Conjugate stabilizers for non-free group actions

A classical assumption when studying group actions on compact spaces is freeness, meaning that all points have trivial stabilizers.
When the action is not free, one may ask how large the set of points with trivial stabilizer is, either from a topological or a measure-theoretic perspective.
This leads to the notions of topologically free and essentially free actions, respectively.
A natural next step is to consider actions in which points have only “few” distinct stabilizers.
In this talk, I will present conditions and results that connect dynamical properties of group actions with this phenomenon.

I will start by discussing minimal equicontinuous actions arising from subgroups of the automorphism group of d-ary trees, and in the second part of the talk, I will turn to group actions with almost normal stabilizers.

This is joint work with M. I. Cortez and O. Lukina.