# On dissociated infinite permutation groups

Rémi Barritault, Colin Jahel, Matthieu Joseph

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Rémi Barritault, Colin Jahel, Matthieu Joseph. On dissociated infinite permutation groups. 2025. hal-05047341

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# On dissociated infinite permutation groups

Rémi BARRITAULT, Colin JAHEL, Matthieu JOSEPH

## Abstract

The goal of this paper is threefold. First, we describe the notion of dissociation for closed subgroups of the group of permutations on a countably infinite set and explain its numerous consequences on unitary representations (classification of unitary representations, Property (T), Howe-Moore property, etc.) and on ergodic actions (non-existence of type III non-singular actions, Stabilizer rigidity, etc.). Some of the results presented here are new, others were proved in different contexts. Second, we introduce a new method to prove dissociation. It is based on a reinforcement of the classical notion of strong amalgamation, where we allow to amalgamate over countable sets. Third, we apply this technique of amalgamation to provide new examples of dissociated closed permutation groups, including isometry groups of some metrically homogeneous graphs, automorphism groups of diversities, and more.

**MSC:** Primary: 22A25, 22F50. Secondary: 37A15, 60G09.

**Keywords:** Infinite permutation groups, unitary representations, type I, Property (T), Howe-Moore property, Stabilizer rigidity, de Finetti theorem.

## Contents

<table><tr><td>1</td><td>About the definition of dissociation</td><td>5</td></tr><tr><td>1.1</td><td>Dissociation: unitary representations and Boolean p.m.p. actions</td><td>5</td></tr><tr><td>1.2</td><td>Lattice of open subgroups and topological simplicity</td><td>7</td></tr><tr><td>2</td><td>Unitary representations of dissociated groups</td><td>8</td></tr><tr><td>2.1</td><td>Classification of unitary representations</td><td>8</td></tr><tr><td>2.2</td><td>Property (T)</td><td>10</td></tr><tr><td>2.3</td><td>Howe-Moore property</td><td>11</td></tr><tr><td>3</td><td>Ergodic theory of dissociated groups</td><td>14</td></tr><tr><td>3.1</td><td>Non-singular actions</td><td>14</td></tr><tr><td>3.2</td><td>Stabilizer rigidity</td><td>16</td></tr><tr><td>3.3</td><td>Exchangeable processes</td><td>17</td></tr></table>
<table><tr><td>4</td><td>How to obtain dissociation</td><td>18</td></tr><tr><td>4.1</td><td>Via approximating sequences</td><td>18</td></tr><tr><td>4.2</td><td>Via an amalgamation property</td><td>18</td></tr><tr><td>4.2.1</td><td>Tail subspaces and dissociation</td><td>18</td></tr><tr><td>4.2.2</td><td>Strong cofinite amalgamation over countable subsets</td><td>21</td></tr><tr><td>4.2.3</td><td>N<sub>0</sub>-categoricity and σ-SAP</td><td>23</td></tr><tr><td>5</td><td>New examples of dissociated permutation groups</td><td>24</td></tr><tr><td>5.1</td><td>Metrically homogeneous graphs</td><td>24</td></tr><tr><td>5.1.1</td><td>Integral metric spaces with no small triangle of odd perimeter</td><td>25</td></tr><tr><td>5.1.2</td><td>Integral metric spaces with no unit simplex</td><td>27</td></tr><tr><td>5.2</td><td>Diversities</td><td>28</td></tr><tr><td>5.3</td><td>Free amalgamation property</td><td>30</td></tr></table>

# Introduction

In this paper, we gather together under a common name two techniques – independence and orthogonality – that were used many times in the past in the study of infinite permutation groups. *Dissociation*, which is the main notion of the present paper, finds its roots in exchangeability theory. In that context, dissociation of a random process corresponds to *independence*. It is a central notion designed to generalize the famous de Finetti’s theorem and has been extensively studied especially in the setting of random arrays [1, 18]. In the last decades, this notion of dissociation has percolated in a domain at the intersection of probability and model theory, which studies random structures that are invariant under the automorphism group of some structure (see e.g. [9, 11, 17]).

Another technique that has been widely used is that of *orthogonality*, that leads to classification results of unitary representations for many automorphism groups of structures [2, 21, 29]. One of the aims of this paper is to explain how these techniques of independence and orthogonality are one and the same, that we call dissociation.

An important motivation for introducing this notion is the fact that this phenomenon has already been observed for the class of $N_0$-categorical structures, yielding many consequences, some ergodic theoretic [16, 17, 28], others on unitary representations [29]. Dissociation beyond the $N_0$-categorical case was already applied to the Urysohn spaces by the authors [2]. We now intend to start a more systematic study of this property.

**NOTATION.** — In this paper, $\Omega$ stands for a countably infinite set and $\text{Sym}(\Omega)$ for the group of all permutations of $\Omega$, equipped with the topology of pointwise convergence. A closed permutation group (on $\Omega$) is a closed subgroup $G$ of $\text{Sym}(\Omega)$. For every $A \subseteq \Omega$, we denote by $G_A$ the pointwise stabilizer of $A$ and by $G_{\{A\}}$ the setwise stabilizer of $A$.
**Dissociation for unitary representations.** Let $G$ be a closed permutation group on $\Omega$ and $\pi: G \to \mathcal{U}(\mathcal{H})$ a unitary representation, that is, a group homomorphism which is continuous for the strong operator topology on the unitary group. For every finite subset $A \subseteq \Omega$, let $\mathcal{H}_A \subseteq \mathcal{H}$ be the space of $G_A$-invariant vectors. We will write $p_A$ for the orthogonal projection onto the subspace $\mathcal{H}_A$ of those vectors that are invariant under the pointwise stabilizer $G_A := \{g \in G: \forall a \in A, g(a) = a\}$. The unitary representation $\pi$ is **dissociated** if for all $A, B \subseteq \Omega$ finite, we have $p_A p_B = p_{A \cap B}$. Geometrically, this means that the subspaces $\mathcal{H}_A \cap (\mathcal{H}_{A \cap B})^{\perp}$ and $\mathcal{H}_B \cap (\mathcal{H}_{A \cap B})^{\perp}$ are orthogonal. One notable feature that we will use in the paper is that there is no real need to reduce our attention to finite subsets of $\Omega$. Indeed, for any subset $A \subseteq \Omega$, one can define:

$$ \mathcal{H}_A := \overline{\bigcup \{\mathcal{H}_B, B \subseteq A \text{ finite}\}}. $$

Dissociation is then equivalent to $p_A p_B = p_{A \cap B}$ for all $A, B \subseteq \Omega$, not necessarily finite.

**Dissociation for Boolean p.m.p. actions** Let $G$ be a closed permutation group on $\Omega$. Let $(X, \mu)$ be a standard probability space and let $\alpha: G \to \mathrm{Aut}(X, \mu)$ be a Boolean p.m.p. action, that is, a homomorphism which is continuous for the weak topology on $\mathrm{Aut}(X, \mu)$. For every finite subset $A \subseteq \Omega$, let $\mathcal{F}_A$ be the $\sigma$-algebra of those measurable subsets $Y \subseteq X$ that are $G_A$-invariant in the sense that $\mu(gY\Delta Y) = 0$ for every $g \in G_A$. The Boolean p.m.p. action $\alpha$ is **dissociated** if for every finite subset $A, B \subseteq \Omega$, the $\sigma$-algebras $\mathcal{F}_A$ and $\mathcal{F}_B$ are independent conditionally on $\mathcal{F}_{A \cap B}$.

**Dissociated groups.** In Section 1, we discuss the relation between dissociation for unitary representations and for Boolean p.m.p. actions. We prove that such an action is dissociated if and only if its Koopman representation is dissociated. We then prove that for a closed permutation group $G$, every unitary representation is dissociated if and only if every Boolean p.m.p. action is dissociated. If this holds, we say that $G$ is **dissociated**.

**Compendium on dissociated groups.** One of the goals of this paper is to collect in one place some results related to dissociation that were either proved in other papers, e.g. [2, 16, 28, 29], or are proved in the present paper.

**THEOREM.** — Let $G$ be a closed permutation group on $\Omega$. If $G$ is dissociated, then the following statements hold.

I. Unitary representations:

1) **Rigidity of unitary representations** (Corollary 2.3). Every irreducible unitary representation of $G$ is induced from an irreducible representation of the setwise stabilizer $G_{\{A\}} := \{g \in G: g(A) = A\}$ for some finite $A \subseteq \Omega$, which is trivial on the pointwise stabilizer $G_A$. Moreover, every unitary representation of $G$ is a direct sum of irreducible ones.
2) **Property (T)** (Theorem 2.6). *G has Kazhdan's Property (T).*

3) **Howe-Moore property** (Theorem 2.12). *If every proper open subgroup of G is coarsely bounded, then G has the Howe-Moore property with respect to the bornology of coarsely bounded sets.*

## II. Ergodic theory:

1) **Rigidity of non-singular actions** (Theorem 3.2). *Every non-singular ergodic action of G is induced by a probability measure-preserving action of the setwise stabilizer $G_{\{A\}}$ for some finite subset $A \subseteq \Omega$. Moreover, every non-singular action of G is a disjoint union of ergodic ones.*

2) **Stabilizer rigidity** (Theorem 3.3). *If G is a proper subgroup of Sym($\Omega$) which acts transitively on $\Omega$, then any Borel p.m.p. ergodic action of G is either essentially free or essentially transitive.*

3) **Rigidity of invariant random processes** (Corollary 3.6). *If G acts transitively on $\Omega$, then every random process $(X_\omega)_{\omega \in \Omega}$ whose law is G-invariant and ergodic is i.i.d.*

**How to obtain dissociation?** To tackle this question, we change our point of view in Section 4 and consider closed subgroups of $Sym(\Omega)$ as groups of automorphisms of countable relational structures and present two general methods for obtaining dissociation. First, through *approximating sequences*, when the automorphism group can be seen as a coherent limit of dissociated groups. Second, by introducing the notion of *strong cofinite amalgamation over countable sets* (abbrev. $\sigma$-SAP, see Definition 4.8). Our main new result in this section is Theorem 4.12, which proves that if a ultrahomogeneous structure satisfies $\sigma$-SAP and weakly eliminates imaginaries then its automorphism group is dissociated.

**New examples of dissociated groups.** Finally, we apply in Section 5 the methods from Section 4 to provide new examples of dissociated groups, such as isometry groups of metrically homogeneous spaces, automorphism groups of diversities and more. In particular, we obtain examples going beyond the more widely studied Roelcke compact case, some being locally Roelcke precompact, others being coarsely bounded but not Roelcke precompact.

**Acknowledgments.** We thank Todor Tsankov for sharing a first version of [28] which inspired many results in the present paper. A few results in that paper (e.g. Theorem 3.2) are even direct adaptations of his results in the aforementioned paper. We also thank Yves Benoist and Christian Rosendal for pointing out to us the similarities between the notion of dissociation and Mackey's work on systems of imprimitivity.
# 1 About the definition of dissociation

## 1.1 Dissociation: unitary representations and Boolean p.m.p. actions

Let $(X, \mu)$ be a standard probability space and $\mathcal{F}_1, \mathcal{F}_2, \mathcal{F}_3$ be three $\sigma$-algebras of measurable sets on $X$ satisfying $\mathcal{F}_2 \subseteq \mathcal{F}_1 \cap \mathcal{F}_3$. We say that $\mathcal{F}_1$ and $\mathcal{F}_3$ are independent conditionally on $\mathcal{F}_2$ if the following holds: for every $\mathcal{F}_3$-measurable function $f \in L^2(X, \mu)$, we have $\mathbb{E}[f | \mathcal{F}_1] = \mathbb{E}[f | \mathcal{F}_2]$, where $\mathbb{E}[\cdot | \mathcal{F}_i]: L^2(X, \mu) \to L^2(X, \mathcal{F}_i, \mu)$ is the conditional expectation.

Let $\mathcal{H}$ be a Hilbert space and $\mathcal{H}_1, \mathcal{H}_2, \mathcal{H}_3$ be three closed subspaces satisfying $\mathcal{H}_2 \subseteq \mathcal{H}_1 \cap \mathcal{H}_3$ and let $p_1, p_2, p_3$ be the associated orthogonal projections. We say that $\mathcal{H}_1$ and $\mathcal{H}_3$ are orthogonal conditionally on $\mathcal{H}_2$, and write:

$$ \mathcal{H}_1 \perp_{\mathcal{H}_2} \mathcal{H}_3 $$

if $p_1p_3 = p_2$, i.e. if the subspaces $\mathcal{H}_1 \cap \mathcal{H}_2$ and $\mathcal{H}_3 \cap \mathcal{H}_2$ are orthogonal. Note that if $\mathcal{H}_1 \perp_{\mathcal{H}_2} \mathcal{H}_3$ then $\mathcal{H}_2 = \mathcal{H}_1 \cap \mathcal{H}_3$.

These two notions of conditional independence/orthogonality are related by the following classical result.

**LEMMA 1.1.** — Let $(X, \mu)$ be a standard probability space and $\mathcal{F}_1, \mathcal{F}_2, \mathcal{F}_3$ be three sub-$\sigma$-algebras on $X$ satisfying $\mathcal{F}_2 \subseteq \mathcal{F}_1 \cap \mathcal{F}_3$. If $L^2(X, \mathcal{F}_i, \mu)$ denotes the space of square-integrable, $\mathcal{F}_i$-measurable complex valued functions on $(X, \mu)$ (for $i = 1, 2, 3$), then the following are equivalent.

* $\mathcal{F}_1$ and $\mathcal{F}_3$ are independent conditionally on $\mathcal{F}_2$.

* $L^2(X, \mathcal{F}_1, \mu)$ and $L^2(X, \mathcal{F}_3, \mu)$ are orthogonal conditionally on $L^2(X, \mathcal{F}_2, \mu)$.

