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

TL;DR

This paper establishes a stabilizer rigidity phenomenon for certain non-archimedean Polish groups, showing that ergodic measure-preserving actions are either essentially free or transitive, and introduces invariant random expansions and subgroups in this context.

Contribution

It introduces the concept of invariant random expansions and studies their properties, extending stabilizer rigidity results to a class of non-locally compact Polish groups.

Findings

Ergodic p.m.p. actions are either essentially free or transitive for the considered groups.

Invariant random expansions are introduced and analyzed in the context of stabilizer rigidity.

Any ergodic invariant random subgroup of the full symmetric group is essentially transitive.

Abstract

Let $G$ be a closed permutation group on a countably infinite set $\Omega$, which acts transitively but not highly transitively. If $G$ is oligomorphic, has no algebraicity and weakly eliminates imaginaries, we prove that any probability measure preserving ergodic action $G\curvearrowright (X,\mu)$ is either essentially free or essentially transitive. As this stabilizers rigidity result concerns a class of non locally compact Polish groups, our methods of proof drastically differ from that of similar results in the realm of locally compact groups. We bring the notion of dissociation from exchangeability theory in the context of stabilizers rigidity by proving that if $G\lneq\mathrm{Sym}(\Omega)$ is a transitive, proper, closed subgroup, which has no algebraicity and weakly eliminates imaginaries, then any dissociated probability measure preserving action of $G$ is either essentially free or essentially transitive. A key notion that we develop in our approach is that of invariant random expansions, which are $G$-invariant probability measures on the space of expansions of the canonical (model theoretic) structure associated with $G$. We also initiate the study of invariant random subgroups for Polish groups and prove that - although the result for p.m.p. ergodic actions fails for the group $\mathrm{Sym}(\Omega)$ of all permutations of $\Omega$ - any ergodic invariant random subgroup of $\mathrm{Sym}(\Omega)$ is essentially transitive.