V)$ has a comeager orbit?

##  3\. The model theory of permutation groups

Even though the Ważewski dendrites are connected spaces, their homeomorphism
groups (and subgroups thereof) are non-archimedean and are best studied as
permutation groups. An important tool for understanding infinite permutation
groups is Fraïssé theory, where the group is viewed as the automorphism group
of a _ultrahomogeneous structure_ and one can study it via the combinatorial
properties of the _age_ of the structure. This is particularly helpful in
topological dynamics, where one can use the Kechris–Pestov–Todorčević
correspondence to study extreme amenability and describe the universal minimal
flows of automorphism groups.

###  3.1. Fraïssé limits and full structures

We recall all relevant definitions and establish some terminology and
notation. Most of them are standard except perhaps for our treatment of
expansions. A _signature_ $\mathcal{L}$ is a collection of relation symbols
$\\{R_{i}\\}_{i}$ and function symbols $\\{F_{j}\\}_{j}$ to each of which is
associated a natural number called its _arity_. An $\mathcal{L}$-structure
${\mathbf{M}}$ consists of a non-empty set $M$, called _the universe of
${\mathbf{M}}$_, together with interpretations of the relation and function
symbols. Each relation symbol of arity $k$ is interpreted as a subset of
$M^{k}$ and each function symbol of arity $k$ is interpreted as a function
$M^{k}\to M$. A function symbol of arity $0$ is called a _constant symbol_. A
signature without function symbols is called _relational_. An _automorphism_
of ${\mathbf{M}}$ is an element of $\operatorname{Sym}(M)$ which respects all
relation and function symbols. A _substructure_ ${\mathbf{M}}^{\prime}$ of
${\mathbf{M}}$ is given by a subset $M^{\prime}\subseteq M$ which is closed
under the functions in the signature (in particular, it contains all
constants). The interpretations of the symbols in ${\mathbf{M}}^{\prime}$ are
inherited from ${\mathbf{M}}$. An _embedding_
${\mathbf{M}}\to{\mathbf{M}}^{\prime}$ is an isomorphism from ${\mathbf{M}}$
onto a substructure of ${\mathbf{M}}^{\prime}$. A structure ${\mathbf{M}}$ is
called _ultrahomogeneous_ if every isomorphism between finitely generated
substructures of ${\mathbf{M}}$ extends to an automorphism of ${\mathbf{M}}$.
In this paper, we will only be interested in countable structures (but in
possibly uncountable signatures).

The _age_ of a countable structure ${\mathbf{M}}$ is given by the isomorphism
classes of all finitely generated substructures of ${\mathbf{M}}$. Every age
is countable, _hereditary_ (closed under taking substructures) and _directed_
(every two structures in the age embed into a third). An age is simply a
collection of isomorphism classes of finitely generated structures with these
three properties. It is easy to see that every age is the age of some
countable structure. If $\mathcal{F}$ is an age, we will say that
${\mathbf{M}}$ is an _$\mathcal{F}$ -structure_ if
$\operatorname{Age}({\mathbf{M}})\subseteq\mathcal{F}$.

Fraïssé isolated an important additional property, called _amalgamation_ ,
which characterizes the ages of ultrahomogeneous structures, and proved that
for every age with amalgamation, there exists a unique countable,
ultrahomogeneous structure with this age, up to isomorphism (often called its
_Fraïssé limit_).

Let $\mathcal{L}$ be a signature. An _atomic formula_ in the variables
$\bar{v}=(v_{1},\ldots,v_{n})$ is of the form
$R(t_{1}(\bar{v}),\ldots,t_{k}(\bar{v}))$, where $R$ is a relation symbol and
$t_{1},\ldots,t_{k}$ are _terms_ , i.e., compositions of functions symbols. A
_quantifier-free formula_ is a Boolean combination of atomic formulas. A
_first-order formula_ (or just a _formula_) is one where we also allow
quantifiers. Note that all quantifier-free formulas are preserved by
embeddings but first-order formulas in general are not. For the purposes of
Fraïssé theory, we may and will simply consider quantifier-free formulas as
part of the signature. If $\theta(\bar{v})$ is a formula and ${\mathbf{M}}$ is
a structure, we will denote by $\theta({\mathbf{M}})$ the set $\\{\bar{a}\in
M^{\bar{v}}:{\mathbf{M}}\models\theta(\bar{a})\\}$. Sets of this form are
called _definable_ (or _$0$ -definable_ when we want to emphasize the absence
of parameters). For a fixed $B\subseteq{\mathbf{M}}$, we can consider the
subsets of $M^{k}$ which are _definable with parameters from $B$_, or _$B$
-definable_, i.e., of the form $\\{\bar{a}\in
M^{\bar{v}}:{\mathbf{M}}\models\phi(\bar{a},\bar{b})\\}$ for some formula
$\phi(\bar{v},\bar{w})$ and $\bar{b}\in B^{\bar{w}}$. A structure _eliminates
quantifiers_ if every $0$-definable set is definable by a quantifier-free
formula.

The _theory of ${\mathbf{M}}$_, denoted by $\operatorname{Th}({\mathbf{M}})$,
is the collection of all sentences true in ${\mathbf{M}}$. Recall that the
space of _$k$ -types_ (for $k\in\mathbb{N}$) of
$\operatorname{Th}({\mathbf{M}})$, denoted by
$\operatorname{S}_{k}(\operatorname{Th}({\mathbf{M}}))$, is the Stone space of
the Boolean algebra of $0$-definable subsets of $M^{k}$. For $A\subseteq M$,
we can also similarly consider the type space with parameters from $A$,
denoted by $\operatorname{S}_{k}(A)$. We also let
$\operatorname{S}_{\omega}=\varprojlim\operatorname{S}_{k}$. These are all
$\mathrm{Aut}({\mathbf{M}})$-flows (if the set of parameters is
$\operatorname{Aut}({\mathbf{M}})$-invariant). They are metrizable if the
signature is countable. For a tuple $\bar{a}\in M^{k}$, the _type of
$\bar{a}$_, denoted by $\operatorname{tp}\bar{a}$ is the element
$p\in\operatorname{S}_{k}(\operatorname{Th}({\mathbf{M}}))$ given by

$p(\bar{v})\models\phi(\bar{v})\iff M\models\phi(\bar{a}),\quad\text{ for
every formula }\phi.$

We say that such types are _realized_ in ${\mathbf{M}}$. Any type
$p\in\operatorname{S}_{k}(\operatorname{Th}({\mathbf{M}}))$ is realized in
some model of $\operatorname{Th}({\mathbf{M}})$.

We will at some point use quantifiers, so we will also need the following
weaker notion of homogeneity: a structure ${\mathbf{M}}$ is _homogeneous_ if
for all tuples $\bar{a}$, $\bar{b}$ from $M$ such that
$\operatorname{tp}\bar{a}=\operatorname{tp}\bar{b}$, there exists
$g\in\operatorname{Aut}({\mathbf{M}})$ such that $g\cdot\bar{a}=\bar{b}$. Of
course, if a model is ultrahomogeneous, then it is homogeneous. A type is
_isolated_ if it is an isolated point in the corresponding type space. A model
that only realizes isolated types is called _atomic_. It is a standard fact
that a countable, atomic model is always homogeneous.

Let now $\mathcal{F}$ be an age in the signature $\mathcal{L}$. An _imaginary
sort_ $\mathcal{S}$ for $\mathcal{F}$ is given by a pair of quantifier-free
formulas $\theta(\bar{v})$, $\epsilon(\bar{u}_{1},\bar{u}_{2})$ such that
$|\bar{v}|=|\bar{u}_{1}|=|\bar{u}_{2}|$ and for every
${\mathbf{A}}\in\mathcal{F}$,
$\epsilon({\mathbf{A}})\cap(\theta({\mathbf{A}})\times\theta({\mathbf{A}}))$
is an equivalence relation on $\theta({\mathbf{A}})$. We define
$\mathcal{S}({\mathbf{A}})$ to be the set of classes of this equivalence
relation.

The following lemma records some basic facts about imaginary sorts and we omit
the routine proof.

######  Lemma 3.1.

Let $\mathcal{F}$ be an age and let $\mathcal{S}$ be an imaginary sort for
$\mathcal{F}$ given by $(\theta,\epsilon)$. Then the following hold:

  1. (i)

For any $\mathcal{F}$-structure ${\mathbf{M}}$,
$\epsilon({\mathbf{M}})\cap(\theta({\mathbf{M}})\times\theta({\mathbf{M}}))$
is an equivalence relation on $\theta({\mathbf{M}})$ and in particular, we can
define

$\mathcal{S}({\mathbf{M}})=\\{[\bar{a}]_{\epsilon({\mathbf{M}})}:\bar{a}\in\theta({\mathbf{M}})\\}.$

  2. (ii)

If $f\colon{\mathbf{M}}\to{\mathbf{N}}$ is an embedding between
$\mathcal{F}$-structures, then $f$ naturally defines an injective map
$\mathcal{S}(f)\colon\mathcal{S}({\mathbf{M}})\to\mathcal{S}({\mathbf{N}})$ by

$\mathcal{S}(f)([\bar{a}]_{\epsilon({\mathbf{M}})})=[f(\bar{a})]_{\epsilon({\mathbf{N}})}\quad\text{
for }\bar{a}\in\theta({\mathbf{M}}).$

Moreover, this correspondence is functorial.

  3. (iii)

In particular, for any $\mathcal{F}$-structure ${\mathbf{M}}$, the group
$\operatorname{Aut}({\mathbf{M}})$ acts on $\mathcal{S}({\mathbf{M}})$.

We will also allow to have symbols from the signature defined on imaginary
sorts. This will be particularly useful for defining expansions.

If one is given a set $M$ and a permutation group
$\Gamma\leq\operatorname{Sym}(M)$, one can define a relational,
ultrahomogeneous structure ${\mathbf{M}}$ with universe $M$ such that
$\operatorname{Aut}({\mathbf{M}})=\Gamma$. There are many such structures and
all of them give rise to equivalent Fraïssé classes, however they do differ
from the point of view of model theory. There are two canonical choices
corresponding to the minimal and the maximal possible Boolean algebras of
definable sets, respectively, under the constraint that the structure is
ultrahomogeneous and $\operatorname{Aut}({\mathbf{M}})=\Gamma$. The minimal
one is obtained by introducing a relation symbol for every $\Gamma$-orbit on
$M^{k}$ for every $k$ and it has the advantage that the resulting signature is
always countable; however, for us, it is the other one which will be more
relevant.

######  Definition 3.2.

Let $M$ be a countable set and let $\Gamma\leq\operatorname{Sym}(M)$ be a
closed subgroup. We will say that a structure ${\mathbf{M}}$ with universe $M$
is _full for $\Gamma$_ if $\operatorname{Aut}({\mathbf{M}})=\Gamma$ and for
every $k\in\mathbb{N}$, every $\Gamma$-invariant subset of $M^{k}$ is
definable in ${\mathbf{M}}$. We will say it is _qf-full_ if every such set is
quantifier-free definable. A structure is _full_ if it is full for its
automorphism group.

Note if $\Gamma\curvearrowright M$ is a permutation group, then all full
structures for $\Gamma$ are interdefinable. If $\Gamma$ is oligomorphic, then
every structure ${\mathbf{M}}$ with automorphism group $\Gamma$ is full; if
moreover ${\mathbf{M}}$ is ultrahomogeneous, then it is qf-full. If $\Gamma$
is not oligomorphic, then every full structure necessarily has a signature of
cardinality continuum. As we will see shortly, a full structure is always
_$\aleph_{0}$ -categorical_ (i.e, the unique countable model of its first-
order theory), which makes it a canonical choice for studying permutation
groups model-theoretically.

######  Theorem 3.3.

Let $M$ be a countable set and let $\Gamma\leq\operatorname{Sym}(M)$ be a
closed subgroup. Let ${\mathbf{M}}$ be a full structure for $\Gamma$ with
universe $M$ and let $T=\operatorname{Th}({\mathbf{M}})$. Then the following
hold:

  1. (i)

$\operatorname{S}_{k}(T)=\beta(M^{k}/\Gamma)$;

  2. (ii)

${\mathbf{M}}$ is an atomic model of $T$;

  3. (iii)

The structure ${\mathbf{M}}$ is $\aleph_{0}$-categorical;

  4. (iv)

If ${\mathbf{M}}$ is qf-full, then ${\mathbf{M}}$ is ultrahomogeneous and
eliminates quantifiers.

###### Proof.

(i) A subset of $M^{k}$ is definable iff it is $\Gamma$-invariant and the
Boolean algebra of $\Gamma$-invariant subsets of $M^{k}$ is isomorphic to the
power set of $M^{k}/\Gamma$.

(ii) The types realized in ${\mathbf{M}}$ correspond to the principal
ultrafilters in $\beta(M^{k}/\Gamma)$. As the principal ultrafilters are
isolated, this implies that ${\mathbf{M}}$ is atomic.

(iii) We will show that any countable model of $T$ is atomic. Then the claim
will follow from the uniqueness of the countable atomic model. The proof is
inspired by a construction of Blass [BlassMathOverflow].

Suppose that $q\in\operatorname{S}_{k}(T)$ is a non-isolated type; our goal is
to show that any model that realizes $q$ is uncountable. Let
$p_{0},p_{1},\ldots$ enumerate the isolated types in $\operatorname{S}_{k}(T)$
and let $\phi_{i}$ be a formula that isolates $p_{i}$. Let
$\\{B_{\alpha}:\alpha\in 2^{\mathbb{N}}\\}$ be a collection of infinite
subsets of $\mathbb{N}$ such that for all $\alpha\neq\alpha^{\prime}$,
$B_{\alpha}\cap B_{\alpha^{\prime}}$ is finite. Let $\psi_{\alpha}(x)$ be the
relation defined in ${\mathbf{M}}$ by $\bigvee_{i\in B_{\alpha}}\phi_{i}(x)$
and let $<$ be the relation defined in ${\mathbf{M}}$ by

$x<y\iff\phi_{i}(x)\land\phi_{j}(y)\land i<j.$

Let now $c$ be a realization of $q$ in some model ${\mathbf{N}}$ and consider
the collection of formulas

$\\{\exists x\ \psi_{\alpha}(x)\land x>c:\alpha\in 2^{\mathbb{N}}\\}.$

We claim that each one of them holds and their solutions are disjoint, showing
that ${\mathbf{N}}$ is uncountable. The first claim follows from the fact that
the sentence $\forall y\exists x\ \psi_{\alpha}(x)\land x>y$ holds in
${\mathbf{M}}$ (because $B_{\alpha}$ is infinite). For the second, note that
for $\alpha\neq\alpha^{\prime}$, the sentence

$\forall x\
\psi_{\alpha}(x)\land\psi_{\alpha^{\prime}}(x)\implies\bigvee_{i\in
B_{\alpha}\cap B_{\alpha^{\prime}}}\phi_{i}(x)$

is true in ${\mathbf{M}}$ and the fact that $q$ is non-principal implies that
$c>y$ for every $y$ for which there is $i$ such that $\phi_{i}(y)$.

