Classification of continuously transitive circle groups

TL;DR

This paper classifies certain transitive subgroups of homeomorphisms of the circle, showing they are conjugate to well-known groups, and confirms a maximality conjecture for PSL(2,R).

Contribution

It provides a complete classification of continuously transitive circle groups containing a non-constant path, confirming a conjecture by Ghys and addressing a question by Bestvina.

Findings

G is conjugate to SO(2,R), PSL(2,R), PSL_k(2,R), Homeo_k(S^1), or Homeo(S^1)

PSL(2,R) is a maximal closed subgroup of Homeo(S^1)

Groups acting transitively on k-tuples with k>3 have closure equal to Homeo(S^1)

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').