Essential holonomy of Cantor actions

TL;DR

This paper explores the concept of essential holonomy in minimal equicontinuous Cantor actions, revealing its relation to group structure, and demonstrating its invariance under certain equivalences, with implications for nilpotent groups.

Contribution

It establishes new links between essential holonomy and group properties, especially for locally quasi-analytic actions, and provides invariance results under return and orbit equivalences.

Findings

Essential holonomy relates to the structure of the acting group.

Nilpotent group actions on Cantor sets have no essential holonomy.

Essential holonomy is preserved under return and orbit equivalences.

Abstract

A group action has essential holonomy if the set of points with non-trivial holonomy has positive measure. If such an action is topologically free, then having essential holonomy is equivalent to the action not being essentially free, which means that the set of points with non-trivial stabilizer has positive measure. In this paper, we investigate the relation between the property of having essential holonomy and structure of the acting group for minimal equicontinuous actions on Cantor sets. We show that if such a group action is locally quasi-analytic and has essential holonomy, then every commutator subgroup in the group lower central series has elements with positive measure set of points with non-trivial holonomy. In particular, this gives a new proof that a minimal equicontinuous Cantor action by a nilpotent group has no essential holonomy. We also show that the property of having essential holonomy is preserved under return equivalence and continuous orbit equivalence of minimal equicontinuous Cantor actions. Finally, we give examples to show that the assumption on the action that it is locally quasi-analytic is necessary.