Classification of continuously transitive circle groups

Giblin, James¹ ; Markovic, Vladimir¹

¹ Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

Geometry & topology, Tome 10 (2006) no. 3, pp. 1319-1346.

Résumé

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 $G<Homeo\left({\mathbb{S}}^1\right)$ 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’).

Détail

DOI : 10.2140/gt.2006.10.1319

Keywords: Circle group, convergence group, transitive group, cyclic cover

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