An alternative for minimal group actions on totally regular curves

Let $G$ be a countable group and $X$ be a totally regular curve. Suppose that $phi:Grightarrow {rm Homeo}(X)$ 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 $X=mathbb S^1$.