Geometry & Topology

Classification of continuously transitive circle groups

James Giblin and Vladimir Markovic

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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’).

Article information

Geom. Topol., Volume 10, Number 3 (2006), 1319-1346.

Received: 12 December 2005
Revised: 22 June 2006
Accepted: 29 July 2006
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37E10: Maps of the circle
Secondary: 22A05: Structure of general topological groups 54H11: Topological groups [See also 22A05]

Circle group convergence group transitive group cyclic cover


Giblin, James; Markovic, Vladimir. Classification of continuously transitive circle groups. Geom. Topol. 10 (2006), no. 3, 1319--1346. doi:10.2140/gt.2006.10.1319.

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