Classification of continuously transitive circle groups

Details Classification of continuously transitive circle groups Classification of continuously transitive circle groups

2009-03-01

arxiv.org/abs/0903.0180v1

Geom. Topol. 10 (2006) 1319-1346

Mathematics

Group Theory

This is the version published by Geometry & Topology on 18 September 2006

Scientific paper

10.2140/gt.2006.10.1319

Let G be a closed transitive subgroup of Homeo(S^1) 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(S^1), Homeo(S^1). This verifies the classification suggested by Ghys [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').

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