*Proof.* Write $\mathcal{H}_i := L^2(X, \mathcal{F}_i, \mu)$ and $p_i := \mathbb{E}[\cdot | \mathcal{F}_i]: L^2(X, \mu) \to \mathcal{H}_i$ for $i = 1, 2, 3$. Notice that $\mathcal{F}_2 \subseteq \mathcal{F}_1 \cap \mathcal{F}_3$ implies $\mathcal{H}_2 \subseteq \mathcal{H}_1 \cap \mathcal{H}_3$. Since $p_i$ is the orthogonal projection onto $\mathcal{H}_i$, the following equivalences hold:

$$ \begin{align*} \mathcal{F}_1 \perp_{\mathcal{F}_3} \mathcal{F}_2 &\Leftrightarrow \forall f \in \mathcal{H}_3, p_1 f = p_2 f \\ &\Leftrightarrow p_1 p_3 = p_2 \\ &\Leftrightarrow \mathcal{H}_1 \perp_{\mathcal{H}_2} \mathcal{H}_3. \end{align*} \quad \square $$

Let $(X, \mu)$ be a probability space and Aut$(X, \mu)$ be equipped with the weak topology. A Boolean p.m.p. action of a topological group $G$ on $(X, \mu)$ is a continuous homomorphism $\alpha: G \to \text{Aut}(X, \mu)$. A measurable set $Y$ is $G$-invariant if every $g \in G$, we have $\mu(\alpha(g)Y \Delta Y) = 0$. The Boolean p.m.p. action $\alpha$ is ergodic if the $\sigma$-algebra $\mathcal{F}$ of $G$-invariant measurable sets is trivial ($Y \in \mathcal{F} \Rightarrow \mu(Y) \in \{0, 1\}$). The following lemma connects $G$-invariant functions with $\mathcal{F}$-measurable functions. While often stated for Borel p.m.p. actions of locally compact groups, its proof readily adapts to Boolean p.m.p actions of topological groups.
LEMMA 1.2. — Let $G$ be a topological group, let $\alpha: G \to \text{Aut}(X, \mu)$ be a Boolean p.m.p. action and let $\mathcal{F}$ be the $\sigma$-algebra of $G$-invariant measurable sets. Then a function $f: X \to \mathbb{C}$ is $\mathcal{F}$-measurable if and only if for every $g \in G$, we have $f \circ g = f \mu$-almost everywhere.

As a consequence, we connect the two notions of dissociation – for unitary representations and Boolean p.m.p. actions – defined in the introduction.

LEMMA 1.3. — Let $G \leq \text{Sym}(\Omega)$ be a closed permutation group and $\alpha: G \to \text{Aut}(X, \mu)$ be a Boolean p.m.p. action. Then $\alpha$ is dissociated if and only if its Koopman representation $\kappa: G \to \text{L}^2(X, \mu)$ is dissociated.

*Proof*. Let $\mathcal{H} := \text{L}^2(X, \mu)$. For every finite subset $C \subseteq \Omega$, Lemma 1.2 implies that $\mathcal{H}_C = \text{L}^2(X, \mathcal{F}_C, \mu)$. Therefore, Lemma 1.1 shows that $\alpha$ is dissociated if and only if $\kappa$ is. $\square$

The following result is standard using Gaussian actions. For a proof in the context of Boolean p.m.p. actions of Polish groups, we refer to Section 8.2 of [8] and the references therein.

THEOREM 1.4. — Let $G$ be a Polish group and $\pi: G \to \mathcal{U}(\mathcal{H})$ be a unitary representation. There exists a standard probability space $(X, \mu)$ and a Boolean p.m.p. action $G \to \text{Aut}(X, \mu)$ whose (complex) Koopman representation contains $\pi$ as a subrepresentation.

We are now ready to prove the following result.

THEOREM 1.5. — Let $G$ be a closed permutation group. Then the following are equivalent.

1. Every unitary representation of $G$ is dissociated.

2. Every Boolean p.m.p. action of $G$ is dissociated.

If this holds, we say that $G$ is dissociated.

*Proof*. Assume that every unitary representation of $G$ is dissociated. Let $\alpha: G \to \text{Aut}(X, \mu)$ be a Boolean p.m.p. action. By assumption, its Koopman representation is dissociated. Therefore, $\alpha$ is dissociated by Lemma 1.3

Assume that every Boolean p.m.p. action of $G$ is dissociated. Let $\pi: G \to \mathcal{U}(\mathcal{H})$ be a unitary representation. By Theorem 1.4, there exists a Boolean p.m.p. action $\alpha: G \to \text{Aut}(X, \mu)$ whose Koopman representation $\kappa: G \to \mathcal{U}(\text{L}^2(X, \mu))$ contains $\pi$ as a subrepresentation. Since $\alpha$ is dissociated, so is $\kappa$ by Lemma 1.3. But dissociation passes to subrepresentations $[2, \text{Lem. 3.4}]$ so $\pi$ is also dissociated. $\square$
## 1.2 Lattice of open subgroups and topological simplicity

Let $G \leq \mathrm{Sym}(\Omega)$ be a closed permutation group. We say that $G$ has no algebraicity if for every finite subset $A \subseteq \Omega$, the orbits of the pointwise stabilizer $G_A$ on $\Omega \setminus A$ are all infinite. We say that $G$ weakly eliminates imaginaries if for every open subgroup $V \leq G$, there exists a finite subset $A \subseteq \Omega$ such that $G_A \leq V$ and $[V : G_A] < +\infty$. The combination of these two properties leads to a precise understanding of the lattice of open subgroups, as explained in the following lemma.

**LEMMA 1.6.** — Let $G$ be a closed subgroup of $\mathrm{Sym}(\Omega)$. Assume that $G$ acts without fixed point on $\Omega$. Then the following are equivalent.

1. $G$ has no algebraicity and weakly eliminates imaginaries.

2. For all finite subsets $A, B \subseteq \Omega$, the subgroup $\langle G_A, G_B \rangle$ generated by $G_A$ and $G_B$ is equal to $G_{A \cap B}$.

3. For every open subgroup $V \leq G$, there exists a unique finite subset $A \subseteq \Omega$ such that $G_A \leq V \leq G_{\{A\}}$.

*Proof.* The equivalence between 1 and 2 is Lemma 3.6 of [16] while 3 is simply an efficient reformulation of the above definitions. Indeed, assume $G$ satisfies 1 and let $V \leq G$ be an open subgroup. By weak elimination of imaginaries, there exists $A \subseteq \Omega$ finite such that $G_A \leq V$ with finite index. Necessarily, every element of $A$ has a finite orbit under the action of $V$. Since $G$ has no algebraicity, $V \leq G_{\{A\}}$.

Assume now $B$ is another finite subset of $\Omega$ such that $G_B \leq V \leq G_{\{B\}}$. Then $G_A \leq V \leq G_{\{B\}}$ and $G_A$ stabilizes $B$. In particular, every element of $B$ has a finite orbit under the action of $G_A$. By the no algebraicity hypothesis, $B \subseteq A$. The reverse inclusion also holds by symmetry.

Conversely, assume 3. Clearly, $G$ has weak elimination of imaginaries. Now let $A \subseteq \Omega$ be finite and assume $G_A \cdot a$ is finite for some $a \in \Omega$. Then $B := A \cup G_A \cdot a$ is finite and stable under the action of $G_A$. Thus:

$$G_B \leq G_A \leq G_{\{B\}}.$$

But we obviously also have $G_A \leq G_A \leq G_{\{A\}}$. By uniqueness in 3, $A = B$ i.e. $a \in A$ and $G$ has no algebraicity. $\square$

In fact, our definition of dissociation is tailored for permutation groups that have no algebraicity and weakly eliminate imaginaries. In particular, we have the following lemma, which appears as Remark 3.2 of [2].

**LEMMA 1.7.** — Let $G \leq \mathrm{Sym}(\Omega)$ be a closed permutation group. Assume that $G$ acts without fixed points on $\Omega$. If $G$ is dissociated, then $G$ has no algebraicity and eliminates weakly imaginaries.

The proof of the following result was explained to us by David Evans (personal communication).
**PROPOSITION 1.8.** — Let $G$ be a transitive, closed subgroup of $\mathrm{Sym}(\Omega)$. If $G$ has no algebraicity and weakly eliminates imaginaries, then $G$ is topologically simple.

*Proof.* Let $H$ be a closed, normal, proper subgroup of $G$. Expressing that $H$ is proper and closed, and therefore not dense in $G$, we get an integer $n \ge 1$ and a $G$-orbit $\Delta \subseteq \Omega^n$ such that $H$ is not transitive on $\Delta$. As $H \trianglelefteq G$, the $H$-orbits form a $G$-equivariant equivalence relation $\sim$ on $\Delta$. Fix $\bar{x} \in \Delta$ and write $[\bar{x}]_{\sim}$ for the equivalence class $H \cdot \bar{x}$ of $\bar{x}$. It is clear that $H$ is contained in the subgroup $V := \{g \in G: g([\bar{x}]_{\sim}) = [\bar{x}]_{\sim}\}$ which is open since it contains the pointwise stabilizer $G_{\bar{x}}$. By Lemma 1.6, there exists a unique finite subset $A \subseteq \Omega$ such that $G_A \le V \le G_{\{A\}}$. Necessarily, $A \ne \emptyset$ otherwise $[\bar{x}]_{\sim} = \Delta$, a contradiction. Now, $H \le G_{\{A\}}$ and, by normality, we get:

$$H \le K := \bigcap_{g \in G} G_{\{g(A)\}}.$$

Let us prove that $K$ is trivial. Since $K$ is normal in $G$ and $G$ acts transitively on $\Omega$, it suffices to prove that $K$ has a fixed point on $\Omega$. Fix any $a \in A$. Since the $G_a$-orbits on $\Omega \setminus \{a\}$ are infinite, there exists by Neumann's lemma [5, Thm. 1] an element $g_0 \in G_a$ such that $A \cap g_0 A = \{a\}$. Thus $K \le G_{\{A\}} \cap G_{\{g_0(A)\}} \le G_a$, which concludes the proof. $\square$

**COROLLARY 1.9.** — Let $G \le \mathrm{Sym}(\Omega)$ be a transitive, closed permutation group. If $G$ is dissociated, then $G$ is topologically simple.

# 2 Unitary representations of dissociated groups

## 2.1 Classification of unitary representations

In order to state the classification result of dissociated unitary representations that we obtained in [2], let us first recall the notion of induced representations. Let $G$ be a separable topological group and $H \le G$ an open subgroup (separability forces the index $[G: H]$ to be at most countable). Let $\sigma: H \to \mathcal{U}(\mathcal{K})$ be a unitary representation. Let $\mathcal{E}$ be the space of maps $f: G \to \mathcal{K}$ such that for every $g \in G$ and $h \in H$ we have $f(gh) = \sigma(h^{-1})f(g)$. Notice that for all $f, f_1, f_2 \in \mathcal{E}$, the maps $g \mapsto \langle f_1(g), f_2(g) \rangle$ and $g \mapsto \|f(g)\|$ are constant on each left $H$-coset. Denote by $\langle f_1, f_2 \rangle (q)$ and $\|f(q)\|$ their respective value on the coset $q \in G/H$. Let $\mathcal{H}$ be the Hilbert space of all $f \in \mathcal{E}$ such that $\sum_{q \in G/H} \|f(q)\|^2 < +\infty$ with inner product given by

$$\langle f_1, f_2 \rangle = \sum_{q \in G/H} \langle f_1, f_2 \rangle (q).$$

The induced representation $\pi := \mathrm{Ind}_H^G(\sigma)$ is the representation of $G$ on $\mathcal{H}$ defined by

$$\pi(g)f: x \mapsto f(g^{-1}x) \quad \text{for all } g \in G, f \in \mathcal{H}.$$
Since $H$ is open, $\mathrm{Ind}_H^G(\sigma): G \to \mathcal{U}(\mathcal{H})$ is indeed continuous.

Induction is a powerful technique whose importance was promoted by Mackey in his seminal work on representation theory of locally compact groups. It turns out that dissociation can be phrased in terms of systems of imprimitivity, a key notion in Mackey's theory. Let $G$ be a topological group, $(X, \mathcal{A})$ be a standard Borel space and $G \times X \to X$ be a Borel action. A system of imprimitivity is a couple $(\pi, p)$, where $\pi: G \to \mathcal{U}(\mathcal{H})$ is a unitary representation and $p: \mathcal{A} \to \mathcal{B}(\mathcal{H})$ is a projection valued measure (see Section 2.5 of [22]) such that for every $A \in \mathcal{A}$ and $g \in G$,

$$ \pi(g)p(A)\pi(g^{-1}) = p(gA). \qquad (1) $$

For a closed permutation group $G \le \mathrm{Sym}(\Omega)$, there is a natural Borel action, that of $G$ on $\Omega$. Moreover, given a unitary representation $\pi: G \to \mathcal{U}(\mathcal{H})$, there is a natural projection valued measure $p_\pi: \mathcal{P}(\Omega) \to \mathcal{B}(\mathcal{H})$, which is defined as follows: for every $A \in \mathcal{P}(\Omega)$ we define $p_\pi(A)$ as the orthogonal projection onto the space $\bigcup_{B \subseteq A \text{ finite}} \mathcal{H}_B$. Notice that equation (1) always holds in that case. Moreover, dissociation easily rephrases as follows.

**FACT 2.1.** — A unitary representation $\pi: G \to \mathcal{U}(\mathcal{H})$ without invariant vector of a closed permutation group $G \le \mathrm{Sym}(\Omega)$ is dissociated if and only if $(\pi, p_\pi)$ is a system of imprimitivity.

In [2, Thm. 3.9], we revisit Mackey's famous Imprimitivity Theorem in the context of closed permutation groups.

**THEOREM 2.2.** — Let $G \le \mathrm{Sym}(\Omega)$ be a closed permutation group without algebraicity.

1. The dissociated irreducible unitary representations of $G$ are exactly the unitary representations isomorphic to one of the form $\mathrm{Ind}_{G_{\{A\}}}^G(\sigma)$ where $A$ ranges over the finite subsets of $\Omega$ and $\sigma$ over the irreducible representations of the finite group $G_{\{A\}}/G_A$.

2. Two such irreducible representations $\mathrm{Ind}_{G_{\{A\}}}^G(\sigma)$ and $\mathrm{Ind}_{G_{\{B\}}}^G(\tau)$ are isomorphic if and only if there exists $g \in G$ such that $gA = B$ and $\sigma^g \simeq \tau$.

3. Every dissociated unitary representation of $G$ splits as direct a sum of irreducible subrepresentations.

In particular, we get the following classification of unitary representations for dissociated groups.

