Stabilizers in group Cantor actions and measures.

ABSTRACT:

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 full measure. On the other hand, there are many examples of group actions, where every point has a non-trivial stabilizer. In this talk, we generalize the notion of an essentially free action to such actions, using the notion of holonomy. For equicontinuous actions of countable groups on Cantor sets, we answer the following question: under what conditions there exists a subgroup H of G, such that the stabilizers of almost all points in X are conjugate to H? These conditions are that the action is locally quasi-analytic and that the action is uniformly non-constant. The notion of a locally quasi-analytic action was previously used by the speaker in the joint works with Hurder to classify equicontinuous actions on Cantor sets. The notion of a uniformly non-constant action is new and was introduced in this work. This is joint work with Maik Groeger.

Posted on Jun 18, 2020 in Dynamical Systems, Seminars