Stabilizers in group Cantor actions and measures

Given a countable group G acting on a Cantor set X by transformations preserving a probability measure, the action is essentially free if the set of points with trivial stabilizers has a full measure. In this talk, we consider actions where no point has a trivial stabilizer, and investigate the properties of the points with non-trivial holonomy. We introduce the notion of a locally non-degenerate action, and show that if an action is locally non-degenerate, then the set of points with trivial holonomy has full measure in X. We discuss applications of this work to the study of invariant random subgroups, induced by actions of countable groups. This is joint work with Maik Gröger.