**COROLLARY 2.3.** — Let $G \le \mathrm{Sym}(\Omega)$ be a closed permutation group. If $G$ is dissociated, then every irreducible unitary representation of $G$ is induced from an irreducible representation of the setwise stabilizer $G_{\{A\}} := \{g \in G: g(A) = A\}$ for some finite $A \subseteq \Omega$, which is trivial on the pointwise stabilizer $G_A$. Moreover, every unitary representation of $G$ is a direct sum of irreducible ones.
Recall that a unitary representation $\pi: G \to \mathcal{U}(\mathcal{H})$ is of type I if the von Neumann algebra $\pi(G)''$ it generates is of type I. A group is of type I if every unitary representation is of type I. We refer to [3, Sec. 6] for a modern reference on type I topological groups. Here is a direct corollary of the above classification of unitary representations of dissociated groups.

**COROLLARY 2.4.** — Every dissociated closed permutation group is of type I.

## 2.2 Property (T)

A topological group $G$ has Property (T) if there exists a compact subset $Q \subseteq G$ and $\varepsilon > 0$ such that every unitary representation $G \to \mathcal{U}(\mathcal{H})$ with a non-zero vector $\xi \in \mathcal{H}$ satisfying

$$ \sup_{g \in Q} \|\pi(g)\xi - \xi\| \le \varepsilon \|\xi\| $$

admits a non-zero invariant vector. The pair $(Q, \varepsilon)$ is called a Kazhdan pair. We describe here a useful criterion for proving Property (T) for closed permutation groups. It is due to Tsankov [29] but is not explicitly formulated. We will not provide a proof, but a careful look at Section 6 (in particular Lemma 6.3, Proposition 6.4 and Lemma 6.5) of Tsankov’s paper allows one to extract the following result.

**THEOREM 2.5.** — Let $G \le \mathrm{Sym}(\Omega)$ be a closed permutation group without algebraicity. Assume that every unitary representation of $G$ is a direct sum of irreducible ones and that every irreducible representation of $G$ is a subrepresentation of $G \curvearrowright \ell^2(\Omega^n)$ for some $n \le 0$. Then $G$ has property (T).

Notice that a dissociated group satisfies the assumption of the above theorem. Indeed, if $G \le \mathrm{Sym}(\Omega)$ is dissociated, then we know by Corollary 2.3 that every unitary representation of $G$ is a direct sum of irreducible ones, and that an irreducible one has the form $\mathrm{Ind}_{G_{\{A\}}}^G(\sigma)$ for some $A \subseteq \Omega$ and some irreducible unitary representation $\sigma: G_{\{A\}} \to \mathcal{U}(\mathcal{K})$ which factors through the finite group $F := G_{\{A\}}/G_A$. If $\lambda_F$ denotes the left-regular representation of $F$, then we have

$$ \begin{aligned} \mathrm{Ind}_{G_{\{A\}}}^G(\sigma) &\le \mathrm{Ind}_{G_{\{A\}}}^G(\lambda_F) \\ &\simeq \mathrm{Ind}_{G_{\{A\}}}^G(\mathrm{Ind}_{G_A}^{G_{\{A\}}}(1_{G_A})) \\ &\simeq \mathrm{Ind}_{G_A}^G(1_{G_A}) \\ &\simeq \ell^2(G/G_A) \\ &\le \ell^2(\Omega^n), \end{aligned} $$

where $n = |A|$. Therefore, Theorem 2.5 applies and yields the following result.

**THEOREM 2.6.** — Every dissociated closed permutation group has Property (T).

A topological group has the strong Property (T) if there exists a Kazhdan pair $(Q, \varepsilon)$ with $Q$ finite. Evans and Tsankov proved in [12] that every dissociated oligomorphic group has strong Property (T).
**QUESTION 2.7.** — Does every dissociated group have strong Property (T)?

To give a positive answer to this question, it would be sufficient to prove that every dissociated group $G \leq \text{Sym}(\Omega)$ contains a free group of finite rank acting freely on $\Omega$, see e.g. [2, Thm. 5.2] or [12].

## 2.3 Howe-Moore property

Let $G$ be a group. A *group bornology* on $G$ is a collection $\mathcal{B} \subseteq \mathcal{P}(G)$ of subsets of $G$ satisfying the following conditions:

1. for every $g \in G$, $\{g\}$ belongs to $\mathcal{B}$,

2. for every $A \in \mathcal{B}$, every subset of $A$ belongs to $\mathcal{B}$,

3. for every $A \in \mathcal{B}$, the set $A^{-1} := \{g^{-1}: g \in A\}$ belongs to $\mathcal{B}$,

4. for every $A, B \in \mathcal{B}$, the sets $A \cup B$ and $AB := \{ab: a \in A, b \in B\}$ belong to $\mathcal{B}$.

An element of $\mathcal{B}$ is called a *bounded subset of $G$*. For a topological group $G$, there are two natural bornologies on it. The first one $\mathcal{K}$ is that of relatively compact subsets of $G$. Another one turns out to be of great interest: the bornology $\mathcal{CB}$ of *coarsely bounded* subsets of $G$. It was introduced by Rosendal (see the book [24]) and is defined as follows: a subset $B \subseteq G$ is *coarsely bounded* if for every continuous, left-invariant pseudometric $d$ on $G$, we have $\text{diam}_d(A) < +\infty$. For every Polish group $G$, the bornology of relatively compact sets is contained in that of coarsely bounded sets whereas for Polish locally compact groups, there is equality [24, Cor. 2.19].

A *topological bornological group* is a couple $(G, \mathcal{B})$ such that $G$ is a topological group and $\mathcal{B}$ is a bornology on $G$ which is contained in the bornology of coarsely bounded sets. A topological bornological group $(G, \mathcal{B})$ is locally bounded if every element of $G$ admits a neighborhood in $\mathcal{B}$, locally bounded second countable (abbrev. 1.b.s.c.) if in addition the topology on $G$ is second countable, and bounded if $G$ belongs to $\mathcal{B}$.

**LEMMA 2.8.** — Let $(G, \mathcal{B})$ be locally bounded. Then $\mathcal{K} \subseteq \mathcal{B}$.

*Proof.* Let $\mathcal{K} \subseteq G$ be a compact subset. For every $x \in \mathcal{K}$, let $U_x \in \mathcal{B}$ be a bounded neighborhood of $x$. By compactness, $\mathcal{K}$ is contained in a finite union of bounded sets. Thus, $\mathcal{K}$ is bounded. Finally, any subset of a bounded set is bounded, so every relatively compact subset of $G$ is bounded. $\square$

One of the interest of bornological group is that there is a notion of divergence to infinity in the group. Let $(G, \mathcal{B})$ be a topological bornological group. A function $f: G \to \mathbb{C}$ *vanishes at infinity* for the bornology $\mathcal{B}$ if for every $\varepsilon > 0$, there exists a bounded set $B \in \mathcal{B}$ such that $g \in G \setminus B \Rightarrow |f(g)| < \varepsilon$. Let $C_0(G, \mathcal{B})$ be the algebra of continuous functions $f: G \to \mathbb{C}$ that vanishes at infinity for the bornology $\mathcal{B}$. Notice that if $(G, \mathcal{B})$ is a 1.b.s.c. group, then $G$ is the union of countably many bounded (and open) sets. In that case, a continuous map $f: G \to \mathbb{C}$ belongs to $C_0(G, \mathcal{B})$ if and only
if for every sequence $(g_n)_{n \ge 0}$ of elements in $G$ which eventually leaves every bounded set (that is, for every $B \in \mathcal{B}$, we have $g_n \notin B$ eventually), we have $f(g_n) \to 0$.

**DEFINITION 2.9.** — A topological bornological group $(G, \mathcal{B})$ has the Howe-Moore property if for every unitary representation $\pi: G \to \mathcal{U}(\mathcal{H})$ without non-zero invariant vector and every $\xi, \eta \in \mathcal{H}$, the matrix coefficient $g \mapsto \langle \pi(g)\xi | \eta \rangle$ belongs to $C_0(G, \mathcal{B})$.

We recover some properties that are well known when $G$ is a locally compact group.

**LEMMA 2.10.** — Let $(G, \mathcal{B})$ be a topological bornological group which has the Howe-Moore property. Then every proper open subgroup of $G$ is bounded.

*Proof.* Let $U \le G$ be a proper open subgroup. Consider the associated quasi-regular representation $\lambda_{G/U}: G \to \mathcal{U}(\ell^2(G/U))$. Note that this representation has non-zero invariant vectors (the constant functions) if and only if $U$ has finite index in $G$. We deal with the two cases separately.

Assume first that $U$ has infinite index in $G$. Write $\delta_U$ for the Dirac function at $U$ on $G/U$. Then $\delta_U \in \ell^2(G/U)$ and, by the Howe-Moore property, there exists a bounded set $B \in \mathcal{B}$ such that:

$$ \forall g \notin B, \quad |\langle \lambda_{G/U}(g)\delta_U | \delta_U \rangle| < 1/2. $$

But for every $g \in G$, $\langle \lambda_{G/U}(g)\delta_U | \delta_U \rangle = 1$ if $g \in U$ and 0 otherwise. The above equation thus rewrites $U \subseteq B$ and shows that $U$ is bounded.

Assume now that $U$ has finite index in $G$ and restrict $\lambda_{G/U}$ to the orthogonal complement $\ell_0^2(G/U)$ of the constant functions. By construction, this representation has no non-zero invariant vector. Denoting again $\delta_U$ the Dirac function at $U$ on $G/U$, define $\xi := \delta_U - 1/[G:U]$. Then $\xi \in \ell_0^2(G/U)$ and by the Howe-Moore property, there exists $B \in \mathcal{B}$ such that:

$$ \forall g \notin B, \quad |\langle \lambda_{G/U}(g)\xi | \xi \rangle| < 1/2. $$

But for every $g \in U$, $\langle \lambda_{G/U}(g)\xi | \xi \rangle = 2 - 2/[G:U] \ge 1$ where the inequality follows from the fact that $U$ is proper. Necessarily, $U \subseteq B$ hence $U$ is bounded. $\square$

Since translates and finite unions of bounded sets are bounded, we get the following consequence.

**COROLLARY 2.11.** — Let $(G, \mathcal{B})$ be a non-bounded topological bornological group which has the Howe-Moore property. Then every proper open subgroup of $G$ has infinite index.

The main result of this section is the following.

**THEOREM 2.12.** — Let $G \le \mathrm{Sym}(\Omega)$ be a closed permutation group which acts without fixed point on $\Omega$. Let $\mathcal{B}$ be a group bornology such that $(G, \mathcal{B})$ is a topological bornological group. If $G$ is dissociated, then the following properties are equivalent.

1. $(G, \mathcal{B})$ has the Howe-Moore property.

2. Every proper open subgroup of $G$ is bounded.
3. For every non-empty finite subset $A \subseteq \Omega$, $G_{\{A\}}$ is bounded.

4. For every non-empty finite subset $A \subseteq \Omega$, $G_A$ is bounded.

5. For every $a \in \Omega$, $G_a$ is bounded.

*Proof.* The implication $1 \Rightarrow 2$ is proved in Lemma 2.10. The equivalences $2 \Leftrightarrow 3 \Leftrightarrow 4 \Leftrightarrow 5$ are straightforward by recalling that for a dissociated group $G$ acting without fixed point on $\Omega$, every open subgroup of $G$ lies between the pointwise stabilizer and the setwise stabilizer of a unique finite subset of $\Omega$ (Lemma 1.6 and 1.7).

Let us finally prove that $3 \Rightarrow 1$. Notice that a direct sum of representations whose matrix coefficients are all in $C_0(G, \mathcal{B})$ have the same property. By the classification of unitary representations for dissociated groups (Corollary 2.3), it suffices to prove that the matrix coefficients of every irreducible unitary representation of $G$ without non-zero invariant vector (equivalently nontrivial irreducible) belong to $C_0(G, \mathcal{B})$.

Let $\pi$ be a nontrivial irreducible representation of $G$ on a Hilbert space $\mathcal{H}$. By Corollary 2.3, there exists $A \subseteq \Omega$ a finite subset and $\sigma: G_{\{A\}} \to \mathcal{U}(\mathcal{K})$ an irreducible unitary representation which is trivial on $G_A$ and such that $\pi \simeq \text{Ind}_{G_{\{A\}}}^{G}(\sigma)$. Note that $A$ is non-empty since $\pi$ is nontrivial. Fix a transversal $(h_i)_{i \in I}$ for $G/G_{\{A\}}$, so that for all $f_1, f_2 \in \mathcal{H}$, the scalar product $\langle f_1 | f_2 \rangle$ is given by:

$$ \langle f_1 | f_2 \rangle = \sum_{i \in I} \langle f_1(h_i) | f_2(h_i) \rangle. $$

Define

$$ D := \{ f \in \mathcal{H} : \text{supp}(f) \text{ is contained in a finite union of left cosets of } G_{\{A\}} \}. $$

Then $D$ is a dense subset of $\mathcal{H}$. Fix a non-zero vector $f \in D$ and let $B := \text{supp}(f)$. Notice that $B$ is bounded by assumption (recall that $A$ is non-empty). Then for all $g \in G$, we have

$$ \begin{aligned} \langle \pi(g)f | f \rangle &= \sum_{i \in I} \langle f(g^{-1}h_i) | f(h_i) \rangle \\ &= \sum_{\substack{i \in I \\ h_i \in B \cap gB}} \langle f(g^{-1}h_i) | f(h_i) \rangle \end{aligned} $$

Therefore, for every $g \notin BB^{-1}$, $\langle \pi(g)f | f \rangle = 0$. Since $BB^{-1}$ is bounded, we obtain that for every $\xi \in D$, the map $g \mapsto \langle \pi(g)\xi | \xi \rangle$ belongs to $C_0(G, \mathcal{B})$. Since $D$ is dense in $\mathcal{H}$, this shows that for all $\xi, \eta \in \mathcal{H}$, the matrix coefficient $g \mapsto \langle \pi(g)\xi | \eta \rangle$ belongs to $C_0(G, \mathcal{B})$. That is, $(G, \mathcal{B})$ has the Howe-Moore property. $\square$

Given a countable, additive subsemigroup of $\mathbb{R}_+$, one can consider the Fraïssé limit of finite metric spaces $(X, d)$ such that $d(X \times X) \subseteq \Delta \cup \{0\}$. This is called the Urysohn $\Delta$-metric space and is denoted by $\mathbb{U}_{\Delta}$. Isometry groups of Urysohn $\Delta$-metric spaces
provide new examples of Polish non-locally compact groups satisfying the Howe-Moore property with respect to the bornology of coarsely bounded sets.

