Classification of continuously transitive circle groups

Abstract

Let G be a closed transitive subgroup of Homeo(S^1) which contains a non-constant continuous path f:[0,1]→G. We show that up to conjugation G is one of the following groups: SO(2,ℝ), PSL(2,ℝ), PSL_k(2,ℝ), Homeo_k(S^1), Homeo(S^1). This verifies the classification suggested by Ghys in [Enseign. Math. 47 (2001) 329-407]. As a corollary we show that the group PSL(2,ℝ) is a maximal closed subgroup of Homeo(S^1) (we understand this is a conjecture of de la Harpe). We also show that if such a group G<Homeo(S^1) acts continuously transitively on k–tuples of points, k>3, then the closure of G is Homeo(S^1) (cf Bestvina’s collection of ‘Questions in geometric group theory’)