Classification of continuously transitive circle groups

Let $G$ be a closed transitive subgroup of $Homeo (S^{1})$ which contains a non-constant continuous path $f : [0, 1] \to G$ . 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’).

Keywords

Circle group, convergence group, transitive group, cyclic cover

Mathematical Subject Classification 2000

Primary: 37E10

Secondary: 22A05, 54H11

References

Forward citations

Publication

Received: 12 December 2005

Revised: 22 June 2006

Accepted: 29 July 2006

Published: 18 September 2006

Proposed: David Gabai

Seconded: Leonid Polterovich, Benson Farb

Authors

James Giblin
Mathematics Institute University of Warwick Coventry, CV4 7AL UK
http://www.maths.warwick.ac.uk/~giblin/
Vladimir Markovic
Mathematics Institute University of Warwick Coventry, CV4 7AL UK
http://www.maths.warwick.ac.uk/~markovic/

Volume 10, issue 3 (2006)

James Giblin and Vladimir Markovic

Abstract