Non-Hausdorff germinal groupoids for actions of countable groups

TL;DR

This paper investigates the conditions under which germinal groupoids from minimal equicontinuous actions of countable groups on Cantor sets are non-Hausdorff, providing new criteria and examples involving self-similar groups.

Contribution

It introduces a novel criterion to identify non-Hausdorff topology in germinal groupoids and applies it to specific classes of self-similar group actions.

Findings

Developed a new obstruction criterion for non-Hausdorff topology.

Identified classes of amenable self-similar groups with non-Hausdorff germinal groupoids.

Provided explicit examples of contracting and non-contracting self-similar groups.

Abstract

We study conditions under which the germinal groupoid associated to a minimal equicontinuous action of a countable group on a Cantor set has non-Hausdorff topology. We develop a new criterion, which serves as an obstruction for the \'etale topology on the groupoid to be non-Hausdorff. We use this and other criteria to study the topology of germinal groupoids for a few classes of actions. In particular, we give examples of families of contracting and non-contracting self-similar groups, which are amenable and whose actions have associated germinal groupoids with non-Hausdorff topology.