An alternative for minimal group actions on totally regular curves

TL;DR

This paper establishes an alternative for minimal group actions on totally regular curves, showing they are either conjugate to circle isometries or contain a free nonabelian subgroup, with new characterizations and lemmas introduced.

Contribution

It introduces a new characterization of totally regular curves via measures and proves an escaping lemma for minimal actions, advancing understanding of group actions on such spaces.

Findings

Actions are either conjugate to circle isometries or contain free nonabelian groups.

New measure-based characterization of totally regular curves.

An improved escaping lemma for minimal group actions.

Abstract

Let $G$ be a countable group and $X$ be a totally regular curve. Suppose that $\phi:G\rightarrow {\rm Homeo}(X)$ is a minimal action. Then we show an alternative: either the action is topologically conjugate to isometries on the circle $\mathbb S^1$ (this implies that $\phi(G)$ contains an abelian subgroup of index at most 2), or has a quasi-Schottky subgroup (this implies that $G$ contains the free nonabelian group $\mathbb Z*\mathbb Z$). In order to prove the alternative, we get a new characterization of totally regular curves by means of the notion of measure; and prove an escaping lemma holding for any minimal group action on infinite compact metric spaces, which improves a trick in Margulis' proof of the alternative in the case that $X=\mathbb S^1$.