# Stabilizers for ergodic actions and invariant random expansions of non-
archimedean Polish groups

Colin JAHEL & Matthieu JOSEPH

###### Abstract

Let $G$ be a closed permutation group on a countably infinite set $\Omega$,
which acts transitively but not highly transitively. If $G$ is oligomorphic,
has no algebraicity and weakly eliminates imaginaries, we prove that any
p.m.p. ergodic action $G\curvearrowright(X,\mu)$ is either essentially free or
essentially transitive. A key notion that we develop in our approach is that
of invariant random expansions, which are $G$-invariant probability measures
on the space of expansions of the canonical (model theoretic) structure
associated with $G$. We also initiate the study of invariant random subgroups
for Polish groups and prove that – although the result for p.m.p. ergodic
actions fails for the group $\mathrm{Sym}(\Omega)$ of all permutation of
$\Omega$ – any ergodic invariant random subgroup of $\mathrm{Sym}(\Omega)$ is
essentially transitive.

MSC: Principal: 37A15, 22F50. Secondary: 03C15, 03C98, 60G09, 03C75.
Keywords: Measure-preserving actions, Polish groups, non-archimedian groups,
invariant random subgroups, model theory, infinitary logic.

###### Contents

  1. 1 Introduction
  2. 2 Dynamically de Finetti groups
  3. 3 Preliminaries on model theory
    1. 3.1 Structures and logic actions
    2. 3.2 First-order and infinitary logics
    3. 3.3 Back-and-forth
  4. 4 Invariant Random Expansions
    1. 4.1 First properties
    2. 4.2 IREs without fixed point
  5. 5 Rigidity for p.m.p. actions of dynamically de Finetti groups
    1. 5.1 Essentially free and essentially transitive actions
    2. 5.2 The proof of the main theorem
  6. 6 Invariant Random Subgroups of Polish groups
    1. 6.1 Definition
    2. 6.2 From IRS to IRE and vice versa
    3. 6.3 Rigidity of ergodic IRSs of $S_{\infty}$
  7. 7 Further discussions

##  1 Introduction

The study of automorphism groups of countable structures is an very active
theme of research which can be approached from a variety of perspectives such
as model theory, Ramsey theory, permutation groups theory, but also
topological dynamics and ergodic theory. The present work contributes to the
study of the ergodic theoretic properties of automorphism groups of countable
structures. More precisely, we will be interested in probability measure
preserving (p.m.p.) actions of automorphism groups of countable structures.

One of the behaviors that we exhibit – essential transitivity of p.m.p.
actions – can be considered as a measure theoretic equivalent to the
topological property for a minimal action of having a comeager orbit.
Topological dynamics of closed subgroups of $\mathrm{Sym}(\Omega)$ is an
extensively studied topic that was kindled by Kechris, Pestov and Todorcevic
in [28]. An especially important result in this area, due Ben-Yaacov,
Melleray, Nguyen Van Thé, Tsankov and Zucker in [14], [31] and [38] is the
classification of groups for which every minimal action on a compact Hausdorff
space admits a comeager orbit. In particular, they show a link between the
existence of those comeager orbits and the metrizability of the universal
minimal flow, both phenomena corresponding to the existence of a suitable
expansion of the canonical structure associated with the group. This mirrors
our own study, as we explore invariant random expansions of structures in
order to study p.m.p. actions.

Before diving into model theoretic considerations, let us discuss our main
result from the point of view of permutation groups theory. We study closed
permutation groups, which are closed subgroups of the symmetric group
$\mathrm{Sym}(\Omega)$ (the group of all the permutations) on a countably
infinite set $\Omega$. Here $\mathrm{Sym}(\Omega)$, and therefore any closed
permutation group, has the natural topology of pointwise convergence which
turns it into a Polish group. A closed permutation group is transitive if the
action $G\curvearrowright\Omega$ is transitive, and proper if
$G\neq\mathrm{Sym}(\Omega)$ (equivalently, $G\curvearrowright\Omega$ is not
highly transitive). Let us now turn to definitions with a model theoretic
flavor. A closed permutation group $G\leq\mathrm{Sym}(\Omega)$ is oligomorphic
if the diagonal action of $G$ on $\Omega^{n}$ has only finitely many orbits
for every $n\geq 1$. The algebraic closure $\mathrm{acl}(A)$ of a finite
subset $A\subseteq\Omega$ is the set of points in $\Omega$ which lies in a
finite orbit of the pointwise stabilizer $G_{A}\coloneqq\\{g\in G\colon\forall
a\in A,g(a)=a\\}$. We say that $G$ has no algebraicity if $\mathrm{acl}(A)=A$
for all finite subset $A\subseteq\Omega$. Finally, we say that $G$ weakly
eliminates imaginaries if every open subgroup of $G$ contains as a finite
index subgroup the pointwise stabilizer $G_{A}$ of a finite subset
$A\subseteq\Omega$. We refer to Example 2.3 for a list of groups having the
three above properties.

In this paper, a p.m.p. action $G\curvearrowright(X,\mu)$ of a closed
permutation group is a Borel action $G\curvearrowright X$ on a standard Borel
space with a $G$-invariant Borel probability measure, i.e. such that
$\mu(Y)=\mu(g\cdot Y)$ for all $Y$ measurable and $g\in G$. A p.m.p. action
$G\curvearrowright(X,\mu)$ is ergodic if any measurable $Y\subseteq X$ that
satisfies $\mu(Y\triangle g\cdot Y)=0$ for all $g\in G$ is either null or
conull. The main goal of this paper is to prove the following result.

######  Theorem 1.1 (see Theorems 2.2 and 5.7). —

Let $G\lneq\mathrm{Sym}(\Omega)$ be a transitive, proper, closed subgroup. If
$G$ is oligomorphic, has no algebraicity and admits weak elimination of
imaginaries, then any p.m.p. ergodic action $G\curvearrowright(X,\mu)$ is
either essentially free or essentially transitive.

A p.m.p. action is essentially free if the stabilizer of almost every point is
trivial, and essentially transitive if there exists a conull orbit.
Surprisingly, this result fails for $\mathrm{Sym}(\Omega)$ as it admits p.m.p.
ergodic actions that are neither essentially free nor essentially ergodic, see
Remark 5.8. Such p.m.p. actions for $\mathrm{Sym}(\Omega)$ have been studied
in-depth from a model theoretic perspective in [4].

Despite this rigidity result, closed permutation groups
$G\leq\mathrm{Sym}(\Omega)$ have no shortage of p.m.p. actions. Indeed, for
any standard probability space $(A,\kappa)$, the generalized Bernoulli shift
$G\curvearrowright(A,\kappa)^{\Omega}$ is a p.m.p. action. If $G$ is proper,
has no algebraicity and weakly eliminates imaginaries, this action is
essentially transitive when $(A,\kappa)$ is purely atomic and essentially free
otherwise, see Lemma 5.5.

Theorem 1.1 applies to a large variety of groups (in fact continuum many),
such as the group $\mathrm{Aut}(\mathbb{Q},<)$ of order-preserving bijections
of $\mathbb{Q}$, the group $\mathrm{Aut}(\mathbb{Q}/\mathbb{Z},<)$ of
bijections of $\mathbb{Q}/\mathbb{Z}$ which preserve the dense cyclic order,
the automorphism group of the Rado graph, and many more. We refer to Example
2.3 for a wider variety of groups covered by our theorem.

The assumptions on the groups in Theorem 1.1 are rather standard. In fact, the
class of closed permutation groups $G\leq\mathrm{Sym}(\Omega)$, which are
oligomorphic, has no algebraicity and admit weak elimination of imaginaries
has been widely studied in various contexts. Tsankov proved that they have
property (T) [36, Thm. 6.6] as a corollary of his classification of unitary
representations for oligomorphic groups [36, Thm. 1.3]. The first author and
Tsankov also have classified ergodic invariant probability measures on product
spaces $X^{\Omega}$ and, in many cases, on the compact space
$\mathrm{LO}(\Omega)$ of linear orders on $\Omega$. They moreover obtain a
property [24, Thm. 3.4], which holds for any p.m.p. action and is strongly
reminiscent of the classical theorem of de Finetti for i.i.d. exchangeable
random variables. This property will be essential in our study and will lead
us to the notion of dynamically de Finetti groups (see Definition 2.1), a
class which contains oligomorphic groups that have no algebraicity and weakly
eliminates imaginaries (see Theorem 2.2). Theorem 5.7 establishes that the
p.m.p. ergodic actions of any transitive, proper, closed permutation group
which is dynamically de Finetti, are either essentially free or essentially
transitive, making it a generalization of Theorem 1.1.

The point of view that we adapt in order to prove Theorem 1.1 comes from model
theory (basic model-theoretic notions will be discussed in Section 3). This is
motivated by the fact that any closed permutation group $G$ is indeed
(isomorphic to) the automorphism group of a countable relational structure
(see [27, § 1.5]), namely the canonical structure associated with $G$. Another
characterization of these groups is they are non-archimedean Polish groups:
they admit a countable basis of neighborhoods of the identity consisting of
open subgroups. Given any countable relational language $\mathcal{L}$ (which
contains the canonical language $\mathcal{L}_{G}$ associated with $G$), we
denote by $\mathrm{Struc}_{\mathcal{L}}^{G}$ the compact space of expansions
of $\mathbf{M}_{G}$ in the language $\mathcal{L}$. These are structures whose
reduct (the structure obtained by removing the relations in
$\mathcal{L}\setminus\mathcal{L}_{G}$) is equal to $\mathbf{M}_{G}$. The space
$\mathrm{Struc}_{\mathcal{L}}^{G}$ carries a continuous $G$-action and we will
study the $G$-invariant Borel probability measures for this action, that we
call invariant random expansions of (the canonical structure associated with)
$G$. Invariant random expansions, IREs for short, of
$\mathrm{Sym}(\mathbb{N})$ were studied from the model theoretic point of view
in a series of papers under different names such as invariant measures,
invariant structures, or ergodic structures [4], [5], [6], [7], [8], [9]. We
prefer here to coin the name IRE as it more accurately describes the objects,
the structures we look at being explicitly expansions. Furthermore, this name
is reminiscent to the acronym for invariant random subgroups (IRS), a topic
that we will discuss in the context of Polish groups below. Let us denote by
$\mathrm{IRE}_{\mathcal{L}}(G)$ the space of invariant random expansions of
$G$ in the language $\mathcal{L}$. An invariant random expansion
$\mu\in\mathrm{IRE}_{\mathcal{L}}(G)$ is concentrated on an orbit if there
exists an orbit $O$ of $G\curvearrowright\mathrm{Struc}_{\mathcal{L}}^{G}$
such that $\mu(O)=1$. Towards proving Theorem 1.1, we prove the following
result.

######  Theorem 1.2 (see Theorems 2.2 and 5.6). —

Let $G\lneq\mathrm{Sym}(\Omega)$ be a transitive, proper, closed permutation
group. Assume that $G$ is oligomorphic, has no algebraicity and admits weak
elimination of imaginaries. Then for any $\mu\in\mathrm{IRE}_{\mathcal{L}}(G)$
ergodic, either $\mathrm{Aut}(\mathbf{M})=\\{1\\}$ for $\mu$-a.e.
$\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}^{G}$, or $\mu$ is concentrated on
an orbit.

Again, this result holds more generally for dynamically de Finetti groups.
Theorem 1.1 is now obtained thanks to Theorem 1.2 by invoking a universality
theorem due to Becker and Kechris, which states that any Borel $G$-action can
be Borel-embedded into $\mathrm{Struc}_{\mathcal{L}}^{G}$, provided that the
language $\mathcal{L}$ is “rich” enough, see [11, Thm. 2.7.4] for a precise
statement.

Another part of our work concerns subgroup dynamics for Polish groups and more
precisely for closed permutation groups equipped with the pointwise
convergence topology. Subgroup dynamics is the study for a topological group
$G$ of its action by conjugation on the set $\mathrm{Sub}(G)$ of its _closed_
subgroups. It turns out that subgroup dynamics is a very active area of
research in the locally compact realm. In the presence of a Polish locally
compact group $G$, the space of closed subgroups $\mathrm{Sub}(G)$ is endowed
with a natural topology, called the Chabauty topology, which turns
$\mathrm{Sub}(G)$ into a compact Hausdorff space. In this setting, subgroup
dynamics turned out to be very fruitful in various contexts of group theory
such as the study of lattices in Lie groups [1], $\mathrm{C}^{*}$-simplicity
[30], or else permutation stability of groups [12]. Yet the picture is not
quite as rosy in the Polish realm. If $G$ is a Polish non locally compact
group, then the Chabauty topology on the space $\mathrm{Sub}(G)$ of closed
subgroups is not Hausdorff in general (and this is indeed not the case when
$G=\mathrm{Sym}(\Omega)$). However, there is a natural $\sigma$-algebra on
$\mathrm{Sub}(G)$ called the Effros $\sigma$-algebra that turns
$\mathrm{Sub}(G)$ into a standard Borel space. This allows to define IRSs for
Polish groups.

An invariant random subgroup (IRS) of a Polish group $G$ is a probability
measure on (the Effros $\sigma$-algebra of) $\mathrm{Sub}(G)$ which is
invariant by conjugation. We denote by

$\mathrm{IRS}(G)\coloneqq\mathrm{Prob}(\mathrm{Sub}(G))^{G}$

the (standard Borel) space of invariant random subgroups of $G$. The theory of
invariant random subgroups as well as the spaces $\mathrm{IRS}(G)$ in the
setting of Polish _locally compact_ groups $G$ have been very recently
extensively studied on their own and the literature in this area is rapidly
growing, see e.g. [2], [10], [15], [16]. On the contrary, invariant random
subgroups of Polish non locally compact groups have not been studied so far to
the best of our knowledge.

One of the foundational results for IRSs of locally compact groups is that
every $\mu\in\mathrm{IRS}(G)$ is obtained as the stabilizer IRS of a p.m.p.
action $G\curvearrowright(X,\mu)$, i.e., the stabilizer of a $\mu$-random
point [1, Thm. 2.6] (see also [2, Prop. 12] for a proof when $G$ is
countable). We prove a similar statement for closed permutation groups.

######  Theorem 1.3 (see Theorem 6.5). —

Let $G\leq\mathrm{Sym}(\Omega)$ be a closed subgroup and let
$\nu\in\mathrm{IRS}(G)$. Then there exists a p.m.p. action
$G\curvearrowright(X,\mu)$ whose stabilizer IRS is equal to $\nu$.

We say that $\nu\in\mathrm{IRS}(G)$ is concentrated on a conjugacy class if
there exists an orbit $O$ of the $G$-action by conjugation on
$\mathrm{Sub}(G)$ such that $\nu(O)=1$. Theorem 1.1 readily implies that if
$G\lneq\mathrm{Sym}(\Omega)$ is a transitive, proper, closed subgroup, which
is oligomorphic, has no algebraicity and weakly eliminates imaginaries, then
any ergodic $\nu\in\mathrm{IRS}(G)$ is concentrated on a conjugacy class. Even
though Theorem 1.1 is false for $\mathrm{Sym}(\Omega)$, we prove that any
ergodic IRS of $\mathrm{Sym}(\Omega)$ is concentrated on a conjugacy class,
therefore obtaining the following result.

######  Theorem 1.4 (see Theorems 2.2 and 6.10). —

Let $G\leq\mathrm{Sym}(\Omega)$ be a transitive closed subgroup. If $G$ is
oligomorphic, has no algebraicity and weakly eliminates imaginaries, then any
ergodic $\nu\in\mathrm{IRS}(G)$ is concentrated on a conjugacy class.

Again, this result holds more generally for dynamically de Finetti groups.

