Geometriae Dedicata

, Volume 134, Issue 1, pp 75–90 | Cite as

McShane’s identity, using elliptic elements

  • Thomas A. Schmidt
  • Mark Sheingorn
Original Paper


We introduce a new method to establish McShane’s Identity. Elliptic elements of order two in the Fuchsian group uniformizing the quotient of a fixed once-punctured hyperbolic torus act so as to exclude points as being highest points of geodesics. The highest points of simple closed geodesics are already given as the appropriate complement of the regions excluded by those elements of order two that factor hyperbolic elements whose axis projects to be simple. The widths of the intersection with an appropriate horocycle of the excluded regions sum to give McShane’s value of 1/2. The remaining points on the horocycle are highest points of simple open geodesics, we show that this set has zero Hausdorff dimension.


McShane’s identity Hyperbolic surface Geodesic length 

Mathematics Subject Classification (2000)

57M50 20H10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Akiyoshi, H., Miyachi, H., Sakuma, M.: A refinement of McShane’s identity for quasifuchsian punctured torus groups. In: In the tradition of Ahlfors and Bers. Contemp. Math., 355, vol. III, pp. 21–40. Amer. Math. Soc., Providence, RI (2004)Google Scholar
  2. 2.
    Beardon A.F., Lehner J., Sheingorn M. (1986) Closed geodesics on a Riemann surface with application to the Markov spectrum. Trans. Am. Math. Soc. 295(2): 635–647CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Bonahon, F., Zhu, X.: The metric space of geodesic laminations on a surface. II. Small surfaces. In: Proceedings of the Casson Fest. Geom. Topol. Monogr., 7, pp. 509–547. Geom. Topol. Publ., Coventry (2004)Google Scholar
  4. 4.
    Bowditch B. (1996) A proof of McShane’s identity via Markoff triples. Bull. London Math. Soc. 28(1): 73–78CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Bowditch B., Maclachlan C., Reid A. (1995) Arithmetic hyperbolic surface bundles. Math. Ann. 302: 31–60CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Cohn H. (1955) Approach to Markoff’s minimal forms through modular functions. Ann. Math. 61(2): 1–12Google Scholar
  7. 7.
    Do, N., Norbury, P.: Weil-Petersson volumes and cone surfaces. Preprint: Arxiv: math.GT/06033406 (2006)Google Scholar
  8. 8.
    Goodman-Strauss C., Rieck Y. (2007) Simple geodesics on a punctured surface. Top. Appl. 154(1): 155–165CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Haas A. (1986) Diophantine approximation on hyperbolic Riemann surfaces. Acta Math. 156: 33–82CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Jørgensen T., Marden A. (1979) Two doubly degenerate groups. Quart. J. Math. 30: 143–156CrossRefGoogle Scholar
  11. 11.
    McShane G. (1998) Simple geodesics and a series constant over Teichmuller space. Invent. Math. 132(3): 607–632CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    McShane G. (2004) Weierstrass points and simple geodesics. Bull. London Math. Soc. 36: 181–187CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    McShane, G.: Length series on Teichmüller space. Preprint: Arxiv: math.GT/0403041 (2004)Google Scholar
  14. 14.
    Mirzakhani M. (2008) Growth of the number of simple closed geodesics on hyperbolic surfaces. Ann. Math. 167(3): 185–215MathSciNetGoogle Scholar
  15. 15.
    Mirzakhani M. (2007) Weil-Petersson volumes and intersection theory on the moduli space of curves. J. Am. Math. Soc. 20(1): 1–23CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Mirzakhani M. (2007) Simple geodesics and Weil-Petersson volumes of the moduli spaces of bordered Riemann surfaces. Invent. Math. 167(1): 179–222CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Rivin I. (2005) A simpler proof of Mirzakhani’s simple curve asymptotics. Geom. Dedicata 114: 229–235CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Schmidt A.L. (1976) Minimum of quadratic forms with respect to Fuchsian groups. I. J. Reine Angew. Math. 286/287: 341–368Google Scholar
  19. 19.
    Schmidt T.A., Sheingorn M. (2003) Parametrizing simple closed geodesy on \({\Gamma^3 \backslash {\mathcal H}}\). J. Aust. Math. Soc. 74(1): 43–60MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Schmidt T.A., Sheingorn M. (2007) Low height geodesics on \({\Gamma^3 \backslash {\mathcal H}}\): height formulas and examples. Int. J. Number Theory 3(3): 475–501CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    Sheingorn M. (1985) Characterization of simple closed geodesics on Fricke surfaces. Duke Math. J. 52: 535–545CrossRefMathSciNetMATHGoogle Scholar
  22. 22.
    Tan S.P., Wong Y.L., Zhang Y. (2006) Generalizations of McShane’s identity to hyperbolic cone-surfaces. J. Differ. Geom. 72(1): 73–112MathSciNetMATHGoogle Scholar
  23. 23.
    Tan S.P., Wong Y.L., Zhang Y. (2008) Generalized Markoff maps and McShane’s identity. Adv. Math. 217: 761–813CrossRefMathSciNetMATHGoogle Scholar
  24. 24.
    Vulakh L.Yu. (1997) The Markov spectra for triangle groups. J. Number Theory 67(1): 11–28CrossRefMathSciNetMATHGoogle Scholar
  25. 25.
    Wolpert S. (1983) On the Kähler form of the moduli space of once punctured tori. Comment. Math. Helv. 58(2): 246–256CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Oregon State UniversityCorvallisUSA
  2. 2.CUNY - Baruch CollegeNew YorkUSA

Personalised recommendations