**COROLLARY 2.13.** — Let $\Delta$ be a countable, additive subsemigroup of $\mathbb{R}_+$. Then Isom($\mathbb{U}_\Delta$) has the Howe-Moore property with respect to the bornology of coarsely bounded sets.

*Proof.* Notice that countable, additive subsemigroup of $\mathbb{R}_+$ are unbounded, so Isom($\mathbb{U}_\Delta$) is locally bounded but not coarsely bounded by [2, Lem. 4.3]. We proved in [2, Thm. 4.12] that Isom($\mathbb{U}_\Delta$) is dissociated. Moreover, pointwise stabilizers Isom($\mathbb{U}_\Delta$)$_A$ of non-empty finite subsets $A \subseteq \mathbb{U}_\Delta$ are coarsely bounded by Theorem 6.31 and Example 6.32 of [24]. The conclusion thus follows from Theorem 2.12. $\square$

A Boolean p.m.p. action $\alpha: G \to \text{Aut}(X, \mu)$ of a topological bornological group $(G, \mathcal{B})$ is *mixing* if for every measurable sets $Y, Z \subseteq X$, the map

$$g \mapsto \mu(Y \cap \alpha(g)Z) - \mu(Y)\mu(Z)$$

belongs to $C_0(G, \mathcal{B})$. One of the main consequence of the Howe-Moore property is that every Boolean p.m.p. ergodic action is mixing. Indeed, a Boolean p.m.p. action $\alpha: G \to \text{Aut}(X, \mu)$ is:

* ergodic if and only if its Koopman representation $\kappa_0: G \to \mathcal{U}(\text{L}^2(X, \mu) \ominus \mathbb{C})$ has no nontrivial invariant vector,

* mixing if and only if for all $\xi, \eta \in \text{L}^2(X, \mu) \ominus \mathbb{C}$, the matrix coefficient $g \mapsto \langle \kappa_0(g)\xi | \eta \rangle$ of $\kappa_0$ belongs to $C_0(G, \mathcal{B})$.

# 3 Ergodic theory of dissociated groups

## 3.1 Non-singular actions

Let $(X, \nu)$ be a $\sigma$-finite nontrivial measure space. We denote by Aut$(X, [\nu])$ the group of bimeasurable bijections which preserve the class of $\nu$, where two such bijections are identified if they coincide on a conull set. Since $\nu$ is equivalent to a probability measure, we may always assume that $\nu$ is a probability measure. The Koopman representation $\kappa: \text{Aut}(X, [\nu]) \to \mathcal{U}(\text{L}^2(X, \nu))$ is the embedding defined as follows: for all $g \in \text{Aut}(X, [\nu])$ and $f \in \text{L}^2(X, \nu)$,

$$\kappa(g)f: x \mapsto \sqrt{\frac{dg_*\nu}{d\nu}}(x)f(g^{-1}x) \qquad (2)$$

where for every $g \in G$, $g_*\nu$ is the push-forward of $\nu$ by $g$ and $dg_*\nu/d\nu$ is the Radon-Nikodym derivative of $g_*\nu$ with respect to $\nu$. The image of $\kappa$ is closed and we equip Aut$(X, [\nu])$ with the topology induced from the strong operator topology on
U(L²(X, μ)). This is a Polish group. For more details on Aut(X, [ν]) as a topological group, we refer to Section 4 of [10].

A non-singular action of a topological group G on a probability space (X, ν) is a continuous homomorphism α: G → Aut(X, [ν]). It is ergodic if every measurable set Y ⊆ X such that for all g ∈ G, ν(α(g)YΔY) = 0 is either null of conull. A non-singular ergodic action G → Aut(X, [ν]) is of type I if ν is purely atomic, of type II if there exists a diffuse, σ-finite, G-invariant measure μ which is equivalent to ν and of type III otherwise.

THEOREM 3.1. — Let $G \leq \mathrm{Sym}(\Omega)$ be a closed permutation group. If $G$ is dissociated, then $G$ admits no ergodic non-singular action of type III.

In fact, a stronger result holds. In order to state it properly, we need the definition of induction in the setting of p.m.p. actions (there is actually a more general definition of induction in the setting of non-singular actions, however it is not relevant for our purpose). Let $G$ be a topological group and $H \leq G$ be an open subgroup (since we are dealing with separable groups, the index of $H$ in $G$ is at most countable). Let $\alpha: H \to \mathrm{Aut}(X, \mu)$ be a Boolean p.m.p. action of $H$ on a standard probability space $(X, \mu)$. Let $\mathrm{MAlg}(X, \mu)$ be the measure algebra of $(X, \mu)$. Define

$$ \mathfrak{A} := \{ f: G \to \mathrm{MAlg}(X, \mu) : \forall g \in G, \forall h \in H, f(gh) = \alpha(h^{-1})f(g) \}. $$

Define two binary operations $\triangle$ and $\cap$ on $\mathfrak{A}$ as follows: for all $f_1, f_2 \in \mathfrak{A}$

$$ f_1 \triangle f_2 : g \mapsto f_1(g) \triangle f_2(g) $$

$$ f_1 \cap f_2 : g \mapsto f_1(g) \cap f_2(g). $$

Let $0_{\mathfrak{A}}$ and $1_{\mathfrak{A}}$ be the functions which are identically equal to $\emptyset$ and $X$ respectively. Notice that for every $f \in \mathfrak{A}$, the map $g \mapsto \mu(f(g))$ is constant on each left $H$-coset. Denote by $\mu(f(q))$ its value on the coset $q \in G/H$. Define the following map $\tilde{\mu}: \mathfrak{A} \to [0, +\infty]$ by

$$ \tilde{\mu}(f) := \sum_{q \in G/H} \mu(f(q)). $$

It is left to the cautious reader to check that $(\mathfrak{A}, \triangle, \cap, 0_{\mathfrak{A}}, 1_{\mathfrak{A}}, \tilde{\mu})$ is a measure algebra, which is separable in its measure-algebra topology. Therefore, there exists a standard probability space $(Y, \nu)$ such that $\mathfrak{A} = \mathrm{MAlg}(Y, \nu)$. The continuous homomorphism $G \to \mathrm{Aut}(\mathfrak{A})$ given by the $G$-action by precomposition on $\mathfrak{A}$ thus provides a Boolean p.m.p. action $G \to \mathrm{Aut}(Y, \nu)$ which is denoted by $\mathrm{Ind}_H^G(\alpha)$ and is called the *Boolean p.m.p. $G$-action induced by $\alpha$*.

The proof of Tsankov's classification of non-singular actions of Roelcke-precompact non-Archimedean Polish groups [28, Thm. 3.4] adapts without effort to the context of dissociated groups. Due to the substantial similarities, we omit the proof of the following result.
**THEOREM 3.2.** — Let $G \leq \mathrm{Sym}(\Omega)$ be a closed permutation group. Assume that $G$ is dissociated. Let $(X, \nu)$ be a standard probability space and $\alpha: G \to \mathrm{Aut}(X, [\nu])$ be a non-singular action. Then $\alpha$ is isomorphic to a countable union $\bigsqcup_{i \in I} \mathrm{Ind}_{G_{\{A_i\}}}^G (\alpha_i)$, where for every $i \in I$, $A_i$ is a finite subset of $\Omega$ and $\alpha_i$ is a Boolean p.m.p. action of $G_{\{A_i\}}$.

## 3.2 Stabilizer rigidity

This section goes over the main result of [16], where are studied Borel p.m.p. actions of closed permutation groups through the perspective of the stabilizers associated to these actions. Here, by a *Borel p.m.p. action* of a Polish group *G* we mean a Borel action *G* × *X* → *X* on a standard Borel space *X* together with a Borel *G*-invariant probability measure *μ*. These are sometimes called spatial p.m.p. actions to emphasize the difference between Boolean p.m.p. actions. To differentiate them, Borel p.m.p. actions will be denoted by *G* ≁ (*X*, *μ*) whereas Boolean p.m.p. actions by *G* → Aut(*X*, *μ*). A Borel p.m.p. action *G* ≁ (*X*, *μ*) always yields a Boolean p.m.p. action *G* → Aut(*X*, *μ*) (continuity is automatic) and we will say that the Borel p.m.p. action *G* ≁ (*X*, *μ*) is dissociated if the associated Boolean p.m.p. action is. A Borel p.m.p. action *G* ≁ (*X*, *μ*) is:

* *essentially transitive* if there exists a *G*-orbit *O* ⊆ *X* such that $\mu(O) = 1$ (recall that orbits are Borel [19, Thm. 15.14]).

* *essentially free* if the set $\{x \in X : \forall g \in G \setminus \{1_G\}, g \cdot x \neq x\}$ is conull.

We keep the results stated as they are in [16], thus we will need to recall one definition before stating the result. A subgroup $G \leq \mathrm{Sym}(\Omega)$ is primitive if it acts transitively on $\Omega$ and there are no $G$-invariant equivalence relation on $\Omega$ apart from equality and $\Omega \times \Omega$. One can easily check that if $G$ has no algebraicity and weakly eliminates imaginaries, then $G$ is primitive (Corollary 3.7 in [16]). We can now state the main result of [16], which is a permutation group variant on Stuck-Zimmer’s Theorem for locally compact groups.

**THEOREM 3.3 (Theorem 1.4 of [16]).** — Let $G < \mathrm{Sym}(\Omega)$ be a proper, closed permutation group. If $G$ has no algebraicity and is primitive, then for every dissociated Borel p.m.p. ergodic action $G \curvearrowleft (X, \mu)$, the following hold:

* either $\mathrm{Stab}(x) \curvearrowleft \Omega$ has a fixed point for $\mu$-almost every $x \in X$ and in this case $G \curvearrowleft (X, \mu)$ is essentially free,

* or $\mathrm{Stab}(x) \curvearrowleft \Omega$ has no fixed point for $\mu$-almost every $x \in X$ and in this case $G \curvearrowleft (X, \mu)$ is essentially transitive.

For dissociated groups, we therefore have the following striking dichotomy, reminiscent of Stuck-Zimmer’s Theorem for higher rank simple Lie group [27].
THEOREM 3.4. — Let G be a proper, transitive, closed permutation group. If G is dissociated, then every Borel p.m.p. ergodic action of G is either essentially transitive or essentially free.

Notice that these results concern Borel p.m.p. actions. Indeed, the notions of essential freeness and essential transitivity make no sense for Boolean p.m.p. action. In fact, there even exists two Borel p.m.p. actions of Sym(Ω), one being essentially free, the other one being essentially transitive, whose associated Boolean p.m.p. actions are Booleanly isomorphic [15, Sec. 4.3].

## 3.3 Exchangeable processes

Let G be a closed permutation group on Ω and let (Z, ζ) be a standard probability space. The Bernoulli shift over G with base space (Z, ζ) is the Borel p.m.p. action of G on (Z, ζ)Ω defined for every g ∈ G and (z_ω)_{ω∈Ω} ∈ ZΩ by

$$g \cdot (z_{\omega})_{\omega \in \Omega} = (z_{g^{-1}(\omega)})_{\omega \in \Omega}.$$

LEMMA 3.5. — Let G ≤ Sym(Ω) be a closed permutation group without algebraicity and let (Z, ζ) be a standard probability space. Then the Bernoulli shift over G with base space (Z, ζ) is dissociated.

Proof. For every subset A ⊆ Ω, let proj_A: Z^Ω → Z^A be the restriction map and let σ Understanding the σ-algebra generated by proj_A.

CLAIM 1. — For every finite subset A ⊆ Ω, the σ-algebra F_A of G_A-invariant subsets coincides (up to null sets) with σ Understanding the σ-algebra generated by proj_A.

Proof of the claim. Fix A ⊆ Ω finite. For every Borel subset Y of Z^A, it is clear that proj_A^{-1}(Y) is G_A-invariant. So the σ-algebra generated by proj_A is contained in F_A. For the converse, fix Y ∈ F_A. Fix H a countable dense subgroup of G and let Y_0 ⊆ Y be a subset of full measure such that for every h ∈ H, the set h · Y_0 is equal to Y_0 (here this is a true equality, not only up to null sets). For every x ∈ Z^A, define

$$Y_0^x := \{\text{proj}_{A^c}((z_n)_{n \ge 0}): (z_n)_{n \ge 0} \in Y_0 \text{ and } (z_a)_{a \in A} = x\}.$$

Then Y_0^x is invariant under H_A for every x ∈ Z^A. Now, H_A remains dense in G_A hence they have the same orbits on Ω \ A. Since G has no algebraicity, all of these orbits are infinite. This implies that H_A acts ergodically on (Z, ζ)^Ω \ A (see e.g. [20, Prop. 2.1] for a proof). Thus, ζ^Ω \ A(Y_0^x) ∈ {0, 1} for every z ∈ Z. If E denotes the set of x ∈ Z^A such that ζ^Ω \ A(Y_0^x) = 1, then Y_0Δproj_A^{-1}(E) has measure 0. This shows the second inclusion. □claim

To prove dissociation of the Bernoulli shift, take A, B ⊆ Ω finite. By the claim, F_A, F_B and F_{A∩B} coincide (up to null sets) respectively with the σ-algebras generated
by $\mathrm{proj}_A$, $\mathrm{proj}_B$ and $\mathrm{proj}_{A\cap B}$. But it is clear that $\sigma(\mathrm{proj}_A)$ and $\sigma(\mathrm{proj}_B)$ are independent conditionally on $\sigma(\mathrm{proj}_{A\cap B})$, which shows that the Bernoulli shift $G \curvearrowright (Z, \zeta)^\Omega$ is dissociated. $\square$

As a direct consequence of Lemma 3.5, we obtain the following corollary.

**COROLLARY 3.6.** — Let $G \le \mathrm{Sym}(\Omega)$ be a transitive, closed permutation group without algebraicity and let $Z$ be a standard Borel space. Let $\mu$ be a $G$-invariant, Borel, ergodic probability measure on $Z^\Omega$. Then the Borel p.m.p. action $G \curvearrowright (Z^\Omega, \mu)$ is dissociated if and only if there exists a Borel probability measure $\lambda$ on $Z$ such that $\mu = \lambda^{\otimes\Omega}$.

