Nilpotent Cantor actions

Abstract

A nilpotent Cantor action is a minimal equicontinuous action $\Phi :\Gamma \times \mathfrak{X}\to \mathfrak{X}$ on a Cantor set $\mathfrak{X}$, where $\Gamma$ contains a finitely-generated nilpotent subgroup ${\Gamma }_0\subset \Gamma$ of finite index. In this note, we show that these actions are distinguished among general Cantor actions: any effective action of a finitely generated group on a Cantor space, which is continuously orbit equivalent to a nilpotent Cantor action, must itself be a nilpotent Cantor action. As an application of this result, we obtain new invariants of nilpotent Cantor actions under continuous orbit equivalence.

Revision corrects assertion and proof of Theorem 1.1, and adds the application in Theorem 1.2. minor additional edits made