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Classification of continuously transitive circle groups

Giblin, James and Markovic, Vladimir (2006) Classification of continuously transitive circle groups. Geometry and Topology, 10 (3). pp. 1319-1346. ISSN 1465-3060. doi:10.2140/gt.2006.10.1319.

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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 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(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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Additional Information:© 2006 Mathematical Sciences Publishers. Received: 12 December 2005; Revised: 22 June 2006; Accepted: 29 July 2006; Published: 18 September 2006.
Subject Keywords:Circle group, convergence group, transitive group, cyclic cover
Issue or Number:3
Classification Code:MSC 2000: Primary: 37E10. Secondary: 22A05, 54H11
Record Number:CaltechAUTHORS:20170508-130256247
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:77260
Deposited By: Tony Diaz
Deposited On:09 May 2017 23:04
Last Modified:15 Nov 2021 17:29

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