Consequently, de Finetti's theorem [13] can be considered as the first historical instance of dissociation, for product actions of the group $\mathrm{Sym}(\Omega)$. Indeed, by Corollary 3.6, de Finetti's theorem is equivalent to saying that for every $\mathrm{Sym}(\Omega)$-invariant ergodic probability measure on $Z^\Omega$, the Borel p.m.p. action $\mathrm{Sym}(\Omega) \curvearrowright (Z^\Omega, \mu)$ is dissociated. A similar result has been obtained by Ryll-Nardzewski for the group $\mathrm{Aut}(\mathbb{Q}, <)$ [25]. It was later generalized to a large class of closed permutation groups, see [17] and [28].

# 4 How to obtain dissociation

## 4.1 Via approximating sequences

In this section, we recall a method to obtain dissociation that we introduced [2] in the context of the automorphism group of the rational Urysohn space.

Let $\Omega$ be a countably infinite set and $G \le \mathrm{Sym}(\Omega)$ be a closed permutation group. Let $\Omega' \subseteq \Omega$ be a countably infinite subset and $H \le \mathrm{Sym}(\Omega')$ be a closed permutation group. An *extension embedding* is an embedding of topological groups $\theta: H \hookrightarrow G$ such that for every $h \in H$ and $x \in \Omega'$, we have $\theta(h)(x) = h(x)$. An *approximating sequence* for $G$ is the data of an increasing sequence $\Omega_0 \subseteq \Omega_1 \subseteq \dots \subseteq \Omega$ of infinite subsets with $\Omega = \bigcup_{n \ge 0} \Omega_n$, a sequence of closed permutation groups $G_n \le \mathrm{Sym}(\Omega_n)$ and a sequence of extension embeddings $\theta_n: G_n \hookrightarrow G_{n+1}$ such that $\bigcup_{n \ge 0} \iota_n(G_n)$ is a dense subgroup of $G$. Here, $\iota_n: G_n \hookrightarrow \mathrm{Sym}(\Omega)$ denotes the natural extension embedding obtained by composing the extension embeddings $\theta_n$ (see [2, Sec. 3.2] for a precise definition of $\iota_n$).

**THEOREM 4.1 ([2, Thm. 3.12])**. — Let $G \le \mathrm{Sym}(\Omega)$ be a closed permutation group with an approximating sequence $G_0 \hookrightarrow G_1 \hookrightarrow \dots \hookrightarrow G$. Assume that for every $n \ge 0$, the closed permutation group $G_n \le \mathrm{Sym}(\Omega_n)$ is dissociated. Then the closed permutation group $G \le \mathrm{Sym}(\Omega)$ is dissociated.

## 4.2 Via an amalgamation property

### 4.2.1 Tail subspaces and dissociation

In this section, we introduce for every finite subset $A \subseteq \Omega$ a tail subspace $\mathcal{T}_A$ associated with a unitary representation of a closed permutation group $G \le \mathrm{Sym}(\Omega)$. Our definition is inspired by the classical definition of the tail $\sigma$-algebra for an exchangeable
random process, see Remark 4.7. The key idea in this section is to obtain dissociation by proving that the tail space $\mathcal{T}_A$ coincides with the space of $G_A$-invariant vectors.

**DEFINITION 4.2.** — Let $G \leq \mathrm{Sym}(\Omega)$ be a closed permutation group and $A \subseteq \Omega$ a finite subset. The tail subspace $\mathcal{T}_A$ associated with a unitary representation $\pi: G \to \mathcal{U}(\mathcal{H})$ is defined by

$$ \mathcal{T}_A := \bigcap_{\substack{C \subseteq \Omega \\ A \subseteq C}} \mathcal{H}_C. $$

Note that by definition, the space $\mathcal{H}_A$ of $G_A$-invariant vectors is always a subspace of the tail subspace $\mathcal{T}_A$.

In the sequel, we will need a finer notion of conditional orthogonality than the one we present in the introduction. Let $\mathcal{H}$ be a Hilbert space. For any closed subspace $\mathcal{G} \subseteq \mathcal{H}$, we denote by $p_{\mathcal{G}} \in \mathcal{B}(\mathcal{H})$ the orthogonal projection onto $\mathcal{G}$.

**DEFINITION 4.3.** — Let $\mathcal{G}, \mathcal{K}, \mathcal{L}$ be three closed subspaces of $\mathcal{H}$. We say that $\mathcal{G}$ and $\mathcal{L}$ are orthogonal conditionally on $\mathcal{K}$ if one of the following equivalent assertions is satisfied:

* $p_{\mathcal{K}^{\perp}}\mathcal{G} \perp p_{\mathcal{K}^{\perp}}\mathcal{L}$.

* $p_{\mathcal{G}}p_{\mathcal{K}}p_{\mathcal{L}} = p_{\mathcal{G}}p_{\mathcal{L}}$.

* $p_{\mathcal{L}}p_{\mathcal{K}}p_{\mathcal{G}} = p_{\mathcal{L}}p_{\mathcal{G}}$.

If this holds, we write $\mathcal{G} \perp_{\mathcal{K}} \mathcal{L}$.

Notice that if $\mathcal{K} \subseteq \mathcal{G} \cap \mathcal{L}$, then we recover the definition given in the introduction: $\mathcal{G} \perp_{\mathcal{K}} \mathcal{L}$ is equivalent to $p_{\mathcal{G}}p_{\mathcal{L}} = p_{\mathcal{L}}p_{\mathcal{G}} = p_{\mathcal{K}}$. In this case, we readily have that $\mathcal{K} = \mathcal{G} \cap \mathcal{L}$.

The following theorem provides a weak form of dissociation from which we will extract a characterization of proper dissociation.

**THEOREM 4.4.** — Let $G \leq \mathrm{Sym}(\Omega)$ be a closed subgroup without algebraicity. Let $\pi: G \to \mathcal{U}(\mathcal{H})$ be a unitary representation. Then $\mathcal{H}_A \perp_{\mathcal{T}_A \cap \mathcal{B}} \mathcal{H}_B$ for all finite subsets $A, B \subseteq \Omega$.

*Proof.* Fix $A, B \subseteq \Omega$ finite and let $\mathcal{G} := \mathcal{H}_A, \mathcal{K} = \mathcal{T}_A \cap \mathcal{B}$ and $\mathcal{L} = \mathcal{H}_B$. Let us prove that $p_{\mathcal{G}}p_{\mathcal{K}}p_{\mathcal{L}} = p_{\mathcal{G}}p_{\mathcal{L}}$. Define $\mathcal{K}' = \mathrm{Vect}(\mathcal{G}, \mathcal{K})$. Notice that if $p_{\mathcal{K}'}p_{\mathcal{L}} = p_{\mathcal{K}}p_{\mathcal{L}}$, then $\mathcal{G} \perp_{\mathcal{K}'} \mathcal{L}$. Indeed

$$ \begin{aligned} (p_{\mathcal{K}'}p_{\mathcal{L}} = p_{\mathcal{K}}p_{\mathcal{L}}) &\Rightarrow (p_{\mathcal{G}}p_{\mathcal{K}}p_{\mathcal{L}} = p_{\mathcal{G}}p_{\mathcal{K}'}p_{\mathcal{L}} = p_{\mathcal{G}}p_{\mathcal{L}}) \\ &\Rightarrow \mathcal{G} \perp_{\mathcal{K}} \mathcal{L}. \end{aligned} $$

where the rightmost equality holds because $\mathcal{G} \subseteq \mathcal{K}'$. So let us prove that for all $\xi \in \mathcal{L}$,

$$ p_{\mathcal{K}'}\xi = p_{\mathcal{K}}\xi. \qquad (3) $$
Since $\mathcal{K}$ is a subspace of $\mathcal{K}'$, notice that (3) is equivalent to $\|p_{\mathcal{K}'\xi}\| = \|p_{\mathcal{K}\xi}\|$. For $\xi \in \mathcal{L}$, the inequality $\|p_{\mathcal{K}\xi}\| \le \|p_{\mathcal{K}'\xi}\|$ is straightforward, so we focus on the converse inequality.

Fix a strictly decreasing sequence $C_0 \supseteq C_1 \supseteq \dots$ of cofinite subsets of $\Omega$ whose intersection $\bigcap_{n \ge 0} C_n$ is $A \cap B$. For every $n \ge 0$, let $\mathcal{K}_n := \mathcal{H}_{C_n}$. Then $(\mathcal{K}_n)_{n \ge 0}$ forms a decreasing sequence of closed subspaces. Moreover, the intersection $\bigcap_{n \ge 0} \mathcal{K}_n$ is equal to $\mathcal{K}$ and $p_{\mathcal{K}_n} \to p_{\mathcal{K}}$ in the strong operator topology. Since $G$ has no algebraicity, Neumann's Lemma [5, Thm. 1] gives for every $n \ge 0$ some element $g_n \in G_B$ such that $g_n(A)$ is contained in $C_n$. In particular, $\pi(g_n)G = \pi(g_n)\mathcal{H}_A = \mathcal{H}_{g_n(A)}$ is contained in $\mathcal{K}_n$.

For all $n \ge 0$, define $\mathcal{K}'_n := \pi(g_n)\mathcal{K}'$. Notice that

$$ \begin{aligned} \mathcal{K}'_n &= \overline{\mathrm{Vect}(\pi(g_n)\mathcal{G}, \pi(g_n)\mathcal{K})} \\ &= \overline{\mathrm{Vect}(\mathcal{H}_{g_n(A)}, \mathcal{K})}, \end{aligned} $$

where the last equality holds since $\mathcal{K}$ is a $G_{A \cap B}$-invariant subspace and $g_n \in G_B \le G_{A \cap B}$. Therefore, $\mathcal{K}'_n$ is a subspace of $\mathcal{K}_n$. For all $\xi \in \mathcal{L}$ and $n \ge 0$, we have

$$ \|p_{\mathcal{K}'\xi}\| = \|\pi(g_n)p_{\mathcal{K}'\xi}\| = \|p_{\mathcal{K}'_n}\pi(g_n)\xi\| = \|p_{\mathcal{K}'_n}\xi\| \le \|p_{\mathcal{K}_n}\xi\| \to \|p_{\mathcal{K}\xi}\|, $$

which concludes the proof. $\square$

Therefore, if $\mathcal{T}_A = \mathcal{H}_A$ for every finite $A \subseteq \Omega$, then the unitary representation is dissociated. We can actually prove the converse of this statement, which will allow us to get a nice equivalence in Corollary 4.6.

**LEMMA 4.5.** — Let $G \le \mathrm{Sym}(\Omega)$ be a closed permutation group. Let $\pi: G \to \mathcal{U}(\mathcal{H})$ be a dissociated unitary representation. Then for every $A \subseteq \Omega$ finite, we have $\mathcal{T}_A = \mathcal{H}_A$.

*Proof.* Let $A \subseteq \Omega$ be a finite subset. Let $(C_n)$ be a decreasing sequence of cofinite subsets of $\Omega$ such that $\bigcap_{n \in \mathbb{N}} C_n = A$. Then $\bigcap_{n \in \mathbb{N}} \mathcal{H}_{C_n} = \mathcal{T}_A$ and $p_{C_n} \to p_{\mathcal{T}_A}$ in the strong operator topology.

Define $B_n := A \cup (\Omega \setminus C_n)$ for every $n \in \mathbb{N}$. Then $(B_n)$ is an increasing sequence of finite sets whose union is $\Omega$. By continuity of $\pi$, we get that $p_{B_n} \to \mathrm{id}$ in the strong operator topology. Moreover, $B_n \cap C_n = A$ for every $n \in \mathbb{N}$. Thus, using dissociation, we have $p_{C_n}p_{B_n} = p_A$ for every $n \in \mathbb{N}$. Note that $p_{C_n}p_{B_n} \to p_{\mathcal{T}_A}$ in the strong operator topology (these operators being projections, they are uniformly bounded). At the limit, we get $p_{\mathcal{T}_A} = p_A$ and $\mathcal{T}_A = \mathcal{H}_A$. $\square$

**COROLLARY 4.6.** — Let $G \le \mathrm{Sym}(\Omega)$ be a closed permutation group without algebraicity. Let $\pi: G \to \mathcal{U}(\mathcal{H})$ be a unitary representation. Then $\pi$ is dissociated if and only if for every finite subset $A \subseteq \Omega$, we have $\mathcal{H}_A = \mathcal{T}_A$.

**REMARK 4.7.** — Every result obtained in that section has an analogue in the context of Boolean p.m.p. actions. Let $G$ be a closed subgroup of $\mathrm{Sym}(\Omega)$ and $G \to \mathrm{Aut}(X, \mu)$ be a Boolean p.m.p. action. For every finite subset $A \subseteq \Omega$, define the following $\sigma$-algebra
$$ \mathcal{T}_A := \bigcap_{C \subseteq \Omega \text{ cofinite}} \sigma(\mathcal{F}_{A \cup B} : B \subseteq C \text{ finite}). $$

When $A$ is the empty set, we call $\mathcal{T}_\emptyset$ the tail $\sigma$-algebra of the action. One checks that the proofs of this section can be adapted in a straightforward way to get analogous results in that context, leading to the following result: if $G$ has no algebraicity, then the Boolean p.m.p. action $\alpha$ is dissociated if and only if for every finite subset $A \subseteq \Omega$, we have $\mathcal{F}_A = \mathcal{T}_A$.

In particular, for a Boolean p.m.p. ergodic, dissociated action, the tail $\sigma$-algebra is trivial. Notice also that for $\mathrm{Sym}(\Omega)$ acting on a product probability space $(Z, \zeta)^\Omega$, triviality of the tail $\sigma$-algebra and of the $\sigma$-algebra of invariant subsets (a.k.a. exchangeable subsets in that context) correspond respectively to Kolmogorov and Hewitt-Savage 0-1 laws.

### 4.2.2 Strong cofinite amalgamation over countable subsets

In this section, we explain a model theoretic condition, which when satisfied by a countable relational structure $\mathbf{M}$, implies that the tail subspace $\mathcal{T}_A$ of every unitary representation of $\mathrm{Aut}(\mathbf{M})$ coincides with the subspace $\mathcal{H}_A$.

