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

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References

  1. Basmajian, A.: The orthogonal spectrum of a hyperbolic manifold. Am. J. Math. 115, 1139–1159 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  2. Birman, J.S., Series, C.: Geodesics with bounded intersection number on surfaces are sparsely distributed. Topology 24, 217–225 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  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 1999

  4. Buser, P.: Geometry and Spectra of Compact Riemann Surfaces. Boston: Birkhäuser 1992

  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 1987

  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 1993

  7. Goldman, W.: The symplectic nature of fundamental groups of surfaces. Adv. Math. 54, 200–225 (1984)

    Article  MATH  Google Scholar 

  8. Harer, J.L., Penner, R.C.: Combinatorics of Train Tracks. Annals of Math. Studies, vol. 125. Princeton, NJ: Princeton University Press 1992

  9. Harris, J., Morrison, I.: Moduli of Curves. Graduate Texts in Mathematics, vol. 187. New York: Springer 1998

  10. Hocking, J., Young, G.: Topology. New York: Dover Publication 1988

  11. Imayoshi, Y., Taniguchi, M.: An Introduction to Teichmüller Spaces. Tokyo: Springer 1992

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  14. Kirwan, F.: Momentum maps and reduction in algebraic geometry. Differ. Geom. Appl. 9, 135–171 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kontsevich, M.: Intersection on the moduli space of curves and the matrix airy function. Commun. Math. Phys. 147, 1–23 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  16. Labourie, F., McShane, G.: Cross ratios and identities for higher Thurston theory. Preprint 2006

  17. Manin, Y., Zograf, P.: Invertible cohomological field theories and Weil-Petersson volumes. Ann. Inst. Fourier 50, 519–535 (2000)

    MATH  MathSciNet  Google Scholar 

  18. McDuff, D.: Introduction to Symplectic Topology. Providence, RI: Am. Math. Soc. 1999

  19. McShane, G.: Simple geodesics and a series constant over Teichmüller space. Invent. Math. 132, 607–632 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  20. Mirzakhani, M.: Growth of the number of simple closed geodesics on a hyperbolic surface. To appear in Ann. Math.

  21. Mirzakhani, M.: Weil-Petersson volumes and intersection theory on the moduli space of curves. To appear in J. Am. Math. Soc.

  22. Nakanishi, T., Näätänen, M.: Areas of two-dimensional moduli spaces. Proc. Am. Math. Soc. 129, 3241–3252 (2001)

    Article  MATH  Google Scholar 

  23. Penner, R.: Weil-Petersson volumes. J. Differ. Geom. 35, 559–608 (1992)

    MATH  MathSciNet  Google Scholar 

  24. Tan, S.P., Wong, Y., Zhang, Y.: Necessary and sufficient conditions for McShane’s identity and variations. Preprint 2004

  25. Tan, S.P., Wong, Y., Zhang, Y.: Generalizations of McShane’s identity to hyperbolic cone-surfaces. J. Differ. Geom. 72, 73–111 (2006)

    MATH  MathSciNet  Google Scholar 

  26. Wolpert, S.: The Fenchel-Nielsen deformation. Ann. Math. 115, 501–528 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  27. Wolpert, S.: On the homology of the moduli space of stable curves. Ann. Math. (2) 118, 491–523 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  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. 1993

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Mirzakhani, M. Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces. Invent. math. 167, 179–222 (2007). https://doi.org/10.1007/s00222-006-0013-2

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