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The Teichmüller distance between finite index subgroups of PSL_2ℤ

Markovic, Vladimir and Šarić, Dragomir (2008) The Teichmüller distance between finite index subgroups of PSL_2ℤ. Geometriae Dedicata, 136 (1). pp. 145-165. ISSN 0046-5755.

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For a given ϵ>0, we show that there exist two finite index subgroups of PSL_2(ℤ) which are (1+ϵ)-quasisymmetrically conjugated and the conjugation homeomorphism is not conformal. This implies that for any ϵ>0 there are two finite regular covers of the Modular once punctured torus T_0 (or just the Modular torus) and a (1+ϵ)-quasiconformal map between them that is not homotopic to a conformal map. As an application of the above results, we show that the orbit of the basepoint in the Teichmüller space T(S^p) of the punctured solenoid S^p under the action of the corresponding Modular group (which is the mapping class group of S^p [6], [7]) has the closure in T(S^p) strictly larger than the orbit and that the closure is necessarily uncountable.

Item Type:Article
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Additional Information:© 2008 Springer Science+Business Media B.V. Received: 27 July 2007; Accepted: 3 July 2008; Published online: 1 August 2008.
Subject Keywords:Modular group; Teichmüller space; Quasiconformal maps; Dilatation; PSL2(Z); Finite index subgroups; Solenoid; Ehrenpreis conjecture
Issue or Number:1
Classification Code:Mathematics Subject Classification (2000): 30F60
Record Number:CaltechAUTHORS:20170508-145545361
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Official Citation:Markovic, V. & Šarić, D. Geom Dedicata (2008) 136: 145. doi:10.1007/s10711-008-9281-x
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:77266
Deposited By: Tony Diaz
Deposited On:15 May 2017 19:53
Last Modified:03 Oct 2019 17:55

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