Let $\mathbf{M}$ be a countable relation structure and let $\sigma\mathrm{Age}(\mathbf{M})$ be the set of isomorphism classes of countable (that is, finite or infinite) substructures of $\mathbf{M}$.

**DEFINITION 4.8.** — We say that $\mathbf{M}$ has the strong cofinite amalgamation property over countable subsets (abbrev. $\sigma$-SAP) if, whenever $\mathbf{A}, \mathbf{B}_1, \mathbf{B}_2 \in \sigma\mathrm{Age}(\mathbf{M})$ and $f_i: \mathbf{A} \to \mathbf{B}_i$ are embeddings with $|\mathrm{dom}(\mathbf{B}_i) \setminus \mathrm{dom}(f_i(\mathbf{A}))| < +\infty$, there exist $\mathbf{C} \in \sigma\mathrm{Age}(\mathbf{M})$ and embeddings $g_i: \mathbf{B}_i \to \mathbf{C}$ so that the following hold:

<figure><img src="image_2.png" /></figure>

2. if there exist $b_1 \in \mathrm{dom}(\mathbf{B}_1)$ and $b_2 \in \mathrm{dom}(\mathbf{B}_2)$ satisfying $g_1(b_1) = g_2(b_2)$, then there exists $a \in \mathrm{dom}(\mathbf{A})$ satisfying $f_1(a) = b_1$ and $f_2(a) = b_2$.

**REMARK 4.9.** — For integral metric spaces, there is a canonical amalgamation over a common non-empty subspace (of any cardinality!). Therefore, several natural Fraïssé limits built out of classes of finite integral metric spaces satisfy $\sigma$-SAP. This is for instance the case for the integral Urysohn space $\mathbb{ZU}$, but other examples will be discussed in Sections 5.1.1 and 5.1.2.

However, it turns out that the rational Urysohn space $\mathbb{QU}$ does not satisfy $\sigma$-SAP. Indeed, there exists an infinite subset $A \subseteq \mathrm{dom}(\mathbb{QU})$ and a point $x \in \mathrm{dom}(\mathbb{QU}) \setminus A$ such that $d(A, x) = 0$. Let $\mathbf{A}$ be the structure generated by $A$ and $\mathbf{B}_1 = \mathbf{B}_2$ be the
structure generated by $A \sqcup \{x\}$. Then any amalgamation $\mathbf{C}$ would identify $\mathbf{B}_1 \setminus A$ and $\mathbf{B}_2 \setminus A$.

**REMARK 4.10.** — $\sigma$-SAP is closely related to *uniform non-algebraicity* introduced by Tsankov in [28] to obtain de Finetti's style results.

**PROPOSITION 4.11.** — Let $\mathbf{M}$ be a countable relational structure. Assume that $\mathbf{M}$ satisfies $\sigma$-SAP and that $\mathrm{Aut}(\mathbf{M})$ weakly eliminates imaginaries. Then for every unitary representation $\pi: \mathrm{Aut}(\mathbf{M}) \to \mathcal{U}(\mathcal{H})$ and every finite subset $A \subseteq \mathrm{dom}(\mathbf{M})$, the subspaces $\mathcal{H}_A$ and $\mathcal{T}_A$ coincide.

*Proof.* Let $G := \mathrm{Aut}(\mathbf{M})$ and $\Omega := \mathrm{dom}(\mathbf{M})$. Fix a unitary representation $\pi: G \to \mathcal{U}(\mathcal{H})$ and a finite subset $A \subseteq \Omega$. The inclusion $\mathcal{H}_A \subseteq \mathcal{T}_A$ always holds, so let us prove the reverse inclusion. We start with the following claim.

**CLAIM 1.** — $\mathcal{T}_A = \overline{\bigcup_{\substack{B \subseteq \Omega \\ \mathrm{finite}}} \mathcal{T}_A \cap \mathcal{H}_B}$.

*Proof of the claim.* Recall that the union of the $\mathcal{H}_B$'s for $B \subseteq \Omega$ finite is dense in $\mathcal{H}$ [2, Lem. 2.6]. Let $\xi \in \mathcal{T}_A$. Then there exists a sequence of finite subsets $(B_n)_{n \ge 0}$ and a sequence of vectors $(\xi_n)_{n \ge 0}$ such that $\xi_n \in \mathcal{H}_{B_n}$ and $\xi_n \to \xi$. Since $\mathcal{H}_{B_n} \subseteq \mathcal{H}_{A \cup B_n}$, we may assume that $A \subseteq B_n$ for every $n \ge 0$. Define $\eta_n := p_{\mathcal{T}_A} \xi_n$. Clearly, $\eta_n \in \mathcal{T}_A$ and $\eta_n \to \xi$. So it suffices to prove that $\eta_n \in \mathcal{H}_{B_n}$. But for $g \in G_{B_n}$, one has $\pi(g)\mathcal{T}_A = \mathcal{T}_A$ (because $G_{B_n} \subseteq G_A$) and therefore

$$ \pi(g)\eta_n = p_{\pi(g)\mathcal{T}_A}\pi(g)\xi_n = p_{\mathcal{T}_A}\xi_n = \eta_n. \quad \square_{\text{claim}} $$

**CLAIM 2.** — For every finite subset $B \subseteq \Omega$, we have $\mathcal{T}_A \cap \mathcal{H}_B \subseteq \mathcal{H}_A$.

*Proof of the claim.* Let $\xi \in \mathcal{T}_A \cap \mathcal{H}_B$. Since $\mathcal{H}_B \subseteq \mathcal{H}_{A \cup B}$, we may assume that $A \subseteq B$. Let $C$ be a cofinite subset disjoint from $B$. Since $\xi$ belongs to $\mathcal{T}_A \subseteq \mathcal{H}_C$, there exists a sequence $(D_n)_{n \ge 0}$ of finite subsets of $C$ and vectors $\xi_n \in \mathcal{H}_{A \cup D_n}$ such that $\xi_n \to \xi$. Let $\mathbf{D}$ be the structure generated by $A \sqcup \bigcup_{n \ge 0} D_n$. Let $\mathbf{E}$ be the structure generated by $B \sqcup \bigcup_{n \ge 0} D_n$. We apply $\sigma$-SAP with $f_1: \mathbf{D} \hookrightarrow \mathbf{E}$ and $f_2: \mathbf{D} \hookrightarrow \mathbf{E}$ both being the embedding induced by the inclusion map $\mathrm{dom}(\mathbf{D}) \subset \mathrm{dom}(\mathbf{E})$. This provides a substructure $\mathbf{F} \subseteq \mathbf{M}$ together with embeddings $g_i: \mathbf{E} \hookrightarrow \mathbf{F}$ so that the two conditions of $\sigma$-SAP hold. Define the following subsets of $\mathrm{dom}(\mathbf{F})$:

$$ \begin{align*} A' &= g_1 \circ f_1(A) = g_2 \circ f_2(A), \\ D'_n &= g_1 \circ f_1(D_n) = g_2 \circ f_2(D_n), \\ B_1 &= g_1(B), \\ B_2 &= g_2(B). \end{align*} $$

By the second condition of $\sigma$-SAP, notice that $B_1 \cap B_2 = A'$. For every $n \ge 0$, $g_2 \circ g_1^{-1}$ induces an isomorphism between the substructure generated by $B_1 \cup D'_n$ and
the substructure generated by $B_2 \cup D'_n$. Moreover, this isomorphism is the identity on $A' \cup D'_n$. Recalling that the amalgamation property implies ultrahomogeneity of $M$, there exists an element $h_n \in G_{A' \cup D'_n}$ such that $h_n(B_1) = B_2$. Fix an element $h \in G$ which extends the partial isomorphism $g_1$ from $B$ to $B_1$. Let $\xi' = \pi(h)\xi$ and $\xi'_n = \pi(h)\xi_n$. Then, for every $n \ge 0$, we have $\xi'_n \in \pi(h)\mathcal{H}_{A \cup D_n} = \mathcal{H}_{h(A) \cup h(D_n)} = \mathcal{H}_{A' \cup D'_n}$ hence $\pi(h_n)\xi'_n = \xi'_n$. Moreover,

$$ \begin{aligned} \|\xi' - \pi(h_n)\xi'\| &\le \|\xi' - \xi'_n\| + \|\xi'_n - \pi(h_n)\xi'_n\| + \|\pi(h_n)\xi'_n - \pi(h_n)\xi'\| \\ &\le 2\|\xi - \xi_n\| \\ &\to 0. \end{aligned} $$

The vector $\xi'$ belongs to $\mathcal{H}_{B_1}$ since $\xi$ belongs to $\mathcal{H}_B$. But $\pi(h_n)\xi'$ belongs to $\mathcal{H}_{h_n(B_1)} = \mathcal{H}_{B_2}$ for every $n \ge 0$ and thus we get that $\xi'$ belongs to $\mathcal{H}_{B_2}$. Finally, $\xi'$ belongs to $\mathcal{H}_{B_1} \cap \mathcal{H}_{B_2}$, which coincides with the space of $\langle G_{B_1}, G_{B_2} \rangle$-invariant vectors i.e., by Lemma 1.6, with $\mathcal{H}_{B_1 \cap B_2} = \mathcal{H}_{A'}$. Thus, $\xi$ belongs to $\mathcal{H}_A$ and this finishes the proof of the claim. $\square$claim

Combining Claim 1 and 2, we readily get that $\mathcal{T}_A \subseteq \mathcal{H}_A$, which finishes the proof. $\square$

As a consequence of the results in this section, we obtain the following theorem.

**THEOREM 4.12.** — Let $M$ be a countable ultrahomogeneous relational structure. If $M$ weakly eliminates imaginaries and satisfies the strong cofinite amalgamation property over countable subsets, then Aut($M$) is dissociated.

### 4.2.3 $\aleph_0$-categoricity and $\sigma$-SAP

The goal of the section is now to give a first application of Theorem 4.12 to $\aleph_0$-categorical ultrahomogeneous structures in order to recover dissociation for oligomorphic groups with weak elimination of imaginaries and no algebraicity (Theorem 3.4 of [17]).

Let $M$ be a countable relational structure. We say that two tuples $\bar{x} = (x_1, \dots, x_n)$ and $\bar{y} = (y_1, \dots, y_n)$ on $M$ have the same isomorphism type if the map $x_i \mapsto y_i$ extends to an isomorphism between the substructures induced by $\bar{x}$ and $\bar{y}$. This defines an equivalence relation on tuples and we denote by $\mathrm{tp}(\bar{x})$ the equivalence class of $\bar{x}$. A $k$-type over $M$ is simply the type of a $k$-tuple in $M$. A structure $M$ is homogeneous if for all (finite) tuple $\bar{x}$ and $\bar{y}$ with $\mathrm{tp}(\bar{x}) = \mathrm{tp}(\bar{y})$, there exists $g \in \mathrm{Aut}(M)$ such that $g(\bar{x}) = \bar{y}$. The tree of isomorphism types of $M$ is a rooted tree defined as follows:

* its vertex set is the collection of all $k$-types over $M$, for every $k \ge 0$,

* for every $k \ge 0$, there is an edge between a $(k+1)$-type $t$ and a $k$-type $s$ if and only if there exists $(x_1, \dots, x_{k+1}) \in M^{k+1}$ such that $t = \mathrm{tp}(x_1, \dots, x_{k+1})$ and $s = \mathrm{tp}(x_1, \dots, x_k)$.

The following result is a direct reformulation of Ryll-Nardzewski's theorem.
LEMMA 4.13. — *The boundary of the tree of isomorphism types of $\mathbf{M}$ is compact if and only if $\mathbf{M}$ is $\aleph_0$-categorical.*

LEMMA 4.14. — Let $\mathbf{M}$ be a countable relational ultrahomogeneous structure which has the strong amalgamation property. If $\mathbf{M}$ is $\aleph_0$-categorical, then $\mathbf{M}$ has the strong amalgamation property over arbitrary subsets.

*Proof.* Let $\mathbf{A}, \mathbf{B}_1, \mathbf{B}_2 \in \sigma\text{Age}(\mathbf{M})$ and embeddings $f_i: \mathbf{A} \to \mathbf{B}_i$ satisfying $\text{dom}(f_i(\mathbf{A})) \subset \text{dom}(f_i(\mathbf{A}))|_{\text{dom}(f_i(\mathbf{A}))} < +\infty$. If $\mathbf{A}$ is finite, there is nothing to prove since $\mathbf{M}$ has the strong amalgamation property. So let us assume that $\mathbf{A}$ is infinite. Fix an enumeration $a_1, \dots, a_n, \dots$ of $\text{dom}(\mathbf{A})$ and let $\mathbf{A}_n$ be the structure generated by $\{a_1, \dots, a_n\}$. Let $\bar{x} = (x_1, \dots, x_k)$ be an enumeration of $\text{dom}(\mathbf{B}_1) \setminus \text{dom}(f_1(\mathbf{A}))$ and $\bar{y} = (y_1, \dots, y_l)$ be an enumeration of $\text{dom}(\mathbf{B}_2) \setminus \text{dom}(f_2(\mathbf{A}))$. Let $\mathbf{C}_n$ be an amalgamation of $\mathbf{B}_1 \setminus f_1(\mathbf{A}) \setminus \mathbf{A}_n)$ and $\mathbf{B}_2 \setminus f_2(\mathbf{A}) \setminus \mathbf{A}_n)$ over $\mathbf{A}_n$ as in Definition 4.8. We identify the domain of $\mathbf{C}_n$ with the union of the domains of $\mathbf{B}_1 \setminus f_1(\mathbf{A}) \setminus \mathbf{A}_n)$ and $\mathbf{B}_2 \setminus f_2(\mathbf{A}) \setminus \mathbf{A}_n)$, and call $t_n$ the type induced by $\mathbf{C}_n$ on $(\bar{x}, \bar{y}, \mathbf{A}_n)$. Let $\xi_n$ be any element of the boundary of the tree of types extending $t_n$. By compactness, $(\xi_n)$ admits a subsequence converging to some point $\xi$. Any infinite tuple $C = (c_1, c_2, \dots) \in \mathbf{M}^\mathbb{N}$ satisfying $(\text{tp}(c_1, \dots, c_n))_{n \ge 0} = \xi$ is as wanted. $\square$

As a consequence, we recover the result [17, Thm. 3.4] of the second author and Tsankov, whose original proof uses the classification of unitary representations for oligomorphic groups due to Tsankov [29]. Recall that a closed permutation group $G \subseteq \text{Sym}(\Omega)$ is *oligomorphic* if for every $n \in \mathbb{N}$, the diagonal action $G \times \Omega^n$ has only finitely many orbits.