######  Convention. —

In this paper, all countable relational structures as well as all closed
permutation groups are defined on $\Omega=\mathbb{N}$. We denote by
$S_{\infty}$ the group $\mathrm{Sym}(\mathbb{N})$ of all permutations on
$\mathbb{N}$. Tuples will be denoted by $\bar{x},\bar{y},\dots$ and
$\mathbb{N}^{<\omega}$ will denote the set of tuples on $\mathbb{N}$.

#### Acknowledgments.

We would like to thank Gianluca Basso, Ronnie Chen, Clinton Conley, David
Evans, François le Maître, Todor Tsankov and Anush Tserunyan for fruitful
discussions related to this work. We also thank Nate Ackerman, Cameron Freer
and Rehana Patel for comments on a draft of this paper. C.J. was partially
funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) – project number 467967530. M.J. was partially supported by a
public grant as part of the Investissement d’avenir project, reference
ANR-11-LABX-0056-LMH, LabEx LMH.

##  2 Dynamically de Finetti groups

If $\mathcal{F},\mathcal{G},\mathcal{H}$ are $\sigma$-fields in a probability
space, we say that $\mathcal{F}$ and $\mathcal{H}$ are _conditionally
independent_ over $\mathcal{G}$, and denote it by
$\mathcal{F}\mathrel{\reflectbox{\rotatebox[origin={c}]{90.0}{$\models$}}}_{\mathcal{G}}\mathcal{H}$
if for all $\mathcal{F}$-measurable random variables $\xi$, we have
$\mathbb{E}[\xi\mid\mathcal{G},\mathcal{H}]=\mathbb{E}[\xi\mid\mathcal{G}]$.
For a p.m.p. action $G\curvearrowright(X,\mu)$ of a closed subgroup $G\leq
S_{\infty}$ and a finite subset $A\subseteq\mathbb{N}$, we denote by
$\mathcal{F}_{A}$ the $\sigma$-algebra of $(X,\mu)$ generated by the
$G_{A}$-invariant measurable subsets of $X$, i.e., measurable subsets
$Y\subseteq X$ such that $\mu(Y\triangle g\cdot Y)=0$ for all $g\in G_{A}$.

######  Definition 2.1. —

A closed subgroup $G\leq S_{\infty}$ is dynamically de Finetti if it has no
algebraicity, admits weak elimination of imaginaries and satisfies the
following property: for all p.m.p. actions $G\curvearrowright(X,\mu)$ and all
$A,B\subseteq\mathbb{N}$ finite subsets, we have
$\mathcal{F}_{A}\mathrel{\reflectbox{\rotatebox[origin={c}]{90.0}{$\models$}}}_{\mathcal{F}_{A\cap
B}}\mathcal{F}_{B}$.

We will discuss the connection between this definition and exchangeability
theory in Section 7. Our main (and so far only) source of dynamically de
Finetti groups is the following result.

######  Theorem 2.2 ([24, Thm. 3.4]). —

Let $G\leq S_{\infty}$ be a closed subgroup. If $G$ has no algebraicity,
admits weak elimination of imaginaries and is oligomorphic, then $G$ is
dynamically de Finetti.

We have strong evidence that there exist dynamically de Finetti groups which
are not oligomorphic (one potential candidate would be the automorphism group
of the universal rational Urysohn space $\mathbb{Q}\mathbb{U}$) and this will
be the topic of a future work. Before stating a key property of dynamically de
Finetti groups, let us discuss examples of such groups.

######  Example 2.3. —

All the examples we will present are obtained as automorphism groups of
Fraïssé limits. A Fraïssé limit $\mathbb{F}$ is an ultrahomogeneous structure
uniquely defined (up to isomorphism) by a Fraïssé class of finite structures.
We refer the reader to [22, § 7.1] for more details on Fraïssé limits. Here is
a non-exhaustive list of some Fraïssé classes whose automorphism groups
satisfy the assumption of Theorem 2.2 and therefore are dynamically de
Finetti.

  1. (i)

The class of finite sets. Its Fraïssé limit is $\mathbb{N}$ and its
automorphism group is $S_{\infty}$.

  2. (ii)

The class of finite linear orders. Its Fraïssé limit is $(\mathbb{Q},<)$ and
its automorphism group is the group $\mathrm{Aut}(\mathbb{Q},<)$ of order-
preserving permutations of $\mathbb{Q}$.

  3. (iii)

The class of finite cyclic orders. Its Fraïssé limit is
$(\mathbb{Q}/\mathbb{Z},<)$ and its automorphism group is the group
$\mathrm{Aut}(\mathbb{Q}/\mathbb{Z},<)$ of bijections of
$\mathbb{Q}/\mathbb{Z}$ which preserves the dense cyclic order.

  4. (iv)

The class of partially ordered finite sets.

  5. (v)

The class of finite simple graphs. Its Fraïssé limit is the Rado graph $R$.

  6. (vi)

The class of $K_{n}$-free finite simple graphs for some $n\geq 3$.

  7. (vii)

The class of $k$-uniform finite hypergraphs for some $k\geq 2$.

  8. (viii)

The class of $k$-uniform finite hypergraphs omitting a complete $k$-uniform
hypergraph for some $k\geq 2$.

  9. (ix)

The class of finite tournaments. A tournament is a directed graph where there
is an oriented edge between any two distinct vertices.

  10. (x)

The class of directed finite graphs omitting a (possibly infinite) set of
finite tournaments. Henson observed [21] that there are continuum many Fraïssé
limits obtained this way.

  11. (xi)

The class of finite metric spaces with distance set $\\{0,1,\dots,n\\}$ for
some fixed $n\geq 1$. For $n=1$, we recover (i) and for $n=2$, we recover (v).

We now explain briefly why the automorphism groups of these structures satisfy
the assumptions of Theorem 2.2.

  * •

In all the examples (i) \- (xi), the automorphism group of the Fraïssé limit
is oligomorphic because there are only finitely many isomorphism types of
finite structures generated by $n$ elements in the corresponding Fraïssé
class.

  * •

In all the examples (i) \- (xi), the automorphism group of the Fraïssé limit
has no algebraicity because the corresponding Fraïssé class has the strong
amalgamation property, see [17, (2.15)] for a definition.

  * •

A closed subgroup $G\leq S_{\infty}$ has the _strong small index property_ if
every subgroup $H\leq G$ of index $<2^{\aleph_{0}}$ lies between the pointwise
and the setwise stabilizer of a finite set $A\subseteq\mathbb{N}$. This
property implies weak elimination of imaginaries and has been verified for the
examples (i) and (v)-(x), see [33]. The fact that the other examples weakly
eliminate imaginaries is rather standard.

Let us close this example by mentioning that the automorphism groups in
(i)-(xi) are pairwise non (abstractly) isomorphic, implying in particular that
there are continuum many dynamically de Finetti groups. We refer to [32] for
the problem of reconstructing a countable structure from its automorphism
group.

One of the main features of dynamically de Finetti groups is that they satisfy
the classical theorem of de Finetti in exchangeable theory.

######  Theorem 2.4. —

Let $G\leq S_{\infty}$ be a closed subgroup, and let $a\in\mathbb{N}$. If $G$
is dynamically de Finetti, then any $G$-invariant probability measure $\mu$ on
$[0,1]^{G\cdot a}$ is mixed i.i.d., that is there exists a probability measure
$\lambda$ on the set $\mathrm{Prob}([0,1])$ such that

$\mu=\int_{\mathrm{Prob}([0,1])}m^{\otimes G\cdot a}d\lambda(m).$

###### Proof.

Assume first that the p.m.p. action $G\curvearrowright([0,1]^{G\cdot a},\mu)$
is ergodic. In this case, the $\sigma$-algebra $\mathcal{F}_{\emptyset}$ of
$G$-invariant measurable subsets of $[0,1]^{G\cdot a}$ is trivial. Therefore,
for any $A,B\subseteq G\cdot a$ finite and disjoint, we have
$\mathcal{F}_{A}\mathrel{\reflectbox{\rotatebox[origin={c}]{90.0}{$\models$}}}\mathcal{F}_{B}$.
This exactly says that $\mu=\otimes_{b\in G\cdot a}m_{b}$ where $m_{b}$ is the
$b^{\text{th}}$-marginal of $\mu$. But $G$ acts transitively on $G\cdot a$, so
all the (one-dimensional) marginals of $\mu$ are equal. Therefore,
$\mu=m^{\otimes G\cdot a}$ for some $m\in\mathrm{Prob}([0,1])$. If $\mu$ is
not ergodic, the ergodic decomposition theorem (see for instance [34, p. 77])
allows to conclude that $\mu$ is mixed i.i.d. ∎

We now state a characterization of having no algebraicity and admitting weak
elimination of imaginaries. This characterization, which will be useful later,
seems to be well known by model theorists (see for instance [35, Lem. 16.17]).
We provide here a proof in the language of permutation groups. If $G\leq
S_{\infty}$ is a closed subgroup, we denote by $\mathrm{Fix}(G)$ the set of
points in $\Omega$ which are fixed by $G$.

######  Lemma 2.5. —

Let $G\leq S_{\infty}$ be a closed subgroup. Then $G$ has no algebraicity and
admits weak elimination of imaginaries if and only if
$\mathrm{Fix}(G)=\emptyset$ and for all $A,B\subseteq\mathbb{N}$ finite, we
have $\langle G_{A},G_{B}\rangle=G_{A\cap B}$.

###### Proof.

$(\Rightarrow)$ First, we readily get $\mathrm{Fix}(G)=\emptyset$ because $G$
has no algebraicity. Fix $A,B\subseteq\mathbb{N}$ two finite subsets and let
$V=\langle G_{A},G_{B}\rangle$. By definition, $V\leq G_{A\cap B}$. Let us
show the reverse inclusion. Since $G$ admits weak elimination of imaginaries,
there exists a finite subset $C\subseteq\mathbb{N}$ such that $G_{C}\leq V$
and $[V:G_{C}]<+\infty$. We will prove that $C\subseteq A\cap B$. The fact
that $G_{C}$ has finite index in $V$ implies that the $V$-orbit of every $c\in
C$ is finite. However, the $G_{A}$-orbit of every $x\in\mathbb{N}\setminus A$
is infinite since $G$ has no algebraicity. Since $G_{A}\leq V$, the $V$-orbit
of every $x\in\mathbb{N}\setminus A$ is infinite. Therefore $C\subseteq A$.
Similarly, $C\subseteq B$. We conclude that $C\subseteq A\cap B$ and thus
$G_{A\cap B}\leq G_{C}\leq V=\langle G_{A},G_{B}\rangle\leq G_{A\cap B}$.

$(\Leftarrow)$ Let us first show that $G$ has no algebraicity. Assume that
this is not the case. Then there exists a finite subset $A\subseteq\mathbb{N}$
such that $G_{A}\curvearrowright\mathbb{N}\setminus A$ has a fixed point
$b\in\mathbb{N}\setminus A$ (indeed, we know that there exists
$A^{\prime}\subseteq\mathbb{N}$ finite such that
$G_{A^{\prime}}\curvearrowright\mathbb{N}\setminus A^{\prime}$ has a finite
orbit, say $\\{b_{1},\dots,b_{n}\\}$, then
$A=A^{\prime}\sqcup\\{b_{1},\dots,b_{n-1}\\}$ works). Set $B=\\{b\\}$. Then we
have $G_{A\sqcup B}=G_{A}$ and $G_{A\cap B}=G$. Therefore, $G=\langle
G_{A},G_{B}\rangle=\langle G_{A\sqcup B},G_{B}\rangle=G_{B}$. This shows that
$B\subseteq\mathrm{Fix}(G)$, which is a contradiction. Thus $G$ has no
algebraicity. We now prove that $G$ admits weak elimination of imaginaries.
Let $V\leq G$ be an open subgroup. The property “$\langle
G_{A},G_{B}\rangle=G_{A\cap B}$ for all $A,B\subseteq\mathbb{N}$ finite”
implies that there exists a unique finite subset $A_{0}\subseteq\mathbb{N}$
which is minimal (for inclusion) among all finite subsets
$A\subseteq\mathbb{N}$ that satisfy $G_{A}\leq V$. Let us show that
$[V:G_{A_{0}}]<+\infty$. Observe that for all $g\in G$, $G_{g(A_{0})}$ is a
subgroup of $gVg^{-1}$ and $g(A_{0})$ is minimal among finite subsets
$A\subseteq\mathbb{N}$ that satisfy $G_{g(A_{0})}\leq gVg^{-1}$. This implies
that for all $g\in V$, we have $g(A_{0})=A_{0}$ and thus
$[V:G_{A_{0}}]<+\infty$. ∎

We close this section with a discussion on topological simplicity. We provide
here a proof of dynamical nature of the fact that dynamically de Finetti
groups are topologically simple. It should however be noted that, as pointed
out to us by David Evans (personal communication), having no algebraicity and
weak elimination of imaginaries is enough to show topological simplicity.

######  Lemma 2.6. —

Let $G\leq S_{\infty}$ be a closed subgroup. Assume that any p.m.p. ergodic
action of $G$ is either essentially free or essentially transitive. Then any
closed non-trivial normal subgroup $N\trianglelefteq G$ is cocompact.

###### Proof.

Assume that $N\trianglelefteq G$ is a closed, non-trivial normal subgroup and
let $\pi:G\twoheadrightarrow G/N$ be the quotient homomorphism. The quotient
group $G/N$ is a closed permutation group by [11, Thm. 1.5.1]. Theorem 6.5
implies that the IRS $\delta_{\\{1_{G/N}\\}}$ of $G/N$ is realized, that is,
$G/N$ admits a p.m.p. essentially free action. By the ergodic decomposition
theorem [34, p. 77], it admits a p.m.p. ergodic essentially free action
$G/N\curvearrowright(X,\mu)$. The action $G\curvearrowright(X,\mu)$ defined by
$g\cdot x\coloneqq\pi(g)\cdot x$ is a p.m.p. ergodic action of $G$ whose
stabilizers are a.s. equal to $N$. Since $N$ is non-trivial, the action is
therefore not essentially free. By assumption on $G$, the action
$G/N\curvearrowright(X,\mu)$ is thus essentially transitive. This means that
there exists a $G$-invariant probability measure $\nu$ on $G/N$ such that
$G\curvearrowright(X,\mu)$ is measurably isomorphic to
$G\curvearrowright(G/N,\nu)$. This implies that $G/N$ is a compact group as
$\nu$ is the Haar probability measure on it. Thus, $N$ is cocompact. ∎

Assume now that $G\lneq S_{\infty}$ is a transitive, closed, proper
permutation group which is dynamically de Finetti. By Theorem 5.7 any p.m.p.
ergodic action of $G$ is either essentially free or essentially transitive. By
Lemma 2.6, any closed non-trivial normal subgroup $N\trianglelefteq G$ is
cocompact. Let us prove that $N=G$. If this is not the case, then $G/N$ is a
non-trivial compact group. But compact closed permutation groups are
profinite, so $G/N$ is a non-trivial profinite group. In particular, $G$
admits a proper finite index subgroup $H$. Since $[G:H]<+\infty$, $H$ is open
in $G$. Since $G$ weakly eliminates imaginaries, there exists
$A\subseteq\mathbb{N}$ finite non-empty, such that $G_{A}\leq H$ and
$[H:G_{A}]<+\infty$. This implies that $[G:G_{A}]<+\infty$. But $G$ has no
algebraicity, so the $G$ orbit of $A$ is infinite and therefore $[G:G_{A}]$ is
infinite, which yields a contradiction. Therefore, dynamically de Finetti
groups are topologically simple.

