Skip to main content
SpringerLink
Log in

On quasiconformal deformations of the universal hyperbolic solenoid

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

We investigate the Teichmüller metric and the complex structure on the Teichmüller space \( \mathcal{T} \)(H ∞) of the universal hyperbolic solenoid H ∞. In particular, a version of the Reich-Strebel inequality for H ∞ is obtained. As a consequence, we show that the Teichmüller type Beltrami coefficients determine unique geodesics in \( \mathcal{T} \)(H ∞), and we compute the infinitesimal form of the Teichmüller metric. In addition, we show that a Beltrami coefficient is Teichmüller extremal if and only if it is infinitesimally extremal. Finally, we show that the Kobayashi metric on \( \mathcal{T} \)(H ∞) equals the Teichmüller metric.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L. V. Ahlfors and L. Bers, Riemann’s Mapping Theorem for variable metrics, Ann. of Math. (2) 72 (1960), 385–404.

    Article  MathSciNet  Google Scholar 

  2. L. Bers, A non-standard integral equation with applications to quasiconformal mappings, Acta Math. 116 (1966), 113–134.

    Article  MATH  MathSciNet  Google Scholar 

  3. L. Bers, An extremal problem for quasiconformal mappings and a theorem by Thurston, Acta Math. 141 (1978), 73–98.

    Article  MATH  MathSciNet  Google Scholar 

  4. I. Biswas and S. Nag, Weil-Petersson geometry and determinant bundles on inductive limits of moduli spaces, Lipa’s Legacy (New York, 1995), Contemp. Math. 211 (1997), 51–80.

  5. I. Biswas and S. Nag, Limit constructions over Riemann surfaces and their parameter spaces, and the commensurability group action, Selecta Math. (N.S.) 6 (2000), 185–224.

    Article  MATH  MathSciNet  Google Scholar 

  6. S. Bonnot, R. C. Penner and D. Šarić, A presentation for the baseleaf preserving mapping class group of the punctured solenoid, Algebr. Geom. Topol. 7 (2007), 1171–1199.

    Article  MATH  MathSciNet  Google Scholar 

  7. A. Candel, Uniformization of surface laminations, Ann. Sci. École Norm. Sup. (4) 26 (1993), 489–516.

    MATH  MathSciNet  Google Scholar 

  8. A. Douady and C. J. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math. 157 (1986), 23–48.

    Article  MATH  MathSciNet  Google Scholar 

  9. C. Earle and F. Gardiner, Geometric isomorphisms between infinite-dimensional Teichmüller spaces, Trans. Amer. Math. Soc. 348 (1996), 1163–1190.

    Article  MATH  MathSciNet  Google Scholar 

  10. C. Earle and I. Kra, On holomorphic mappings between Teichmüller spaces, in Contributions to Analysis, Academic Press, New York, 1974, pp. 107–124.

    Google Scholar 

  11. C. Earle and V. Markovic, Isometries Between the Spaces of L 1 Holomorphic quadratic differentials on Riemann surfaces of finite type, Duke Math. J. 120 (2003), 433–440.

    Article  MATH  MathSciNet  Google Scholar 

  12. A. Epstein, V. Markovic and D. Šarić, Extremal maps for the universal hyperbolic solenoid, preprint, available at arxiv: http://www.arxiv.org/abs/math.DS/0604411.

  13. A. Fathi, F. Laudenbach, V. Poenaru, Travaux de Thurston sur les surfaces, Astérisque 66–67 Société Mathématique de France, 1979.

  14. F. Gardiner, Teichmüller Theory and Quadratic Differentials, John Wiley & Sons, Inc., New York, 1987.

    MATH  Google Scholar 

  15. F. Gardiner, On Teichmüller contraction, Proc. Amer. Math. Soc. 118 (1993), 865–875.

    Article  MATH  MathSciNet  Google Scholar 

  16. F. Gardiner and N. Lakic, Efficient smooth quasiconformal mappings, Complex Manifolds and Hyberbolic Geometry, Contemp. Math. 311 (2002), 197–205.

