Non-Compact Stabilizers in Infinite Permutation Groups

• Non-Compact Stabilizers are defined for p.m.p. actions of non-locally-compact, oligomorphic permutation groups on infinite sets, exhibiting a clear dichotomy between essentially free and essentially transitive actions.
• This framework uses combinatorial techniques and model-theoretic properties like weak elimination of imaginaries and absence of algebraicity to bypass classical Haar measure methods.
• The approach extends to invariant random subgroups and expansions, unifying ergodic theory and logic to rigorously classify stabilizer behaviors in infinite permutation dynamics.

Non-compact stabilizers arise in the context of probability-measure-preserving (p.m.p.) actions of non-locally-compact, closed permutation groups on countably infinite sets. Unlike locally compact groups, where the classical stabilizer rigidity results rely on geometric constructions and the existence of Haar measure, stabilizer properties for these groups require fundamentally different, combinatorial and model-theoretic methods. Recent advances have established dichotomies and rigidity theorems for such stabilizers, particularly for oligomorphic Polish groups acting transitively with no algebraicity and weak elimination of imaginaries. The central theme is the sharp restriction on the possible stabilizer distributions in ergodic actions and invariant random subgroups, demonstrating a collapse to either trivial (free) or essentially transitive types.

1. Oligomorphic Permutation Groups and Their Model-Theoretic Structure

Let $G < \mathrm{Sym}(\Omega)$ be a closed, transitive subgroup acting on a countably infinite set $\Omega$. The group $G$ is oligomorphic if for each integer $n$, the diagonal action $G \curvearrowright \Omega^n$ has finitely many orbits. Additionally, $G$ admits no algebraicity if, for every finite $A \subset \Omega$, the pointwise stabilizer $G_A$ acts on $\Omega \setminus A$ with no finite orbits. Weak elimination of imaginaries means that every open subgroup of $G$ contains $G_A$ as a finite-index subgroup for some finite $A \subset \Omega$. These properties ensure $G$ is highly transitive on large spaces but constrained by model-theoretic characteristics, permitting combinatorial control rather than topological or measure-theoretic tools associated with locally compact cases (Jahel et al., 2023).

2. Stabilizer Rigidity and the Main Dichotomy

The central result is a dichotomy theorem for the stabilizers in ergodic p.m.p. actions $G \curvearrowright (X, \mu)$ under the above hypotheses. Specifically, for such $G$, any ergodic p.m.p. action is either:

• Essentially free: $$\mu(\{ x \in X : \operatorname{Stab}_G(x) = \{e\} \}) = 1$$
• Essentially transitive: There exists $x_0 \in X$ such that $$\mu(\{ x \in X : \exists g \in G,\, g \cdot x_0 = x \}) = 1$$

Here, $$\operatorname{Stab}_G(x) = \{ g \in G : g \cdot x = x \}$$ denotes the stabilizer of $x$. No intermediate case arises under these structural constraints, establishing a strong form of stabilizer rigidity (Jahel et al., 2023).

3. Dissociated Actions and Rigidity for Proper Subgroups

The notion of dissociation, adapted from exchangeability theory, is crucial to extending the dichotomy to wider settings. Define, for each finite $A \subset \Omega$, $F_A$ as the $\sigma$-algebra generated by $G_A$-invariant measurable subsets of $X$. An action $G \curvearrowright (X, \mu)$ is dissociated if for all disjoint finite $A, B \subset \Omega$, the $\sigma$-algebras $F_A$ and $F_B$ are independent; equivalently, the joint law factorizes over disjoint coordinate sets.

For any closed, transitive, proper subgroup $G \subsetneq \mathrm{Sym}(\Omega)$ that is primitive, especially when $G$ has no algebraicity and weakly eliminates imaginaries, every dissociated ergodic p.m.p. action is either essentially free or essentially transitive. This result demonstrates that non-compact stabilizers are tightly constrained even beyond the maximally symmetric group, provided the action is dissociated in the sense defined above (Jahel et al., 2023).

