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 ϕ:G→Homeo(X) is a minimal action. Then we show an alternative: either the action is topologically conjugate to isometries on the circle S1 (this implies that ϕ(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 Z⁎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=S1.