  17. F. Gardiner and N. Lakic, Quasiconformal Teichmüller Theory, Amer. Math. Soc., Providence, RI, 2000.

    MATH  Google Scholar 

  18. N. Lakic, An isometry theorem for quadratic differentials on Riemann surfaces of finite genus, Trans. Amer. Math. Soc. 349 (1997), 2951–2967.

    Article  MATH  MathSciNet  Google Scholar 

  19. M. Lyubich and Y. Minsky, Laminations in holomorphic dynamics, J. Differential Geometry. 47 (1997), 17–94.

    MATH  MathSciNet  Google Scholar 

  20. V. Markovic, Biholomorphic maps between Teichmüller spaces, Duke Math. J. 120 (2003), 405–431.

    Article  MATH  MathSciNet  Google Scholar 

  21. V. Markovic and D. Šarić, Teichmüller mapping class group for the universal hyperbolic solenoid, Trans. Amer. Math. Soc. 358 (2006), 2637–2650.

    Article  MATH  MathSciNet  Google Scholar 

  22. V. Markovic and D. Šarić, The Teichmüller distance between finite index subgroups of PSL 2 (ℤ), Geom. Dedicata, to appear, available at arXiv: 0707.0308.

  23. S. Nag and D. Sullivan, Teichmüller theory and the universal period mappings via quantum calculus and the H 1/2 space of the circle, Osaka J. Math. 32 (1995), 1–34.

    MATH  MathSciNet  Google Scholar 

  24. C. Odden, The baseleaf preserving mapping class group of the universal hyperbolic solenoid, Trans. Amer. Math. Soc. 357 (2005), 1829–1858.

    Article  MATH  MathSciNet  Google Scholar 

  25. R. C. Penner and D. Šarić, Teichmüller theory of the punctured solenoid, Geom. Dedicata 132 (2008), 179–212.

    Article  MATH  MathSciNet  Google Scholar 

  26. E. Reich, On criteria for unique extremality of Teichmüller mappings, Indiana Univ. Math. J. 30 (1981), 289–301.

    Article  Google Scholar 

  27. H. Royden, Automorphisms and isometries of Teichmüller space, 1971 Advances in the Theory of Riemann Surfaces, Princeton Univ. Press, Princeton, NJ, pp. 369–383

  28. E. Reich and K. Strebel, Extremal quasiconformal mappings with given boundary values, in Contributions to Analysis, Academic Press, New York, 1974, pp. 373–391.

    Google Scholar 

  29. W. Rudin, Real and Complex Analysis, third edition, McGraw-Hill, Inc., New York, 1987.

    MATH  Google Scholar 

  30. W. Rudin, Functional Analysis, second edition, McGraw-Hill, Inc., New York, 1991.

    MATH  Google Scholar 

  31. D. Sullivan, Linking the universalities of Milnor-Thurston, Feigenbaum and Ahlfors-Bers, in Topological Methods in Modern Mathematics, Publish or Perish, Houston, TX, 1993, pp. 543–564.

    Google Scholar 

  32. W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), 417–431.

    Article  MATH  MathSciNet  Google Scholar 

  33. L. Zhong, Nonuniqueness of geodesics in infinite dimensional teichmüller spaces, Complex Variables Theory Appl. 16 (1991), 261–272.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Author notes

  1. Dragomir Šarić

    Present address: Department of Mathematics, Queens College of CUNY, 65-30 Kissena Blvd., Flushing, NY, 11367, USA

Authors and Affiliations

  1. Department of Mathematics, University of Southern California, 1042 W. 36th Place, DRB 210, Los Angeles, CA, 90089-1113, USA

    Dragomir Šarić

Authors
  1. Dragomir Šarić
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dragomir Šarić.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Šarić, D. On quasiconformal deformations of the universal hyperbolic solenoid. J Anal Math 105, 303–343 (2008). https://doi.org/10.1007/s11854-008-0038-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-008-0038-0

Keywords

Search

Navigation