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 [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').
Cite
@article{arxiv.0903.0180,
title = {Classification of continuously transitive circle groups},
author = {James Giblin and Vladimir Markovic},
journal= {arXiv preprint arXiv:0903.0180},
year = {2009}
}
Comments
This is the version published by Geometry & Topology on 18 September 2006