Geometriae Dedicata

, Volume 115, Issue 1, pp 1–18 | Cite as

The Weil–Petersson Visual Sphere

  • Jeffrey F. Brock


We formulate and describe a visual compactification of the Teichmüller space by Weil–Petersson geodesic rays emanating from a point X. We focus on analogies with Bers’s compactification: due to noncompleteness, finite rays correspond to cusps, and such cusps are dense in the visual sphere. By analogy with a result of Kerckhoff and Thurston, we show the natural action of the mapping class group does not extend continuously to the visual compactification. We conclude with examples that distinguish the visual boundary from Bers’s boundary for Teichmüller space


Teichmüller space Weil–Petersson metric hyperbolic 3-manifold 

Mathematics Subject Classification (2000)



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  1. 1.
    Abikoff, W. 1977Degenerating families of Riemann surfacesAnn. of Math.1052944Google Scholar
  2. 2.
    Bers, L. 1960Simultaneous uniformizationBull. Amer. Math. Soc.669497Google Scholar
  3. 3.
    Bers, L. 1970On boundaries of Teichmüller spaces and on kleinian groups: IAnn. of Math.91570600Google Scholar
  4. 4.
    Bers L. Spaces of degenerating Riemann surfaces, In: Discontinuous Groups and Riemann Surfaces, Ann. of Math Stud. 76, Princeton University Press, 1974, pp. 43–55.Google Scholar
  5. 5.
    Bridson, M., Haefliger, A. 1999Metric Spaces of Non-Positive CurvatureSpringer-VerlagNew YorkGoogle Scholar
  6. 6.
    Brock, J. 2001Boundaries of Teichmüller spaces and geodesic laminationsDuke Math. J.106527552CrossRefGoogle Scholar
  7. 7.
    Brock, J. 2001Iteration of mapping classes and limits of hyperbolic 3-manifoldsInvent. Math.143523570Google Scholar
  8. 8.
    Brock, J. 2003The Weil–Petersson metric and volumes of 3-dimensional hyperbolic convex coresJ. Amer. Math. Soc.16495535CrossRefGoogle Scholar
  9. 9.
    Buser, P. 1992Geometry and Spectra of Compact Riemann SurfacesBirkhauserBostonGoogle Scholar
  10. 10.
    Daskolopoulos, G., Wentworth, R. 2003Classification of Weil–Petersson isometriesAmer. J. Math.125941975Google Scholar
  11. 11.
    Harvey, W.J.: Boundary structure of the modular group, In: Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference, Ann. of Math Stud. 97, Princeton University Press, 1981.Google Scholar
  12. 12.
    Imayoshi, Y., Taniguchi, M. 1992An Introduction to Teichmüller SpacesSpringer-VerlagNew YorkGoogle Scholar
  13. 13.
    Kerckhoff, S. 1980The asymptotic geometry of Teichmüller spaceTopology192341CrossRefGoogle Scholar
  14. 14.
    Kerckhoff, S., Thurston, W. 1990Non-continuity of the action of the modular group at Bers’ boundary of Teichmüller spaceInvent. Math.1002548CrossRefGoogle Scholar
  15. 15.
    Masur, H., Thurston, W. 1976The extension of the Weil–Petersson metric to the boundary of Teichmüller spaceDuke. Math.43623635CrossRefGoogle Scholar
  16. 16.
    Masur, H., Minsky, Y. 1999Geometry of the complex of curves I: hyperbolicityInvent. Math.138103149CrossRefGoogle Scholar
  17. 17.
    McMullen, C. 1991Cusps are denseAnn. Math.133217247Google Scholar
  18. 18.
    McMullen, C.: Renormalization and 3-Manifolds Which Fiber Over the Circle, Ann. of Math. Stud. 142, Princeton University Press, 1996Google Scholar
  19. 19.
    McMullen, C. 2000The moduli space of Riemann surfaces is Kähler hyperbolicAnn. of Math.151327357Google Scholar
  20. 20.
    Wolpert, S. 1975Noncompleteness of the Weil–Petersson metric for Teichmüller spacePacific J. Math.61573577Google Scholar
  21. 21.
    Wolpert, S. 1977The finite Weil–Petersson diameter of Riemann spacePacific J. Math.70281288Google Scholar
  22. 22.
    Wolpert, S. 1987Geodesic length functions and the Nielsen problemJ. Differential Geom.25275296Google Scholar
  23. 23.
    Wolpert, S. 1990The hyperbolic metric and the geometry of the universal curveJ. Differential Geom.31417472Google Scholar
  24. 24.
    Wolpert, S.: Geometry of the Weil–Petersson completion of Teichmüller space, In: Surveys in Differential Geometry, Vol. VIII (Boston, MA, 2002), International Press, Somerville, MA, 2003, pp. 357–393.Google Scholar
  25. 25.
    Yamada S. Weil–Petersson completion of Teichmüller spaces and mapping class group actions. Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Department of MathematicsBrown UniversityProvidenceU.S.A

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