##  3 Preliminaries on model theory

###  3.1 Structures and logic actions

A relational language $\mathcal{L}=(R_{i})_{i\in I}$ is a countable collection
of relation symbols, each of which has a given arity $r_{i}$. A structure
$\mathbf{M}$ in the language $\mathcal{L}$ (with domain $\mathbb{N}$) is a
collection of subsets $R_{i}^{\mathbf{M}}\subseteq\mathbb{N}^{r_{i}}$ for each
$i\in I$, which are interpretations of the abstract relation symbols $R_{i}$.

Examples of structures include simple graphs, where the language is a single
binary relation, whose interpretation is the set of edges of the graph.
Similarly, $k$-hypergraphs are structures, the language being a single $k$-ary
relation. We can also see linear (as well as partial) orders as structures
with a single binary relation.

We denote by

$\mathrm{Struc}_{\mathcal{L}}\coloneqq\prod_{i\in
I}\\{0,1\\}^{\mathbb{N}^{r_{i}}}$

the space of structures in the language $\mathcal{L}$. This is a compact space
with the product topology and there is a natural continuous
$S_{\infty}$-action on it called the logic action: for $g\in S_{\infty}$ and
$\mathbf{M}$ a structure, $g\cdot\mathbf{M}$ is the structure $\mathbf{N}$
defined by

$\forall i\in I,(R_{i}^{\mathbf{N}}(x_{1},\dots,x_{r_{i}})=1\Leftrightarrow
R_{i}^{\mathbf{M}}(g^{-1}(x_{1},\dots,x_{r_{i}}))=1).$

The automorphism group of $\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}$ is the
stabilizer of $\mathbf{M}$ for the logic action, i.e.,

$\mathrm{Aut}(\mathbf{M})\coloneqq\\{g\in S_{\infty}\colon
g\cdot\mathbf{M}=\mathbf{M}\\}.$

Let $G$ be a closed subgroup of $S_{\infty}$. For all $n\geq 1$, let $J_{n}$
be the set of orbits of the diagonal action $G\curvearrowright\mathbb{N}^{n}$
and let $J=\bigcup_{n\geq 1}J_{n}$. We denote by
$\mathcal{L}_{G}\coloneqq(R_{j})_{j\in J}$ the canonical language associated
with $G$, where $R_{j}$ is of arity $n$ for all $j\in J_{n}$. The canonical
structure associated with $G$ is the structure
$\mathbf{M}_{G}\coloneqq(R_{j}^{G})_{j\in J}$, where
$R_{j}^{G}=j\subseteq\mathbb{N}^{n}$ for all $j\in J_{n}$. It is easy to check
that $\mathrm{Aut}(\mathbf{M}_{G})=G$. In this article we will deal with
structures which expand the canonical structure associated with a closed
subgroup $G\leq S_{\infty}$. For a language $\mathcal{L}$ which contains
$\mathcal{L}_{G}$, an element $\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}$ is
an expansion of $\mathbf{M}_{G}$ if the structure
$\mathbf{M}_{\upharpoonright\mathcal{L}_{G}}\in\mathrm{Struc}_{\mathcal{L}_{G}}$
(called the $\mathcal{L}_{G}$-reduct of $\mathbf{M}$) obtained from
$\mathbf{M}$ by removing the relation symbols from
$\mathcal{L}\setminus\mathcal{L}_{G}$ is equal to $\mathbf{M}_{G}$.

######  Definition 3.1. —

Let $G\leq S_{\infty}$ be a closed subgroup and let $\mathcal{L}$ be a
language which contains $\mathcal{L}_{G}$. A $G$-structure in the language
$\mathcal{L}$ is an element of $\mathrm{Struc}_{\mathcal{L}}$ which is an
expansion of the canonical structure $\mathbf{M}_{G}$.

We denote by $\mathrm{Struc}_{\mathcal{L}}^{G}$ the space of $G$-structures in
the language $\mathcal{L}$. This is a closed subset of
$\mathrm{Struc}_{\mathcal{L}}$ and there is a natural continuous $G$-action on
$\mathrm{Struc}_{\mathcal{L}}^{G}$ which is induced from the $G$-action on
$\mathrm{Struc}_{\mathcal{L}}$ and is called the relativized logic action [11,
§2.7]. With this terminology, for any language $\mathcal{L}$, there is a
canonical homeomorphism between $\mathrm{Struc}_{\mathcal{L}}$ and
$\mathrm{Struc}^{S_{\infty}}_{\mathcal{L}\sqcup\mathcal{L}_{S_{\infty}}}$ and
we will therefore consider elements of $\mathrm{Struc}_{\mathcal{L}}$ as
$S_{\infty}$-structures. We will use the relativized logic action in an
essential way in Section 5 through a universality theorem of Becker and
Kechris [11, Thm. 2.7.4], which states that whenever
$\mathcal{L}\setminus\mathcal{L}_{G}$ contains relations of arbitrarily high
arity, then for any Borel action $G\curvearrowright X$ on a standard Borel
space, there exists a Borel $G$-equivariant injective map
$X\to\mathrm{Struc}_{\mathcal{L}}^{G}$.

###  3.2 First-order and infinitary logics

In this section, we fix a countable relational language
$\mathcal{L}=(R_{i})_{i\in I}$. An atomic formula in the language
$\mathcal{L}$ is an expression of the form $R_{i}(v_{1},\dots,v_{r_{i}})$ for
some $i\in I$, where $v_{1},\dots,v_{r_{i}}$ are free variables and $r_{i}$ is
the arity of $R_{i}$. The set of quantifier-free formulas is the smallest set
that contains atomic formulas and is closed under negation, finite conjunction
and finite disjunction. We denote by $\mathcal{L}_{\omega_{1},\omega}$ the
infinitary logic in the language $\mathcal{L}$, which is the smallest set
containing atomic formulas and closed under negation, under universal and
existential quantification and under conjunction and disjunction of any
countable family of formulas with a common finite set of free variables. We
denote by $\mathcal{L}_{\omega,\omega}$ the standard finitary first-order
logic in the language $\mathcal{L}$, which consists in first-order formulas,
that is, formulas in $\mathcal{L}_{\omega_{1},\omega}$ with only finitely many
disjunctions and conjunctions. A formula in $\mathcal{L}_{\omega_{1},\omega}$
without free variables is called a sentence. For an atomic formula
$\phi(v_{1},\dots,v_{r_{i}})=R_{i}(v_{1},\ldots,v_{r_{i}})$ and
$\bar{x}=(x_{1},\ldots,x_{r_{i}})\in\mathbb{N}^{r_{i}}$, we say that
$\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}$ satisfies $\phi(\bar{x})$ if
$R_{i}^{\mathbf{M}}(x_{1},\ldots,x_{r_{i}})=1$. The interpretation of any non-
atomic formula $\phi(v_{1},\dots,v_{n})\in\mathcal{L}_{\omega_{1},\omega}$ is
defined inductively, the interpretation of each symbol corresponding to its
usual use in mathematics. For any tuple $\bar{x}\in\mathbb{N}^{n}$, we write
$\mathbf{M}\models\phi(\bar{x})$ whenever $\mathbf{M}$ satisfies
$\phi(\bar{x})$. The following lemma is straightforward and will be used many
times in the sequel.

######  Lemma 3.2. —

Let $\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}$,
$\phi(v_{1},\dots,v_{n})\in\mathcal{L}_{\omega_{1},\omega}$ and
$x\in\mathbb{N}^{<\omega}$.

  1. (i)

The set
$\\{\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}\colon\mathbf{M}\models\phi(\bar{x})\\}$
is a Borel subset of $\mathrm{Struc}_{\mathcal{L}}$.

  2. (ii)

For all $g\in S_{\infty}$, we have
$\mathbf{M}\models\phi(\bar{x})\Leftrightarrow
g\cdot\mathbf{M}\models\phi(g(\bar{x}))$.

A fragment in $\mathcal{L}_{\omega_{1},\omega}$ is a set which contains
$\mathcal{L}_{\omega,\omega}$ and is closed under subformula, finite
conjunction and disjunction, negation, universal and existential
quantification, and substitution of free variables. Any subset
$\Sigma\leq\mathcal{L}_{\omega_{1},\omega}$ is contained in a least fragment
denoted by $\langle\Sigma\rangle$, which is countable whenever $\Sigma$ is. By
definition, $\langle\emptyset\rangle=\mathcal{L}_{\omega,\omega}$. Let $F$ be
a fragment. Given $n\geq 1$, the set of formulas $\phi(v_{1},\dots,v_{n})\in
F$ with $n$-free variables is a Boolean algebra and we denote by $S^{n}_{F}$
its Stone space, which is the space of $F$-types in $n$ variables. By Stone
duality, $S^{n}_{F}$ is a compact Hausdorff totally disconnected space, which
is moreover metrizable if and only if $F$ is a countable fragment. A basis of
clopen sets of $S^{n}_{F}$ is given by the sets $[\phi]\coloneqq\\{p\in
S^{n}_{F}\colon\phi\in p\\}$ for $\phi\in F$. Given
$\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}$ and $\bar{x}\in\mathbb{N}^{n}$ we
denote by $\mathrm{tp}^{\mathbf{M}}_{F}(\bar{x})$ the $F$-type in $S^{n}_{F}$
defined by

$\mathrm{tp}^{\mathbf{M}}_{F}(\bar{x})\coloneqq\\{\phi\in
F\colon\mathbf{M}\models\phi(\bar{x})\\}.$

The following lemma is the equivalent of Lemma 3.2 for $F$-types.

######  Lemma 3.3. —

Let $F$ be a fragment and $\bar{x}\in\mathbb{N}^{n}$. The following holds

  1. (i)

The map $\mathrm{tp}_{F}(\bar{x}):\mathrm{Struc}_{\mathcal{L}}\to S^{n}_{F}$
is Borel.

  2. (ii)

For all $\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}$ and $g\in S_{\infty}$, we
have
$\mathrm{tp}_{F}^{\mathbf{M}}(\bar{x})=\mathrm{tp}_{F}^{g\cdot\mathbf{M}}(g(\bar{x}))$.

  3. (iii)

For all $\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}$ and all
$\bar{y}\in\mathrm{Aut}(\mathbf{M})\cdot\bar{x}$, we have
$\mathrm{tp}_{F}^{\mathbf{M}}(\bar{x})=\mathrm{tp}_{F}^{\mathbf{M}}(\bar{y})$.

###### Proof.

Let us prove (i). For all $\phi\in F$, observe that

$\\{\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}\colon\mathrm{tp}^{\mathbf{M}}_{F}(\bar{x})\in[\phi]\\}=\\{M\in\mathrm{Struc}_{\mathcal{L}}^{G}\colon\mathbf{M}\models\phi(\bar{x})\\},$

which is Borel by Lemma 3.2 (i). Therefore, the map
$\mathrm{tp}_{F}(\bar{x}):\mathrm{Struc}_{\mathcal{L}}\to S^{n}_{F}$ is Borel.
The proof of (ii) is a straightforward consequence of Lemma 3.2 (ii). Finally,
(iii) is a particular case of (ii). ∎

We finally need the notion of quantifier-free types. The space of quantifier-
free types in $n$ variables is the Stone space of the Boolean algebra of
quantifier-free formulas with $n$ free variables. Given
$\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}$ and $\bar{x}\in\mathbb{N}^{n}$, we
denote by $\mathrm{qftp}^{\mathbf{M}}(\bar{x})$ the quantifier-free type
defined by

$\mathrm{qftp}^{\mathbf{M}}(\bar{x})\coloneqq\\{\phi(v_{1},\dots,v_{n})\text{
quantifier-free formula}\colon\mathbf{M}\models\phi(\bar{x})\\}.$

An important remark is that if $\mathbf{M}$ is a $G$-structure for some closed
subgroup $G\leq S_{\infty}$, then for all
$\bar{x},\bar{y}\in\mathbb{N}^{<\omega}$,
$\mathrm{qftp}^{\mathbf{M}}(\bar{x})=\mathrm{qftp}^{\mathbf{M}}(\bar{y})$
implies that $\bar{x}$ and $\bar{y}$ are in the same $G$-orbit. An immediate
corollary is that for any countable fragment $F$,
$\mathrm{tp}^{\mathbf{M}}_{F}(\bar{x})=\mathrm{tp}^{\mathbf{M}}_{F}(\bar{y})$
implies that $\bar{x}$ and $\bar{y}$ are in the same $G$-orbit.

###  3.3 Back-and-forth

Let us fix in this section a closed subgroup $G\leq S_{\infty}$ and a
countable relational language $\mathcal{L}$ which contains $\mathcal{L}_{G}$.
The following lemma, which is almost tautological, is at the heart of the
technique of back-and-forth that we explain next.

######  Lemma 3.4. —

Let $\mathbf{M},\mathbf{N}\in\mathrm{Struc}_{\mathcal{L}}^{G}$. Assume that
there exist two sequences $A_{0}\subseteq A_{1}\subseteq\dots$,
$B_{0}\subseteq B_{1}\subseteq\dots$ of finite subsets of $\mathbb{N}$ and a
sequence $g_{0}\subseteq g_{1}\subseteq\dots$ of bijections $g_{n}:A_{n}\to
B_{n}$, which are restrictions of elements in $G$, such that
$\bigcup_{n}A_{n}=\bigcup_{n}B_{n}=\mathbb{N}$ and for all $n\geq 0$, for all
$R\in\mathcal{L}$ of arity $r$, for all $x_{1},\dots,x_{r}\in A_{n}$, we have

$\displaystyle R^{\mathbf{M}}(x_{1},\dots,x_{r})=1\Leftrightarrow
R^{\mathbf{N}}(g_{n}(x_{1},\dots,x_{r}))=1.$ ($*$)

Then $g=\bigcup_{n}g_{n}$ belongs to $G$ and satisfies
$g\cdot\mathbf{M}=\mathbf{N}$. If $\mathbf{M}=\mathbf{N}$, then
$g\in\mathrm{Aut}(\mathbf{M})$.

The way we will use this lemma in our proofs is by inductively constructing
$A_{n},B_{n}$ and $g_{n}$. Assume for the initial step that we have
$A_{0},B_{0}$ with same quantifier-free type and $g_{0}$ a bijection between
$A_{0}$ and $B_{0}$ preserving the relations. The strategy is to take
$(x_{i})_{i\geq 1}$ and $(y_{j})_{j\geq 1}$ enumerations of $\mathbb{N}$ and
to ensure that $A_{n}$ contains $x_{1},\ldots,x_{n}$ and $B_{n}$ contains
$y_{1},\ldots,y_{n}$. The key is to construct both sets in order to ensure
that for any $x\in\mathbb{N}$ there is $y\in\mathbb{N}$ such that
$\mathrm{qftp}^{\mathbf{M}}(A_{n},x)=\mathrm{qftp}^{\mathbf{N}}(B_{n},y)$ and
for all $y^{\prime}\in\mathbb{N}$ there is $x^{\prime}\in\mathbb{N}$ such that
$\mathrm{qftp}^{\mathbf{N}}(B_{n},y^{\prime})=\mathrm{qftp}^{\mathbf{M}}(A_{n},x^{\prime})$.

