Stabilizers for Ergodic Actions and Invariant Random Expansions of Non-Archimedean Polish Groups

Abstract Let $G$ be a closed, proper subgroup of $S_\infty $, the group of all permutations of ${\mathbb{N}}$. If $G$ is oligomorphic, has no algebraicity, and weakly eliminates imaginaries, we prove that any probability measure-preserving ergodic Borel action $G\curvearrowright (X,\mu )$ is either essentially free or essentially transitive. We bring the notion of dissociation from exchangeability theory in the context of stabilizer rigidity by proving that if $G$ is a closed, proper subgroup of $S_\infty $, which has no algebraicity and is primitive, then any probability measure-preserving ergodic Borel action of $G$ that is dissociated 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 probability measure-preserving ergodic Borel actions fails for the group $ S_\infty $—any ergodic invariant random subgroup of $ S_\infty $ is essentially transitive.