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Geometriae Dedicata

, Volume 136, Issue 1, pp 145–165 | Cite as

The Teichmüller distance between finite index subgroups of \({PSL_2(\mathbb{Z})}\)

  • Vladimir Markovic
  • Dragomir Šarić
Original Paper
  • 53 Downloads

Abstract

For a given \({\epsilon > 0}\) , we show that there exist two finite index subgroups of \({PSL_2(\mathbb{Z})}\) which are \(({1+\epsilon})\) -quasisymmetrically conjugated and the conjugation homeomorphism is not conformal. This implies that for any \({\epsilon > 0}\) there are two finite regular covers of the Modular once punctured torus T 0 (or just the Modular torus) and a \({(1+\epsilon)}\) -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.

Keywords

Modular group Teichmüller space Quasiconformal maps Dilatation \({PSL_2(\mathbb{Z})}\) Finite index subgroups Solenoid Ehrenpreis conjecture 

Mathematics Subject Classification (2000)

30F60 

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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of MathematicsStony Brook UniversityStony BrookUSA
  2. 2.Department of MathematicsQueens College of CUNYFlushingUSA

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