If such is the case, assume that we have constructed $A_{n},B_{n}$ as in the
former paragraph and $g_{n}:A_{n}\to B_{n}$ a bijection which is a restriction
of an element in $G$. Let $i$ be the smallest index such that $x_{i}\notin
A_{n}$ and $j$ the smallest index such that $y_{j}\notin B_{n}$ (by
construction, $i>n$ and $j>n$). By assumption, there exists $y\in\mathbb{N}$
and $x\in\mathbb{N}$

$\displaystyle\mathrm{qftp}^{\mathbf{M}}(A_{n},x_{i})$
$\displaystyle=\mathrm{qftp}^{\mathbf{N}}(B_{n},y),$ (1)
$\displaystyle\mathrm{qftp}^{\mathbf{N}}(B_{n},y,y_{j})$
$\displaystyle=\mathrm{qftp}^{\mathbf{M}}(A_{n},x_{i},x).$ (2)

We then set $A_{n+1}=A_{n}\cup\\{x_{i},x\\}$,
$B_{n+1}=B_{n}\cup\\{y,y_{j}\\}$. Since $x_{i}\notin A_{n}$ and $y_{j}\notin
B_{n}$, the equalities of the quantifier-free types in (1) and (2) imply that
$x\notin A_{n}$ and $y\notin B_{n}$. This allows to extend $g_{n}$ into a
bijection $g_{n+1}:A_{n+1}\to B_{n+1}$ by setting $g_{n+1}(x_{i})=y$ and
$g_{n+1}(x)=y_{j}$. Equations (1) and (2) and the definition of quantifier-
free type ensure that $g_{n+1}$ is indeed the restriction of an element in $G$
and that the condition ($*$ ‣ 3.4) in Lemma 3.4 is satisfied.

##  4 Invariant Random Expansions

In this section we fix a closed subgroup $G\leq S_{\infty}$ and a countable
relational language $\mathcal{L}$ which contains the canonical language
$\mathcal{L}_{G}$. Recall that $\mathrm{Struc}_{\mathcal{L}}^{G}$ denotes the
compact space of expansions of the canonical structure $\mathbf{M}_{G}$, which
are the structures $\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}$ whose
$\mathcal{L}_{G}$-reduct $\mathbf{M}_{\upharpoonright\mathcal{L}_{G}}$ is
equal to $\mathbf{M}_{G}$. We now turn our attention to invariant random
expansions.

######  Definition 4.1. —

A invariant random expansion of $G$ in the language $\mathcal{L}$ is a Borel
probability measure on $\mathrm{Struc}_{\mathcal{L}}^{G}$ which is invariant
under the relativized logic action.

We denote by $\mathrm{IRE}(G)$ (or $\mathrm{IRE}_{\mathcal{L}}(G)$ if we want
to emphasize the language) the space of invariant random expansions of $G$. We
will also use $G$-IRE to refer to an element of $\mathrm{IRE}(G)$. Let us give
concrete examples of invariant random expansions.

######  Example 4.2. —

  1. (i)

For all $p\in]0,1[$, the random simple graph whose vertex set is $\mathbb{N}$
and the edges are i.i.d. with distribution $\mathrm{Ber}(p)$ is an
$S_{\infty}$-IRE.

  2. (ii)

Let $\mathrm{LO}(\mathbb{N})\subseteq\\{0,1\\}^{\mathbb{N}\times\mathbb{N}}$
be the space of linear orders on $\mathbb{N}$. There is a unique
$S_{\infty}$-invariant probability measure $\mu$ on $\mathrm{LO}(\mathbb{N})$
and it is defined by its values on the cylinders $\\{x_{1}<\dots<x_{n}\\}$ by
$\mu(\\{x_{1}<\dots<x_{n}\\})=1/n!$. This gives an $S_{\infty}$-IRE.

  3. (iii)

The structure we consider in this case is a particular $3$-hypergraph. A
$2$-graph is a 3-hypergraph such that there is an even number of hyperedges
between any four vertices. A simple way to produce a $2$-graph is to take a
graph, put an hyperedge between three vertices if there is an even number of
edges between them, and then remove the edges. Any $2$-graph can be obtained
this way and we call a graph producing a given $2$-graph $H$ a graphing of
$H$. Consider $\mathbf{N}$ the Fraïssé limit of finite $2$-graphs, which can
be obtained by the above construction starting from a Rado graph. Denote by
$G$ the automorphism group of $\mathbf{N}$. There is a $G$-IRE concentrated on
the space of graphings of $\mathbf{N}$, see [23, Chap. 2]. One notable feature
is that this IRE doesn’t come from an $S_{\infty}$-IRE, see Section 7 for a
more in-depth discussion.

###  4.1 First properties

In this section, we state several lemmas about IREs of groups with no
algebraicity. Recall that $G$ has no algebraicity if for all finite subset
$A\subseteq\mathbb{N}$, we have $\mathrm{acl}(A)=A$. We will not prove the
measurability of sets and maps in this section, all of them being
straightforward, save for $\mathbf{M}\mapsto\mathrm{Aut}(\mathbf{M})$. The
measurability of this map is the object of Lemma 6.4 (ii).

######  Lemma 4.3. —

Let $\mu\in\mathrm{IRE}(G)$. Then for $\mu$-a.e.
$\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}^{G}$, the set
$\mathrm{Fix}(\mathrm{Aut}(\mathbf{M}))$ is either empty or infinite.

###### Proof.

Assume that there exists a finite subset $A\subseteq\mathbb{N}$ such that

$\mu(\\{\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}^{G}\colon\mathrm{Fix}(\mathrm{Aut}(\mathbf{M}))=A\\})>0.$

Since $G$ has no algebraicity, there exists infinitely many pairwise disjoints
sets $A_{n}$ in the $G$-orbit of $A$ by Neumann’s lemma [22, Cor. 4.2.2]. By
$G$-invariance of $\mu$, the sets
$\\{\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}^{G}\colon\mathrm{Fix}(\mathrm{Aut}(\mathbf{M}))=A_{n}\\}$
all have the same measure, which is positive. This is absurd since they are
pairwise disjoint. ∎

The following lemma shows that if $\mu$ is an ergodic invariant random
expansion of $G$ which has no algebraicity, then $\mu$-a.s.,
$\mathrm{Aut}(\mathbf{M})$ has no algebraicity, apart from the fixed points
$\mathrm{Fix}(\mathrm{Aut}(\mathbf{M}))$.

######  Lemma 4.4. —

Assume that $G$ has no algebraicity and let $\mu\in\mathrm{IRE}(G)$. Then for
all tuples $\bar{x}\in\mathbb{N}^{<\omega}$, the following holds:

  1. (i)

for $\mu$-a.e. $\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}^{G}$, we have
$\mathrm{Fix}(\mathrm{Aut}(\mathbf{M})_{\bar{x}})=\mathrm{Fix}(\mathrm{Aut}(\mathbf{M}))\cup\bar{x}$.

  2. (ii)

for $\mu$-a.e. $\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}^{G}$, the
$\mathrm{Aut}(\mathbf{M})_{\bar{x}}$-orbits on $\mathbb{N}$ are either of size
$1$ or infinite.

###### Proof.

Let us prove (i). It is clear that $\mu$-a.s., we have
$\mathrm{Fix}(\mathrm{Aut}(\mathbf{M}))\cup\bar{x}\subseteq\mathrm{Fix}(\mathrm{Aut}(\mathbf{M})_{\bar{x}})$.
We now prove the reverse inclusion. Assume by contradiction that there exists
$a\in\mathbb{N}\setminus\bar{x}$ such that

$\mu(\\{\mathrm{Fix}(\mathrm{Aut}(\mathbf{M})_{\bar{x}})\setminus\mathrm{Fix}(\mathrm{Aut}(\mathbf{M}))\text{
contains }a\\})>0.$

Therefore, there exists $\bar{y}\in\mathbb{N}^{<\omega}$ such that

$\mu(\\{\mathrm{Fix}(\mathrm{Aut}(\mathbf{M})_{\bar{x}})\text{ contains
}a\text{ and }\exists g\in\mathrm{Aut}(\mathbf{M})\colon
g(\bar{x})=\bar{y},g(a)\neq a\\})>0.$

Conditionally on the set
$\\{\mathrm{Fix}(\mathrm{Aut}(\mathbf{M})_{\bar{x}})\text{ contains }a\\}$, we
have that for $\mu$-a.e. $\mathbf{M}$, for all
$g,h\in\mathrm{Aut}(\mathbf{M})$, if $g(\bar{x})=h(\bar{x})=\bar{y}$ then
$g(a)=h(a)$. If we denote by $\bar{z}$ the tuple $(\bar{x},\bar{y},a)$, then
for all $b\in\mathbb{N}\setminus\bar{z}$, the $\mu$-measure of the set

$\\{\mathrm{Fix}(\mathrm{Aut}(\mathbf{M})_{\bar{x}})\text{ contains }a\text{
and for any }g\in\mathrm{Aut}(\mathbf{M})\text{ s.t.
}g(\bar{x})=\bar{y},g(a)=b\\}$

is strictly positive and constant along the $G_{\bar{z}}$-orbit of $b$. These
orbits are infinite since $G$ has no algebraicity, and the former sets are
pairwise disjoint, which is a contradiction.

Let us now prove (ii). Assume by contradiction that there exists a tuple
$\bar{y}=(y_{1},\dots,y_{n})$ of size $n\geq 2$ such that
$\mu(\\{\bar{y}\text{ is an orbit of
}\mathrm{Aut}(\mathbf{M})_{\bar{x}}\\})>0$. Let
$\bar{z}\coloneqq(\bar{x},y_{1},\dots,y_{n-1})$. Then
$\mu(\\{\mathrm{Fix}(\mathrm{Aut}(\mathbf{M})_{\bar{z}})\setminus\mathrm{Fix}(\mathrm{Aut}(\mathbf{M}))\text{
contains }y_{n}\\})>0$, which contradicts (i). ∎

The following result is a version of Neumann’s lemma for IRE.

######  Lemma 4.5. —

Assume that $G$ has no algebraicity and let $\mu\in\mathrm{IRE}(G)$. Let
$\bar{x}\in\mathbb{N}^{<\omega}$. Then for $\mu$-a.e.
$\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}^{G}$, either $\bar{x}$ contains an
element of $\mathrm{Fix}(\mathrm{Aut}(\mathbf{M}))$ or the
$\mathrm{Aut}(\mathbf{M})$-orbit of $\bar{x}$ contains an infinite set of
pairwise disjoint tuples.

###### Proof.

We know by Lemma 4.4 that $\mu$-a.s., the orbits of
$\mathrm{Aut}(\mathbf{M})_{\bar{x}}$ are either of size $1$ or infinite. Then
Neumann’s lemma [22, Cor. 4.2.2] gives us the desired conclusion. ∎

###  4.2 IREs without fixed point

If $\mu\in\mathrm{IRE}(G)$, we say that $\mu$ has no fixed point if for
$\mu$-a.e. $\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}^{G}$, the set
$\mathrm{Fix}(\mathrm{Aut}(\mathbf{M}))$ is empty. We now focus our attention
on IREs without fixed points under the additional assumption that the group
$G$ is dynamically de Finetti.

######  Lemma 4.6. —

Let $G\leq S_{\infty}$ be a closed subgroup, which is dynamically de Finetti.
Let $\mu\in\mathrm{IRE}(G)$ be ergodic and let $F$ be a countable fragment. If
$\mu$ has no fixed point, then for all $\bar{x}\in\mathbb{N}^{n}$, the
probability measure $\mathrm{tp}_{F}(\bar{x})_{*}\mu$ on $S^{n}_{F}$ is purely
atomic.

###### Proof.

First of all, observe that by Lemma 3.3 (ii), for all
$\bar{x}\in\mathbb{N}^{n}$, the random variable
$\mathrm{tp}_{F}(\bar{x}):(\mathrm{Struc}_{\mathcal{L}}^{G},\mu)\to S^{n}_{F}$
is $G_{\bar{x}}$-invariant and therefore $\mathcal{F}_{\bar{x}}$-measurable,
where $\mathcal{F}_{\bar{x}}$ denotes the $\sigma$-algebra of
$G_{\bar{x}}$-invariant measurable subsets of
$\mathrm{Struc}_{\mathcal{L}}^{G}$. Since $\mu$ is ergodic and $G$ is
dynamically de Finetti, we deduce that for all
$\bar{x},\bar{y}\in\mathbb{N}^{n}$ disjoint, the random variables
$\mathrm{tp}_{F}(\bar{x})$ and $\mathrm{tp}_{F}(\bar{y})$ are independent.

If $\mu$ has no fixed point, then by Lemma 4.5, for $\mu$-a.e. $\mathbf{M}$,
there exists $\bar{y}$ disjoint from $\bar{x}$ in the
$\mathrm{Aut}(\mathbf{M})$-orbit of $\bar{x}$. This implies that $\mu$-a.s.,
there is $\bar{y}\in\mathbb{N}^{n}$ disjoint from $\bar{x}$ such that
$\mathrm{tp}_{F}^{\mathbf{M}}(\bar{x})=\mathrm{tp}_{F}^{\mathbf{M}}(\bar{y})$.
Assume that $\mathrm{tp}_{F}(\bar{x})_{*}\mu$ is not purely atomic. This means
that there exists a measurable subset
$X\subseteq\mathrm{Struc}_{\mathcal{L}}^{G}$, $\mu(X)>0$, such that
$\mathrm{tp}_{F}(\bar{x})_{*}\mu(\cdot\mid X)$ is a diffuse measure. There
exists $\bar{y}\in\mathbb{N}^{n}$ (deterministic) disjoint from $\bar{x}$ and
a measurable subset $X^{\prime}\subseteq X$ of positive measure such that for
all $\mathbf{M}\in X^{\prime}$,
$\mathrm{tp}_{F}^{\mathbf{M}}(\bar{x})=\mathrm{tp}_{F}^{\mathbf{M}}(\bar{y})$.
By independence of $\mathrm{tp}_{F}(\bar{x})$ and $\mathrm{tp}_{F}(\bar{y})$
and using the fact that $\mathrm{tp}_{F}(\bar{x})_{*}\mu(\cdot\mid
X^{\prime})$ is diffuse, we get that

$0=\mu(\\{\mathbf{M}\in
X^{\prime}\colon\mathrm{tp}_{F}^{\mathbf{M}}(\bar{x})=\mathrm{tp}_{F}^{\mathbf{M}}(\bar{y})\\})=\mu(X^{\prime})>0,$

which yields a contradiction. Thus, $\mathrm{tp}_{F}(\bar{x})_{*}\mu$ is
purely atomic. ∎

######  Remark 4.7. —

If $\mu$ has fixed points, then one can prove with a similar argument that the
probability measure $\mathrm{tp}_{F}(\bar{x})_{*}\tilde{\mu}$ is purely
atomic, where $\tilde{\mu}$ is the conditional measure
$\mu(\cdot\mid\\{\bar{x}\cap\mathrm{Fix}(\mathrm{Aut}(\mathbf{M}))=\emptyset\\})$.

For all $n\geq 0$, we denote by $S^{n}_{F}(\mu)$ the countable set of $p\in
S^{n}_{F}$ for which there exists $\bar{x}\in\mathbb{N}^{n}$ such that
$\mu(\\{\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}^{G}\colon\mathrm{tp}_{F}^{\mathbf{M}}(\bar{x})=p\\})>0$.
In order to analyze in details IRE with no fixed points of dynamically de
Finetti groups, we need the following result, a version for $S_{\infty}$-IRE
of which is contained in [4, Lem. 4.6]. Moreover, the proof we present here
contains no new argument compared to the proof of the aforementioned result.

######  Theorem 4.8. —

Assume that $G$ is dynamically de Finetti. Let $\mu\in\mathrm{IRE}(G)$ be
ergodic. If $\mu$ has no fixed point, then there exists a countable fragment
$F$ such that for all $p\in S^{n}_{F}(\mu)$, $q\in S^{n+1}_{F}(\mu)$ and
$(\bar{x},z)\in\mathbb{N}^{n+1}$ satisfying
$\mu(\\{\mathbf{M}\colon\mathrm{tp}_{F}^{\mathbf{M}}(\bar{x})=p\text{ and
}\mathrm{tp}_{F}^{\mathbf{M}}(\bar{x},z)=q\\})>0$, the following holds
$\mu$-a.s.

