Geometry & Topology 10 (2006) 1319–1346

DOI: 10.2140/gt.2006.10.1319

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,R), PSL(2,R), PSL_k(2,R), Homeo_k(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,R) 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

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