The No-Core Principle for Stationary Actions and Ends of Stationary Random Subgroups

Abstract

We prove a No-Core Principle for stationary actions of countable groups. Namely, if a Borel set intersects almost every orbit in finitely many points and has positive measure, then it is supported, modulo null sets, on the finite-orbit part of the action. This extends to stationary actions a basic regularity phenomenon known for measure-preserving actions. We apply this principle to the geometry of Stationary Random Subgroups. For a finitely generated group, we prove that the Schreier graph of a stationary random subgroup has almost surely 0,1,2, or infinitely many ends. Finally, we contrast this probabilistic regularity with the topological notion of Boomerang subgroups: for every $k\geq 3$, including $k=\aleph_0$, we construct a Boomerang subgroup of $\mathbb{F}_3$ whose Schreier graph has exactly $k$ ends.