$\forall\bar{y}\in\mathbb{N}^{n},(\mathrm{tp}_{F}^{\mathbf{M}}(\bar{y})=p\Rightarrow\exists
z^{\prime}\colon\mathrm{tp}_{F}^{\mathbf{M}}(\bar{y},z^{\prime})=q).$

###### Proof.

Let $\omega_{1}$ be the first uncountable ordinal. We will construct a family
of countable fragments $(F_{\alpha})_{\alpha<\omega_{1}}$ depending on $\mu$,
indexed by countable ordinals and show that for some ordinal
$\alpha<\omega_{1}$, the countable fragment $F_{\alpha}$ is as wanted. We
define $F_{\alpha}$ by transfinite induction:

$\displaystyle F_{0}$ $\displaystyle=\mathcal{L}_{\omega,\omega},$
$\displaystyle F_{\alpha+1}$
$\displaystyle=\Big{\langle}F_{\alpha},\bigwedge_{\phi\in
p}\phi(v_{1},\dots,v_{n})\text{ for all }p\in S^{n}_{F_{\alpha}}(\mu)\text{
and }n\geq 0\Big{\rangle},$ $\displaystyle F_{\beta}$
$\displaystyle=\bigcup_{\alpha<\beta}F_{\alpha}\text{ if }\beta\text{ is a
limit ordinal}.$

For all $n\geq 0$ the set $S^{n}_{F}(\mu)$ is countable, therefore
$F_{\alpha}$ is a countable fragment for all ordinal $\alpha$.

######  Lemma 4.9. —

There is an ordinal $\alpha$ such that for $\alpha<\beta<\omega_{1}$, all
$\bar{x}\in\mathbb{N}^{n}$ and $r\in S^{n}_{F_{\beta}}(\mu)$, if
$\mu(\\{\mathbf{M}\colon\mathrm{tp}_{F_{\beta}}^{\mathbf{M}}(\bar{x})=r\\})>0$,
then

$\displaystyle\mu(\\{\mathbf{M}\colon\mathrm{tp}_{F_{\beta}}^{\mathbf{M}}(\bar{x})=r\\})=\mu(\\{\mathbf{M}\colon\mathrm{tp}_{F_{\alpha}}^{\mathbf{M}}(\bar{x})=s\\})$
(3)

where $s\subseteq r$ denotes the restriction of $r$ to the fragment
$F_{\alpha}$.

###### Proof.

We reproduce the argument from the proof of [4, Lem. 4.5]. Fix
$\bar{x}\in\mathbb{N}^{n}$. For $\alpha<\omega_{1}$ an ordinal, let us denote
by $\mathrm{Sp}(\alpha)(\bar{x})$ the set of $p\in S^{n}_{F_{\alpha}}(\mu)$
such that there exists $\beta>\alpha$ and $q\in S^{n}_{F_{\beta}}(\mu)$ whose
restriction to the fragment $F_{\alpha}$ is $p$, satisfying

$\displaystyle
0<\mu(\\{\mathbf{M}\colon\mathrm{tp}_{F_{\beta}}^{\mathbf{M}}(\bar{x})=q\\})<\mu(\\{\mathbf{M}\colon\mathrm{tp}_{F_{\alpha}}^{\mathbf{M}}(\bar{x})=p\\}).$

Assume that for all $\alpha<\omega_{1}$, $\mathrm{Sp}(\alpha)(\bar{x})$ is
non-empty. We construct a sequence $(\alpha_{\delta})_{\delta<\omega_{1}}$ of
ordinals and we prove that
$r_{\alpha_{\delta}}(\bar{x})\coloneqq\sup_{p\in\mathrm{Sp}(\alpha)(\bar{x})}\mu(\\{\mathbf{M}\colon\mathrm{tp}_{F_{\alpha_{\delta}}}^{\mathbf{M}}(\bar{x})=p\\})$
is a strictly decreasing sequence of reals of lenght $\omega_{1}$, which can
not exist.

Assume $\delta=\gamma+1$ and that we have constructed $\alpha_{\gamma}$. Then
$r_{\alpha_{\gamma}}$ is realized by a finite number of types
$p_{1},\ldots,p_{k}\in\mathrm{Sp}(\alpha_{\gamma})(\bar{x})$. Let us take
$\beta>\alpha_{\gamma}$ such that for all $i\leq k$ there is $q_{i}\in
S^{n}_{F_{\beta}}(\mu)$ whose restriction to $F_{\alpha_{\gamma}}$ is $p_{i}$
and satisfies

$\displaystyle
0<\mu(\\{\mathbf{M}\colon\mathrm{tp}_{F_{\beta}}^{\mathbf{M}}(\bar{x})=q_{i}\\})<\mu(\\{\mathbf{M}\colon\mathrm{tp}_{F_{\alpha_{\gamma}}}^{\mathbf{M}}(\bar{x})=p_{i}\\}).$

We set $\alpha_{\delta}=\beta$. We have $r_{\beta}<r_{\alpha_{\gamma}}$,
indeed if $q$ realizes $r_{\beta}$, take $p$ its restriction to
$F_{\alpha_{\gamma}}$. If $p\in\\{p_{1},\ldots,p_{k}\\}$ then we are done by
definition of $\beta$ and if not, then
$\mu(\\{\mathbf{M}\colon\mathrm{tp}_{F_{\beta}}^{\mathbf{M}}(\bar{x})=q\\})\leq\mu(\\{\mathbf{M}\colon\mathrm{tp}_{F_{\alpha_{\gamma}}}^{\mathbf{M}}(\bar{x})=p\\})<r_{\alpha_{\gamma}}$.

If $\delta$ is a limit ordinal, then set
$\alpha_{\delta}=\sup_{\gamma<\delta}\alpha_{\gamma}$. If $\gamma<\delta$ then
$\gamma+1<\delta$ and $r_{\alpha_{\gamma}}<r_{\alpha_{\gamma+1}}\leq
r_{\delta}$.

This is enough to conclude that there must be $\alpha(\bar{x})$ such that
$\mathrm{Sp}(\alpha)(\bar{x})$ is empty. Take
$\alpha=\sup_{\bar{x}\in\mathbb{N}^{<\omega}}\alpha(\bar{x})$, then
$F_{\alpha}$ is as wanted. ∎

Let us show that $F\coloneqq F_{\alpha}$ is a countable fragment satisfying
the conclusion of the theorem. Fix $p\in S^{n}_{F}(\mu)$, $q\in
S^{n+1}_{F}(\mu)$ and $(\bar{x},z)\in\mathbb{N}^{n+1}$ satisfying
$\mu(\\{\mathbf{M}\colon\mathrm{tp}_{F_{\alpha}}^{\mathbf{M}}(\bar{x})=p\text{
and }\mathrm{tp}_{F_{\alpha}}^{\mathbf{M}}(\bar{x},z)=q\\})>0$. Let us define

$\psi(v_{1},\dots,v_{n})\coloneqq\exists v,\bigwedge_{\phi\in
q}\phi(v_{1},\dots,v_{n},v).$

Since $q\in S^{n+1}_{F_{\alpha}}(\mu)$, the formula $\bigwedge_{\phi\in
q}\phi(v_{1},\dots,v_{n+1})$ belongs to $F_{\alpha+1}$ Therefore, $\psi\in
F_{\alpha+1}$. Since $\mu$ has no fixed, we can fix by Lemma 4.6, a type $r\in
S^{n}_{F_{\alpha+1}}$ which is an atom of the probability measure
$\mathrm{tp}_{F_{\alpha+1}}(\bar{x})_{*}\tilde{\mu}$, where $\tilde{\mu}$ is
the conditional measure
$\mu(\cdot\mid\mathrm{tp}_{F_{\alpha}}^{\mathbf{M}}(\bar{x})=p)$. Then
$\mu(\\{\mathbf{M}\colon\mathrm{tp}_{F_{\alpha+1}}^{\mathbf{M}}(\bar{x})=r\\})>0$
and the restriction of $r$ to the fragment $F_{\alpha}$ is exactly $p$. By
equation (3), we therefore get that
$\mu(\\{\mathbf{M}\colon\mathrm{tp}_{F_{\alpha}}^{\mathbf{M}}(\bar{x})=p\\})=\mu(\\{\mathbf{M}\colon\mathrm{tp}_{F_{\alpha+1}}^{\mathbf{M}}(\bar{x})=r\\})$.
Let us prove that $\psi\in r$. For any $\mathbf{M}$ such that
$\mathrm{tp}_{F_{\alpha}}^{\mathbf{M}}(\bar{x})=p$ and
$\mathrm{tp}_{F_{\alpha}}^{\mathbf{M}}(\bar{x},z)=q$ (which is a set of
positive measure by assumption), we have that
$\mathbf{M}\models\psi(\bar{x})$. This shows that $\psi\in r$ as otherwise we
would have
$\mu(\\{\mathbf{M}\colon\mathrm{tp}_{F_{\alpha}}^{\mathbf{M}}(\bar{x})=p\\})>\mu(\\{\mathbf{M}\colon\mathrm{tp}_{F_{\alpha+1}}^{\mathbf{M}}(\bar{x})=r\\})$.
To conclude, we obtain that $\mu$-a.s., if
$\mathrm{tp}_{F_{\alpha}}^{\mathbf{M}}(\bar{x})=p$, then there exists
$z^{\prime}\in\mathbb{N}$ such that
$\mathrm{tp}_{F_{\alpha}}^{\mathbf{M}}(\bar{x},z^{\prime})=q$. The same
conclusion holds for any $\bar{y}\in\mathbb{N}^{n}$ by $G$-invariance of the
measure $\mu$. ∎

We can now prove the main theorem of this section. An invariant random
expansion $\mu\in\mathrm{IRE}(G)$ is concentrated on an orbit if there exists
an orbit $O$ of $G\curvearrowright\mathrm{Struc}_{\mathcal{L}}^{G}$ such that
$\mu(O)=1$. This definition is legitimate as orbits of Borel actions are
indeed Borel, see [26, Thm. 15.14]. The following lemma will be useful to
prove that p.m.p. actions are essentially free.

######  Lemma 4.10. —

Let $G$ be a Polish group and $G\curvearrowright(X,\mu)$ be a p.m.p ergodic
action. If for $\mu\otimes\mu$-a.e. $(x,y)\in X\times X$, $x$ and $y$ are in
the same orbit, then there is a conull orbit of the action.

###### Proof.

If we denote by $E$ the set of $(x,y)$ such that $x\in G\cdot y$, we have

$\displaystyle 1=\mu\otimes\mu(E)$
$\displaystyle=\int_{X}\left(\int_{X}\mathds{1}_{G\cdot
x}(y)d\mu(y)\right)d\mu(x)$ $\displaystyle=\int_{X}\mu(G\cdot x)d\mu(x),$

By ergodicity of $\mu$, for a.e. $x\in X$, the value of $\mu(G\cdot x)$ is
either $0$ or $1$. Therefore, there exists an orbit $O$ of the action
$G\curvearrowright X$ such that $\mu(O)=1$ ∎

######  Theorem 4.11. —

Let $G$ be a dynamically de Finetti group. Let $\mu\in\mathrm{IRE}(G)$ be
ergodic. If $\mu$ has no fixed point, then $\mu$ is concentrated on an orbit.

###### Proof.

For $\mu\otimes\mu$-a.e.
$\mathbf{M},\mathbf{N}\in\mathrm{Struc}_{\mathcal{L}}^{G}$, we will construct
an element $g\in G$ such that $g\cdot\mathbf{M}=\mathbf{N}$, i.e., such that
for all $\bar{x}\in\mathbb{N}^{<\omega}$, we have

$\mathrm{qftp}^{\mathbf{M}}(\bar{x})=\mathrm{qftp}^{\mathbf{N}}(g(\bar{x})).$

This will imply that $\mu$ is concentrated on an orbit by Lemma 4.10.

Let us use Theorem 4.8 to build $g$ via a back-and-forth. We use the notation
of Lemma 3.4 and its following discussion. Take $(x_{i})$ and $(y_{j})$ two
enumerations of $\mathbb{N}$. Let $F$ be a fragment given by Theorem 4.8.

We start our back-and-forth by defined $A_{0}=B_{0}=\emptyset$. Assume that
$A_{n}$ and $B_{n}$ have been built, each containing the first $n$ elements of
each enumeration respectively. Let us build $A_{n+1}$ and $B_{n+1}$. Let $i$
be the smallest index such that $x_{i}\notin A_{n}$ and $j$ the smallest index
such that $y_{j}\notin B_{n}$ (by construction, $i>n$ and $j>n$). By Lemma 4.6
and Theorem 4.8, $\mu\otimes\mu$-a.s. there is $y\in\mathbb{N}$ such that

$\mathrm{tp}_{F}^{\mathbf{M}}(A_{n},x_{i})=\mathrm{tp}_{F}^{\mathbf{N}}(B_{n},y),$

implying that

$\mathrm{qftp}^{\mathbf{M}}(A_{n},x_{i})=\mathrm{qftp}^{\mathbf{N}}(B_{n},y_{i}).$

Similarly, there is $\mu\otimes\mu$-a.s. $x\in\mathbb{N}$ such that

$\mathrm{tp}_{F}^{\mathbf{M}}(A_{n},x_{i},x)=\mathrm{tp}_{F}^{\mathbf{N}}(B_{n},y,y_{j}).$

The a.s. existence of $x$ is also a consequence of Theorem 4.8. We set
$A_{n+1}\coloneqq A_{n}\cup\\{x_{i},x\\}$, $B_{n+1}\coloneqq
B_{n}\cup\\{y,y_{j}\\}$ and $g_{n+1}(x_{i})\coloneqq y$, $g_{n+1}(x)\coloneqq
y_{j}$ and $g_{n+1}(a)=g_{n}(a)$ for any $a\in A_{n}$. The way we constructed
$g_{n}$ using an enumeration ensures it converges for the pointwise topology
to an element of $G$. Moreover, its limit $g$ satisfies a.s.

$\mathrm{qftp}^{\mathbf{M}}(\bar{x})=\mathrm{qftp}^{\mathbf{N}}(g(\bar{x})).\qed$

##  5 Rigidity for p.m.p. actions of dynamically de Finetti groups

The aim of this section is to give a proof of Theorem 1.1. Let us first recall
the notions of essential freeness and essential transitivity for p.m.p.
actions of Polish groups. The free part of a p.m.p. action
$G\curvearrowright(X,\mu)$ of a Polish group $G$ is the set $\\{x\in
X\colon\forall g\in G\setminus\\{1_{G}\\},g\cdot x\neq x\\}$. This is a
$\mu$-measurable $G$-invariant set (see Lemma 6.4 (ii)). We say that
$G\curvearrowright(X,\mu)$ is

  * •

essentially free if the free part is a conull set,

  * •

essentially transitive if there exists a conull set on which the action is
transitive,

  * •

properly ergodic if it is ergodic and every orbit has measure $0$.

###  5.1 Essentially free and essentially transitive actions

We will provide concrete examples both of essentially free and of essentially
transitive p.m.p. ergodic actions in Lemma 5.5. Before that, let us prove that
if $G$ is Polish non-compact, p.m.p. ergodic actions cannot be both.

######  Lemma 5.1. —

Let $G$ be a Polish non-compact group. If $G\curvearrowright(X,\mu)$ is a
p.m.p. essentially transitive action, then it is not essentially free.

