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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 ŠarićEmail author
Original Paper
  • 61 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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References

  1. 1.
    Douady A., Earle C.J.: Conformally natural extension of homeomorphisms of the circle. Acta Math. 157(1–2), 23–48 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Gendron, T.: The Ehrenpreis conjecture and the moduli-rigidity gap. Complex manifolds and hyperbolic geometry (Guanajuato, 2001), pp. 207–229. Contemp. Math., vol. 311. Amer. Math. Soc., Providence, RI (2002)Google Scholar
  3. 3.
    Long D., Reid A.: Pseudomodular surfaces. Reine Angew J. Math. 552, 77–100 (2002)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Markovic V., Šarić D.: The Teichmüller mapping class group of the universal hyperbolic solenoid. Trans. Am. Math. Soc. 358(6), 2637–2650 (2006)zbMATHCrossRefGoogle Scholar
  5. 5.
    McMullen C.: Amenability, Poincaré series and quasiconformal maps. Invent. Math. 97(1), 95–127 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Nag S., Sullivan D.: Teichmüller theory and the universal period mapping via quantum calculus and the H 1/2 space on the circle. Osaka J. Math. 32(1), 1–34 (1995)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Odden C.: The baseleaf preserving mapping class group of the universal hyperbolic solenoid. Trans. Am. Math Soc. 357, 1829–1858 (2004)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Penner R.C.: Bounds on least dilatations. Proc. Am. Math. Soc. 113(2), 443–450 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Penner R.C.: Universal constructions in Teichmüller theory. Adv. Math. 98, 143–215 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Penner R.C., Šarić D.: Teichmüller theory of the punctured solenoid. Geom. Dedicata 132, 179–212 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Sullivan, D.: Linking the universalities of Milnor-Thurston, Feigenbaum and Ahlfors-Bers. In: Goldberg, L., Phillips, A. (eds.) Milnor Festschrift Topological Methods in Modern Mathematics, pp. 543–563. Publish or Perish (1993)Google Scholar

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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