(iv) This follows from the fact that all definable sets in ${\mathbf{M}}$ are
$\Gamma$-invariant, so quantifier-free definable. ∎

###  3.2. Expansions as types

Starting from [Kechris2005], metrizable universal minimal flows of
automorphism groups of a countable structure ${\mathbf{M}}$ are often
represented as spaces of expansions of ${\mathbf{M}}$ in a suitable relational
signature. In principle, this can always be done and it works well when the
expansion is simple (say, by a linear ordering) but in more complicated
situations, a more expressive formalism may be convenient. For example, in
order to describe the UMF of kaleidoscopic groups, we are naturally led to
consider expansions by a constant not belonging to the original structure,
which, while possible to deal with in the traditional KPT setting (see
[Kwiatkowska2018]), require a lot of coding and are less transparent.

Our definition of expansions is more general than the usual one on two counts:
first, we allow expansions by function symbols (in particular, constants) that
may add new elements to the structure, and second, we also allow new symbols
on the imaginary sorts. Formally, let ${\mathbf{M}}$ be an
$\mathcal{L}$-structure and let $\mathcal{L}^{\prime}\supseteq\mathcal{L}$. An
_$\mathcal{L}^{\prime}$ -expansion_ of ${\mathbf{M}}$ is an
$\mathcal{L}^{\prime}$-structure ${\mathbf{N}}$ together with a map
$\iota\colon M\to N$ such that:

  * •

$\iota$ is an $\mathcal{L}$-embedding;

  * •

$N=\operatorname{dcl}^{\mathcal{L}^{\prime}}\iota[M]$;

  * •

$\iota[M]$ is $\operatorname{Aut}({\mathbf{N}})$-invariant.

Moreover, we allow some of the new symbols in $\mathcal{L}^{\prime}$ to apply
to the imaginary sorts of ${\mathbf{N}}$. As usual, $\operatorname{dcl}$
denotes the _definable closure_ , i.e., $\operatorname{dcl}A$ is the
collection of elements $b\in N$ such that the singleton $\\{b\\}$ is
$A$-definable. The main property that our definition ensures is that an
expansion gives rise to a closed subgroup of the automorphism group, which, as
the proof of 3.5 shows, is all that matters.

######  Proposition 3.4.

Let ${\mathbf{M}}$ be an $\mathcal{L}$-structure and ${\mathbf{N}}$ be a
$\mathcal{L}^{\prime}$-expansion of ${\mathbf{M}}$ with embedding $\iota\colon
M\to N$. Then the restriction map
$f\colon\operatorname{Aut}({\mathbf{N}})\to\operatorname{Aut}({\mathbf{M}})$
defined by $f(g)=g|_{\iota[M]}$ is a topological group embedding.

###### Proof.

We may assume that $\iota$ is the identity, so that ${\mathbf{M}}$ is a
substructure of the $\mathcal{L}$ reduct of ${\mathbf{N}}$. Since $M$ is
$\operatorname{Aut}({\mathbf{N}})$-invariant,
$g|_{M}\in\mathrm{Aut}({\mathbf{M}})$.

Suppose that $f(g)=\operatorname{id}_{M}$. Then $g(b)=b$ for any
$b\in\operatorname{dcl}(M)=N$, so $g=\operatorname{id}_{N}$.

To show that $f$ is an embedding, let $(g_{n})$ be a sequence of elements of
$\operatorname{Aut}({\mathbf{N}})$ such that
$g_{n}|_{M}\to\operatorname{id}_{M}$. Fix $b\in N$ in order to prove that
eventually $g_{n}(b)=b$. Let $\phi(v,\bar{a})$ define $\\{b\\}$; then for any
$h\in\operatorname{Aut}({\mathbf{M}})$, which fixes $\bar{a}$, we have
$h(b)=b$. But $g_{n}\to\operatorname{id}_{M}$, so eventually
$g_{n}\cdot\bar{a}=\bar{a}$ and we are done. ∎

We say that an expansion is _relational_ if $N=M$ and
$\mathcal{L}^{\prime}\setminus\mathcal{L}$ is relational.

Next we explain how to represent any flow of the form
$\operatorname{Sa}(G/H)$, where $G=\operatorname{Aut}({\mathbf{M}})$ and
$H\leq G$ is an arbitrary closed subgroup, as a space of types for an
appropriate expansion. The following is just a restatement of Fact 2.4 in a
model-theoretic setting but being able to use arbitrary first order formulas
is convenient in applications and also makes the action of the group $G$ more
explicit.

If ${\mathbf{N}}$ is an expansion of ${\mathbf{M}}$, we denote

$\operatorname{Exp}({\mathbf{M}},{\mathbf{N}})=\\{p\in\operatorname{S}_{\omega}(\operatorname{Th}({\mathbf{N}})):p|_{\mathcal{L}}=\operatorname{tp}_{\mathcal{L}}\bar{a}\\},$

where $\bar{a}$ is some fixed enumeration of ${\mathbf{M}}$. It will also
often be convenient to identify the variables of the type with the elements of
$M$ via this fixed enumeration. Note that $\operatorname{Aut}({\mathbf{M}})$
acts naturally on $\operatorname{Exp}({\mathbf{M}},{\mathbf{N}})$ by permuting
the variables.

######  Proposition 3.5.

Let ${\mathbf{M}}$ be a countable structure in a signature $\mathcal{L}$,
which is an atomic model of its first-order theory. Suppose that
${\mathbf{N}}$ is an expansion of ${\mathbf{M}}$ in a signature
$\mathcal{L}^{\prime}$, and that ${\mathbf{N}}$ is full. Denote
$G=\operatorname{Aut}({\mathbf{M}})$ and $H=\operatorname{Aut}({\mathbf{N}})$.
Then

$\operatorname{Sa}(G/H)\cong\operatorname{Exp}({\mathbf{M}},{\mathbf{N}}).$

###### Proof.

From Fact 2.4, we have that

$\operatorname{Sa}(G/H)=\varprojlim\beta(V\mkern 2.5mu\reflectbox{/}\mkern
1.0muG/H)=\varprojlim\beta(G_{\bar{b}}\mkern 2.5mu\reflectbox{/}\mkern
1.0muG/H),$

where the second limit is taken over all finite tuples $\bar{b}$ in $M$. The
double coset space $G_{\bar{b}}\mkern 2.5mu\reflectbox{/}\mkern 1.0muG/H$ can
be identified with the set of orbits of the action $H\curvearrowright
G\cdot\bar{b}$. Now, since ${\mathbf{M}}$ is atomic, it is homogeneous, so

$G\cdot\bar{b}=\\{\bar{c}\in
M^{|\bar{b}|}:\operatorname{tp}_{\mathcal{L}}\bar{c}=\operatorname{tp}_{\mathcal{L}}\bar{b}\\}$

is an $\mathcal{L}$-definable set and by the fullness of ${\mathbf{N}}$, every
$H$-invariant subset of $G\cdot\bar{b}$ is $\mathcal{L}^{\prime}$-definable,
so we obtain a natural identification between the power set of
$G_{\bar{b}}\mkern 2.5mu\reflectbox{/}\mkern 1.0muG/H$ and the algebra of
$\mathcal{L}^{\prime}$-definable subsets of $G\cdot\bar{b}$. Passing to the
respective Stone spaces, we obtain the identification

$\beta(G_{\bar{b}}\mkern 2.5mu\reflectbox{/}\mkern
1.0muG/H)=\\{p\in\operatorname{S}_{|\bar{b}|}(\operatorname{Th}({\mathbf{N}})):p|_{\mathcal{L}}=\operatorname{tp}_{\mathcal{L}}\bar{b}\\}.$

Finally, taking the inverse limit over all finite tuples $\bar{b}$ in
${\mathbf{M}}$ completes the proof. ∎

As a simple illustration, we show how to represent model-theoretically the
flow $\operatorname{Sa}(G/V)$ for an open subgroup $V\leq G$.

######  Corollary 3.6.

Let $\Gamma\leq\operatorname{Sym}(M)$ be a transitive permutation group and
let ${\mathbf{M}}$ be a full structure for $\Gamma$. Let $c\in M$. Then
$\operatorname{Sa}(\Gamma/\Gamma_{c})\cong\operatorname{S}_{1}(M)$ as
$\Gamma$-flows.

###### Proof.

Apply Proposition 3.5 to the expansion by a single constant symbol
$\mathbf{c}$ which is interpreted as an element $c\in M$ (it does not matter
which one because of transitivity). This expansion is a full structure for
$\Gamma_{c}$ since any $\Gamma_{c}$-invariant set $D\subseteq M^{k}$ is
defined by $\phi(\mathbf{c},\bar{v})$, where $\phi(u,\bar{v})$ is a formula
defining $\Gamma\cdot(\\{c\\}\times D)$ in ${\mathbf{M}}$. ∎

We recall a combinatorial criterion for presyndeticity. If ${\mathbf{M}}$ is
an ultrahomogeneous structure and ${\mathbf{N}}$ is an ultrahomogeneous,
relational expansion of ${\mathbf{M}}$, we will say that ${\mathbf{N}}$ is a
_minimal expansion_ of ${\mathbf{M}}$ if for every
${\mathbf{B}}^{\prime}\in\operatorname{Age}({\mathbf{N}})$, there exists
${\mathbf{A}}\in\operatorname{Age}({\mathbf{M}})$ such that for every
${\mathbf{A}}^{\prime}\in\operatorname{Age}({\mathbf{N}})$ which is an
expansion of ${\mathbf{A}}$, there exists an embedding of
${\mathbf{B}}^{\prime}$ into ${\mathbf{A}}^{\prime}$. Zucker considered this
condition in a slightly different context in [Zucker2018a]. A similar property
had already been studied in the literature under the name of the _ordering
property_ (see, e.g., [Kechris2005]), when the expansion is by a linear order,
or the _expansion property_ for general relational expansions (see
[NguyenVanThe2013]). The definition above is equivalent to the expansion
property for a precompact expansion but not in general. The following is well-
known and implied in [Zucker2018a] but we could not find a direct reference,
so we include the proof of the direction we need.

######  Proposition 3.7.

Let ${\mathbf{M}}$ be a countable, ultrahomogeneous structure and let
${\mathbf{N}}$ be an ultrahomogeneous, relational expansion of ${\mathbf{M}}$.
Then the following are equivalent:

  1. (i)

${\mathbf{N}}$ is a minimal expansion of ${\mathbf{M}}$;

  2. (ii)

$\operatorname{Aut}({\mathbf{N}})$ is presyndetic in
$\operatorname{Aut}({\mathbf{M}})$.

###### Proof.

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

Write $G=\operatorname{Aut}({\mathbf{M}})$ and
$H=\operatorname{Aut}({\mathbf{N}})$. Let $\bar{b}$ be a finite tuple in $M$
and denote by ${\mathbf{B}}$ and ${\mathbf{B}}^{\prime}$ the substructure
generated by $\bar{b}$ in ${\mathbf{M}}$ and in ${\mathbf{N}}$, respectively.
(Note that because the expansion is relational, ${\mathbf{B}}$ and
${\mathbf{B}}^{\prime}$ are structures with the same underlying set $B$.) Let
${\mathbf{A}}\in\operatorname{Age}({\mathbf{M}})$ be a witness of minimality
for ${\mathbf{B}}^{\prime}$ and realize ${\mathbf{A}}$ as a substructure of
${\mathbf{M}}$. Let $F$ be a finite subset of $G$ such that
$\binom{{\mathbf{A}}}{{\mathbf{B}}}=\\{f|_{B}:f\in F\\}$. Our goal is to show
that $G=HG_{B}F^{-1}$. To that end, let $g\in G$ be arbitrary. We want to find
$h\in H$ and $f\in F$ such that $hgf$ fixes $B$. Let ${\mathbf{A}}^{\prime}$
be the substructure of ${\mathbf{N}}$ with underlying set $g\cdot A$. By
minimality of the expansion and the choice of $F$, there exists $f\in F$ such
that $gf\cdot{\mathbf{B}}^{\prime}\cong{\mathbf{B}}^{\prime}$. Now we conclude
by ultrahomogeneity of ${\mathbf{N}}$.

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

The proof is similar and we omit it. ∎

##  4\. Kaleidoscopic and root-kaleidoscopic groups

###  4.1. Ważewski dendrites and kaleidoscopic groups

An important property of topological trees is that they are _uniquely arcwise
connected_ , i.e., between every two points there is a unique arc connecting
them. (An _arc_ is just a homeomorphic copy of the unit interval $[0,1]$.)
Dendrites are topological spaces, generalizing trees, where branch points are
allowed to be dense. More formally, a _dendrite_ is a compact, metrizable,
locally connected, uniquely arcwise connected, topological space.

Let $D$ be a dendrite and for a point $x\in D$, denote by $\hat{x}$ the set of
connected components of $D\setminus\\{x\\}$. The points of $D$ are classified
by the cardinality of $\hat{x}$: $x$ is called an _endpoint_ if
$\lvert\hat{x}\rvert=1$; it is called a _regular point_ if
$\lvert\hat{x}\rvert=2$; and it is called a _branch point_ if
$\lvert\hat{x}\rvert\geq 3$. If $x$ is a branch point, the cardinal
$\lvert\hat{x}\rvert$ is called the _order_ of $x$. The fact that $D$ is
second countable and locally connected implies that the order of every point
is countable. We will denote by $\mathrm{End}(D)$, $\mathrm{Reg}(D)$, and
$\operatorname{Br}(D)$ the sets of endpoints, regular points, and branch
points of $D$, respectively. The set $\operatorname{Br}(D)$ is always
countable and $\mathrm{End}(D)$ is a $G_{\delta}$ subset of $D$.

We will denote

(4.1)
$\widehat{\operatorname{Br}(D)}=\bigsqcup_{x\in\operatorname{Br}(D)}\hat{x}.$

When $a\in\hat{x}$ will say that $a$ is a component _around_ $x$. For a set
$Z$, denote by $\Delta_{Z}$ the diagonal $\\{(z_{1},z_{2})\in
Z^{2}:z_{1}=z_{2}\\}$. Define the map

$\Phi\colon(\operatorname{Br}(D)\times
D)\setminus\Delta_{\operatorname{Br}(D)}\to\widehat{\operatorname{Br}(D)}$

by

(4.2) $\Phi(x,y)=a\iff a\in\hat{x}\text{ and }y\in a.$

In words, $\Phi(x,y)$ is the component around $x$ that contains $y$.

If $x,y\in D$ with $x\neq y$, we will denote by $[x,y]$ the unique arc
connecting $x$ and $y$ in $D$ and we let $(x,y)=[x,y]\setminus\\{x,y\\}$. We
define the function $K\colon D^{3}\to D$ by

(4.3) $\\{K(x,y,z)\\}=[x,y]\cap[y,z]\cap[x,z]$

and we say that $K(x,y,z)$ is the _center of $x$, $y$, and $z$_. Note that if
$x$, $y$, and $z$ are branch points or distinct endpoints, then $K(x,y,z)$ is
always a branch point.

An arc $[x,y]$ with $x,y\in\operatorname{Br}(D)$ is called _free_ if its
interior $(x,y)$ is an open set in $D$; equivalently, if $(x,y)$ does not
contain any branch points. The following theorem of Ważewski [Wazewski1923]
(see also Charatonik [Charatonik1980]) is the basis for much of what follows.

######  Theorem 4.1 (Ważewski).

For every $n\in\\{3,4,\ldots,\infty\\}$, there exists a unique, up to
homeomorphism, dendrite $W_{n}$ such that:

  * •

every branch point of $W_{n}$ has order $n$;

  * •

$W_{n}$ has no free arcs.