###### Proof.

By contradiction, assume that $G\curvearrowright(X,\mu)$ is essentially free
and essentially transitive. Thus, there exists a $G$-invariant conull set
$A\subseteq X$ on which the action is free and transitive. In other words,
there exists a Borel probability measure $m$ on $G$ which is invariant by left
translation, such that $G\curvearrowright(G,m)$ is measurably isomorphic to
$G\curvearrowright(X,\mu)$. This shows that $G$ has a probability Haar
measure. Thus $G$ is compact and this finishes the proof. ∎

######  Corollary 5.2. —

Any p.m.p. ergodic essentially free action of a Polish non-compact group is
properly ergodic.

In the next section, we will prove that apart from $S_{\infty}$, the converse
of Corollary 5.2 holds for any dynamically de Finetti group. The following
lemma will be useful to characterize proper subgroups of $S_{\infty}$. A
permutation $\tau\in S_{\infty}$ is a transposition if $\tau$ is a $2$-cycle.

######  Lemma 5.3. —

Let $G\leq S_{\infty}$ be a transitive, closed subgroup, which has no
algebraicity and weakly eliminates imaginaries. If $G$ contains a
transposition, then $G=S_{\infty}$.

###### Proof.

We define a $G$-invariant equivalence relation $E$ on $\mathbb{N}$ as follows:

$xEy\Leftrightarrow x=y\text{ or the transposition which exchange }x\text{ and
}y\text{ belongs to }G.$

Let us prove that $E=\mathbb{N}\times\mathbb{N}$. By assumption,
$E\neq\\{(x,x)\colon x\in\mathbb{N}\\}$, so take $x\in\mathbb{N}$ whose
$E$-class $[x]_{E}$ satisfies $\lvert[x]_{E}\rvert\geq 2$. Let $V$ be the
stabilizer of $[x]_{E}$, that is, $V\coloneqq\\{g\in G\colon
g\cdot[x]_{E}=[x]_{E}\\}$. Then for any $y\in[x]_{E}$, we have $G_{y}\leq V$.
By Lemma 2.5, we get that $G=V$ and therefore $[x]_{E}=G\cdot x$. Since $G$ is
transitive, this shows that $E=\mathbb{N}\times\mathbb{N}$. ∎

From the previous lemma, we get the following corollary.

######  Corollary 5.4. —

Let $G\lneq S_{\infty}$ be a transitive, proper, closed subgroup. If $G$ has
no algebraicity and weakly eliminates imaginaries, then for all
$x,y\in\mathbb{N}$ distinct, there exists infinitely many disjoint tuples
$\bar{z}$, all disjoint from $x$ and $y$, such that $(\bar{z},x)$ and
$(\bar{z},y)$ lies in different $G$-orbits.

###### Proof.

Assume that there is no such tuple $\bar{z}$. Fix an enumeration
$z_{0},z_{1},\dots$ of $\mathbb{N}\setminus\\{x,y\\}$. Then there exists a
sequence $(g_{n})_{n\geq 0}$ of elements in $G$ such that $g_{n}(z_{i})=z_{i}$
for all $i\leq n$ and $g_{n}(x)=y$. The sequence $(g_{n})_{n\geq 0}$ converges
to the transposition that exchanges $x$ and $y$. But $G$ is a proper subgroup
of $S_{\infty}$, so $G$ contain no transposition by Lemma 5.3. This yields a
contradiction. So there exists such a tuple $\bar{z}$. But $G$ has no
algebraicity, so there exists infinitely many disjoint such tuples by
Neumann’s lemma. ∎

We can now provides examples both of essentially free and of essentially
transitive p.m.p. ergodic actions.

######  Lemma 5.5. —

Let $G\lneq S_{\infty}$ be a proper, closed subgroup, which has no
algebraicity and weak elimination of imaginaries. Let $(A,\kappa)$ be a
standard probability space. Then the p.m.p. action
$G\curvearrowright(A,\kappa)^{\otimes\mathbb{N}}$ is ergodic. If moreover
$(A,\kappa)$ is purely atomic, then
$G\curvearrowright(A,\kappa)^{\otimes\mathbb{N}}$ is essentially transitive.
Else, $G\curvearrowright(A,\kappa)^{\otimes\mathbb{N}}$ is essentially free.

###### Proof.

The ergodicity of $G\curvearrowright(A,\kappa)^{\otimes\mathbb{N}}$ is proved
verbatim in [29, Prop. 2.1], the only assumption needed is that the orbits of
$G\curvearrowright\mathbb{N}$ are infinite, which is weaker than having no
algebraicity. Assume that $(A,\kappa)$ is purely atomic and let us prove that
$G\curvearrowright(A,\kappa)^{\otimes\mathbb{N}}$ is essentially transitive.
Let $\mu=\kappa^{\otimes\mathbb{N}}$. In order to prove that
$G\curvearrowright(A,\kappa)^{\otimes\mathbb{N}}$ is essentially transitive,
we will prove that for $\mu\otimes\mu$-a.e.
$((a_{n})_{n\in\mathbb{N}},(b_{n})_{n\in\mathbb{N}})\in A^{\mathbb{N}}\times
A^{\mathbb{N}}$, there exists a random element $g\in G$ such that
$g\cdot(a_{n})_{n\in\mathbb{N}}=(b_{n})_{n\in\mathbb{N}}$. For this we run a
back-and forth argument.

Fix $x_{1},x_{2},\dots$ and $y_{1},y_{2},\dots$ two deterministic enumerations
of $\mathbb{N}$. We will inductively construct $g$, ensuring that, at the
$n$-th step, its domain contains $x_{1},\ldots,x_{n}$ and that its image
contains $y_{1},\ldots,y_{n}$. To initiate the back-and-forth we define $g$ on
the empty set and define its image as the empty set. Let us now assume that we
have defined $g$ as wanted with domain $A_{n}$ containing $x_{1},\ldots,x_{n}$
and image $B_{n}$ contains $y_{1},\ldots,y_{n}$. Let us define $g(x_{n+1})$
and $g^{-1}(y_{n+1})$. If $x_{n+1}\in A_{n}$, $g(x_{n+1})$ is already defined.
Otherwise, since $G$ has no algebraicity, the $G$-orbit of $x_{n+1}$ is
infinite. Therefore, for $\mu\otimes\mu$-a.e.
$((a_{n})_{n\in\mathbb{N}},(b_{n})_{n\in\mathbb{N}})\in A^{\mathbb{N}}\times
A^{\mathbb{N}}$, there exists $\tilde{x}_{n+1}$ in the $G$-orbit of $x_{n+1}$
such that $a_{x_{n+1}}=b_{\tilde{x}_{n+1}}$. Set $g(x_{n+1})=\tilde{x}_{n+1}$.
If $y_{n+1}\in B_{n}$, then $g^{-1}(y_{n+1})$ is already defined. Otherwise,
by the same argument for the existence of $\tilde{x}_{n+1}$,
$\mu\otimes\mu$-a.s. there exists $\tilde{y}_{n+1}$ such that
$a_{\tilde{y}_{n+1}}=b_{y_{n+1}}$ and we set $g(\tilde{y}_{n+1})=y_{n+1}$.
This construction provides for $\mu\otimes\mu$-a.e. couple of sequences
$(a_{n})_{n\in\mathbb{N}}$ and $(b_{n})_{n\in\mathbb{N}}$ an element $g$,
which belongs to $G$ as a limit of elements in $G$, such that
$g\cdot(a_{n})=(b_{n})$. This shows that
$G\curvearrowright(A,\kappa)^{\otimes\mathbb{N}}$ is essentially transitive by
Lemma 4.10.

We now tackle the case when $(A,\kappa)$ is not purely atomic. Let $O$ be a
measurable set such that $\kappa(O)>0$ and $\kappa(\\{x\\})=0$ for all $x\in
O$. For $\mu$-a.e. $a\in A$, infinitely many coordinates of $a$ are in $O$.
Let us denote by $a_{O}$ the set of those coordinates. Since a.s. for any
$i,j\in a_{O}$, $a_{i}\neq a_{j}$, any $g\in G$ such that $g\cdot a=a$
stabilizes $a_{O}$ pointwise. However, for any $x,y\in\mathbb{N}$ distinct, by
Corollary 5.4, there must be $\bar{z}$ contained in $a_{O}$ such that
$(\bar{z},x)$ and $(\bar{z},y)$ are in different $G$-orbits. Therefore for any
$x,y\in\mathbb{N}$ distinct, no $g\in G\setminus\\{1_{G}\\}$ satisfying
$g\cdot a=a$ can send $x$ to $y$ for any $x,y$ distinct. Therefore
$G\curvearrowright(A,\kappa)^{\otimes\mathbb{N}}$ is essentially free. ∎

###  5.2 The proof of the main theorem

Towards proving Theorem 1.1, we first prove a version of it for IREs.

######  Theorem 5.6. —

Let $G\lneq S_{\infty}$ be a proper, transitive, closed subgroup. Let
$\mathcal{L}$ be a countable relational language which contains the canonical
language $\mathcal{L}_{G}$. If $G$ is dynamically de Finetti, then for any
$\mu\in\mathrm{IRE}_{\mathcal{L}}(G)$ ergodic, either
$\mathrm{Aut}(\mathbf{M})=\\{1\\}$ for $\mu$-a.e.
$\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}^{G}$, or $\mu$ is concentrated on
an orbit.

###### Proof.

Assume that $\mu$ is not concentrated on an orbit. By Theorem 4.11, for
$\mu$-a.e. $\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}^{G}$, the set
$\mathrm{Fix}(\mathrm{Aut}(\mathbf{M}))$ is non-empty. Let us prove the
following claim.

######  Claim. —

For $\mu$-a.e. $\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}^{G}$, we have
$G_{\mathrm{Fix}(\mathrm{Aut}(\mathbf{M}))}=\\{1_{G}\\}$.

###### Proof.

Fix $x,y\in\mathbb{N}$ distinct. The set
$\mathrm{Fix}(\mathrm{Aut}(\mathbf{M}))$ is a $G$-invariant ergodic random
subset of $\mathbb{N}$. By Theorem 2.4, its law is therefore i.i.d. since $G$
is dynamically Finetti. Therefore, we can use Corollary 5.4 to get for
$\mu$-a.e. $\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}^{G}$ a tuple
$\bar{z}\in\mathbb{N}^{<\omega}$ disjoint from $x$ and $y$ and which is
contained in $\mathrm{Fix}(\mathrm{Aut}(\mathbf{M}))$, such that $(\bar{z},x)$
and $(\bar{z},y)$ lies in different $G$-orbits. This shows that $\mu$-a.s.,
$x$ and $y$ lies in different
$G_{\mathrm{Fix}(\mathrm{Aut}(\mathbf{M}))}$-orbits, which proves the claim. ∎

So for $\mu$-a.e. $\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}^{G}$, we obtain
that $\mathrm{Aut}(\mathbf{M})$ is a subgroup of
$G_{\mathrm{Fix}(\mathrm{Aut}(\mathbf{M}))}$, which is trivial by the above
claim. Therefore, $\mathrm{Aut}(\mathbf{M})=\\{1_{G}\\}$, $\mu$-a.s., which
concludes the proof. ∎

We are now ready to prove our main result.

######  Theorem 5.7. —

Let $G\lneq S_{\infty}$ be a proper, transitive, closed subgroup. If $G$ is
dynamically de Finetti, then any p.m.p. ergodic action of $G$ is either
essentially free or essentially transitive.

###### Proof.

Let $G\curvearrowright(X,\nu)$ be a p.m.p. ergodic action which is not
essentially transitive. Fix $\mathcal{L}$ a countable relational language
which contains the canonical language $\mathcal{L}_{G}$ and such that
$\mathcal{L}\setminus\mathcal{L}_{G}$ contains relations of arbitrarily high
arity. By [11, Thm. 2.7.4], the relativized logic action
$G\curvearrowright\mathrm{Struc}_{\mathcal{L}}^{G}$ is Borel-universal. That
is, every Borel $G$-action can be Borel-embedded in
$\mathrm{Struc}_{\mathcal{L}}^{G}$. In particular, this implies that there
exists a $G$-invariant Borel probability measure $\mu$ on
$\mathrm{Struc}_{\mathcal{L}}^{G}$ such that the p.m.p. action
$G\curvearrowright(X,\nu)$ is measurably isomorphic to
$G\curvearrowright(\mathrm{Struc}_{\mathcal{L}}^{G},\mu)$. Thus, $\mu$ is a
ergodic IRE of $G$, which is not concentrated on an orbit. By Theorem 5.6, we
obtain that $\mathrm{Aut}(\mathbf{M})=\\{1\\}$ for $\mu$-a.e.
$\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}^{G}$. Since the automorphism group
of $\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}^{G}$ coincide with the
stabilizer of $\mathbf{M}$ for the relativized logic action, this exactly
means that $\mu$ is essentially free. Therefore, $G\curvearrowright(X,\nu)$ is
essentially free. ∎

######  Remark 5.8. —

The group $S_{\infty}$ admits p.m.p. properly ergodic actions that are not
essentially free. In particular, they are neither essentially free nor
essentially transitive. An example can be constructed as follows. Fix an
i.i.d. sequence $(X_{n})_{n\in\mathbb{N}}$ of uniform random variables on
$[0,1]$ and an i.i.d. sequence $(Y_{n})_{n\in\mathbb{N}}$ of $\mathrm{Ber}(p)$
random variables. Let $A\subseteq\mathbb{N}$ be a random
$S_{\infty}$-invariant non-empty subset, independent of
$(X_{n})_{n\in\mathbb{N}}$ and $(Y_{n})_{n\in\mathbb{N}}$. Then define the
random variables $(Z_{n})_{n\in\mathbb{N}}$ by setting $Z_{n}=X_{n}$ if $n\in
A$ and $Z_{n}=Y_{n}$ if $n\notin A$. If $\mu$ denotes the law of
$(Z_{n})_{n\in\mathbb{N}}$, then
$S_{\infty}\curvearrowright([0,1]^{\mathbb{N}},\mu)$ is properly ergodic but
not essentially free.

##  6 Invariant Random Subgroups of Polish groups

###  6.1 Definition

In this section we define the notion of invariant random subgroups for Polish
groups. Let $G$ be a Polish group. We denote by $\mathrm{Sub}(G)$ the space of
closed subgroups of $G$. The Effros $\sigma$-algebra is the $\sigma$-algebra
on $\mathrm{Sub}(G)$ generated by the sets

$\\{H\in\mathrm{Sub}(G)\colon H\cap U\neq\emptyset\\},$

where $U$ varies over open subsets of $G$. The following lemma is probably
well-known but we were not able to locate a proof in the literature.

######  Lemma 6.1. —

If $G$ is a Polish group, then $\mathrm{Sub}(G)$ equipped with the Effros
$\sigma$-algebra is a standard Borel space.

