The notion of continuous orbit equivalence for group actions on Cantor sets was introduced by Boyle in his thesis. Boyle also proved the first rigidity result in this area, namely, that two actions of integers on a Cantor set C are continuously orbit equivalent if and only if they are flip conjugate.

More recently, continuous orbit equivalence for equicontinuous actions of finitely generated groups on a Cantor set C was studied by Li and by Cortez and Medynets. In both works, an important assumption is that the actions are topologically free, which means that the set of points with trivial stabilizers is dense in C.

In this talk, we concentrate on actions of finitely generated groups on a Cantor set C which are not topologically free. Such actions are ubiquitous in mathematics, arising, for example, as actions of iterated monodromy groups in geometric group theory, actions associated to arboreal representations in arithmetic dynamics, and in other contexts. For some such actions, the induced holonomy action on a sufficiently small clopen subset of C is topologically free. We call such actions stable. Other actions never stabilize, and we call such actions wild.

For actions which are not topologically free, we discuss an appropriate notion of rigidity, and introduce two group-theoretical invariants of continuous orbit equivalence. We investigate these invariants, and their relation to the topology of the associated etale groupoids, for stable and wild actions. We show that these invariants are non-equal if the topology of the groupoid associated to the action is non-Hausdorff.

Joint work with Steve Hurder.