The second condition of the theorem is equivalent to the statement that
$\operatorname{Br}(W_{n})$ is dense in $W_{n}$. It is not hard to see that
$\mathrm{Reg}(W_{n})$ and $\mathrm{End}(W_{n})$ are also dense in $W_{n}$. The
dendrite $W_{n}$ is called the _Ważewski dendrite of order $n$_.

Throughout the paper, we will work with a fixed $n\in\\{3,4,\ldots,\infty\\}$
and we will write $W$ instead of $W_{n}$ to avoid clutter in the notation.

The homeomorphism group $\operatorname{Homeo}(W)$ acts transitively on each of
the sets $\operatorname{Br}(W)$, $\mathrm{Reg}(W)$, and $\mathrm{End}(W)$ and,
by density, a homeomorphism $g$ is determined uniquely by its action on each
of these sets. The action
$\operatorname{Homeo}(W)\curvearrowright\operatorname{Br}(W)$ defines an
injective continuous homomorphism
$\operatorname{Homeo}(W)\to\operatorname{Sym}(\operatorname{Br}(W))$, where
$\operatorname{Sym}(\operatorname{Br}(W))$ is equipped with the pointwise
convergence topology with $\operatorname{Br}(W)$ considered as a discrete set.
It was shown in [Duchesne2019] that this homomorphism is a topological group
embedding and, consequently, the image is closed and $\operatorname{Homeo}(W)$
is a non-archimedean Polish group.

Next we explain the construction of kaleidoscopic permutation groups carried
out in [Duchesne2019a]. To lighten notation, we will denote by $X$ the set
$\operatorname{Br}(W)$ and as in (4.1), we write:

$\widehat{X}=\bigsqcup_{x\in X}\hat{x}.$

For the rest of the section and throughout the paper, fix a set $M$ of
cardinality $n$. A function $\kappa\colon\widehat{X}\to M$ is called a
_coloring_ if for each $x\in X$, the restriction
$\kappa_{x}\coloneqq\kappa|_{\hat{x}}$ is a bijection.

######  Definition 4.2 ([Duchesne2019a]).

A coloring $\kappa$ is _kaleidoscopic_ if for all $x\neq y\in X$ and all
$a\neq b\in M$, there exists $z\in X\cap(x,y)$ such that $\kappa(\Phi(z,x))=a$
and $\kappa(\Phi(z,y))=b$.

It is shown in [Duchesne2019a] that a kaleidoscopic coloring exists and is
unique up to homeomorphism of $W$. Any coloring $\kappa$ also defines a
cocycle $\alpha_{\kappa}\colon\operatorname{Homeo}(W)\times
X\to\operatorname{Sym}(M)$ by

(4.4) $\alpha_{\kappa}(g,x)=\kappa_{g\cdot x}\circ
g|_{\hat{x}}\circ\kappa_{x}^{-1},$

called _local action of $g$ at $x$_, which satisfies the usual cocycle
identity:

(4.5) $\alpha_{\kappa}(gg^{\prime},x)=\alpha_{\kappa}(g,g^{\prime}\cdot
x)\alpha_{\kappa}(g^{\prime},x).$

Now let $\Gamma$ be a closed subgroup of $\operatorname{Sym}(M)$ and fix a
kaleidoscopic coloring $\kappa$ of $\widehat{X}$. Define the _kaleidoscopic
group_ $\mathcal{K}(\Gamma)$ by

(4.6)
$\mathcal{K}(\Gamma)=\\{g\in\operatorname{Homeo}(W):\alpha_{\kappa}(g,x)\in\Gamma\text{
for all }x\in X\\}.$

It follows from the uniqueness of the kaleidoscopic coloring that
$\mathcal{K}(\Gamma)$ does not depend on $\kappa$, up to conjugation by an
element of $\operatorname{Homeo}(W)$.

Below we collect several facts from [Duchesne2019a] that we will use.

######  Fact 4.3.

Let $\Gamma\leq\operatorname{Sym}(M)$ be a closed subgroup and let
$\mathcal{K}(\Gamma)$ be defined as above. Then the following hold:

  1. (i)

$\mathcal{K}(\Gamma)$ is a closed subgroup of $\operatorname{Homeo}(W)$ (and
therefore also of $\operatorname{Sym}(X)$);

  2. (ii)

The action $\mathcal{K}(\Gamma)\curvearrowright X$ is oligomorphic iff the
action $\Gamma\curvearrowright M$ is oligomorphic;

  3. (iii)

The actions of $\mathcal{K}(\Gamma)$ on $\mathrm{End}(W),\mathrm{Reg}(W),X$
are transitive. In particular, the action $\mathcal{K}(\Gamma)\curvearrowright
W$ is minimal.

  4. (iv)

For each $x\in X$, the map
$\alpha_{\kappa}(\cdot,x)\colon\mathcal{K}(\Gamma)_{x}\to\Gamma$ is a
surjective homomorphism.

  5. (v)

The action $\mathcal{K}(\Gamma)\curvearrowright\mathrm{End}(W)$ is primitive,
so the stabilizer $\mathcal{K}(\Gamma)_{\xi}$ is a maximal subgroup of
$\mathcal{K}(\Gamma)$.

###### Proof.

We point to the statements in [Duchesne2019a]. Items (i) and (ii) are Theorem
1.9 and Theorem 1.4, respectively. Item (iii) is Proposition 5.5, (iv) is
Proposition 5.1, and (v) is Corollary 5.11. ∎

###  4.2. Rooted kaleidoscopic groups

An important role in the theory of kaleidoscopic groups is played by the
stabilizer $\mathcal{K}(\Gamma)_{\xi}$ of an endpoint $\xi\in\mathrm{End}(W)$.
By Fact 4.3, this stabilizer does not depend on $\xi$, up to conjugation. In
this subsection we consider a rooted variant of the kaleidoscopic construction
and we describe the relation between this construction and the endpoint
stabilizer for a transitive $\Gamma$.

Let $\xi\in\mathrm{End}(W)$ be a fixed endpoint which we call the _root_. The
root allows us to define two other functions which will be useful: $\rho\colon
X\to\widehat{X}$ by

$\rho(x)=a\iff a\in\hat{x}\text{ and }\xi\in a$

and the _meet_ , ${\mathbin{\wedge}}\colon X^{2}\to X$ by

$x\mathbin{\wedge}y=K(x,y,\xi).$

Since the meet is associative, we can write $\bigwedge\bar{x}$ as a shorthand
for $x_{1}\mathbin{\wedge}\cdots\mathbin{\wedge}x_{k}$. The meet also defines
a partial order $\preceq$ on $X$ by

(4.7) $x\preceq y\iff x\mathbin{\wedge}y=x,$

which makes $X$ a meet semi-lattice.

We define root-kaleidoscopic colorings, the appropriate analogue of
kaleidoscopic colorings (cf. Definition 4.2) for rooted groups. They have
similar genericity properties but satisfy the additional condition that around
each branch point the color of the component containing the root is always the
same element $c$.

######  Definition 4.4.

Let $c\in M$ be a distinguished element. A _rooted coloring_ of $W$ is a
function $\kappa^{*}\colon\widehat{X}\to M$ such that for each $x$,
$\kappa^{*}_{x}\coloneqq\kappa^{*}|_{\hat{x}}$ is a bijection and
$\kappa^{*}(\rho(x))=c$. A rooted coloring $\kappa^{*}$ is called _root-
kaleidoscopic_ if for all $x\in X$, $y\in\left(\xi,x\right)$ and $a\neq c$ in
$M$, there is $z\in(y,x)$ with $\kappa^{*}(\Phi(z,x))=a$.

Concerning the existence of root-kaleidoscopic colorings, it is possible to
follow the approach of [Duchesne2019a] and show that the generic rooted
coloring is root-kaleidoscopic. However, we prefer to give a construction
based on a given kaleidoscopic coloring, which will be useful later.

Let $\kappa$ be a kaleidoscopic coloring (as in Definition 4.2). Let
$\sigma\colon M\to\operatorname{Sym}(M)$ be a function such that
$\sigma(c)=1_{\operatorname{Sym}(M)}$ and $\sigma(a)\cdot a=c$ for all $a\in
M$. Define

(4.8) $\kappa^{*}(a)=\sigma(\kappa(\rho(x)))\cdot\kappa(a),\quad\text{where
}a\in\hat{x}.$

######  Lemma 4.5.

The function $\kappa^{*}$ defined by (4.8) is a root-kaleidoscopic coloring of
$W$.

###### Proof.

That $\kappa^{*}$ is a rooted coloring is clear from the definition. To check
that it is root-kaleidoscopic, let $x$ and $y$ with $y\in(\xi,x)$ be given. By
kaleidoscopicity of $\kappa$, there is $z\in(y,x)$ such that
$\kappa(\Phi(z,x))=a$ and $\kappa(\Phi(z,y))=c$. But $\rho(z)=\Phi(z,y)$, so
$\sigma(\kappa(\rho(z)))=\sigma(c)$ is the identity. Therefore
$\kappa^{*}(\Phi(z,x))=\kappa(\Phi(z,x))=a$ and
$\kappa^{*}(\Phi(z,y))=\kappa(\Phi(z,y))=c$. ∎

Root-kaleidoscopic colorings are also unique up to translation by an element
of $\operatorname{Homeo}(W)_{\xi}$ (cf. Corollary 5.9). As in the non-rooted
case, $\kappa^{*}$ defines a cocycle
$\alpha_{\kappa^{*}}\colon\operatorname{Homeo}(W)_{\xi}\times
X\to\operatorname{Sym}(n)_{c}$ using the same formula (4.4).

Now let $\Delta\curvearrowright M$ be a permutation group which fixes some
$c\in M$. Fix a root-kaleidoscopic coloring $\kappa^{*}$. We define the _root-
kaleidoscopic group_ ${\mathcal{K}}^{*}(\Delta)$ by

(4.9)
${\mathcal{K}}^{*}(\Delta)=\\{g\in\operatorname{Homeo}(W)_{\xi}:\alpha_{\kappa^{*}}(g,x)\in\Delta\text{
for all }x\in X\\}.$

It follows from the uniqueness of the root-kaleidoscopic coloring that
${\mathcal{K}}^{*}(\Delta)$ does not depend on $\kappa^{*}$, up to conjugation
by an element of $\operatorname{Homeo}(W)_{\xi}$. When $\Gamma$ is transitive,
the root-kaleidoscopic group of a point stabilizer is equal to the stabilizer
of $\xi$ in $\mathcal{K}(\Gamma)$ as the next proposition shows.

######  Proposition 4.6.

Let $\Gamma\curvearrowright M$ be a transitive permutation group and let $c\in
M$. Then

${\mathcal{K}}^{*}(\Gamma_{c})=\mathcal{K}(\Gamma)_{\xi}.$

###### Proof.

By transitivity of $\Gamma$, there is $\sigma\colon M\to\Gamma$ such that
$\sigma(c)=1_{\Gamma}$ and $\sigma(a)\cdot a=c$ for all $a\in M$. Suppose that
$\kappa$ is a kaleidoscopic coloring and $\kappa^{*}$ is defined by (4.8) from
$\kappa$ and $\sigma$. It follows from the uniqueness of root-kaleidoscopic
colorings that $\kappa^{*}$ gives rise to ${\mathcal{K}}^{*}(\Gamma_{c})$ by
(4.9). From the definition (4.8) it is clear that for all
$g\in\operatorname{Homeo}(W)$ and $x\in X$,

$\alpha_{\kappa^{*}}(g,x)\in\Gamma_{c}\text{ if and only if
}\alpha_{\kappa}(g,x)\in\Gamma.$

Now, looking at the definitions (4.6) and (4.9), we obtain the desired
conclusion. ∎

##  5\. Structures for kaleidoscopic groups and endpoint stabilizers

In this section and the next lies the combinatorial core of this paper. The
goal is to prove the following statements, from which we will derive our
description of the UMFs of kaleidoscopic groups. If $\Gamma\curvearrowright M$
is a transitive permutation group and $\Delta\leq\Gamma_{c}$ is a closed
subgroup, for some $c\in M$, then:

  * •

if $\Delta$ is presyndetic in $\Gamma_{c}$, then ${\mathcal{K}}^{*}(\Delta)$
is presyndetic in ${\mathcal{K}}^{*}(\Gamma_{c})=\mathcal{K}(\Gamma)_{\xi}$
(5.14).

  * •

if $\Delta$ is oligomorphic then ${\mathcal{K}}^{*}(\Delta)$ is oligomorphic
(5.10).

  * •

if $\Delta$ is extremely amenable, so is ${\mathcal{K}}^{*}(\Delta)$ (6.9).

The first step is to reduce them to combinatorial problems, by representing
$\mathcal{K}(\Gamma)$, ${\mathcal{K}}^{*}(\Gamma_{c})$, and
${\mathcal{K}}^{*}(\Delta)$ as automorphism groups of appropriate
ultrahomogeneous structures.

###  5.1. Structures for the homeomorphism groups and endpoint stabilizers

First we explain how to view the full homeomorphism group of the Ważewski
dendrite $W_{n}$ as the automorphism group of an appropriate ultrahomogeneous
structure. A similar approach was already used by Kwiatkowska
[Kwiatkowska2018]; our structure is somewhat different as we use a signature
with function symbols rather than a relational one, and it is better adapted
to our treatment of kaleidoscopic groups.

For the rest of this section, we fix $n\in\\{3,4,\ldots,\infty\\}$ and as
before, we will write $W$ instead of $W_{n}$. We consider a signature
$\mathcal{L}_{\mathbf{X}}$ with one ternary function symbol $K$ and we define
a structure ${\mathbf{X}}$ with universe $X\coloneqq\operatorname{Br}(W)$ by
interpreting $K$ as the (restriction of) the center function given by (4.3).
It is a particular case of 5.4 that ${\mathbf{X}}$ is ultrahomogeneous.

If $A\subseteq X$ is a finite set, the substructure of ${\mathbf{X}}$
generated by $A$ is the smallest center-closed subset of $X$ containing $A$;
it can be visualized as a combinatorial tree whose geometric realization
embeds as a subdendrite of $W$. A finitely generated substructure of
${\mathbf{X}}$ is always finite. We denote
$\mathcal{F}_{\mathbf{X}}=\operatorname{Age}({\mathbf{X}})$.

The function $K$ allows us to define (via a quantifier-free formula) the
_betweenness relation_ $B$ by:

$B(x,y,z)\iff K(x,y,z)=y.$

Conversely, $K$ can be defined from $B$ by

$K(x_{1},x_{2},x_{3})=y\iff B(x_{1},y,x_{2})\text{ and }B(x_{1},y,x_{3})\text{
and }B(x_{2},y,x_{3}).$

Thus for checking whether a certain map between
$\mathcal{F}_{\mathbf{X}}$-structures is an embedding, it suffices to see that
it preserves $B$. This observation allows us to see that our structure is
enough to recover the dendrite, at least as far as automorphisms are
concerned.

######  Proposition 5.1.

$\operatorname{Aut}({\mathbf{X}})=\operatorname{Homeo}(W)$.