THEOREM 4.15. — Let $G \subseteq \text{Sym}(\Omega)$ be a closed subgroup. Assume that $G$ is oligomorphic, has no algebraicity, and weakly eliminates imaginaries. Then $G$ is dissociated.

*Proof.* Let $\mathbf{M}_G$ be the canonical structure associated with $G$. It is a countable relational ultrahomogeneous structure, which satisfies the strong amalgamation property. Moreover, by Ryll-Nardzewski's theorem, $\mathbf{M}_G$ is $\aleph_0$-categorical, see [7, (2.10)]. Thus by Lemma 4.14, $\mathbf{M}_G$ satisfies $\sigma$-SAP. Finally, Theorem 4.12 allows us to conclude that $\text{Aut}(\mathbf{M}_G) = G$ is dissociated. $\square$

# 5 New examples of dissociated permutation groups

## 5.1 Metrically homogeneous graphs

In this section, we consider some classes of countable metrically homogeneous connected graphs of infinite diameter for which the automorphism group satisfies the assumptions of Theorem 4.12. A metric space $(X, \mathcal{d})$ is *metrically homogeneous* if every surjective isometry between two finite subsets of $X$ extends to a surjective isometry of $X$. A graph $\Gamma$ is metrically homogeneous if the metric space $\Gamma$ equipped with the graph metric is metrically homogeneous. When $\Gamma$ is countable, we equip $\text{Aut}(\Gamma)$ with the permutation
group topology. Our interest in countable metrically homogeneous graph is twofold. First, we will illustrate in this section how to use the techniques developed in Section 4 to show that the automorphism group of some countable metrically homogeneous graphs is dissociated. Second, this section provides new examples of locally Roelcke precompact, non-Roelcke precompact, closed permutation groups that are dissociated. Indeed, Rosendal proved that the automorphism group Aut(Γ) of a countable metrically homogeneous connected graph is locally Roelcke precompact [24, Theorem 3.5], but not coarsely bounded if the graph has infinite diameter.

An integral metric space is a metric space $(X, d)$ such that $d(X \times X) \subseteq \mathbb{N}$. To such a metric space, we associate the distance-1 graph of the metric space, which is a graph whose vertex set is the metric space and where there is an edge between two points if and only if the distance between them is one. The countable metrically homogeneous graphs that we will work with in this section are all obtained as the distance-1 graph of integral metric spaces which are ultrahomogeneous (when viewed as a structure in the language of integral metric spaces). Every countable integral metric space $(X, d)$ may be considered as a countable relational structure in the language $\mathcal{L} = (R_n)_{n \ge 0}$ where each $R_n$ is a relation of arity 2 whose interpretation $R_n^X$ in $X$ is given by

$$R_n^X := \{(x, y) \in X^2 : d(x, y) = n\}.$$

Notice that $X$ is ultrahomogeneous (as a countable structure) if and only if $X$ is metrically homogeneous. For integral metric spaces, there is a canonical amalgamation over non-empty subspaces that will be useful later.

**DEFINITION 5.1.** — Let $(X_1, d_1)$ and $(X_2, d_2)$ be two integral metric spaces such that the intersection $Y := X_1 \cap X_2$ is non-empty. Assume that $d_1$ and $d_2$ coincide on $Y \times Y$. Let $X := X_1 \cup X_2$ and define a map $d: X \times X \to \mathbb{N}$, which restricts to $d_1$ on $X_1 \times X_1$, to $d_2$ on $X_2 \times X_2$ and such that for all $x_1 \in X_1 \setminus Y$ and $x_2 \in X_2 \setminus Y$,

$$d(x_1, x_2) = \min\{d(x_1, y) + d(y, x_2) : y \in Y\}.$$

Then $(X, d)$ is an integral metric space. It is called the metric amalgam of $(X_1, d_1)$ and $(X_2, d_2)$ over $Y$ and is denoted by $(X_1, d_1) *_Y (X_2, d_2)$.

### 5.1.1 Integral metric spaces with no small triangle of odd perimeter

Let $p \ge 1$ be an integer. A metric space $(X, d)$ has no triangle of odd perimeter less than $p$ if the following holds: for every $(x, y, z) \in X^3$, if $d(x, y) + d(y, z) + d(z, x)$ is odd, then $d(x, y) + d(y, z) + d(z, x) > p$.

**LEMMA 5.2.** — Let $p \ge 1$ be an integer. Let $(X_1, d_1)$ and $(X_2, d_2)$ be two integral metric spaces which contain no triangle of odd perimeter less than $p$. If $X_1 \cap X_2$ is non-empty, then the metric amalgam of $(X_1, d_1)$ and $(X_2, d_2)$ over $X_1 \cap X_2$ contains no triangle of odd perimeter less than $p$.
*Proof.* Let $X := X_1 \cup X_2$ and $Y := X_1 \cap X_2$. Fix $x_1 \in X_1 \setminus Y$, $x_2 \in X_2 \setminus Y$ and $x \in X$ such that $d(x_1, x_2) + d(x_2, x) + d(x, x_1)$ is odd. There are two cases to check.

Assume that $x \in Y$. Let $y \in Y$ be such that $d(x_1, x_2) = d(x_1, y) + d(y, x_2)$. Then $d(x_1, y) + d(y, x_2) + d(x_2, x) + d(x, x_1)$ is odd. Therefore, there is $i \in \{1, 2\}$ such that $d(x_i, y) + d(y, x) + d(x, x_i)$ is odd. But this last quantity is $> p$ since $(x_i, y, x) \in X_i \times X_i \times X_i$. Therefore,

$$d(x_1, x_2) + d(x_2, x) + d(x, x_1) = d(x_1, y) + d(y, x_2) + d(x_2, x) + d(x, x_1) \\ \geq d(x_i, y) + d(y, x) + d(x, x_i) \\ > p,$$

which is what we wanted.

Assume now that $x \in X \setminus Y$. Without loss of generality, we may assume that $x \in X_1 \setminus Y$. Let $y, z \in Y$ be such that $d(x_1, x_2) = d(x_1, y) + d(y, x_2)$ and $d(x_2, x) = d(x_2, z) + d(z, x)$. Then $d(x_1, y) + d(y, x_2) + d(x_2, z) + d(z, x) + d(x, x_1)$ is odd. Therefore, among the following three quantities

$$c_1 := d(x, x_1) + d(x_1, y) + d(y, x),$$

$$c_2 := d(x, y) + d(y, z) + d(z, x),$$

$$c_3 := d(z, y) + d(y, x_2) + d(x_2, z),$$

at least one of them is odd. But if $c_i$ is odd, then $c_i > p$. By the triangle inequality, we obtain that

$$d(x_1, x_2) + d(x_2, x) + d(x, x_1) \geq \max(c_1, c_2, c_3) > p,$$

which concludes the proof.

<figure><img src="image_3.png" /></figure>

**LEMMA 5.3.** — Let $p \geq 1$ be an integer. Then the class of finite integral metric spaces with no triangle of odd perimeter less than $p$ is a Fraïssé class satisfying the strong amalgamation property.

*Proof.* It is clear that this class satisfies the hereditary property. The joint embedding property will follow from the amalgamation property since the language is relational and there is no constant relation. Let us show the strong amalgamation property. For this, fix three finite integral metric spaces $(A, d_A)$, $(B_1, d_1)$ and $(B_2, d_2)$ with no triangle of odd perimeter less than $p$. Assume that $(B_1, d_1)$ and $(B_2, d_2)$ contain an isometric copy of $A$. Let $C$ be the set obtained by identifying both copies of $A$ in the disjoint union $B_1 \cup B_2$. We define a new metric $d$ on $C$ as follows. There are two cases to check.

If $A$ is non-empty, then let $d$ be the metric amalgam of $(B_1, d_1)$ and $(B_2, d_2)$ over $(A, d_A)$. Then $(C, d)$ is a finite integral metric space with no triangle of odd perimeter less than $p$ by Lemma 5.2.

If $A$ is empty, then let $d$ be the metric on $C$ which restricts to $d_i$ on $X_i \times X_i$ and such that for all $x_1 \in B_1$, $x_2 \in B_2$,

$$d(x_1, x_2) = \max(p, \text{diam}(B_1, d_1), \text{diam}(B_2, d_2)).$$
Then $(C, d)$ is an integral metric space, with no triangle of odd perimeter less than $p$.

In both cases, we have found a strong amalgam of $(B_1, d_1)$ and $(B_2, d_2)$ over $(A, d_A)$,
which concludes the proof. □

In the sequel, given an integer $p \ge 1$, we denote by $M_p$ the Fraïssé limit of the class of finite integral metric spaces with no triangle of odd perimeter less than $p$. Notice that when $p = 1$, $M_p$ is the integral Urysohn space $\mathbb{ZU}$.

**LEMMA 5.4.** — $M_p$ satisfies $\sigma$-SAP.

*Proof.* Since the Fraïssé class of finite integral metric spaces with no triangle of odd perimeter less than $p$ satisfies the strong amalgamation property, it remains to prove the existence of a strong cofinite amalgam over a countable metric space. The existence of such an amalgam is provided by Lemma 5.2. □

Using standard techniques (see for instance [26]) one can prove the following result. We do not provide a proof as some technicalities are involved, but we will do so in an upcoming version of this paper.

**LEMMA 5.5.** — Aut$(M_p)$ weakly eliminates imaginaries.

As a consequence of the results obtained in this section, Theorem 4.12 applies.

**THEOREM 5.6.** — Aut$(M_p)$ is dissociated.

We close this section by discussing the geometric nature of Aut$(M_p)$.

**PROPOSITION 5.7.** — Aut$(M_p)$ is locally Roelcke precompact but not coarsely bounded.

*Proof.* It is locally Roelcke precompact because of [24, Thm. 3.5]. However, it is not bounded because it acts transitively by isometries on the unbounded space $M_p$. □

Finally, we can show that this group satisfies the Howe-Moore property.

**PROPOSITION 5.8.** — Aut$(M_p)$ has the Howe-Moore property with respect to the bornology of coarsely bounded sets.

*Proof.* The proof of [24, Thm. 3.5] shows that for every $a \in \text{dom}(M_p)$, the point stabilizer Aut$(M_p)_a$ is Roelcke precompact and therefore coarsely bounded. Since Aut$(M_p)$ is dissociated, Theorem 2.12 implies that Aut$(M_p)$ has the Howe-Moore property with respect to coarsely bounded sets. □

**5.1.2 Integral metric spaces with no unit simplex**

Let $r \ge 3$ be an integer. A unit $(r-1)$ simplex in an integral metric space $(X, d)$ is a subset of $X$ consisting of $r$ points that lie mutually at distance 1.

**LEMMA 5.9.** — Let $r \ge 3$ be an integer. Let $(X_1, d_1)$ and $(X_2, d_2)$ be two integral metric spaces that contain no unit $(r-1)$-simplex. If $X_1 \cap X_2$ is non-empty, then the metric amalgam of $(X_1, d_1)$ and $(X_2, d_2)$ over $X_1 \cap X_2$ contains no unit $(r-1)$-simplex.
*Proof.* Let $Y := X_1 \cap X_2$ and let $X := (X_1, d_1) *_Y (X_2, d_2)$ be the metric amalgam. Let $F \subseteq X$ be a subset consisting of $r$ points. If $F \subseteq X_1$ or $F \subseteq X_2$, then $F$ is not a unit $(r-1)$ simplex by assumption. Else, there exist two points $x_1, x_2 \in F$ with $x_1 \in X_1 \setminus Y$ and $x_2 \in X_2 \setminus Y$. Let $y \in Y$ be such that $d(x_1, x_2) = d(x_1, y) + d(y, x_2)$. Then $d(x_1, x_2) > 1$ and thus $F$ is not a unit $(r-1)$ simplex. $\square$

**LEMMA 5.10.** — Let $n \ge 1$ be an integer. Then the class of finite integral metric spaces with no unit $(r-1)$-simplex is a Fraïssé class satisfying the strong amalgamation property.

*Proof.* The proof is identical to that of Lemma 5.3. $\square$

<figure><img src="image_4.png" /></figure>

In the sequel, given an integer $r \ge 3$, we denote by $N_r$ the Fraïssé limit of the class of finite integral metric spaces with no unit $(r-1)$-simplex.

**LEMMA 5.11.** — $N_r$ satisfies $\sigma$-SAP.

*Proof.* As in Lemma 5.4, the proof follows from the fact that the metric amalgam over a non-empty subset preserves the property of having no unit $(r-1)$-simplex, see Lemma 5.9. $\square$

<figure><img src="image_5.png" /></figure>

The same techniques as the ones mentioned in the previous section allow us to prove the following result. Again, a proof will be provided in an upcoming version of this paper.

**LEMMA 5.12.** — Aut($N_r$) weakly eliminates imaginaries.

Therefore, Theorem 4.12 applies.

**THEOREM 5.13.** — Aut($N_r$) is dissociated.

As in the previous section, we obtain the following two results. Since the proofs are identical, we omit them.

**PROPOSITION 5.14.** — Aut($N_r$) is locally Roelcke precompact but not coarsely bounded.

**PROPOSITION 5.15.** — Aut($N_r$) has the Howe-Moore property with respect to the bornology of coarsely bounded sets.

## 5.2 Diversities

Given a set $X$, we denote by $P_{\text{fin}}(X)$ the set of finite subsets of $X$. A diversity is a couple $(X, \delta)$ where $X$ is a set and $\delta: P_{\text{fin}}(X) \to \mathbb{R}_+$ is a map satisfying:

* $\delta(A) = 0$ if and only if $|A| \le 1$,

* if $B \ne \emptyset$, then $\delta(A \cup C) \le \delta(A \cup B) + \delta(B \cup C)$.
We call the map $\delta$ a diversity map. These axioms imply that a diversity map $\delta$ is monotonous: if $A \subseteq B$, then $\delta(A) \le \delta(B)$. Moreover, it is sublinear on sets with non-empty intersection: if $A \cap B \ne \emptyset$, then $\delta(A \cup B) \le \delta(A) + \delta(B)$. We say that a diversity is integral/rational if the diversity map takes only integer/rational values. In this section, we will discuss diversities constructed by means of a Fraïssé limit. We first focus on diversities with integral values.