###### Proof.

If $X$ is a standard Borel space and $\mathcal{F}(X)$ denotes the space of
closed subsets of $X$, then $\mathcal{F}(X)$ is a standard Borel space when
equipped with the $\sigma$-algebra generated by the sets
$\\{F\in\mathcal{F}(X)\colon F\cap U\neq\emptyset\\}$ where $U$ varies over
open subsets of $X$ [18]. Therefore, in our case $\mathcal{F}(G)$ is standard
Borel. So it remains to show that $\mathrm{Sub}(G)$ is Borel in
$\mathcal{F}(G)$. By the selection theorem of Kuratowski and Ryll-Nardzewski
[26, Thm. 12.13], there exists a countable sequence of Borel maps
$d_{i}:\mathcal{F}(G)\to G$ with $i\in I$ such that for all nonempty
$F\in\mathcal{F}(G)$, the set $\\{d_{i}(F)\colon i\in I\\}$ is dense in $F$.
But a closed subset $F\in\mathcal{F}(G)$ belongs to $\mathrm{Sub}(G)$ if and
only if $1_{G}\in F$ and $d_{i}(F)d_{j}(F)^{-1}\in F$ for all $i,j\in I$. This
implies that $\mathrm{Sub}(G)$ is Borel in $\mathcal{F}(G)$ and thus
$\mathrm{Sub}(G)$ is a standard Borel space. ∎

######  Remark 6.2. —

If $G$ is a Polish locally compact group, then $\mathrm{Sub}(G)$ is usually
endowed with the Chabauty topology, which is the topology generated by the
sets

$\\{H\in\mathrm{Sub}(G)\colon H\cap U\neq\emptyset,H\cap K=\emptyset\\}$

where $U$ varies over open subsets of $G$ and $K$ over compact subsets of $G$.
In this case, $\mathrm{Sub}(G)$ is a compact Hausdorff space and its Borel
$\sigma$-algebra is the Effros $\sigma$-algebra. However, for Polish non
locally compact groups, the Chabauty topology is not Hausdorff in general (one
can adapt the proof of [13, Thm. 4.4.12] to get that this is indeed not the
case for many Polish groups including $S_{\infty}$).

The $G$-action by conjugation on $\mathrm{Sub}(G)$ is Borel and we are
interested in the probability measures invariant under this action.

######  Definition 6.3. —

Let $G$ be a Polish group. An Invariant Random Subgroup of $G$ is a Borel
probability measure on $\mathrm{Sub}(G)$ that is invariant by conjugation.

We denote by $\mathrm{IRS}(G)=\mathrm{Prob}(\mathrm{Sub}(G))^{G}$ the space of
invariant random subgroups of $G$. This is a standard Borel space equipped
with the $\sigma$-algebra generated by the maps $\mu\mapsto\mu(A)$ with $A$
varying over Borel subsets of $\mathrm{Sub}(G)$ [26, Thm. 17.23 and 17.24]. We
say that $\nu\in\mathrm{IRS}(G)$ is concentrated on a conjugacy class if there
exists an orbit $O$ of the $G$-action by conjugation on $\mathrm{Sub}(G)$ such
that $\nu(O)=1$. Recall that orbits of Borel actions are indeed Borel [26,
Thm. 15.14] so this definition makes sense.

We now explain how to construct IRSs. Let $G$ be a Polish group and let
$G\curvearrowright(X,\mu)$ be a p.m.p. action of $G$. Recall that for us, a
p.m.p. action of a Polish group is a Borel action on some standard Borel space
with a Borel invariant probability measure. For $x\in X$, let
$\mathrm{Stab}(x)\coloneqq\\{g\in G\colon g\cdot x=x\\}$ denotes the
stabilizer of $x$. We will prove that the law of the stabilizer of a
$\mu$-random point is indeed an IRS of $G$. For this, we need the following
lemma.

######  Lemma 6.4. —

Let $G$ be a Polish group and $G\curvearrowright(X,\mu)$ a p.m.p. action. Then

  1. (i)

for all $x\in X$, $\mathrm{Stab}(x)$ is a closed subgroup,

  2. (ii)

the map $\mathrm{Stab}:x\in X\mapsto\mathrm{Stab}(x)\in\mathrm{Sub}(G)$ is
$\mu$-measurable.

###### Proof.

By [11, Thm. 5.2.1], there exists a Polish topology on $X$ whose Borel
$\sigma$-algebra is that of $X$, such that the action $G\curvearrowright X$ is
continuous. Let us fix such a topology. The proof of (i) is obvious:
stabilizers are closed because the action is continuous. Let us prove (ii).
Let $U\subseteq G$ be open and let $B_{U}\coloneqq\\{H\in\mathrm{Sub}(G)\colon
H\cap U\neq\varnothing\\}$. Let us prove that $\mathrm{Stab}^{-1}(B_{U})$ is
analytic (the continuous image of a Borel set in a Polish space). Then [26,
Theorem 21.10] will allow us to conclude that $\mathrm{Stab}$ is
$\mu$-measurable. First, we have

$\displaystyle\mathrm{Stab}^{-1}(B_{U})$ $\displaystyle=\\{x\in X\colon\exists
g\in U,g\cdot x=x\\}$ $\displaystyle=\pi(B),$

where $B=\\{(x,g)\in X\times U\colon g\cdot x=x\\}$ and $\pi:X\times U\to X$
denotes the projection onto the first coordinate. So we need to prove that $B$
is Borel. But $B$ is the preimage under the continuous map $(x,g)\in X\times
U\to(x,g\cdot x)\in X\times X$ of the diagonal set $\\{(x,x)\colon x\in X\\}$,
which is Borel (because $X$ is Polish). ∎

Therefore, p.m.p. actions of Polish groups produce IRSs: if
$G\curvearrowright(X,\mu)$ is a p.m.p. action of a Polish group, then
$\mathrm{Stab}_{*}\mu$ is an IRS of $G$. In the next theorem, we prove that
for closed permutation group groups, every IRS arises this way.

######  Theorem 6.5. —

Let $G$ be a closed subgroup of $S_{\infty}$ and let $\nu\in\mathrm{IRS}(G)$.
Then there exists a p.m.p. action $G\curvearrowright(X,\mu)$ such that
$\mathrm{Stab}_{*}\mu=\nu$.

###### Proof.

Fix $\nu\in\mathrm{IRS}(G)$ and let us prove that $\nu$ is a stabilizer IRS.
Let $X$ be the standard Borel space defined by

$X\coloneqq\mathrm{Sub}(G)\times[0,1]^{\mathbb{N}^{<\omega}}.$

Recall that $\mathbb{N}^{<\omega}$ stands for the disjoint union of
$\mathbb{N}^{n}$ for $n\geq 1$. The action of $G$ on $\mathbb{N}^{<\omega}$
induces a Borel action of $G$ on $[0,1]^{\mathbb{N}^{<\omega}}$. Let us now
construct a $G$-invariant probability measure $\mu$ on $X$ such that
$\mathrm{Stab}_{*}\mu=\nu$.

Given $H\in\mathrm{Sub}(G)$, a coloring of $H$ is a map
$c:\mathbb{N}^{<\omega}\to[0,1]$ which is constant on each orbit of the action
$H\curvearrowright\mathbb{N}^{<\omega}$. Let $H\in\mathrm{Sub}(G)$. Then there
exists a unique Borel probability measure $\lambda^{H}$ on
$[0,1]^{\mathbb{N}^{<\omega}}$ concentrated on colorings of $H$ such that if
$c$ is a random variable with law $\lambda^{H}$ and
$\bar{x_{1}},\dots,\bar{x_{n}}\in\mathbb{N}^{<\omega}$ are tuples whose
$H$-orbits are pairwise disjoint, then $c(\bar{x_{1}}),\dots,c(\bar{x_{n}})$
are i.i.d. uniform random variables. By uniqueness, we have that for all $g\in
G$, $g_{*}\lambda^{H}=\lambda^{gHg^{-1}}$. Therefore the probability measure
on $X$

$\mu=\int_{X}\lambda^{H}(c)d\nu(H)$

is $G$-invariant.

We will now prove that for $\mu$-a.e. $(H,c)\in X$, we have
$\mathrm{Stab}(H,c)=H$. Remark first that if $g\in\mathrm{Stab}(H,c)$ then
$g\in N_{G}(H)$. The following claim is what we need to conclude that
$\mathrm{Stab}_{*}\mu=\nu$.

######  Claim. —

Let $H\in\mathrm{Sub}(G)$ and let $g\in N_{G}(H)$ be such that any orbit of
the action $H\curvearrowright\mathbb{N}^{<\omega}$ is invariant by $g$. Then
$g\in H$.

###### Proof of the claim.

Fix an enumeration $x_{1},x_{2},\dots$ of $\mathbb{N}$. For all $n\geq 1$, the
$H$-orbit of $(x_{1},\dots,x_{n})$ is preserved by $g$. Since $g\in N_{G}(H)$,
this implies that

$H(x_{1},\dots,x_{n})=gH(x_{1},\dots,x_{n})=Hg(x_{1},\dots,x_{n}).$

Therefore there exists $h_{n}\in H$ such that
$h_{n}(x_{1},\dots,x_{n})=g(x_{1},\dots,x_{n})$. This means that $h_{n}\to g$
as $n\to+\infty$. Since $H$ is closed, we obtain that $g\in H$. ∎

∎

Versions of these results for Polish locally compact groups already appeared
in the literature. When $G$ is Polish locally compact, Lemma 6.4 (i) was
proved by Varadarajan [37]. Theorem 6.5 was proved for discrete groups by
Abért, Glasner and Virag [2] and for Polish locally compact groups in [1].

######  Remark 6.6. —

Theorem 6.5 is false in full generality for Polish groups, as there exist
Polish groups, such as $\mathrm{Aut}(X,\mu)$, which admits no non-trivial
p.m.p. action, [19], [20]. These groups therefore have no essentially free
p.m.p. actions. For such groups $G$, the IRS $\delta_{\\{1_{G}\\}}$ is not
realized. We thank Anush Tserunyan and Ronnie Chen for pointing us toward such
examples. Nonetheless, it would be interesting to understand which Polish
groups do realize their IRSs.

###  6.2 From IRS to IRE and vice versa

We will draw a connection between invariant random subgroups and invariant
random expansions for closed permutation groups.

#### From IRS to IRE.

Fix $G\leq S_{\infty}$ a closed subgroup. Let us recall the definition of the
canonical language. For all $n\geq 1$, let $J_{n}$ be the set of orbits of the
diagonal action $G\curvearrowright\mathbb{N}^{n}$ and let $J=\bigcup_{n\geq
1}J_{n}$. The canonical language associated with $G$ is
$\mathcal{L}_{G}\coloneqq(R_{j})_{j\in J}$ where $R_{j}$ is a relation symbol
of arity $n$ for all $j\in J_{n}$. We define a new language
$\mathcal{L}_{dyn}\coloneqq(T_{n})_{n\geq 1}\sqcup\mathcal{L}_{G}$ that we
call the dynamical language of $G$, where $T_{n}$ is a relation symbol of
arity $n$ for each integer $n\geq 1$.

To any $H\in\mathrm{Sub}(G)$ we associate an expansion
$\mathbf{M}_{G}(H)\in\mathrm{Struc}_{\mathcal{L}_{dyn}}^{G}$ of the canonical
structure $\mathbf{M}_{G}$ in the language $\mathcal{L}_{dyn}$ as follows:

$\mathbf{M}_{G}(H)\coloneqq((\mathcal{R}_{H\curvearrowright\mathbb{N}^{n}})_{n\in\mathbb{N}},(R_{j}^{G})_{j\in
J}),$

where $R_{j}^{G}=j\subseteq\mathbb{N}^{n}$ for all $j\in J_{n}$ and where
$\mathcal{R}_{H\curvearrowright\mathbb{N}^{n}}\coloneqq\\{(\bar{x},\bar{y})\in\mathbb{N}^{n}\times\mathbb{N}^{n}\colon\bar{y}\in
H\bar{x}\\}$ is the orbit equivalence relation of the $H$-action on
$n$-tuples.

######  Lemma 6.7. —

Let $G\leq S_{\infty}$ be a closed subgroup. The following hold.

  1. (i)

The map
$H\in\mathrm{Sub}(G)\mapsto\mathbf{M}_{G}(H)\in\mathrm{Struc}_{\mathcal{L}_{\text{dyn}}}^{G}$
is Borel, $G$-equivariant and injective.

  2. (ii)

For all $H\in\mathrm{Sub}(G)$, we have
$\mathrm{Aut}(\mathbf{M}_{G}(H))=N_{G}(H)$.

###### Proof.

We only show that the map is Borel, the rest being straightforward. For this,
it suffices to show that for all $n\geq 1$ and all
$\bar{x},\bar{y}\in\mathbb{N}^{n}$, the set
$\\{H\in\mathrm{Sub}(G)\colon(\bar{x},\bar{y})\in\mathcal{R}_{H\curvearrowright\mathbb{N}^{n}}\\}$
is Borel. But this is exactly $\\{H\in\mathrm{Sub}(G)\colon H\cap
U\neq\emptyset\\}$ where $U\subseteq G$ is the open subset consisting of the
elements $g\in G$ such that $g(\bar{x})=\bar{y}$. This last set belongs to the
Effros $\sigma$-algebra, therefore $H\mapsto\mathbf{M}_{G}(H)$ is Borel. ∎

To any $\nu\in\mathrm{IRS}(G)$ we can therefore associate an IRE of $G$ as the
law of the random expansion $\mathbf{M}_{G}(H)$ where $H$ is a $\nu$-random
closed subgroup of $G$.

######  Remark 6.8. —

This construction is in fact underlying in the proof of Theorem 6.5, in which
we implicitly use this IRE that we expand using colorings of each orbit.

#### From IRE to IRS.

Let $G$ be a closed subgroup of $S_{\infty}$ and fix a countable relational
language $\mathcal{L}$ which contains $\mathcal{L}_{G}$. Let
$\mu\in\mathrm{IRE}(G)$. If $\mathbf{M}$ is a $\mu$-random expansion, then the
law of $\mathrm{Aut}(\mathbf{M})$ is an IRS of $G$. It may happen that the IRS
obtained this way is trivial (that is equal to $\delta_{\\{1_{G}\\}}$) whereas
the IRE is a nontrivial object of interest. We give details of such an IRE in
the next example.

######  Example 6.9 (The kaleidoscope random graph). —

Let $\mathcal{L}=(R_{n})_{n\in\mathbb{N}}$ be the language consisting in
countably many binary relations. Denote by $\mathcal{P}_{2}(\mathbb{N})$ the
set of subsets $A\subseteq\mathbb{N}$ with $\lvert A\rvert=2$. Consider the
random element $\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}$ obtained by first
picking an $S_{\infty}$-invariant random non-empty subset
$A_{\\{i,j\\}}\subseteq\mathbb{N}$, independently for each
$\\{i,j\\}\in\mathcal{P}_{2}(\mathbb{N})$ and then setting
$R_{n}^{\mathbf{M}}(i,j)=1$ if and only if $n\in A_{\\{i,j\\}}$. The law of
$\mathbf{M}$ is indeed an IRE of $S_{\infty}$. This IRE can be thought as the
union of countably many random graphs on $\mathbb{N}$, each of which having
its edges labeled by a different color. The theory of such an IRE is studied
in [4, Ex. 3.2] and [6, §5.1] . This is an example of what they call a
properly ergodic structure, which is the main object of study in [4]. If $\mu$
denotes the law of the IRE $\mathbf{M}$, then the p.m.p. action
$S_{\infty}\curvearrowright(\mathrm{Struc}_{\mathcal{L}},\mu)$ is measurably
conjugate to
$S_{\infty}\curvearrowright([0,1],\mathrm{Leb})^{\otimes\mathcal{P}_{2}(\mathbb{N})}$
which is easily seen to be a properly ergodic p.m.p. action.

###  6.3 Rigidity of ergodic IRSs of $S_{\infty}$

We have already proved in Theorem 5.7 that the p.m.p. ergodic actions of any
proper, transitive, closed subgroup $G\lneq S_{\infty}$ which is dynamically
de Finetti, are either essentially free or essentially transitive. In
particular, this implies that for any such group $G$, any ergodic
$\nu\in\mathrm{IRS}(G)$ is concentrated on a conjugacy class. On the other
hand, we have seen in Remark 5.8 that $S_{\infty}$ admits p.m.p. ergodic
actions that are neither essentially free nor essentially transitive. However,
we prove that such a behavior never appears for ergodic IRSs of $S_{\infty}$.
We therefore obtain the following result.

