#### Volume 10, issue 3 (2006)

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Classification of continuously transitive circle groups

### James Giblin and Vladimir Markovic

Geometry & Topology 10 (2006) 1319–1346
 arXiv: 0903.0180
##### Abstract

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

##### Keywords
Circle group, convergence group, transitive group, cyclic cover
##### Mathematical Subject Classification 2000
Primary: 37E10
Secondary: 22A05, 54H11