###### Proof.

By [Duchesne2019]*Proposition 2.4 and [Duchesne2019a]*Proposition 2.4, the
group of homeomorphisms of $W$ is equal to the group of permutations of $X$
which preserve $B$. ∎

If ${\mathbf{T}}\in\mathcal{F}_{{\mathbf{X}}}$ and $x\in T$, we can define, as
before, $\hat{x}$ as the set of connected components of $T\setminus\\{x\\}$
(where we view $T$ as a geometric tree) and $\widehat{T}=\bigsqcup_{x\in
T}\hat{x}$. This construction can be captured as an imaginary sort for
$\mathcal{F}_{\mathbf{X}}$. Indeed, consider the formulas

$\theta(x,y)\equiv\big{(}x\neq y\big{)}\text{ and
}\epsilon(x_{1},y_{1},x_{2},y_{2})=\big{(}x_{1}=x_{2}\land\neg
B(y_{1},x_{1},y_{2})\big{)}.$

It is clear that $\epsilon$ is precisely the equivalence relation
“$x_{1}=x_{2}$ and $y_{1}$ and $y_{2}$ are in the same component of
$T\setminus\\{x_{1}\\}$”. For a general $\mathcal{F}_{\mathbf{X}}$-structure
${\mathbf{Y}}$, we will denote by $\widehat{Y}$ the elements of this sort.

Now we describe a structure whose automorphism group is the stabilizer
$\operatorname{Homeo}(W)_{\xi}$ for an endpoint $\xi\in\mathrm{End}(W)$, which
we call the root, as in Section 4.2. Let $\mathcal{L}^{*}_{\mathbf{X}}$ be the
signature $\mathcal{L}_{\mathbf{X}}\sqcup\\{\boldsymbol{\xi}\\}$, where
$\boldsymbol{\xi}$ is a constant symbol. Let ${\mathbf{X}}^{*}$ be the
$\mathcal{L}^{*}_{\mathbf{X}}$-expansion of ${\mathbf{X}}$ obtained by taking
$X^{*}=X\sqcup\\{\xi\\}$, interpreting $\boldsymbol{\xi}$ as $\xi$, and
interpreting the function $K$ as usual by (4.3). Notice that this respects our
definition of expansion since $\xi\in\operatorname{dcl}(\emptyset)$ and $X$ is
$\operatorname{Aut}({\mathbf{X}}^{*})$-invariant. Recall that the root allows
us to define $\rho\colon X\to\widehat{X}$, the meet
${\mathbin{\wedge}}\colon(X^{*})^{2}\to X^{*}$ and the partial order $\prec$
on $X^{*}$. All of their definitions are by quantifier-free formulas, so they
are also valid for the age of ${\mathbf{X}}^{*}$.

Similarly as before, we have the following.

######  Proposition 5.2.

$\operatorname{Aut}({\mathbf{X}}^{*})=\operatorname{Homeo}(W)_{\xi}$.

###### Proof.

The $\supseteq$ inclusion follows from Proposition 5.1 and the definition of
${\mathbf{X}}^{*}$. For the other, suppose that
$g\in\operatorname{Aut}({\mathbf{X}}^{*})$, viewed as a homeomorphism of $W$
via the identification
$\operatorname{Aut}({\mathbf{X}}^{*})\subseteq\operatorname{Aut}({\mathbf{X}})=\operatorname{Homeo}(W)$,
moves $\xi$ to another endpoint $\xi^{\prime}$. Let $x\neq y\in X$ be such
that $\xi,x,y,\xi^{\prime}$ lie on an arc, in this order. Then, on the one
hand, we have that $B(x,y,\xi^{\prime})$ and as $g$ is a homeomorphism, we
obtain that $B(g^{-1}\cdot x,g^{-1}\cdot y,\xi)$. On the other, $g$ is an
automorphism of ${\mathbf{X}}^{*}$, so it preserves the binary relation
$B(\cdot,\cdot,\xi)$ and the fact that $B(y,x,\xi)$ implies that
$B(g^{-1}\cdot y,g^{-1}\cdot x,\xi)$, contradiction. ∎

###  5.2. Structures for kaleidoscopic groups

Next we turn to kaleidoscopic groups. Let $\mathcal{L}_{0}$ be a relational
signature, let ${\mathbf{M}}$ be an ultrahomogeneous
$\mathcal{L}_{0}$-structure with $|M|=n$ and let
$\Gamma=\operatorname{Aut}({\mathbf{M}})$. (As we explained earlier, this
entails no loss of generality: any closed subgroup
$\Gamma\leq\operatorname{Sym}(M)$ can be represented in this fashion.)

Fix $\kappa\colon\widehat{X}\to M$, a kaleidoscopic coloring of $W$. We use
$\kappa$ to define an expansion $\mathcal{K}({\mathbf{M}})$ of ${\mathbf{X}}$.

######  Definition 5.3.

Let $\mathcal{K}({\mathbf{M}})$ be the relational expansion of ${\mathbf{X}}$
in the signature $\mathcal{L}_{\mathbf{X}}\sqcup\mathcal{L}_{0}$, where the
relations in $\mathcal{L}_{0}$ are interpreted on the sort $\widehat{X}$, as
follows. For a $k$-ary relation $R\in\mathcal{L}_{0}$ and
$\bar{a}=(a_{0},\ldots,a_{k-1})\in\widehat{X}^{k}$, we define

$R^{\mathcal{K}({\mathbf{M}})}(\bar{a})\iff\exists x\in X\
a_{0},\ldots,a_{k-1}\in\hat{x}\text{ and
}R^{\mathbf{M}}(\kappa_{x}(\bar{a})).$

In other words, for all $x\in X$, $\kappa_{x}\colon\hat{x}\to M$ is an
$\mathcal{L}_{0}$-isomorphism.

It follows easily from the results of [Duchesne2019a] that this structure
captures the kaleidoscopic construction.

######  Proposition 5.4.

The groups $\mathcal{K}(\Gamma)$ and
$\operatorname{Aut}(\mathcal{K}({\mathbf{M}}))$ coincide. Moreover, the
structure $\mathcal{K}({\mathbf{M}})$ is ultrahomogeneous and does not depend
on the choice of kaleidoscopic coloring, up to isomorphism.

###### Proof.

The first assertion follows from the definitions.

Next we prove ultrahomogeneity. Let ${\mathbf{S}}$ and ${\mathbf{T}}$ be two
finite substructures of $\mathcal{K}({\mathbf{M}})$ and let
$f\colon{\mathbf{S}}\to{\mathbf{T}}$ be an isomorphism. For every $s\in S$,

$\kappa_{f(s)}\circ
f\circ\kappa_{s}^{-1}\colon\kappa_{s}[\hat{s}]\to\kappa_{f(s)}[\widehat{f(s)}]$

is an isomorphism of finite substructures of ${\mathbf{M}}$. As ${\mathbf{M}}$
is ultrahomogeneous, this isomorphism extends to some
$\gamma_{s}\in\operatorname{Aut}({\mathbf{M}})$. Now it follows from
[Duchesne2019a]*Lemma 5.8 that there is
$g\in\mathcal{K}(\Gamma)=\operatorname{Aut}(\mathcal{K}({\mathbf{M}}))$ which
extends $f$.

Finally, let $\kappa^{\prime}\colon\widehat{X}\to M$ be some other
kaleidoscopic coloring. By [Duchesne2019a]*Theorem 3.4, there is
$h\in\operatorname{Homeo}(W)$ such that $\kappa=\kappa^{\prime}\circ h$ and it
is easy to see that $h$ defines an isomorphism between
$\mathcal{K}_{\kappa}({\mathbf{M}})$ and
$\mathcal{K}_{\kappa^{\prime}}({\mathbf{M}})$. ∎

###  5.3. Structures for rooted kaleidoscopic groups

Let now $\mathcal{L}_{0}$ be a relational signature as before and let
${\mathbf{N}}$ be an ultrahomogeneous structure in the signature
$\mathcal{L}_{0}\sqcup\\{\mathbf{c}\\}$, where $\mathbf{c}$ is a constant
symbol. As usual, let $N$ denote the universe of ${\mathbf{N}}$ and suppose
that $|N|=n$. We denote by $c\in N$ the interpretation of the constant symbol
$\mathbf{c}$.

We also fix some root-kaleidoscopic coloring $\kappa^{*}$ of $W$, with
$\kappa^{*}(\rho(x))=c$ for all $x\in X$. We use it to define the expansion
${\mathcal{K}}^{*}({\mathbf{N}})$ of the structure ${\mathbf{X}}^{*}$.

######  Definition 5.5.

Let ${\mathcal{K}}^{*}({\mathbf{N}})$ be the relational expansion of
${\mathbf{X}}^{*}$ to the signature
$\mathcal{L}^{*}_{\mathbf{X}}\sqcup\mathcal{L}_{0}$ defined as follows: for
each $R\in\mathcal{L}_{0}$ of arity $k$, we interpret $R$ as a relation on
$\widehat{X}$, which for $\bar{a}=(a_{0},\ldots,a_{k-1})\in\widehat{X}^{k}$,
is given by

$R^{{\mathcal{K}}^{*}({\mathbf{N}})}(\bar{a})\iff\exists x\in X\
a_{0},\ldots,a_{k-1}\in\hat{x}\text{ and
}R^{\mathbf{N}}(\kappa^{*}(\bar{a})).$

It will be convenient to establish some combinatorial notation and
terminology. For us, a _combinatorial tree_ is just an element of the age of
${\mathbf{X}}$. Two vertices of a tree are _adjacent_ if there is no other
vertex between them. Similarly, we call a _rooted tree_ an element of the age
of ${\mathbf{X}}^{*}$. Note that every combinatorial tree as well as every
rooted tree is finite. The _root_ of a rooted tree is the interpretation of
$\boldsymbol{\xi}$. A _leaf_ of a rooted tree is a $\prec$-maximal element.
The _height_ of an element $t$ of a rooted tree ${\mathbf{T}}$ is the number
$\lvert\\{s\in T:s\prec t\text{ and }s\neq t\\}\rvert$. The _height_ of the
tree is the maximum of the heights of its elements. If $t\in T$, we denote by
$\operatorname{Succ}(t)$ the set of $\prec$-immediate successors of $t$.
Notice that the immediate successors are adjacent to $t$. If $A\subseteq T$,
we denote by $\bigwedge A$ the $\prec$-greatest lower bound of $A$. Every
rooted tree ${\mathbf{T}}$ with more than one vertex also has another
distinguished vertex apart from the root, namely
$\bigwedge(T\setminus\boldsymbol{\xi})$, the unique immediate successor of
$\boldsymbol{\xi}$. As we will use this vertex frequently, we denote it by
$r_{{\mathbf{T}}}$. Note, however, that $r_{{\mathbf{T}}}$ is not preserved by
embeddings. For $t\in T$, let

${\mathbf{T}}(t)\coloneqq\\{s\in T:t\preceq s\\}\cup\\{\xi\\}.$

So $t=r_{{\mathbf{T}}(t)}$.

Our next step is to describe a cofinal family in the age of
${\mathcal{K}}^{*}({\mathbf{N}})$ that is easier to work with.

######  Definition 5.6.

Let ${\mathbf{A}}$ be a finite structure in the signature
$\mathcal{L}_{0}\cup\\{\mathbf{c}\\}$ and let $h$ be a natural number. We
define ${\mathbf{A}}[h]$ to be the finite
$\mathcal{L}^{*}_{\mathbf{X}}\sqcup\mathcal{L}_{0}$-structure such that:

  * •

the reduct of ${\mathbf{A}}[h]$ to $\mathcal{L}^{*}_{\mathbf{X}}$ is a rooted
tree whose every leaf has height $h+1$ and such that every non-leaf different
from $\boldsymbol{\xi}$ has exactly $|A|-1$ immediate successors;

  * •

for any $s\in A[h]$ which is not a leaf and not equal to $\boldsymbol{\xi}$,
the $\mathcal{L}_{0}\sqcup\\{\mathbf{c}\\}$-structure on $\hat{s}$, given by
interpreting $\mathbf{c}$ as $\rho(s)$, is isomorphic to ${\mathbf{A}}$. We
denote by $\lambda_{s}$ this isomorphism.

[scale=0.7]drawings/structures-Ah-BW

Figure 1. A graphical representation of $A[h]$.

######  Proposition 5.7.

The family of structures

$\\{{\mathbf{A}}[h]:{\mathbf{A}}\in\operatorname{Age}({\mathbf{N}}),h=0,1,\ldots\\}$

is cofinal in $\operatorname{Age}({\mathcal{K}}^{*}({\mathbf{N}}))$.

###### Proof.

First we show that for any ${\mathbf{A}}\in\operatorname{Age}({\mathbf{N}})$
and $h\in\mathbb{N}$, there is a substructure of
${\mathcal{K}}^{*}({\mathbf{N}})$ isomorphic to ${\mathbf{A}}[h]$. We do so by
fixing ${\mathbf{A}}\subseteq{\mathbf{N}}$ and arguing by induction on $h$.
For $h=0$, choose any $x\in X$. Then the generated substructure $\langle
x\rangle=\\{x,\xi\\}$ of ${\mathbf{X}}^{*}$ is isomorphic to
${\mathbf{A}}[0]$. Now suppose we have found
${\mathbf{S}}\subseteq{\mathcal{K}}^{*}({\mathbf{N}})$ which is isomorphic to
${\mathbf{A}}[h-1]$, for some $h>0$. For each leaf $x\in S$, and each $a\in
A\setminus\\{c\\}$, pick a point $x_{a}\in(\kappa^{*}_{x})^{-1}(a)$. Let
${\mathbf{S}}^{\prime}$ be the substructure of $\mathcal{K}^{*}({\mathbf{N}})$
consisting of ${\mathbf{S}}$ and all of these new points $x_{a}$. Notice that
this set is closed under $K$. It is also easy to see that
${\mathbf{S}}^{\prime}$ is isomorphic to ${\mathbf{A}}[h]$.

Conversely, let ${\mathbf{S}}$ be a finite substructure of
${\mathcal{K}}^{*}({\mathbf{N}})$. Let $A=\\{\kappa^{*}(a):a\in\widehat{S}\\}$
and let ${\mathbf{A}}$ be the substructure of ${\mathbf{N}}$ with universe
$A$. We also let $h$ be the height of ${\mathbf{S}}$ and we define an
embedding $f$ of ${\mathbf{S}}$ into ${\mathbf{A}}[h]$ by induction as
follows. First we let $f(\xi)=\xi$ and
$f(r_{\mathbf{S}})=r_{{\mathbf{A}}[h]}$. Then, supposing that for some non-
leaf $s\in S$, $f(s)$ has been defined, we let

$f|_{\operatorname{Succ}(s)}=\lambda_{f(s)}^{-1}\circ\kappa_{s}^{*}.$

It is easy to check that $f$ is an embedding. ∎

Now let $\Delta\curvearrowright N$ be a permutation group which fixes some
$c\in N$. Let ${\mathbf{N}}$ be an ultrahomogeneous structure in the signature
$\mathcal{L}_{0}\cup\\{\mathbf{c}\\}$, where $\mathbf{c}$ is a constant
symbol, interpreted as $c$, such that
$\operatorname{Aut}({\mathbf{N}})=\Delta$.

######  Theorem 5.8.

The groups ${\mathcal{K}}^{*}(\Delta)$ and
$\operatorname{Aut}({\mathcal{K}}^{*}({\mathbf{N}}))$ coincide. Moreover,
${\mathcal{K}}^{*}({\mathbf{N}})$ is ultrahomogeneous and does not depend on
the choice of $\kappa^{*}$, up to isomorphism.

