Inventiones mathematicae

, Volume 167, Issue 1, pp 179–222 | Cite as

Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces

Article

Keywords

Modulus Space Riemann Surface Boundary Component Recursive Formula Mapping Class Group 

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References

  1. 1.
    Basmajian, A.: The orthogonal spectrum of a hyperbolic manifold. Am. J. Math. 115, 1139–1159 (1993)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Birman, J.S., Series, C.: Geodesics with bounded intersection number on surfaces are sparsely distributed. Topology 24, 217–225 (1985)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bismut, J.-M., Labourie, F.: Symplectic geometry and the Verlinde formulas. In: Surveys in Differential Geometry: Differential Geometry Inspired by String Theory, vol. 5 of Surv. Differ. Geom., pp. 97–331. Boston, MA: Int. Press 1999Google Scholar
  4. 4.
    Buser, P.: Geometry and Spectra of Compact Riemann Surfaces. Boston: Birkhäuser 1992Google Scholar
  5. 5.
    Canary, R.D., Epstein, D.B.A., Green, P.: Notes on notes of Thurston. In: Analytical and Geometric Aspects of Hyperbolic Space, pp. 3–92. Cambridge :Cambridge University Press 1987Google Scholar
  6. 6.
    Donaldson, S.: Gluing techniques in the cohomology of moduli spaces. In: Topological Methods in Modern Mathematics, pp. 137–170. Houston, TX: Publish or Perish 1993Google Scholar
  7. 7.
    Goldman, W.: The symplectic nature of fundamental groups of surfaces. Adv. Math. 54, 200–225 (1984)MATHCrossRefGoogle Scholar
  8. 8.
    Harer, J.L., Penner, R.C.: Combinatorics of Train Tracks. Annals of Math. Studies, vol. 125. Princeton, NJ: Princeton University Press 1992Google Scholar
  9. 9.
    Harris, J., Morrison, I.: Moduli of Curves. Graduate Texts in Mathematics, vol. 187. New York: Springer 1998Google Scholar
  10. 10.
    Hocking, J., Young, G.: Topology. New York: Dover Publication 1988Google Scholar
  11. 11.
    Imayoshi, Y., Taniguchi, M.: An Introduction to Teichmüller Spaces. Tokyo: Springer 1992Google Scholar
  12. 12.
    Jeffrey, L., Kirwan, F.: Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann surface. Ann. Math. (2) 148, 109–196 (1998)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Kaufmann, R., Manin, Y., Zagier, D.: Higher Weil-Petersson volumes of moduli spaces of stable n-pointed curves. Commun. Math. Phys. 181, 736–787 (1996)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kirwan, F.: Momentum maps and reduction in algebraic geometry. Differ. Geom. Appl. 9, 135–171 (1998)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Kontsevich, M.: Intersection on the moduli space of curves and the matrix airy function. Commun. Math. Phys. 147, 1–23 (1992)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Labourie, F., McShane, G.: Cross ratios and identities for higher Thurston theory. Preprint 2006Google Scholar
  17. 17.
    Manin, Y., Zograf, P.: Invertible cohomological field theories and Weil-Petersson volumes. Ann. Inst. Fourier 50, 519–535 (2000)MATHMathSciNetGoogle Scholar
  18. 18.
    McDuff, D.: Introduction to Symplectic Topology. Providence, RI: Am. Math. Soc. 1999Google Scholar
  19. 19.
    McShane, G.: Simple geodesics and a series constant over Teichmüller space. Invent. Math. 132, 607–632 (1998)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Mirzakhani, M.: Growth of the number of simple closed geodesics on a hyperbolic surface. To appear in Ann. Math.Google Scholar
  21. 21.
    Mirzakhani, M.: Weil-Petersson volumes and intersection theory on the moduli space of curves. To appear in J. Am. Math. Soc.Google Scholar
  22. 22.
    Nakanishi, T., Näätänen, M.: Areas of two-dimensional moduli spaces. Proc. Am. Math. Soc. 129, 3241–3252 (2001)MATHCrossRefGoogle Scholar
  23. 23.
    Penner, R.: Weil-Petersson volumes. J. Differ. Geom. 35, 559–608 (1992)MATHMathSciNetGoogle Scholar
  24. 24.
    Tan, S.P., Wong, Y., Zhang, Y.: Necessary and sufficient conditions for McShane’s identity and variations. Preprint 2004Google Scholar
  25. 25.
    Tan, S.P., Wong, Y., Zhang, Y.: Generalizations of McShane’s identity to hyperbolic cone-surfaces. J. Differ. Geom. 72, 73–111 (2006)MATHMathSciNetGoogle Scholar
  26. 26.
    Wolpert, S.: The Fenchel-Nielsen deformation. Ann. Math. 115, 501–528 (1982)MATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Wolpert, S.: On the homology of the moduli space of stable curves. Ann. Math. (2) 118, 491–523 (1983)MATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Zograf, P.: The Weil-Petersson volume of the moduli space of punctured spheres. In: Mapping Class Groups and Moduli Spaces of Riemann Surfaces. Contemp. Math., vol. 150, pp. 367–372. Providence, RI: Am. Math. Soc. 1993Google Scholar

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© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA

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