4. Invariant Random Expansions and the Logic Action

Invariant random expansions (IREs) offer a universal model for constructing and classifying p.m.p. actions. Fix a countable relational language $L \supset L_G$, with $L_G$ encoding the orbits of $G$ on $\Omega^n$. Consider the compact space $\operatorname{Struc}_L^G$ of all expansions of the canonical $G$-structure. The logic action of $G$ on $\operatorname{Struc}_L^G$ is continuous, and an IRE is a $G$-invariant Borel probability measure on $\operatorname{Struc}_L^G$. By the Becker–Kechris universality theorem, every Borel $G$-action embeds into a logic action, so all p.m.p. actions arise (up to isomorphism) from such expansions. IREs thus serve as canonical models for measure-preserving actions and provide a systematic approach to analyzing the stabilizer structure of non-compact group actions (Jahel et al., 2023).

5. Invariant Random Subgroups and Rigidity in the Symmetric Group

For any Polish group $G$, $\operatorname{Sub}(G)$, equipped with the Effros $\sigma$-algebra, is the space of closed subgroups. An invariant random subgroup (IRS) is a conjugation-invariant probability measure on $\operatorname{Sub}(G)$. For closed $G \leq \mathrm{Sym}(\Omega)$, every IRS can be realized as the stabilizer IRS of some p.m.p. action, $\nu = \operatorname{Stab}_* \mu$. However, for $\mathrm{Sym}(\Omega)$, although there exist p.m.p. ergodic actions that are neither essentially free nor transitive, every ergodic IRS is concentrated on a single conjugacy class of subgroups—achieving essential transitivity for IRS under conjugation.

Property	Dissociated Rigidity (Proper $G$)	IRS Rigidity ($\mathrm{Sym}(\Omega)$)
Ergodic p.m.p. actions	Free or transitive if dissociated	Not necessarily free or transitive
Ergodic IRSs	Realizable by stabilizer distributions	Only single conjugacy class (transitive)

This table summarizes the contrast between the behavior of stabilizers and invariant random subgroups for non-compact and maximally compact groups (Jahel et al., 2023).

6. Combinatorial and Model-Theoretic Proof Techniques

The proof approach bypasses classical locally compact techniques—such as the use of Haar measure and geometric methods on homogeneous spaces $G/H$—which are unavailable in the non-locally compact, non-Archimedean Polish context. Instead, the analysis relies on:

• Exchangeability and dissociation: De Finetti-type properties equate ergodicity of $G$-invariant measures with dissociation.
• Model-theoretic imaginaries: Weak elimination of imaginaries reduces open subgroups to pointwise stabilizers of finite sets. No algebraicity ensures infinite orbits outside finite sets, together ensuring primitivity and allowing back-and-forth arguments on quantifier-free and infinitary types.
• Back-and-forth with orbit full measure: Almost every pair of points can be related by a $G$-element extendable via partial isomorphisms, implying that measure is concentrated on the orbit relation and establishing essential transitivity.
• Combinatorial universality: The logic-action viewpoint and random expansions enable the transfer of ergodic properties and the dichotomy to the entire space of p.m.p. actions accessible to $G$.

These methods demonstrate that stabilizer rigidity in non-compact settings is governed by combinatorial and model-theoretic properties, establishing a dichotomy analogous to the Stuck–Zimmer theorem in locally compact groups, but proved via fundamentally different methods (Jahel et al., 2023).

7. Summary and Significance

The study of non-compact stabilizers within the framework of Polish groups acting on countable sets has led to a dichotomy theorem for ergodic p.m.p. actions, showing the absence of intermediate stabilizer types outside essentially free or essentially transitive cases. The framework extends to invariant random subgroups, characterizing all ergodic IRSs for $\mathrm{Sym}(\Omega)$ as concentrated on single conjugacy classes. Mapping these phenomena via invariant random expansions highlights the role of combinatorial and model-theoretic structure in driving stabilizer rigidity beyond the range of classical topological group techniques. These results underscore the deep connections between infinite permutation group theory, probability, and model theory, and establish robust boundaries for the behavior of non-compact stabilizers in infinite permutation dynamics (Jahel et al., 2023).