###### Proof.

The first assertion follows from the definitions.

To verify ultrahomogeneity, we apply the standard back-and-forth argument.
Given an isomorphism $f\colon{\mathbf{S}}\to{\mathbf{T}}$ between finite
substructures of ${\mathcal{K}}^{*}({\mathbf{N}})$ and a point
$x\in{\mathcal{K}}^{*}({\mathbf{N}})\setminus S$, it suffices to extend $f$ to
an embedding $f^{\prime}\colon\langle
S,x\rangle\to{\mathcal{K}}^{*}({\mathbf{N}})$.

Denote ${\mathbf{S}}^{\prime}=\langle S,x\rangle$ and let $y$ be the immediate
$\prec$-predecessor of $x$ in ${\mathbf{S}}^{\prime}$. There are three cases
depending on the position of $y$.

Case 1. $y\in S$. Let $a=\Phi(y,x)$. The isomorphism $f$ induces an
isomorphism
$\tilde{f}\colon\kappa^{*}_{y}[\widehat{S}]\to\kappa^{*}_{f(y)}[\widehat{T}]$
between finite substructures of ${\mathbf{N}}$. Since ${\mathbf{N}}$ is
ultrahomogeneous, there is $\tilde{g}\in\operatorname{Aut}({\mathbf{N}})$ that
extends $\tilde{f}$. Let
$b=(\kappa^{*}_{f(y)})^{-1}(\tilde{g}(\kappa^{*}(a)))$. Finally, choose any
$x^{\prime}\in b\setminus T$ and let $f^{\prime}=f\cup\\{(x,x^{\prime})\\}$.

Case 2. $y\in(\xi,r_{\mathbf{S}})$. Let $a=\Phi(y,r_{\mathbf{S}})$ and notice
that $\kappa^{*}(a)\neq c$. By root-kaleidoscopicity of $\kappa^{*}$, there is
$y^{\prime}\in(\xi,f(r_{\mathbf{S}}))$ such that
$\kappa^{*}(\Phi(y^{\prime},f(r_{\mathbf{S}})))=\kappa^{*}(a)$. We let
$f^{\prime}=f\cup\\{(y,y^{\prime})\\}$. If $x=y$, we are done; if not, we have
reduced to Case 1.

Case 3. There are $z_{1},z_{2}\in S$ adjacent in ${\mathbf{S}}$, with
$z_{1}\in(\xi,z_{2})$ and such that $y\in(z_{1},z_{2})$. Let $a=\Phi(y,z_{2})$
and note that $\kappa^{*}(a)\neq c$. By root-kaleidoscopicity of $\kappa^{*}$,
there is $y^{\prime}\in(f(z_{1}),f(z_{2}))$ such that
$\kappa^{*}(\Phi(y^{\prime},f(z_{2})))=\kappa^{*}(a)$. We let
$f^{\prime}=f\cup\\{(y,y^{\prime})\\}$. If $x=y$, we are done; otherwise, we
have again reduced to Case 1. This concludes the proof of ultrahomogeneity.

If ${\kappa^{*}}^{\prime}$ is another root-kaleidoscopic coloring, then
${\mathcal{K}}^{*}_{\kappa^{*}}({\mathbf{N}})$ and
${\mathcal{K}}^{*}_{{\kappa^{*}}^{\prime}}({\mathbf{N}})$ are two
ultrahomogeneous structures with the same age (by Proposition 5.7), so they
are isomorphic by Fraïssé’s theorem. ∎

By considering a structure ${\mathbf{N}}$ with trivial automorphism group, we
obtain the following.

######  Corollary 5.9.

If $\kappa^{*}_{1}$ and $\kappa^{*}_{2}$ are two root-kaleidoscopic colorings,
then there exists $g\in\operatorname{Homeo}(W)_{\xi}$ such that
$\kappa^{*}_{2}=\kappa^{*}_{1}\circ g$.

###### Proof.

If $g$ is an isomorphism between
${\mathcal{K}}^{*}_{\kappa^{*}_{1}}({\mathbf{N}})$ and
${\mathcal{K}}^{*}_{\kappa^{*}_{2}}({\mathbf{N}})$, then $g$ preserves
betweenness on $X$ and $\xi$, so $g\in\operatorname{Homeo}(W)_{\xi}$. Since
${\mathbf{N}}$ has trivial automorphism group, it follows that
$\kappa^{*}_{2}=\kappa^{*}_{1}\circ g$. ∎

######  Corollary 5.10.

Let $\Delta\curvearrowright N$ be a permutation group which fixes some $c\in
N$. Then ${\mathcal{K}}^{*}(\Delta)\curvearrowright X$ is oligomorphic if and
only if $\Delta\curvearrowright N$ is.

###### Proof.

$(\Leftarrow)$ Let ${\mathbf{N}}$ be an ultrahomogeneous structure such that
$\operatorname{Aut}({\mathbf{N}})=\Delta$, as above. The hypothesis means that
we can choose a signature for ${\mathbf{N}}$ that contains, apart from
$\mathbf{c}$, finitely many relation symbols in each arity. Theorem 5.8
implies that the conclusion is equivalent to: for every $k$, there are only
finitely many $k$-generated structures in
$\operatorname{Age}({\mathcal{K}}^{*}({\mathbf{N}}))$, up to isomorphism. This
follows from the fact that a $k$-generated substructure of ${\mathbf{X}}^{*}$
has at most $2k$ elements and that a finite substructure
${\mathbf{T}}\subseteq{\mathcal{K}}^{*}({\mathbf{N}})$ is determined by its
reduct to the signature of ${\mathbf{X}}^{*}$ and the isomorphism types of
$\hat{t}$ for $t\in T$, as elements of $\operatorname{Age}({\mathbf{N}})$.

$(\Rightarrow)$ Suppose there is $k>0$ such that there are infinitely many
pairwise non-isomorphic ${\mathbf{A}}_{1},{\mathbf{A}}_{2},\ldots$ elements of
$\operatorname{Age}({\mathbf{N}})$ with $k+1$ elements (including
$\mathbf{c}$). Then ${\mathbf{A}}_{1}[1],{\mathbf{A}}_{2}[1],\ldots$ are
$k$-generated non-isomorphic elements of
$\operatorname{Age}({\mathcal{K}}^{*}({\mathbf{N}}))$. ∎

Now fix $x\in X$ and a component $a\in\hat{x}$ which does not contain the root
$\xi$. Then $a\cup\\{x\\}$ is a dendrite homeomorphic to $W$ (by Theorem 4.1)
and $\kappa^{*}$ restricted to it is again a root-kaleidoscopic coloring, this
time with root $x$. By Corollary 5.9, there exists a homeomorphism
$h_{a}\colon W\to a\cup\\{x\\}$ mapping $\xi$ to $x$ and such that
$\kappa^{*}=\kappa^{*}\circ h_{a}$. Using these homeomorphisms, one can run
the argument in [Duchesne2019a]*Proposition 5.1 to obtain the following.

######  Corollary 5.11.

Let $\Delta\curvearrowright n$ be a permutation group which fixes some $c\in
n$ and let $x\in X$. Then the homomorphism
$\alpha_{\kappa^{*}}(\cdot,x)\colon{\mathcal{K}}^{*}(\Delta)_{x}\to\Delta$ is
_split-surjective_ , i.e., there exists a continuous homomorphism
$\phi\colon\Delta\to{\mathcal{K}}^{*}(\Delta)_{x}$ with a closed image such
that $c_{\kappa}^{*}(\cdot,x)\circ\phi=\operatorname{id}_{\Delta}$.

###  5.4. The endpoint stabilizer and root-kaleidoscopic groups

Let $\mathcal{L}_{0}$ be a relational signature, let ${\mathbf{M}}$ be an
ultrahomogeneous $\mathcal{L}_{0}$-structure with universe $M$ and let
$\Gamma=\operatorname{Aut}({\mathbf{M}})$. Assume that the action
$\Gamma\curvearrowright M$ is transitive. Let $c\in M$ and let $\Delta$ be a
closed subgroup of $\Gamma_{c}$. In this subsection, we explain how to
represent ${\mathcal{K}}^{*}(\Delta)$ as the automorphism group of
${\mathcal{K}}^{*}({\mathbf{N}})$ for an appropriate expansion ${\mathbf{N}}$
of the structure ${\mathbf{M}}$ satisfying
$\operatorname{Aut}({\mathbf{N}})=\Delta$.

Let $\kappa$ be some fixed kaleidoscopic coloring of $W$. We have seen in
Proposition 5.4 that
$\mathcal{K}(\Gamma)=\operatorname{Aut}(\mathcal{K}({\mathbf{M}}))$. Let
${\mathbf{M}}_{\mathbf{c}}$ be the expansion of ${\mathbf{M}}$ to the
signature $\mathcal{L}_{0}\sqcup\\{\mathbf{c}\\}$ obtained by interpreting the
symbol $\mathbf{c}$ as $c$. Then
$\operatorname{Aut}({\mathbf{M}}_{\mathbf{c}})=\Gamma_{c}$. Let ${\mathbf{N}}$
be an expansion of ${\mathbf{M}}_{\mathbf{c}}$ to a signature
$\mathcal{L}_{0}\sqcup\mathcal{L}_{1}\sqcup\\{\mathbf{c}\\}$ such that
$\mathcal{L}_{1}$ is relational, ${\mathbf{N}}$ is ultrahomogeneous, and
$\operatorname{Aut}({\mathbf{N}})=\Delta$.

We have that ${\mathcal{K}}^{*}(\Gamma_{c})=\mathcal{K}(\Gamma)_{\xi}$ by 4.6,
and therefore that
${\mathcal{K}}^{*}(\Delta)\leq{\mathcal{K}}^{*}(\Gamma_{c})\leq\mathcal{K}(\Gamma)$.
We see how this is reflected at the level of structures and expansions.

######  Proposition 5.12.

Let $\Gamma\curvearrowright M$ be transitive and let ${\mathbf{M}}$,
${\mathbf{M}}_{\mathbf{c}}$ and ${\mathbf{N}}$ be defined as above. Then the
following hold:

  1. (i)

${\mathcal{K}}^{*}({\mathbf{N}})$ is a relational expansion of
${\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}})$ and
${\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}})$ is an expansion of
$\mathcal{K}({\mathbf{M}})$;

  2. (ii)

${\mathcal{K}}^{*}(\Delta)=\operatorname{Aut}({\mathcal{K}}^{*}({\mathbf{N}}))$
and
${\mathcal{K}}^{*}(\Gamma_{c})=\operatorname{Aut}({\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}}))=\mathcal{K}(\Gamma)_{\xi}$.

###### Proof.

(i) It is clear that ${\mathcal{K}}^{*}({\mathbf{N}})$ is a relational
expansion of ${\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}})$. The only symbol
of the signature of ${\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}})$ which is
not in the signature of $\mathcal{K}({\mathbf{M}})$ is the constant symbol
$\boldsymbol{\xi}$ and the respective universes of $\mathcal{K}({\mathbf{M}})$
and ${\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}})$ are $X$ and
$X\cup\\{\xi\\}$. Thus to check the claim, it suffices to prove that the
identity on $X$ is an embedding, that is, that the interpretations of the
other symbols agree on $X$ and $\widehat{X}$. We have already checked that
${\mathbf{X}}^{*}$ is an expansion of ${\mathbf{X}}$. For a relation symbol
$R$ in $\mathcal{L}_{0}$ and $\bar{a}\in\hat{x}^{k}$, we have

$R^{{\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}})}(\bar{a})\iff
R^{{\mathbf{M}}_{\mathbf{c}}}(\kappa^{*}(\bar{a}))\iff
R^{\mathbf{M}}(\kappa(\bar{a}))\iff R^{\mathcal{K}({\mathbf{M}})}(\bar{a}).$

The middle equivalence follows from the definition (4.8) of $\kappa^{*}$ and
the fact that we can use $\sigma$ which take values in
$\Gamma=\operatorname{Aut}({\mathbf{M}})$.

(ii) This is 4.6 and Theorem 5.8 applied to
$\operatorname{Aut}({\mathbf{M}}_{\mathbf{c}})=\Gamma_{c}$. ∎

Using the identification in point (ii) above and 5.10, we obtain the
following.

######  Corollary 5.13.

If $\Gamma\curvearrowright M$ is transitive, then $\mathcal{K}(\Gamma)_{\xi}$
is co-precompact in $\mathcal{K}(\Gamma)$ if and only if $\Gamma$ is
oligomorphic.

######  Proposition 5.14.

If $\Gamma\curvearrowright M$ is a transitive permutation group and
$\Delta\leq\Gamma_{c}$ is a closed presyndetic subgroup, for some $c\in M$,
then ${\mathcal{K}}^{*}(\Delta)$ is presyndetic in $\mathcal{K}(\Gamma)_{\xi}$
and in $\mathcal{K}(\Gamma)$.

###### Proof.

Let ${\mathbf{M}}$, ${\mathbf{M}}_{\mathbf{c}}$ and ${\mathbf{N}}$ be defined
as above. We assume that ${\mathbf{N}}$ is a minimal expansion of
${\mathbf{M}}_{\mathbf{c}}$ and prove that ${\mathcal{K}}^{*}({\mathbf{N}})$
is a minimal expansion of ${\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}})$. Fix
${\mathbf{B}}^{\prime}\in\operatorname{Age}({\mathbf{N}})$ and let
${\mathbf{A}}\in\operatorname{Age}({\mathbf{M}}_{\mathbf{c}})$ be a witness of
minimality for ${\mathbf{B}}^{\prime}$. We show that ${\mathbf{A}}[h]$ is a
witness of minimality for ${\mathbf{B}}^{\prime}[h]$, for each
$h\in\mathbb{N}$. By 5.7 and Proposition 3.7, this will be enough.

So fix an expansion ${\mathbf{S}}$ of ${\mathbf{A}}[h]$. Then
${\mathbf{S}}\upharpoonright\mathcal{L}_{\mathbf{X}}={\mathbf{A}}[h]\upharpoonright\mathcal{L}_{\mathbf{X}}$.
For each $s\in{\mathbf{S}}$, which is not a leaf nor $\boldsymbol{\xi}$, the
structure $\kappa^{*}[\hat{s}]$ is an expansion of ${\mathbf{A}}$, and thus
there is an embedding
$g_{s}\colon{\mathbf{B}}^{\prime}\to\kappa^{*}[\hat{s}]$. Therefore there is a
level-preserving embedding $f$ of ${\mathbf{B}}^{\prime}[h]$ into
${\mathbf{S}}$ which is given by mapping the $r_{{\mathbf{B}}[h]}$ to
$r_{{\mathbf{S}}}$ and, if $f$ is defined for $p$ and $s$ is an immediate
successor of $p$, letting $f(s)$ be the immediate successor of $f(p)$ which
belongs to the component $(\kappa^{*}_{f(p)})^{-1}\circ
g_{f(p)}\circ\kappa^{*}_{p}(\Phi(p,s))$.

