An alternative for minimal group actions on totally regular curves

Abstract

Let $G$ be a countable group and $X$ be a totally regular curve. Suppose that $\varphi :G\to \mathrm{Homeo}(X)$ 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 $\varphi (G)$ contains an abelian subgroup of index at most 2), or has a quasi-Schottky subgroup (this implies that $G$ contains the free nonabelian group $\mathbb{Z}\ast \mathbb{Z}$). 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 $X={\mathbb{S}}^1$.

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