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

Let G be a closed, proper subgroup of S_∞, the group of all permutations of N. If G is oligomorphic, has no algebraicity, and weakly eliminates imaginaries, we prove that any probability measure-preserving ergodic Borel action G (X,μ ) 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_∞, 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_∞—any ergodic invariant random subgroup of S_∞ is essentially transitive.