Skip to main content
SpringerLink
Log in

Criteria for the divergence of pairs of Teichmüller geodesics

  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

We study the asymptotic geometry of Teichmüller geodesic rays. We show that, when the transverse measures to the vertical foliations of the quadratic differentials determining two different rays are topologically equivalent, but are not absolutely continuous with respect to each other, the rays diverge in Teichmüller space.

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. Ahlfors, L.: Lectures on Quasiconformal Mappings. University Lecture Series, AMS, 2nd edn. (2006)

  2. Bers L.: Spaces of degenerating Riemann surfaces. Discontinuous groups and Riemann surfaces Annals of Mathematics Studies 79. Princeton University Press, Princeton, New Jersey (1974)

    Google Scholar 

  3. Choi, Y., Rafi, K., Series, C.: Lines of minima and Teichmüller geodesics To appear, Geometric and Functional Analysis

  4. Fathi, A., Laudenbach, F., Poenaru, V.: Travaux de Thurston sur les surfaces. Asterisque. 66–67 (1979)

  5. Ivanov, N.: Isometries of Teichmüller spaces from the point of view of Mostow rigidity. In: Turaev, V., Vershik, A. (eds.) Topology, Ergodic Theory, Real Algebraic Geometry, pp. 131–149. Amer. Math. Soc. Transl. Ser. 2, V. 202, American Mathematical Society (2001)

  6. Kerckhoff S.: The asymptotic geometry of Teichmüller space. Topology. 19, 23–41 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  7. Masur H.: On a class of geodesics in Teichmüller space. Ann. Math. 102(2), 205–221 (1975)

    Article  MathSciNet  Google Scholar 

  8. Masur H.: Uniquely ergodic quadratic differentials. Comment. Math. Helvetici 55, 255–266 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  9. Masur H.: Hausdorff dimension of the set of nonergodic foliations of a quadratic differential. Duke Math. J. 66, 387–442 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  10. Masur H., Tabachnikov S.: Rational billiards and flat structures. In: Hasselblatt B., Katok, A. (eds.) Handbook of Dynamical Systems, vol. 1A, pp. 1015–1089. Elsevier, Amsterdam (2002)

    Chapter  Google Scholar 

  11. Masur H.: Interval exchange transformations and measured foliations. Ann. Math. 115, 169–200 (1982)

    Article  MathSciNet  Google Scholar 

  12. Maskit B.: Comparison of hyperbolic and extremal lengths. Ann. Acad. Sci. Fenn. Ser. A I Math. 10, 381–386 (1985) MR 87c:30062

    MATH  MathSciNet  Google Scholar 

  13. Minsky Y.: Extremal length estimates and product regions in Teichmüller space. Duke Math. J. 83(2), 249–286 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  14. Minsky Y.: Harmonic maps, length, and energy in Teichmüller space. J. Differ. Geom. 35, 151–217 (1992)

    MATH  MathSciNet  Google Scholar 

  15. Rafi K.: Thick–thin decomposition for quadratic differentials. Math. Res. Lett 14(2), 333–341 (2007)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA

    Anna Lenzhen

  2. Department of Mathematics, University of Chicago, Chicago, IL, 60637, USA

    Howard Masur

Authors
  1. Anna Lenzhen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Howard Masur
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Howard Masur.

Additional information

The second author is supported in part by the NSF.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lenzhen, A., Masur, H. Criteria for the divergence of pairs of Teichmüller geodesics. Geom Dedicata 144, 191–210 (2010). https://doi.org/10.1007/s10711-009-9397-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-009-9397-7

Keywords

Mathematics Subject Classification (2000)

Search

Navigation