**LEMMA 5.16.** — The class of finite diversities whose diversity map takes only integral values is a Fraïssé class, the limit of which we denote by $\mathbb{ZD}$.

We refer to [14, Prop. 3.10] for a proof of this lemma. The only non-obvious part is the amalgamation property. As for integral metric spaces, it turns out that there is a natural way of amalgamating integral diversities (of any cardinality). Let $X$ be a set. A collection $E_1, \dots, E_n$ of finite subsets of $X$ is connected if the intersection graph associated with the $E_i$'s is connected.

**DEFINITION 5.17.** — Let $(X_1, \delta_1)$ and $(X_2, \delta_2)$ be two integral diversities such that the intersection $Y := X_1 \cap X_2$ is non-empty. Assume that $d_1$ and $d_2$ coincide on $\mathcal{P}_{\text{fin}}(Y)$. Let $X = X_1 \cup X_2$ and define a map $\delta: \mathcal{P}_{\text{fin}}(X) \to \mathbb{N}$ as follows:

$$ \delta(A) := \min\{\sum_{i=1}^{n} \delta_{k_i}(E_i) : E_1, \dots, E_n \text{ is connected, } A \subseteq \bigcup_{i=1}^{n} E_i \text{ and } E_i \subseteq X_{k_i}\}. $$

Then $(X, \delta)$ is an integral diversity such that the restriction of $\delta$ to $\mathcal{P}_{\text{fin}}(X_i) \subseteq \mathcal{P}_{\text{fin}}(X)$ coincides with $\delta_i$ for every $i \in \{1, 2\}$. It is called the diversity amalgam of $(X_1, \delta_1)$ and $(X_2, \delta_2)$ over $Y$.

As before, we use this amalgamation to obtain $\sigma$-SAP.

**LEMMA 5.18.** — $\mathbb{ZD}$ satisfies $\sigma$-SAP.

*Proof.* This is a consequence of the existence of the amalgam in Definition 5.17 $\square$

**REMARK 5.19.** — If $\mathbb{QD}$ denotes the Fraïssé limit of finite diversities whose diversity map takes only rational values, then $\mathbb{QD}$ does not satisfy $\sigma$-SAP. The reason is the same as the one discussed in Remark 4.9 for the rational Urysohn space $\mathbb{QU}$.

Again, weak elimination of imaginaries is technical, but standard methods allow us to prove that it holds for $\mathbb{ZD}$ and $\mathbb{QD}$. We will provide the details in an upcoming version of the paper.

**LEMMA 5.20.** — Aut($\mathbb{ZD}$) and Aut($\mathbb{QD}$) weakly eliminate imaginaries.

Therefore, Theorem 4.12 applies to $\mathbb{ZD}$.

**THEOREM 5.21.** — Aut($\mathbb{ZD}$) is dissociated.
Recall that we denote by $\mathbb{QD}$ the Fraïssé limit of finite diversities whose diversity map only takes rational values. Even though $\mathbb{QD}$ does not satisfy $\sigma$-SAP, we can still obtain dissociation via approximating sequences. Indeed, by using the analogue of Katětov functions for diversities developed in [6], one can build an approximating sequence for $\mathbb{QD}$ consisting of dissociated groups and therefore obtain the following result by Theorem 4.1.

**THEOREM 5.22.** — Aut($\mathbb{QD}$) is dissociated.

**PROPOSITION 5.23.** — Aut($\mathbb{ZD}$) and Aut($\mathbb{QD}$) have the Howe-Moore property with respect to the bornology of coarsely bounded sets.

*Proof.* Let $G$ be either Aut($\mathbb{ZD}$) or Aut($\mathbb{QD}$). Then $G$ is dissociated by Theorems 5.21 and 5.22. We leave to the cautious reader the proof of the fact that the amalgamation of finite diversities (the above definition applies to *finite* diversities, even if the diversity map takes non-integer values) is a functorial amalgamation in the sense of Rosendal [24, Def. 6.30]. Therefore, by Theorem 6.31 of [24], $G_A$ is coarsely bounded for every non-empty finite subset $A$ of either $\mathbb{ZD}$ or $\mathbb{QD}$. Finally, Theorem 2.12 allows us to conclude that $G$ has the Howe-Moore property with respect to the bornology of coarsely bounded sets. $\square$

## 5.3 Free amalgamation property

The aim of this section is to provide two different proofs that the automorphism group of a Fraïssé limit with the free amalgamation property is dissociated. In the first proof, we show that such structures have $\sigma$-SAP. The second proof is done using approximating sequences.

**DEFINITION 5.24.** — A Fraïssé limit $\mathbf{M}$ in a language $\mathcal{L}$ satisfies the free amalgamation property (FAP) if for all $\mathbf{A}, \mathbf{B}_1, \mathbf{B}_2 \in \text{Age}(\mathbf{M})$ and all embeddings $f_i: \mathbf{A} \to \mathbf{B}_i$, there exists $\mathbf{C} \in \text{Age}(\mathbf{M})$ and embeddings $g_i: \mathbf{B}_i \to \mathbf{C}$ such that

<figure><img src="image_6.png" /></figure>

* if there exist $b_i \in \mathbf{B}_i$ satisfying $g_1(b_1) = g_2(b_2)$, then there exists $a \in \mathbf{A}$ satisfying $f_i(a) = b_i$,

* for every relation $R \in \mathcal{L}$ and every tuple $\bar{c}$ of elements in $\mathbf{C}$ such that $R^c(\bar{c})$ holds, then $\bar{c}$ belongs either to $g_1(\mathbf{B}_1)$ or to $g_2(\mathbf{B}_2)$.
We refer to [23, Example 2.3] for some examples of Fraïssé limits with the free amalgamation property. Since all of them are $\aleph_0$-categorical (and therefore Roelcke-precompact), we give below an example of a Fraïssé limit satisfying (FAP) whose automorphism group is not Roelcke-precompact.

**EXAMPLE 5.25.** — Given a simplicial graph $\Gamma$, denote by $E(G)$ the set of its edges. Suppose $\Gamma$ is endowed with a map $c: E(\Gamma) \to \mathbb{N}$, thought of as a coloring of its edges. It can then be viewed as a structure in the language $\mathcal{L} = (R_n)_{n \ge 0}$, where each $R_n$ is a relation of arity 2 whose interpretation $R_n^\Gamma$ is given by

$$R_n^\Gamma := \{(x, y) \in V(\Gamma)^2 : \{x, y\} \in E(\Gamma) \text{ and } c(\{x, y\}) = n\}.$$

Let $\mathcal{C}$ be the class of all structures $(\Gamma, c)$ such that $\Gamma$ is a finite simplicial graph and $c: E(\Gamma) \to \mathbb{N}$. Then $\mathcal{C}$ is a Fraïssé class with the free amalgamation property. Let $\mathbf{M}$ be its Fraïssé limit. Let us show that Aut($\mathbf{M}$) is not Roelcke precompact. Since $\mathbf{M}$ is ultrahomogeneous and one-point substructures of $\mathbf{M}$ are all isomorphic, then Aut($\mathbf{M}$) acts transitively on its domain. However, there are countably many isomorphism types of two-point substructures in $\mathbf{M}$ which are given by the values of $c$. Therefore, there are infinitely many orbits for the action of Aut($\mathbf{M}$) on pairs of points. Thus, the action of Aut($\mathbf{M}$) on dom($\mathbf{M}$) is not oligomorphic. This shows that Aut($\mathbf{M}$) is not Roelcke precompact by [29, Thm. 2.4].

However, the above example is coarsely bounded as a consequence of the following.

**LEMMA 5.26.** — Let $\mathbf{M}$ be a Fraïssé limit with the free amalgamation property. Then Aut($\mathbf{M}$) is coarsely bounded.

*Proof.* Let $\mathcal{C}$ be the Fraïssé class whose limit is $\mathbf{M}$. It is straightforward to check that the free amalgamation provides a functorial amalgamation of $\mathcal{C}$ over the empty structure $\emptyset$ in the sense of Rosendal [24, Def. 6.30]. Therefore, Aut($\mathbf{M}$) is coarsely bounded by Theorem 6.31 of [24]. □

**LEMMA 5.27.** — Let $\mathbf{M}$ be a Fraïssé limit with the free amalgamation property. Then Aut($\mathbf{M}$) has no algebraicity and weakly eliminates imaginaries.

*Proof.* This is the combination of Lemma 2.5 and 2.7 in [23] and Lemma 1.6. □

**THEOREM 5.28.** — Let $\mathbf{M}$ be a Fraïssé limit satisfying the free amalgamation property. Then $\mathbf{M}$ satisfies $\sigma$-SAP.

*Proof.* Fix $\mathbf{A}, \mathbf{B}_1, \mathbf{B}_2 \in \sigma\text{Age}(\mathbf{M})$ and embeddings $f_i: \mathbf{A} \hookrightarrow \mathbf{B}_i$. Consider $\mathbf{C}$ the free amalgamation of $\mathbf{B}_1$ and $\mathbf{B}_2$ over $\mathbf{A}$, i.e. the union of $\mathbf{B}_1$ and $\mathbf{B}_2$ identifying $\mathbf{A}$ and where all the relations are the ones from $\mathbf{B}_1$ and $\mathbf{B}_2$. $\mathbf{C}$ is well-defined and is in $\sigma\text{Age}(\mathbf{M})$ as all of its finite substructures are in $\text{Age}(\mathbf{M})$. □

**THEOREM 5.29.** — If $\mathbf{M}$ is a Fraïssé limit satisfying the free amalgamation property, then Aut($\mathbf{M}$) is dissociated.
*Proof.* Let $\mathbf{M}$ be a Fraïssé limit satisfying the free amalgamation property. Then $\mathbf{M}$ weakly eliminates imaginaries by Lemma 5.27 and satisfies $\sigma$-SAP. Finally, Theorem 4.12 yields that $\text{Aut}(\mathbf{M})$ is dissociated. $\square$

We finally give an alternative proof of Theorem 5.29 using the method from Section 4.1.

*Proof of Theorem 5.29.* Let $\mathcal{C}$ be a Fraïssé class in some countable relational language $\mathcal{L}$ whose Fraïssé limit is $\mathbf{M}$. Write $\mathcal{L} = (R_n)_{n\ge 0}$. For every $n \ge 0$, let $\mathcal{C}_n$ be class of those elements $\mathbf{A} \in \mathcal{C}$ such that for every $m > n$, the interpretation $R_m^{\mathbf{A}}$ is empty. Since $\mathcal{C}$ has the free amalgamation property, it is clear that $\mathcal{C}_n$ is a Fraïssé class with the free amalgamation property. For every $n \ge 0$, let $\mathbf{M}_n$ be the Fraïssé limit of $\mathcal{C}_n$. By construction, the age of $\mathbf{M}_n$ is contained in the age of $\mathbf{M}_{n+1}$, so by Theorem 3.9 of [4], we can assume that $\mathbf{M}_n$ is a substructure of $\mathbf{M}_{n+1}$ so that there exists an extension embedding $\theta_n: \text{Aut}(\mathbf{M}_n) \hookrightarrow \text{Aut}(\mathbf{M}_{n+1})$. Set $\Omega = \bigcup_{n\ge 0} \text{dom}(\mathbf{M}_n)$. Then the structure with domain $\Omega$ generated by the increasing union of the substructures $\mathbf{M}_n$ is isomorphic to $\mathbf{M}$. Indeed, it is clearly ultrahomogeneous and its age is exactly $\mathcal{C}$.

We claim that the sequence of extension embeddings $\theta_n: \text{Aut}(\mathbf{M}_n) \hookrightarrow \text{Aut}(\mathbf{M}_{n+1})$ forms an approximating sequence for $\text{Aut}(\mathbf{M})$. Indeed, the only thing left to check is that $\bigcup_{n\ge 0} \iota_n(\text{Aut}(\mathbf{M}_n))$ is a dense subgroup of $\text{Aut}(\mathbf{M})$. Fix $g \in \text{Aut}(\mathbf{M})$ and let $x_1, \dots, x_k \in \Omega$ be pairwise distinct. Then $x_1, \dots, x_k, g(x_1), \dots, g(x_k)$ belong to $\text{dom}(\mathbf{M}_n)$ for some $n \ge 0$. Moreover, $g$ induces an isomorphism between the structures generated by $x_1, \dots, x_n$ and $g(x_1), \dots, g(x_n)$. By ultrahomogeneity of $\mathbf{M}_n$, there exists an element $g_n \in \text{Aut}(\mathbf{M}_n)$ such that $g_n(x_i) = g(x_i)$ for every $i \in \{1, \dots, k\}$. Since $\iota_n$ is an extension embedding, then $\iota_n(g_n)(x_i) = g_n(x_i) = g(x_i)$ for every $i \in \{1, \dots, k\}$, which proves the desired density.

Finally, $\mathbf{M}_n$ is oligomorphic (because the language is finite), has no algebraicity and weakly eliminates imaginaries. So $\text{Aut}(\mathbf{M}_n) \le \text{Sym}(\text{dom}(\mathbf{M}_n))$ is dissociated by Theorem 4.15. Therefore, we conclude that $\text{Aut}(\mathbf{M}) \le \text{Sym}(\text{dom}(\mathbf{M}))$ is dissociated by Theorem 3.12 of [2]. $\square$

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R. Barritault, UNIVERSITE CLAUDE BERNARD LYON 1, CNRS, CENTRALE LYON, INSA LYON, UNIVERSITÉ JEAN MONNET, ICJ UMR5208, 69622 VILLEURBANNE, FRANCE.

E-mail address: barritault@math.univ-lyon1.fr

C. Jahel, INSTITUT FUR ALGEBRA, TECHNISCHE UNIVERSITÄT DRESDEN,
DRESDEN, GERMANY

E-mail address: colin.jahel@tu-dresden.de

M. Joseph, UNIVERSITÉ PARIS-SACLAY, CNRS, LABORATOIRE DE MATHÉMATIQUES D'ORSAY,
ORSAY, FRANCE

E-mail address: matthieu.joseph@universite-paris-saclay.fr