######  Theorem 6.10. —

Let $G\leq S_{\infty}$ be a transitive closed subgroup. If $G$ is dynamically
de Finetti, then any ergodic $\nu\in\mathrm{IRS}(G)$ is concentrated on a
conjugacy class.

###### Proof.

As mentioned in the above discussion, if $G$ is a proper subgroup of
$S_{\infty}$, this is a consequence of Theorem 5.7, therefore the proof is
dedicated to the case when $G=S_{\infty}$. Let $\mathcal{L}_{dyn}$ be the
dynamical language of $S_{\infty}$. Let us define $\mu\in\mathrm{IRE}(G)$ in
the language $\mathcal{L}_{dyn}$ as the pushforward measure of $\nu$ by the
map $H\mapsto\mathbf{M}_{G}(H)$. Let us prove that $\mu$ has no fixed point.
First, for $\nu$-a.e. $H\in\mathrm{Sub}(G)$,
$\mathrm{Aut}(\mathbf{M}_{G}(H))=N_{S_{\infty}}(H)$ by Lemma 6.7 (ii). But for
all $H\in\mathrm{Sub}(G)$, the group $N_{S_{\infty}}(H)$ acts transitively on
$\mathrm{Fix}(H)$. Indeed, for any distinct $x,y\in\mathrm{Fix}(H)$, the
transposition that exchanges $x$ and $y$ belongs to $N_{S_{\infty}}(H)$. This
shows that $\mu$ has no fixed point and concludes the proof. ∎

##  7 Further discussions

#### On the existence of IREs.

The main result in the present paper (Theorem 5.7 and its corresponding
version for IRE in Theorem 5.6) suggests examining both essentially free and
essentially transitive p.m.p. actions of dynamically de Finetti groups.
Essentially free actions of $S_{\infty}$ have been analyzed in depth from a
model theoretic perspective in [4] through a notion that is called properly
ergodic structures, but such a study for other groups is lacking. On the other
hand, Ackerman, Freer and Patel have obtained a great understanding of
essentially transitive p.m.p. actions of $S_{\infty}$. In their seminal paper
[8], they proved that for any countable relational language $\mathcal{L}$ and
any $\mathbf{M}\in\mathrm{Struc}_{\mathcal{L}}$ with no algebraicity, there
exists an IRE of $S_{\infty}$ supported on the orbit of $\mathbf{M}$. In
another paper with Kwiatkowska [5], they moreover characterize the cardinality
of the set of such measures. A natural generalization of these works would
then be:

######  Question 7.1. —

Is there a natural condition on an expansion $\mathbf{N}$ of the canonical
structure $\mathbf{M}_{G}$ associated with a given closed subgroup $G\leq
S_{\infty}$, such that there exists a $G$-IRE concentrated on the $G$-orbit of
$\mathbf{N}$?

This question has been answered in some special cases in [3] and [6]. The
following observations may help to answer this question, however they also
suggest to us that this problem might be difficult.

  1. 1)

For any closed $G\leq S_{\infty}$ and any $S_{\infty}$-IRE $\mu$ in a language
$\mathcal{L}$ (disjoint from $\mathcal{L}_{G}$), one readily gets a $G$-IRE in
the language $\mathcal{L}\sqcup\mathcal{L}_{G}$. Indeed, take $\mathbf{M}$ a
random structure with law $\mu$ and define a $G$-IRE $\mathbf{N}$ by

$R^{\mathbf{N}}(\bar{x})\Leftrightarrow\left\\{\begin{array}[]{c}R^{\mathbf{M}}(\bar{x})\text{
whenever }R\in\mathcal{L}.\\\ R^{\mathbf{M}_{G}}(\bar{x})\text{ whenever
}R\in\mathcal{L}_{G}.\end{array}\right.$

  2. 2)

There are groups that admit IREs not produced as in 1). This is the case for
the IRE of the automorphism group of the Fraïssé limit of 2-graphs considered
in Example 4.2 (iii).

  3. 3)

There are expansions which orbits can not be the support of an IRE. We give
two examples.

    1. i)

Any expansion of the generic poset into a linear order. The existence of such
an IRE would contradict the non-amenability of the automorphism groups of the
generic poset.

    2. ii)

The expansion $\mathbf{N}$ of $(\mathbb{Q},<)$ given by adding a unary
relation $R$ and for a fixed irrational $\alpha$, setting for all
$q\in\mathbb{Q}$, $R^{\mathbf{N}}(q)$ if and only if $q<\alpha$. If the orbit
of this expansion was the support of an ergodic
$\mathrm{Aut}(\mathbb{Q},<)$-IRE, we would get a non product
$\mathrm{Aut}(\mathbb{Q},<)$-invariant ergodic measure on
$\\{0,1\\}^{\mathbb{Q}}$ which does not exist by [24].

#### On the definition of dynamically de Finetti groups.

Let us discuss the different hypotheses in Definition 2.1. We classify them by
their natures:

  1. a)

the model-theoretic hypotheses: no algebraicity and weak elimination of
imaginaries,

  2. b)

the dynamical hypothesis: for any p.m.p. action $G\curvearrowright(X,\mu)$, we
have
$\mathcal{F}_{A}\mathrel{\reflectbox{\rotatebox[origin={c}]{90.0}{$\models$}}}_{\mathcal{F}_{A\cap
B}}\mathcal{F}_{B}$ for all finite $A,B\subseteq\mathbb{N}$.

Both are heavily used in our proofs, however, our result might hold under
weaker hypotheses.

######  Question 7.2. —

Let $G\lneq S_{\infty}$ be a proper, transitive, closed subgroup satisfying
b). Are all its ergodic p.m.p. actions either essentially free or essentially
transitive?

One way to go about answering positively this question would be to prove that
b) implies a). The converse is most likely not true however, as we have strong
evidence, in an ongoing project of the first author and Perruchaud, of the
existence of a group satisfying a) and not b).

Moreover, Item b) closely resembles the notion of dissociation, which is
relevant in exchangeability theory and is related to the Aldous-Hoover-
Kallenberg representation Theorem, see [25, Lem.7.35]. We say that a p.m.p.
action is dissociated if
$\mathcal{F}_{A}\mathrel{\reflectbox{\rotatebox[origin={c}]{90.0}{$\models$}}}\mathcal{F}_{B}$
for all finite disjoint $A,B\subseteq\mathbb{N}$. This name suggests that we
call strongly dissociated a p.m.p. action such that
$\mathcal{F}_{A}\mathrel{\reflectbox{\rotatebox[origin={c}]{90.0}{$\models$}}}_{\mathcal{F}_{A\cap
B}}\mathcal{F}_{B}$ for all finite $A,B\subseteq\mathbb{N}$. One can check
carefully the proofs in this paper and notice that we never used fully strong
dissociation for dynamically de Finetti groups. Instead we only use that every
p.m.p ergodic action is dissociated. We do not know whether dissociation of
every p.m.p. ergodic action is actually more general than strong dissociation
of every p.m.p. action. We preferred to use strong dissociation in our
definition as it more accurately describes the behavior of dynamically de
Finetti oligomorphic groups, see [24, Thm. 3.4].

## References

  * [1] Miklós Abért et al.  “On the growth of $L^{2}$-invariants for sequences of lattices in Lie groups”  In _Ann. of Math. (2)_ 185.3, 2017, pp. 711–790  DOI: 10.4007/annals.2017.185.3.1
  * [2] Miklós Abért, Yair Glasner and Bálint Virág  “Kesten’s theorem for invariant random subgroups.”  In _Duke Math. J._ 163.3, 2014, pp. 465–488  DOI: 10.1215/00127094-2410064
  * [3] Nathanael Ackerman  “Representations of $\mathrm{Aut}(M)$-Invariant Measures”, 2021  arXiv:1509.06170
  * [4] Nathanael Ackerman, Cameron Freer, Alex Kruckman and Rehana Patel  “Properly ergodic structures”, 2017  arXiv:1710.09336
  * [5] Nathanael Ackerman, Cameron Freer, Aleksandra Kwiatkowska and Rehana Patel  “A classification of orbits admitting a unique invariant measure”  In _Ann. Pure Appl. Logic_ 168.1, 2017, pp. 19–36  DOI: 10.1016/j.apal.2016.08.003
  * [6] Nathanael Ackerman, Cameron Freer, Jaroslav Nešetřil and Rehana Patel  “Invariant measures via inverse limits of finite structures”  In _Eur. J. Comb._ 52, 2016, pp. 248–289  DOI: 10.1016/j.ejc.2015.07.006
  * [7] Nathanael Ackerman, Cameron Freer and Rehana Patel  “Countable infinitary theories admitting an invariant measure”, 2017  arXiv:1710.06128
  * [8] Nathanael Ackerman, Cameron Freer and Rehana Patel  “Invariant measures concentrated on countable structures”  In _Forum Math. Sigma_ 4 Cambridge University Press, 2016, pp. e17  DOI: 10.1017/fms.2016.15
  * [9] Nathanael Ackerman, Cameron Freer and Rehana Patel  “The Entropy Function of an Invariant Measure”  In _Proceedings of the $14^{\text{th}}$ and $15^{\text{th}}$ Asian Logic Conferences_, pp. 3–34  DOI: 10.1142/9789813237551_0001
  * [10] Uri Bader, Bruno Duchesne, Jean Lécureux and Phillip Wesolek  “Amenable invariant random subgroups”  In _Isr. J. Math._ 213, 2016, pp. 399–422  DOI: 10.1007/s11856-016-1324-7
  * [11] Howard Becker and Alexander S. Kechris  “The descriptive set theory of Polish group actions” 232, Lond. Math. Soc. Lect. Note Ser.  Cambridge: Cambridge University Press, 1996
  * [12] Oren Becker, Alexander Lubotzky and Andreas Thom  “Stability and invariant random subgroups”  In _Duke Math. J._ 168.12, 2019, pp. 2207–2234  DOI: 10.1215/00127094-2019-0024
  * [13] Gerald Beer  “Topologies on closed and closed convex sets” 268, Math. Appl., Dordr.  Dordrecht: Kluwer Academic Publishers, 1993
  * [14] Itaï Ben Yaacov, Julien Melleray and Todor Tsankov  “Metrizable universal minimal flows of Polish groups have a comeagre orbit”  In _Geom. Funct. Anal._ 27.1, 2017, pp. 67–77  DOI: 10.1007/s00039-017-0398-7
  * [15] Lewis Bowen  “Invariant random subgroups of the free group”  In _Groups Geom. Dyn._ 9.3, 2015, pp. 891–916  DOI: 10.4171/GGD/331
  * [16] Lewis Bowen, Rostislav Grigorchuk and Rostyslav Kravchenko  “Invariant random subgroups of lamplighter groups”  In _Isr. J. Math._ 207, 2015, pp. 763–782  DOI: 10.1007/s11856-015-1160-1
  * [17] Peter J. Cameron  “Oligomorphic permutation groups” 152, Lond. Math. Soc. Lect. Note Ser.  Cambridge: Cambridge University Press, 1990
  * [18] Edward G. Effros  “Convergence of Closed Subsets in a Topological Space”  In _Proc. Amer. Math. Soc._ 16.5 American Mathematical Society, 1965, pp. 929–931
  * [19] Eli Glasner, Boris Tsirelson and Benjamin Weiss  “The automorphism group of the Gaussian measure cannot act pointwise”  In _Isr. J. Math._ 148, 2005, pp. 305–329  DOI: 10.1007/BF02775441
  * [20] Eli Glasner and Benjamin Weiss  “Spatial and non-spatial actions of Polish groups”  In _Ergod. Theory Dyn. Syst._ 25.5, 2005, pp. 1521–1538  DOI: 10.1017/S0143385705000052
  * [21] C. Henson  “Countable homogeneous relational structures and $\aleph_{0}$-categorical theories”  In _J. Symb. Log._ 37, 1972, pp. 494–500  DOI: 10.2307/2272734
  * [22] Wilfrid Hodges  “Model Theory”, Encyclopedia of Mathematics and its Applications  Cambridge University Press, 1993  DOI: 10.1017/CBO9780511551574
  * [23] Colin Jahel  “Some progress on the unique ergodicity problem”, 2021
  * [24] Colin Jahel and Todor Tsankov  “Invariant measures on products and on the space of linear orders”  In _J. Ec. Polytech. Math._ 9, 2022, pp. 155–176
  * [25] Olav Kallenberg  “Probabilistic symmetries and invariance principles.”, Probab. Appl.  New York, NY: Springer, 2005  DOI: 10.1007/0-387-28861-9
  * [26] Alexander S. Kechris  “Classical descriptive set theory”  In _Grad. Texts Math._ 156 Berlin: Springer-Verlag, 1995
  * [27] Alexander S. Kechris  “Dynamics of non-archimedean Polish groups”  In _European Congress of Mathematics. Proceedings of the 6th ECM congress, Kraków, Poland, July 2–7 July, 2012_ Zürich: European Mathematical Society (EMS), 2013, pp. 375–397  DOI: 10.4171/120-1/22
  * [28] Alexander S. Kechris, Vladimir G. Pestov and Stevo Todorcevic  “Fraï ssé limits, Ramsey theory, and topological dynamics of automorphism groups”  In _Geom. Funct. Anal._ 15.1, 2005, pp. 106–189  DOI: 10.1007/s00039-005-0503-1
  * [29] Alexander S. Kechris and Todor Tsankov  “Amenable actions and almost invariant sets”  In _Proc. Amer. Math. Soc._ 136.2, 2008, pp. 687–697  DOI: 10.1090/S0002-9939-07-09116-2
  * [30] Matthew Kennedy  “An intrinsic characterization of C*-simplicity”  In _Ann. Sci. Éc. Norm. Supér. (4)_ 53.5, 2020, pp. 1105–1119  DOI: 10.24033/asens.2441
  * [31] Julien Melleray, Lionel Nguyen Van Thé and Todor Tsankov  “Polish groups with metrizable universal minimal flows”  In _Int. Math. Res. Not._ , 2016, pp. 1285–1307  DOI: 10.1093/imrn/rnv171
  * [32] Gianluca Paolini and Saharon Shelah  “Reconstructing structures with the strong small index property up to bi-definability”  In _Fund. Math._ 247.1, 2019, pp. 25–35  DOI: 10.4064/fm640-9-2018
  * [33] Gianluca Paolini and Saharon Shelah  “Research Trends in Contemporary Logic”  College Publications, forthcoming
  * [34] Robert R. Phelps  “Lectures on Choquet’s theorem” 1757, Lect. Notes Math.  Berlin: Springer, 2001  DOI: 10.1007/b76887
  * [35] Bruno Poizat  “A course in model theory. An introduction to contemporary mathematical logic. Transl. from the French by Moses Klein”, Universitext  New York, NY: Springer, 2000
  * [36] Todor Tsankov  “Unitary representations of oligomorphic groups”  In _Geom. Funct. Anal._ 22.2, 2012, pp. 528–555  DOI: 10.1007/s00039-012-0156-9
  * [37] Veeraualli S. Varadarajan  “Geometry of Quantum Theory”  New York, NY: Springer-Verlag, 1985  DOI: 10.1007/978-1-4612-5082-9
  * [38] Andy Zucker  “Topological dynamics of automorphism groups, ultrafilter combinatorics, and the generic point problem”  In _Trans. Amer. Math. Soc._ 368.9, 2016, pp. 6715–6740  DOI: 10.1090/tran6685

C. Jahel, Institut fur Algebra, Technische Universität Dresden, GERMANY

E-mail address<EMAIL_ADDRESS>
M. Joseph, Université Paris-Saclay, CNRS, Laboratoire de mathématiques
d’Orsay, 91405, Orsay, France

E-mail address<EMAIL_ADDRESS>