## Geometry & Topology

### Classification of continuously transitive circle groups

#### 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: $SO(2,ℝ)$, $PSL(2,ℝ)$, $PSLk(2,ℝ)$, $Homeok(S1)$, $Homeo(S1)$. This verifies the classification suggested by Ghys in [Enseign. Math. 47 (2001) 329-407]. As a corollary we show that the group $PSL(2,ℝ)$ 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’).

#### Article information

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

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

https://projecteuclid.org/euclid.gt/1513799768

Digital Object Identifier
doi:10.2140/gt.2006.10.1319

Mathematical Reviews number (MathSciNet)
MR2255499

Zentralblatt MATH identifier
1126.37025

#### Citation

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. https://projecteuclid.org/euclid.gt/1513799768

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