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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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(S1) 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, ), PSL(2, ), PSLk(2, ), Homeok(S1), Homeo(S1). This verifies the classification suggested by Ghys in [Enseign. Math. 47 (2001) 329-407]. As a corollary we show that the group PSL(2, ) is a maximal closed subgroup of Homeo(S1) (we understand this is a conjecture of de la Harpe). We also show that if such a group G < Homeo(S1) acts continuously transitively on k–tuples of points, k > 3, then the closure of G is Homeo(S1) (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
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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/