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Lengths of geodesics on non-orientable hyperbolic surfaces

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Abstract

We give an identity involving sums of functions of lengths of simple closed geodesics, known as a McShane identity, on any non-orientable hyperbolic surface with boundary which generalises Mirzakhani’s identities on orientable hyperbolic surfaces with boundary.

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References

  1. Bowditch, B.H.: A proof of McShane’s identity via Markoff triples. Bull. London Math. Soc. 28, 73–78 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bowditch, B.H.: Markoff triples and quasi-Fuchsian groups. Proc. London Math. Soc. 77(3), 697–736 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. Jost, J. : Minimal surfaces and Teichmüller theory. Tsing Hua lectures on geometry and analysis (Hsinchu, 1990–1991), pp. 149–211. International Press, Cambridge, MA (1997)

  4. McShane, G.: A remarkable identity for lengths of curves. Ph.D. Thesis, University of Warwick (1991)

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Mirzakhani, M.: Growth of the number of simple closed geodesics on hyperbolic surfaces. Ann. Math. (to appear)

  7. Mirzakhani, M.: Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces. Invent. Math. 167, 179–222 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  8. Mirzakhani, M.: Weil-Petersson volumes and intersection theory on the moduli space of curves. J. Am. Math. Soc. 20, 1–23 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Mondello, G.: Triangulated Riemann surfaces with boundary and the Weil-Petersson Poisson structure. arXiv:math/0610698

  10. Penner, R.C.: The decorated Teichmüller space of punctured surfaces. Comm. Math. Phys. 113, 299–339 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  11. Seppälä, M.: Teichmüller spaces of Klein surfaces. Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 15, 37 pp. (1978)

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  13. Wahl, N.: Homological stability for the mapping class groups of non-orientable surfaces. Preprint (2006)

  14. Wolpert, S.: On the Weil-Petersson geometry of the moduli space of curves. Am. J. Math. 107, 969–997 (1985)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Melbourne, Melbourne, 3010, Australia

    Paul Norbury

  2. Boston University, 111 Cummington St, Boston, MA, 02215, USA

    Paul Norbury

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  1. Paul Norbury
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Correspondence to Paul Norbury.

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Norbury, P. Lengths of geodesics on non-orientable hyperbolic surfaces. Geom Dedicata 134, 153–176 (2008). https://doi.org/10.1007/s10711-008-9251-3

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  • DOI: https://doi.org/10.1007/s10711-008-9251-3

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