Finally, note that as the action $\mathcal{K}(\Gamma)\curvearrowright W$ is
minimal and the orbit $\mathcal{K}(\Gamma)\cdot\xi$ is $G_{\delta}$, it
follows from Proposition 2.7 and the fact that highly proximal extensions
preserve minimality that the action
$\mathcal{K}(\Gamma)\curvearrowright\operatorname{Sa}(\mathcal{K}(\Gamma)/\mathcal{K}(\Gamma)_{\xi})$
is minimal. Thus, by [Zucker2021]*Proposition 6.6, $\mathcal{K}(\Gamma)_{\xi}$
is presyndetic in $\mathcal{K}(\Gamma)$ and by Proposition 2.2 and what we
already proved, ${\mathcal{K}}^{*}(\Delta)$ is also presyndetic in
$\mathcal{K}(\Gamma)$. ∎

###  5.5. Rendering structures full

In our representation theorems of flows as type spaces, it will be important
that we deal with full structures. It is unfortunately not true in general
that $\mathcal{K}({\mathbf{M}})$ is full, even if we start with a full
structure ${\mathbf{M}}$, unless $\operatorname{Aut}({\mathbf{M}})$ is
oligomorphic (or, equivalently, ${\mathbf{M}}$ is $\aleph_{0}$-categorical in
a countable language). So our remedy is to render them qf-full by expanding
the language to include relations for all
$\operatorname{Aut}(\mathcal{K}({\mathbf{M}}))$-invariant sets of tuples. We
denote this full expansion by $\overline{\mathcal{K}}({\mathbf{M}})$ and its
signature by $\overline{\mathcal{L}}$ and note that by Theorem 3.3, it
eliminates quantifiers (and, of course, its automorphism group is equal to
$\operatorname{Aut}(\mathcal{K}({\mathbf{M}}))$). If
$\operatorname{Aut}({\mathbf{M}})$ is transitive and $c\in M$, we define
$\overline{\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}})$ as the relational
expansion of ${\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}})$ to the signature
$\overline{\mathcal{L}}_{\boldsymbol{\xi}}\coloneqq\overline{\mathcal{L}}\sqcup\\{\boldsymbol{\xi}\\}$.
Note that this expansion does not change the automorphism group of
${\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}})$ and that for any quantifier-
free $\overline{\mathcal{L}}$-formula $\phi(\bar{v})$ and $\bar{x}\in
X^{\bar{v}}$, we have that
$\overline{\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}})\models\phi(\bar{x})\iff\overline{\mathcal{K}}({\mathbf{M}})\models\phi(\bar{x})$.

We will call a formula $\phi(\bar{v})$ _isolating_ if it isolates a type,
i.e., $\phi(\overline{\mathcal{K}}({\mathbf{M}}))$ is a single
$\operatorname{Aut}(\mathcal{K}({\mathbf{M}}))$-orbit. Then every
$\overline{\mathcal{L}}$-formula is equivalent in
$\overline{\mathcal{K}}({\mathbf{M}})$ to (a possibly infinite) disjunction of
isolating formulas.

######  Lemma 5.15.

Suppose that $\operatorname{Aut}({\mathbf{M}})\curvearrowright M$ is
transitive, ${\mathbf{M}}$ is ultrahomogeneous, and $c\in M$, and define
$\mathcal{K}({\mathbf{M}})$ and ${\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}})$
as before. Then the structure
$\overline{\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}})$ is full and, moreover,
every formula is equivalent in
$\overline{\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}})$ to an existential one.

###### Proof.

Let $G=\operatorname{Aut}(\mathcal{K}({\mathbf{M}}))$ and recall that then
$G_{\xi}=\operatorname{Aut}({\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}}))$.
Let $D\subseteq(X\cup\\{\xi\\})^{k}$ be a $G_{\xi}$-invariant set which we
want to prove is definable. By taking a conjunction with a formula stating
which of the $k$ variables are equal to $\boldsymbol{\xi}$, we can assume that
tuples in $D$ do not contain $\xi$, i.e., $D\subseteq X^{k}$. Let us write
$D=\bigsqcup_{\alpha}D_{\alpha}$, where each $D_{\alpha}$ is a
$G_{\xi}$-orbit.

By ultrahomogeneity, for each $\alpha$, all the tuples in $D_{\alpha}$ have
the same type, so there is some
$\overline{\mathcal{L}}_{\boldsymbol{\xi}}$-formula $\theta_{\alpha}(\bar{v})$
which isolates it. We may assume that $\Phi(\xi,v_{i})$ does not occur in
$\theta_{\alpha}$, for any $1\leq i\leq k$, since no atomic formulas
mentioning the unique component around $\xi$ hold in
$\overline{\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}})$.

Let $\phi_{\alpha}(u,\bar{v})$ be the $\overline{\mathcal{L}}$-formula
obtained by substituting each occurrence of $\boldsymbol{\xi}$ in
$\theta_{\alpha}$ by $u$.

######  Claim.

For any $\bar{x}\in X^{k}$,
${\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}})\models\theta_{\alpha}(\bar{x})$
if and only for some (equivalently, for all) $z\in\rho(\bigwedge\bar{x})$, it
holds that $\mathcal{K}({\mathbf{M}})\models\phi_{\alpha}(z,\bar{x})$

###### Proof.

Since $\Phi(\boldsymbol{\xi},v_{i})$ does not occur in $\theta_{\alpha}$,
neither does $\Phi(u,v_{i})$ in $\phi_{\alpha}$. Therefore it is only a matter
of noting that for any $z\in\rho(\bigwedge\bar{x})$ and $1\leq i,j\leq k$, we
have $K(z,x_{i},x_{j})=K(\xi,x_{i},x_{j})$ and
$\Phi(x_{i},z)=\Phi(x_{i},\xi)$. ∎

Using the qf-fullness of $\overline{\mathcal{K}}({\mathbf{M}})$, let
$\phi(u,\bar{v})$ be the quantifier free $\overline{\mathcal{L}}$-formula
equivalent to the possibly infinite disjunction
$\bigvee_{\alpha}\phi_{\alpha}(u,\bar{v})$ in
$\overline{\mathcal{K}}({\mathbf{M}})$. Then we claim that the formula

(5.1) $\psi(\bar{v})\coloneqq\exists u\in\rho(\bigwedge\bar{v})\
\phi(u,\bar{v})$

defines $D$. If $\bar{x}\in D$, then there is $\alpha$ such that $\bar{x}\in
D_{\alpha}$, so
${\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}})\models\theta_{\alpha}(\bar{x})$.
The Claim tells us that for any $z\in\rho(\bigwedge\bar{v})$,
$\overline{\mathcal{K}}({\mathbf{M}})\models\phi_{\alpha}(z,\bar{x})$. As
$\phi_{\alpha}$ is quantifier-free, this implies that
$\overline{\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}})\models\phi_{\alpha}(z,\bar{x})$,
so
$\overline{\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}})\models\psi(\bar{x})$.

Conversely, suppose that
$\overline{\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}})\models\psi(\bar{x})$.
It follows that there are $z\in\rho(\bigwedge\bar{x})$ and $\alpha$ such that
$\overline{\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}})\models\phi_{\alpha}(z,\bar{x})$.
Since $\phi_{\alpha}$ doesn’t mention $\boldsymbol{\xi}$,
$\overline{\mathcal{K}}({\mathbf{M}})\models\phi_{\alpha}(z,\bar{x})$, which,
by the Claim, shows that
$\overline{\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}})\models\theta_{\alpha}(\bar{x})$,
so $\bar{x}\in D_{\alpha}\subseteq D$. ∎

##  6\. A Ramsey interlude

The goal of this section is to prove a transfer Ramsey theorem that will be
our main combinatorial tool for calculating universal minimal flows of
kaleidoscopic groups. First we recall the definition of the Ramsey property.

If ${\mathbf{A}}$ and ${\mathbf{B}}$ are two structures in the same signature,
we denote by $\binom{{\mathbf{B}}}{{\mathbf{A}}}$ the set of embeddings of
${\mathbf{A}}$ into ${\mathbf{B}}$. For three structures
${\mathbf{A}},{\mathbf{B}},{\mathbf{C}}$ and $k\in\mathbb{N}$, we denote by
${\mathbf{C}}\to({\mathbf{B}})_{\mathbf{A}}^{k}$ the statement that
$\binom{{\mathbf{C}}}{{\mathbf{A}}}\neq\emptyset$ and for any coloring
$\gamma\colon\binom{{\mathbf{C}}}{{\mathbf{A}}}\to k$, there exists
$f\in\binom{{\mathbf{C}}}{{\mathbf{B}}}$ such that $\gamma$ is constant on the
set $f\circ\binom{{\mathbf{B}}}{{\mathbf{A}}}$.

Let $\mathcal{F}$ be an age. We will say that ${\mathbf{A}}\in\mathcal{F}$ is
a _Ramsey structure_ for $\mathcal{F}$ if for every
${\mathbf{B}}\in\mathcal{F}$, there exists ${\mathbf{C}}\in\mathcal{F}$ such
that ${\mathbf{C}}\to({\mathbf{B}})_{\mathbf{A}}^{k}$. The age $\mathcal{F}$
has the _Ramsey property_ if all structures in $\mathcal{F}$ are Ramsey for
$\mathcal{F}$. We will say that an ultrahomogeneous structure ${\mathbf{N}}$
has the Ramsey property if its age does.

The reason the Ramsey property is important in topological dynamics is the
following fundamental result of Kechris, Pestov, and Todorčević.

######  Theorem 6.1 (Kechris–Pestov–Todorčević [Kechris2005]).

Let ${\mathbf{N}}$ be an ultrahomogeneous structure. Then the following are
equivalent:

  * •

${\mathbf{N}}$ has the Ramsey property;

  * •

$\operatorname{Aut}({\mathbf{N}})$ is extremely amenable.

There is also a local version of the KPT theorem, which we will use. If $G$ is
any topological group, $V\leq G$ is an open subgroup and $Z$ is a compact
space, then $G$ acts naturally on $Z^{G/V}$ by

$(g\cdot z)(hV)=z(g^{-1}hV).$

We will say that $V\leq G$ is a _Ramsey subgroup of $G$_ if the only minimal
subflows of $2^{G/V}$ (equivalently, of $Z^{G/V}$ for any zero-dimensional
compact $Z$) are the fixed points, that is, the constant functions $G/V\to 2$.
Note that if $V\leq G$ is Ramsey and $\phi\in\operatorname{Aut}(G)$, then
$\phi[V]$ is also Ramsey.

Let now ${\mathbf{N}}$ be ultrahomogeneous and let
$G=\operatorname{Aut}({\mathbf{N}})$. If ${\mathbf{A}}$ is a finitely
generated substructure of ${\mathbf{N}}$, we will denote by $G_{\mathbf{A}}$
the pointwise stabilizer of ${\mathbf{A}}$ in $G$. This is an open subgroup of
$G$ and it is well-known and easy to see that $G_{{\mathbf{A}}}$ is a Ramsey
subgroup of $G$ iff ${\mathbf{A}}$ is a Ramsey substructure of ${\mathbf{N}}$.
One equivalent way to state Theorem 6.1 is that a non-archimedean $G$ is
extremely amenable iff every open subgroup of $G$ is Ramsey.

It is well-known that extreme amenability is stable under taking group
extensions. Below we prove a local version of this fact.

######  Proposition 6.2.

Let $G$ be a topological group, $H\lhd G$ be a closed, normal subgroup and let
$K=G/H$. Denote by $\pi\colon G\to K$ the quotient map. Let $V\leq G$ be an
open subgroup of $G$. If $V\cap H$ is Ramsey in $H$ and $\pi(V)$ is Ramsey in
$K$, then $V$ is Ramsey in $G$.

###### Proof.

Let $z_{0}\in 2^{G/V}$. Our goal is to show that $\overline{G\cdot z_{0}}$
contains a fixed point. Let $T\subseteq G$ be a set of representatives for the
double $(H,V)$-cosets in $G$, i.e., such that $G=\bigsqcup_{t\in T}HtV$. There
is a natural $H$-equivariant bijection $\theta\colon G/V\to\bigsqcup_{t\in
T}H/(V^{t^{-1}}\cap H)$ given by

$\theta(gV)=hV^{t^{-1}},\quad\text{ where }h\in H,t\in T,gV=htV,$

which allows to identify $2^{G/V}$ and $\prod_{t\in T}2^{H/(V^{t^{-1}}\cap
H)}$, with the diagonal action, as $H$-spaces. Our assumption that $V\cap H$
is Ramsey in $H$ tells us that the set $Z$ of $H$-fixed points in
$\overline{G\cdot z_{0}}$ is non-empty. Indeed, the only minimal $H$-subflows
of each factor $2^{H/(V^{t^{-1}}\cap H)}$ are the fixed points, so we can take
an element of the product all of whose coordinates are fixed points. We can
define an injective map $\phi\colon Z\to 2^{K/\pi(V)}$ by

$\phi(z)(\pi(g)\pi(V))=z(gV).$

To conclude, note that any $K$-fixed point in $\phi(Z)$ gives us a $G$-fixed
point in $Z$. ∎

Note also that $G$ is always a Ramsey subgroup of itself.

######  Corollary 6.3.

Let $\\{G_{i}:i\in I\\}$ be a family of topological groups and let
$G=\prod_{i\in I}G_{i}$. Let $V_{i}\leq G_{i}$ be an open Ramsey subgroup for
each $i\in I$ with $V_{i}=G_{i}$ for all but finitely many $i$. Then
$\prod_{i}V_{i}$ is a Ramsey subgroup of $G$.

###### Proof.

The case where $I$ is finite follows by induction on $|I|$ from Proposition
6.2. For an arbitrary $I$, let $F=\\{i\in I:V_{i}\neq G_{i}\\}$ and apply
Proposition 6.2 to the extension

$1\to\prod_{i\in F}G_{i}\to G\to\prod_{i\in I\setminus F}G_{i}\to 1.\qed$

As elements of the age of ${\mathcal{K}}^{*}({\mathbf{N}})$ are rooted trees,
to prove Theorem 6.8, we will be led to do inductive arguments on the height
of the tree. In order to facilitate this, we introduce an auxiliary class of
structures.

Let $\mathcal{L}_{0}$ be a relational signature and let $\mathbf{c}$ be a
constant symbol. Let ${\mathbf{N}}$ be an ultrahomogeneous structure in the
signature $\mathcal{L}_{0}\cup\\{\mathbf{c}\\}$ and denote
$\Delta=\operatorname{Aut}({\mathbf{N}})$. Let
${\mathcal{K}}^{*}({\mathbf{N}})$ be constructed as in Section 5, so
$\operatorname{Aut}({\mathcal{K}}^{*}({\mathbf{N}}))={\mathcal{K}}^{*}(\Delta)$.
Denote $\mathcal{N}=\operatorname{Age}({\mathbf{N}})$ and
${\mathcal{K}}^{*}(\mathcal{N})=\operatorname{Age}({\mathcal{K}}^{*}({\mathbf{N}}))$,
and let $\mathcal{L}^{*}$ be the language of
${\mathcal{K}}^{*}({\mathbf{N}})$. The main goal of this section is to show
that if $\mathcal{N}$ has the Ramsey property, then so does
${\mathcal{K}}^{*}(\mathcal{N})$. An important role in the proof is played by
the following notion. If
${\mathbf{T}},{\mathbf{S}}\in{\mathcal{K}}^{*}(\mathcal{N})$, an embedding
$f\colon{\mathbf{T}}\to{\mathbf{S}}$ is called _rooted_ if
$f(r_{\mathbf{T}})=r_{\mathbf{S}}$. We can introduce a new constant
$\mathbf{r}$ to the language and consider the class
${\mathcal{K}}^{*}_{\mathbf{r}}(\mathcal{N})$ of
$\mathcal{L}^{*}\cup\\{\mathbf{r}\\}$-structures ${\mathbf{T}}_{\mathbf{r}}$,
obtained from ${\mathbf{T}}\in{\mathcal{K}}^{*}(\mathcal{N})$ by interpreting
$\mathbf{r}$ as $r_{{\mathbf{T}}}$.

Fix some $r\in X$, and let $C_{\xi}$ denote the component $\rho(r)$ considered
as a subset of $X$. Let ${\mathcal{K}}^{*}_{\mathbf{r}}({\mathbf{N}})$ denote
the substructure of ${\mathcal{K}}^{*}({\mathbf{N}})$ whose domain is
$(X\setminus C_{\xi})\cup\\{\xi\\}$ with the new constant $\mathbf{r}$
interpreted as $r$. In order to study the structure
${\mathcal{K}}^{*}_{\mathbf{r}}({\mathbf{N}})$ and its automorphism group, we
will repeatedly make use of the following lemma from [Duchesne2019a], which
allows us to patch automorphisms from pieces.

######  Lemma 6.4 ([Duchesne2019a]*Lemma 2.3).

Let $\mathcal{U}$ be a family of disjoint open subsets of a dendrite $D$. For
each $U\in\mathcal{U}$, let $f_{U}$ be a homeomorphism of $D$ which is the
identity outside of $U$. Then the map $f\colon D\to D$ given by $f_{U}$ on
each $U$ and the identity outside of $\bigcup\mathcal{U}$ is a homeomorphism
of $D$.

An immediate consequence of Lemma 6.4 is that
$\operatorname{Aut}({\mathcal{K}}^{*}_{\mathbf{r}}({\mathbf{N}}))$ can be
identified with the pointwise stabilizer of $C_{\xi}$ in
${\mathcal{K}}^{*}(\Delta)$. The following are the basic properties of
${\mathcal{K}}^{*}_{\mathbf{r}}({\mathbf{N}})$ which we will use.

######  Proposition 6.5.

Let ${\mathcal{K}}^{*}_{\mathbf{r}}({\mathbf{N}})$ be defined as above. Then
the following hold:

  1. (i)

${\mathcal{K}}^{*}_{\mathbf{r}}({\mathbf{N}})$ is ultrahomogeneous and does
not depend on the choice of $r\in X$, up to conjugation by an element of
${\mathcal{K}}^{*}({\mathbf{N}})$;

  2. (ii)

$\operatorname{Age}({\mathcal{K}}^{*}_{\mathbf{r}}({\mathbf{N}}))={\mathcal{K}}^{*}_{\mathbf{r}}(\mathcal{N})$;

  3. (iii)

$\\{{\mathbf{A}}[h+1]_{\mathbf{r}}:{\mathbf{A}}\in\mathcal{N},h\in\mathbb{N}\\}$
is cofinal in ${\mathcal{K}}^{*}_{\mathbf{r}}(\mathcal{N})$.

###### Proof.

(i) Let $f\colon{\mathbf{T}}\to{\mathbf{T}}^{\prime}$ be an isomorphism
between finite substructures of
${\mathcal{K}}^{*}_{\mathbf{r}}({\mathbf{N}})$. Then, by ultrahomogeneity of
${\mathcal{K}}^{*}({\mathbf{N}})$, there is
$g\in\operatorname{Aut}({\mathcal{K}}^{*}({\mathbf{N}}))$ such that
$g|_{\mathbf{T}}=f$; in particular, $g(r)=r$. We can then use Lemma 6.4 to
obtain an automorphism of ${\mathcal{K}}^{*}_{\mathbf{r}}({\mathbf{N}})$ by
patching $g|_{X\setminus(C_{\xi}\cup\\{r\\})}$ with the identity on $C_{\xi}$.
The second statement is a consequence of the transitivity of the action
${\mathcal{K}}^{*}({\mathbf{N}})\curvearrowright X$, which follows from
Theorem 5.8.

(ii) This is clear.

(iii) This follows from Proposition 5.7 and (ii). ∎

Now we are able to state one of the main ingredients used in the proof of the
Ramsey theorem.

######  Lemma 6.6.

Let ${\mathbf{A}}\in\mathcal{N}$ and $h\in\mathbb{N}$. If ${\mathbf{A}}[h]$ is
Ramsey in ${\mathcal{K}}^{*}(\mathcal{N})$ and ${\mathbf{A}}$ is Ramsey in
$\mathcal{N}$ then ${\mathbf{A}}[h+1]_{\mathbf{r}}$ is Ramsey in
${\mathcal{K}}^{*}_{\mathbf{r}}(\mathcal{N})$.

###### Proof.

To simplify notation, let
$G=\operatorname{Aut}({\mathcal{K}}^{*}_{\mathbf{r}}({\mathbf{N}}))$,
$H=\operatorname{Aut}({\mathcal{K}}^{*}({\mathbf{N}}))$, and recall that
$\Delta=\operatorname{Aut}({\mathbf{N}})$. For any $a\in N\setminus\\{c\\}$,
denote by $C_{a}$ the set of points in $X$ in the component
$(\kappa^{*}_{r})^{-1}(a)$ and let $H_{(a)}$ be the pointwise stabilizer of
$X\setminus C_{a}$ in $H$. Note that, by the remarks preceding 5.11, $H_{(a)}$
is isomorphic to $H$. By ultrahomogeneity of ${\mathbf{N}}$ and
${\mathcal{K}}^{*}({\mathbf{N}})$, the hypotheses translate to $\Delta_{A}$
and $H_{{\mathbf{A}}[h]}$ being Ramsey subgroups of $\Delta$ and $H$,
respectively.

We have the following short exact sequence:

(6.1) $1\to\prod_{a\in N\setminus\\{c\\}}H_{(a)}\to
G\overset{\pi}{\to}\Delta\to 1,$

where $\pi(g)=\alpha_{\kappa^{*}}(g,r)$. Indeed, it is clear that $\pi$ is a
homomorphism and it is surjective by Corollary 5.11. If we let $K=\ker\pi$, we
can define a homomorphism $\theta_{a}\colon K\to H_{(a)}$ by
$\theta_{a}(g)=g|_{C_{a}}\cup\operatorname{id}|_{X\setminus C_{a}}$. It is
clear that the product $\prod_{a\in N\setminus\\{c\\}}\theta_{a}$ is injective
and it is onto $\prod_{a\in N\setminus\\{c\\}}H_{(a)}$ by Lemma 6.4.

Now, realize ${\mathbf{A}}[h+1]_{\mathbf{r}}$ as a substructure of
${\mathcal{K}}^{*}_{\mathbf{r}}({\mathbf{N}})$, with
$\kappa^{*}_{r}\lambda_{\mathbf{r}}^{-1}(a)=a$ for all $a\in{\mathbf{A}}$.
Consider the open subgroup $V\coloneqq G_{{\mathbf{A}}[h+1]_{\mathbf{r}}}$ of
$G$. Then

$U\coloneqq V\cap\prod_{a\in N\setminus\\{c\\}}H_{(a)}=\prod_{a\in
N\setminus\\{c\\}}(H_{(a)}\cap V).$

This follows from the fact that for all $a$, $\theta_{a}(V\cap K)=H_{(a)}\cap
V$. Note that $H_{(a)}\cap V=H_{(a)}$ for all $a\notin A$. For $a\in
A\setminus\\{c\\}$, the $\mathcal{L}^{*}$-substructure of
${\mathbf{A}}[h+1]_{\mathbf{r}}$ with domain $C_{a}\cup\\{\xi\\}$ is
isomorphic to ${\mathbf{A}}[h]$, so $H_{(a)}\cap V$ is conjugate to
$H_{{\mathbf{A}}[h]}$, which is a Ramsey subgroup of $H$, by hypothesis. Thus,
by Corollary 6.3, $U$ is a Ramsey subgroup of $K$. Also, $\pi[V]=\Delta_{A}$,
so $\pi[V]$ is a Ramsey subgroup of $\Delta$. Thus, we can apply Proposition
6.2 and conclude that $V$ is a Ramsey subgroup of $G$, i.e., that
${\mathbf{A}}[h+1]_{\mathbf{r}}$ is Ramsey in
${\mathcal{K}}^{*}_{\mathbf{r}}(\mathcal{N})$. ∎

######  Remark 6.7.

To understand better the group
$\operatorname{Aut}({\mathcal{K}}^{*}_{\mathbf{r}}({\mathbf{N}}))$, as well as
our strategy for proving the Ramsey theorem below, it will be perhaps helpful
to observe that the exact sequence (6.1) splits (by Corollary 5.11), so
$\operatorname{Aut}({\mathcal{K}}^{*}_{\mathbf{r}}({\mathbf{N}}))$ is in fact
isomorphic to the permutational wreath product

$\operatorname{Aut}({\mathcal{K}}^{*}({\mathbf{N}}))^{N\setminus\\{c\\}}\wr\Delta.$

The main theorem of this section is the following.

######  Theorem 6.8.

Let $\mathcal{L}_{0}$ be a relational signature and let $\mathbf{c}$ be a
constant symbol. Let ${\mathbf{N}}$ be an ultrahomogeneous
$\mathcal{L}_{0}\sqcup\\{\mathbf{c}\\}$-structure with the Ramsey property.
Then ${\mathcal{K}}^{*}({\mathbf{N}})$ has the Ramsey property.

Combining this with Theorem 6.1, we obtain:

######  Corollary 6.9.

Let $\Delta\curvearrowright M$ be a permutation group, let $c\in M$, and
suppose that $\Delta$ stabilizes $c$. If $\Delta$ is extremely amenable, then
so is ${\mathcal{K}}^{*}(\Delta)$.

###### Proof of Theorem 6.8.

We denote

$\mathcal{F}=\\{{\mathbf{A}}[h]:{\mathbf{A}}\in\operatorname{Age}({\mathbf{N}}),h=0,1,\ldots\\}.$

By Proposition 5.7, $\mathcal{F}$ is cofinal in
$\operatorname{Age}({\mathcal{K}}^{*}({\mathbf{N}}))$. By ultrahomogeneity of
${\mathcal{K}}^{*}({\mathbf{N}})$, it thus suffices to show that each
${\mathbf{A}}[h]$ is a Ramsey structure in $\mathcal{F}$. We do so by fixing
${\mathbf{A}}$ and carrying out induction on $h$.

For the base case, we prove that ${\mathbf{A}}[0]$ is Ramsey. To that end,
take any ${\mathbf{B}}[h]\in\mathcal{F}$. Notice that, for any
${\mathbf{T}}\in\mathcal{F}$, coloring the embeddings of ${\mathbf{A}}[0]$ in
${\mathbf{T}}$ corresponds to coloring the points of ${\mathbf{T}}$. Moreover,
since ${\mathbf{B}}$ is rigid, embeddings from ${\mathbf{B}}[h]$ to
${\mathbf{B}}[k]$ correspond to subtrees of ${\mathbf{B}}[k]$ isomorphic to
${\mathbf{B}}[h]$. Therefore, this is a statement about trees, and we can
apply [Deuber1975]*Lemma 5 to get that
${\mathbf{B}}[2h-1]\to({\mathbf{B}}[h])_{2}^{{\mathbf{A}}[0]}$.

Next we do the induction step. Let $\ell>0$ and suppose that
${\mathbf{A}}[\ell-1]$ is Ramsey. Let ${\mathbf{U}}\in\mathcal{F}$ be
arbitrary such that ${\mathbf{A}}[\ell]\subseteq{\mathbf{U}}$; we will find
${\mathbf{T}}\in\mathcal{F}$ such that
${\mathbf{T}}\to({\mathbf{U}})^{{\mathbf{A}}[\ell]}_{2}$ holds.

We denote by $\overset{r}{\to}$ the arrow relation for rooted embeddings. It
follows from Lemma 6.6 and the induction hypothesis that ${\mathbf{A}}[\ell]$
is Ramsey for rooted embeddings.

By the base of induction we know that there is ${\mathbf{D}}[h]\in\mathcal{F}$
such that ${\mathbf{D}}[h]\to({\mathbf{U}})^{{\mathbf{A}}[0]}_{2}$. Let
${\mathbf{C}}_{h}\coloneqq{\mathbf{D}}$ and $n_{h}\coloneqq 0$. Suppose we
have defined ${\mathbf{C}}_{i+1}\in\mathcal{N}$ and $n_{i+1}\in\mathbb{N}$,
for some $0\leq i<h$. Let
${\mathbf{C}}_{i}\supseteq{\mathbf{C}}_{i+1},n_{i}\geq n_{i+1}$ be such that

${\mathbf{C}}_{i}[n_{i}]\overset{r}{\to}({\mathbf{C}}_{i+1}[n_{i+1}+1])_{2}^{{\mathbf{A}}[\ell]}.$

Let ${\mathbf{T}}\coloneqq{\mathbf{C}}_{0}[n_{0}]$. We show that
${\mathbf{T}}\to({\mathbf{U}})_{2}^{{\mathbf{A}}[\ell]}$. To that end, fix a
coloring $\gamma\colon\binom{{\mathbf{T}}}{{\mathbf{A}}[\ell]}\to 2$. Let
$p=r_{{\mathbf{A}}[\ell]}$. We construct a substructure ${\mathbf{S}}$ of
${\mathbf{T}}$ such that any two embeddings
$f,f^{\prime}\colon{\mathbf{A}}[\ell]\to{\mathbf{S}}$ that agree on $p$
receive the same color.

Let ${\mathbf{S}}_{0}\coloneqq{\mathbf{T}}$. For $0\leq i<h$, suppose we have
found ${\mathbf{S}}_{i}\subseteq{\mathbf{T}}$ such that

  1. (i)

${\mathbf{S}}_{i}$ has height $i+n_{i}$;

  2. (ii)

if $0\leq j<i$, then ${\mathbf{S}}_{i}\upharpoonright
j={\mathbf{S}}_{j}\upharpoonright j$;

  3. (iii)

for any $s\in{\mathbf{S}}_{i}$ of height $i$,
${\mathbf{S}}_{i}(s)\cong{\mathbf{C}}_{i}[n_{i}]$;

  4. (iv)

for any $f,f^{\prime}\colon{\mathbf{A}}[\ell]\to{\mathbf{S}}_{i}$ such that
$f(p)=f^{\prime}(p)$ has height less than $i$, it holds that
$\gamma(f)=\gamma(f^{\prime})$.

It is clear that ${\mathbf{S}}_{0}$ satisfies the conditions. We suppose that
${\mathbf{S}}_{i}$ has been constructed and we show how to construct
${\mathbf{S}}_{i+1}\subseteq{\mathbf{S}}_{i}$. First, we let
${\mathbf{S}}_{i+1}\upharpoonright i={\mathbf{S}}_{i}\upharpoonright i$, which
ensures that ${\mathbf{S}}_{i+1}$ satisfies point (ii) above. Now fix $s$ of
height $i$ in ${\mathbf{S}}_{i}$. Since
${\mathbf{S}}_{i}(s)\cong{\mathbf{C}}_{i}[n_{i}]$, we have that
${\mathbf{S}}_{i}(s)\overset{r}{\to}({\mathbf{C}}_{i+1}[n_{i+1}+1])_{2}^{{\mathbf{A}}[\ell]}$,
so there is a $\gamma$-monochromatic rooted embedding of
${\mathbf{C}}_{i+1}[n_{i+1}+1]$ into ${\mathbf{S}}_{i}(s)$. We let
${\mathbf{S}}_{i+1}(s)$ be the image of this embedding, so
${\mathbf{S}}_{i+1}$ satisfies point (iii). Then ${\mathbf{S}}_{i+1}$ has
height $i+n_{i+1}+1$, so point (i) is taken care of. Suppose that
$f,f^{\prime}\colon{\mathbf{A}}[\ell]\to{\mathbf{S}}_{i+1}$ are such that
$s\coloneqq f(p)=f^{\prime}(p)$ has height $j\leq i$. If $j<i$, then $s$ has
height $j$ in ${\mathbf{S}}_{j}$ by point (ii), so
$\gamma(f)=\gamma(f^{\prime})$ by (iv) for
${\mathbf{S}}_{j}\supseteq{\mathbf{S}}_{i}$. Otherwise, $f$ and $f^{\prime}$
are rooted embeddings of ${\mathbf{A}}[\ell]$ into ${\mathbf{S}}_{i+1}(s)$, so
by construction, they receive the same $\gamma$-color. This takes care of
point (iv) and completes the inductive construction.

Let ${\mathbf{S}}={\mathbf{S}}_{h}$. By (i) it has height $h+n_{h}=h$. By
points (ii) and (iii), for any $s\in S$ of height less than $h$,
${\mathbf{D}}[1]$ embeds into ${\mathbf{S}}(s)\upharpoonright 1$, so
${\mathbf{D}}[h]$ embeds into ${\mathbf{S}}$. It follows that
${\mathbf{S}}\to({\mathbf{U}})_{2}^{{\mathbf{A}}[0]}$. By point (iv), for each
$s\in S$ there is $\delta(s)\in\\{0,1\\}$ such that $\gamma(f)=\delta(s)$ for
each $f\colon{\mathbf{A}}[\ell]\to{\mathbf{S}}$ such that $f(p)=s$. (Note
that, as $\ell\geq 1$, there is no embedding
$f\colon{\mathbf{A}}[\ell]\to{\mathbf{S}}$ with $f(p)$ of height $h$.) By
recalling that coloring points is equivalent to coloring embeddings of
${\mathbf{A}}[0]$, we find a copy ${\mathbf{U}}_{0}$ of ${\mathbf{U}}$ in
${\mathbf{S}}\subseteq{\mathbf{T}}$ which is $\delta$-monochromatic. It
follows that ${\mathbf{U}}_{0}$ is $\gamma$-monochromatic as well and we are
done. ∎

######  Remark 6.10.

It follows from the representation given in Remark 6.7 that if $\Delta$ is
extremely amenable, then so is
$\operatorname{Aut}({\mathcal{K}}^{*}_{\mathbf{r}}({\mathbf{N}}))$.

##  7\. The universal highly proximal extension of the action on $W$

As a first application of the tools we have developed, we give a description
of the universal highly proximal extension of the flow
$\mathcal{K}(\Gamma)\curvearrowright W$ and characterize when it is
metrizable.

######  Theorem 7.1.

Let $\Gamma\curvearrowright M$ be transitive and let ${\mathbf{M}}$ be an
ultrahomogeneous, relational structure for $\Gamma$ with universe $M$. Then

(7.1)
$\operatorname{\mathcal{S}}_{\mathcal{K}(\Gamma)}(W)\cong\\{p\in\operatorname{S}_{1}(\overline{\mathcal{K}}({\mathbf{M}})):\forall
x,y\in X\ p(v)\models\neg B(x,v,y)\\},$

with the action of $\mathcal{K}(\Gamma)$ on the right-hand side given by
permuting the parameters. The MHP factor map
$\pi\colon\operatorname{\mathcal{S}}_{\mathcal{K}(\Gamma)}(W)\to W$ is given
by

(7.2) $\pi(p)=\zeta\iff\forall x\in X\ p(v)\models\Phi(x,v)=\Phi(x,\zeta).$

###### Proof.

Write $G=\mathcal{K}(\Gamma)$ and let $\xi\in W$ be an endpoint. By Fact 4.3,
the orbit $G\cdot\xi=\mathrm{End}(W)$ is comeager in $W$, so from Proposition
2.7, we have that
$\operatorname{\mathcal{S}}_{G}(W)\cong\operatorname{Sa}(G/G_{\xi})$. Let
$\mathbf{c}$ be a new constant symbol, fix some $c\in M$ and let
${\mathbf{M}}_{\mathbf{c}}$ be the expansion of ${\mathbf{M}}$ where one
interprets $\mathbf{c}$ as $c$, as in Subsection 5.4. Proposition 5.12 gives
us a representation of $G_{\xi}$ as the automorphism group
$\operatorname{Aut}({\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}}))$. Denote by
$\overline{\mathcal{L}}$ the qf-full signature of
$\overline{\mathcal{K}}({\mathbf{M}})$ and let
$\overline{\mathcal{L}}_{\boldsymbol{\xi}}\coloneqq\overline{\mathcal{L}}\sqcup\\{\boldsymbol{\xi}\\}$.
By Lemma 5.15, $\overline{\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}})$, the
expansion of ${\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}})$ to
$\overline{\mathcal{L}}_{\boldsymbol{\xi}}$, is full. Now we can apply
Proposition 3.5 to obtain that

(7.3)
$\operatorname{\mathcal{S}}_{G}(W)\cong\operatorname{Exp}(\mathcal{K}({\mathbf{M}}),\overline{\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}})).$

Let $\mathcal{S}$ and $\mathcal{S}^{\prime}$ denote the right-hand sides of
(7.3) and (7.1), respectively. Next we explain how to identify $\mathcal{S}$
and $\mathcal{S}^{\prime}$. We define the map
$\Theta\colon\mathcal{S}\to\operatorname{S}_{1}(\overline{\mathcal{K}}({\mathbf{M}}))$
by

$\Theta(p)(v)\models\phi(v,\bar{x})\iff p\models\exists
z\in\rho(\bigwedge\bar{x})\ \phi(z,\bar{x})$

for every quantifier-free $\overline{\mathcal{L}}$-formula $\phi$.

That $\Theta$ is well-defined follows from the fact that
$\overline{\mathcal{K}}({\mathbf{M}})$ eliminates quantifiers in
$\overline{\mathcal{L}}$. It is obvious that $\Theta$ is continuous and
$G$-equivariant. To see that it is injective, note that if $p_{1}\neq
p_{2}\in\mathcal{S}$, then the proof of Lemma 5.15 and (5.1) give us a
quantifier-free $\overline{\mathcal{L}}$-formula $\phi$ which witnesses that
$\Theta(p_{1})\neq\Theta(p_{2})$.
$\mathrm{Im\,}\Theta\subseteq\mathcal{S}^{\prime}$ because the sentence

$\forall x_{1},x_{2}\ \forall z\in\rho(x_{1}\mathbin{\wedge}x_{2})\ \neg
B(x_{1},z,x_{2})$

holds in $\overline{\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}})$, so
$\Theta(p)(v)\models\neg B(x_{1},v,x_{2})$. Next we see that if
$\phi(v,\bar{x})$ is an isolating, quantifier-free
$\overline{\mathcal{L}}$-formula and $\exists v\
\phi(v,\bar{x})\land\bigwedge_{i,j}\neg B(x_{i},v,x_{j})$ is true in
$\overline{\mathcal{K}}({\mathbf{M}})$, then the formula $\exists
z\in\rho(\bigwedge\bar{x})\ \phi(z,\bar{x})$ is consistent with
$\operatorname{Th}(\overline{\mathcal{K}}^{*}({\mathbf{M}}_{\mathbf{c}}))$.
Notice that for such a $v$ there is a single component $a\in\hat{v}$ such that
$\bar{x}$ is contained in $a$. Now, it suffices to choose as
$\boldsymbol{\xi}$ an endpoint in a component around $v$ which is not $a$ (and
recall that for all choices of an endpoint $\xi$ the obtained models are
isomorphic). Indeed, for any such $\xi$, we have $v\in\rho(\bigwedge\bar{x})$,
so we can take $z=v$. This shows $\mathrm{Im\,}\Theta$ is dense in
$\mathcal{S}^{\prime}$, so by compactness, it is all of it.

It remains to check the final assertion of the theorem. First, we see that
(7.2) indeed defines a function. If $\zeta_{1}\neq\zeta_{2}$, then there
exists $x\in X\cap(\zeta_{1},\zeta_{2})$, so
$\Phi(x,\zeta_{1})\neq\Phi(x,\zeta_{2})$. That $\pi$ is continuous follows
from the fact that the sets of the form $\\{\zeta\in W:\Phi(x,\zeta)=a\\}$,
where $x$ varies in $X$ and $a$ varies in $\hat{x}$, form a subbasis for the
topology of $W$. That $\pi$ is $G$-equivariant is obvious. Finally the
uniqueness of the factor map (Proposition 2.7) implies that $\pi$ is indeed
the MHP factor map. ∎

We also have the following corollary (from 5.13).

######  Corollary 7.2.

Let $\Gamma\curvearrowright M$ be transitive. Then the following are
equivalent:

  1. (i)

$\operatorname{\mathcal{S}}_{\mathcal{K}(\Gamma)}(W)$ is metrizable;

  2. (ii)

$\Gamma$ is oligomorphic.

Since $\operatorname{\mathcal{S}}_{\mathcal{K}(\Gamma)}(W)$ is minimal, this
shows that if $\Gamma$ is not oligomorphic then $\mathcal{K}(\Gamma)$ is not
CAP.

######  Example 7.3.

As an illustration of Theorem 7.1, we give a description of the preimages
$\pi^{-1}(\zeta)$ (where
$\pi\colon\operatorname{\mathcal{S}}_{\mathcal{K}(\Gamma)}(W)\to W$ is the
factor map given by (7.2)) for the different types of points $\zeta\in W$ in
the case when the action $\Gamma\curvearrowright M$ is $3$-transitive. Let
${\mathbf{M}}$ be a qf-full structure for $\Gamma$ with universe $M$. We show
that the preimages of endpoints and regular points are singletons and for each
$x\in X$, we have $\pi^{-1}(x)\cong\operatorname{S}_{1}({\mathbf{M}})$. Recall
that
$\operatorname{S}_{1}({\mathbf{M}})\cong\operatorname{Sa}(\Gamma/\Gamma_{c})$,
for some $c\in M$, by 3.6.

If $\xi\in W$ is an endpoint, $\pi^{-1}(\xi)$ is always a singleton, by 2.7.

Now fix a regular point $\zeta\in W$. We show that $\pi^{-1}(\zeta)$ is also a
singleton. Let $p\in\operatorname{S}_{1}(\mathcal{K}({\mathbf{M}}))$ be such
that $\pi(p)=\zeta$ and let $T\subseteq X$ be a finite subtree such that
$\zeta$ belongs to the subdendrite generated by $T$. Let $x_{0},x_{1}\in T$ be
such that $\zeta\in(x_{0},x_{1})$ and there are no vertices of $T$ in
$(x_{0},x_{1})$. Let $y$ be a realization of $p|_{T}$ and let
$z=K(x_{0},x_{1},y)$. Note that the tree generated by $T$ and $y$ is
$T^{\prime}\coloneqq T\cup\\{y,z\\}$. By quantifier elimination, the type
$\operatorname{tp}(y/T)$ is determined by the isomorphism type of the
substructure ${\mathbf{T}}^{\prime}\subseteq\mathcal{K}({\mathbf{M}})$ with
domain $T^{\prime}$. The isomorphism type of ${\mathbf{T}}^{\prime}$ as a tree
(in the language $\mathcal{L}_{\mathbf{X}}$) is already determined by $\zeta$
and the rest is decided by

(7.4)
$\operatorname{tp}_{{\mathbf{M}}}(\kappa(\Phi(z,y)),\kappa(\Phi(z,x_{0})),\kappa(\Phi(z,x_{1}))),$

which is unique by the $3$-transitivity assumption.

Now consider a branching point $x_{0}\in W$. We show that the map
$\Psi\colon\pi^{-1}(x_{0})\to\operatorname{S}_{1}({\mathbf{M}})$ given by

$\Psi(p)(v)\models\theta(v,\bar{a})\iff
p(v)\models\theta(\Phi(x_{0},v),\kappa_{x_{0}}^{-1}(\bar{a})),$

for any quantifier-free formula $\theta$ in the language of ${\mathbf{M}}$ and
any tuple $\bar{a}$ from $M$, is a homeomorphism. Continuity and surjectivity
are clear, while we need to use $3$-transitivity again to prove that $\Psi$ is
injective. We do so by showing that $\Psi^{-1}(q)$ is a singleton for any
$q\in\operatorname{S}_{1}({\mathbf{M}})$. Let $p$ be any element of
$\Psi^{-1}(q)$. We have two cases.

Case 1. The type $q$ is realized in ${\mathbf{M}}$ by an element $a_{0}$. Let
$T$ be a finite subtree of $X$ containing $x_{0}$ and having a point in the
component $\kappa_{x_{0}}^{-1}(a_{0})$. Let $x_{1}\in T$ be the point in this
component closest to $x_{0}$. Take a realization $y$ of $p|_{T}$ and let
$z=K(x_{0},x_{1},y)$. Then the substructure of $\mathcal{K}({\mathbf{M}})$
with domain $T^{\prime}\coloneqq T\cup\\{y,z\\}$ again has isomorphism type
determined by $a_{0}$ and the $3$-type (7.4). So we conclude by
$3$-transitivity as before.

Case 2. The type $q$ is not realized in ${\mathbf{M}}$. Let again $T$ be a
finite subtree of $X$ containing $x_{0}$ and let $y$ be a realization of
$p|_{T}$. Now $T\cup\\{y\\}$ is already closed under $K$, so the isomorphism
type of the substructure of $\mathcal{K}({\mathbf{M}})$ with domain
$T\cup\\{y\\}$ is determined by
$\operatorname{tp}_{\mathbf{M}}(\kappa_{x_{0}}^{-1}(\Phi(x_{0},y))/\kappa_{x_{0}}^{-1}(\hat{x}_{0}\cap\widehat{T}))$,
which is decided by $q$.

We conclude by two remarks about the analysis above. First, one can describe
$\operatorname{\mathcal{S}}_{\mathcal{K}(\Gamma)}(W)$ even without the
$3$-transitivity assumption but the description is more complex as it has to
involve the space of $3$-types
$\operatorname{S}_{3}(\operatorname{Th}({\mathbf{M}}))$. Second, one obtains a
particularly simple description when $\mathcal{K}(\Gamma)$ is the full
homeomorphism group of $W$ (which will be used in Section 9 below). Namely, in
that case, $\operatorname{S}_{1}({\mathbf{M}})$ can simply be identified with
$M$ when $M$ is finite and with the one-point compactification of $M$ when $M$
is infinite.

##  8\. Universal minimal flows of kaleidoscopic groups

###  8.1. The UMFs of endpoint stabilizers

With the Ramsey theorem from Section 6 in hand, we start by describing the