Advertisement

Almost simple geodesics on the triply-punctured sphere

  • Moira Chas
  • Curtis T. McMullen
  • Anthony PhillipsEmail author
Article
  • 24 Downloads

Abstract

In this paper we study closed hyperbolic geodesics \(\gamma \) on the triply-punctured sphere \(M = {\widehat{{\mathbb C}}}- \{0,1,\infty \}\) that are almost simple, in the sense that the difference \(\delta = I(\gamma )-L(\gamma )\) between the self-intersection number of \(\gamma \) and its combinatorial (word) length is fixed. We show that for each fixed \(\delta \), the number of almost simple geodesics with \(L(\gamma )=L\) is given by a quadratic polynomial \(p_\delta (L)\), provided \(L \ge \delta + 4\).

Mathematics Subject Classification

30F60 57M05 

Notes

References

  1. 1.
    Bonahon, F.: The geometry of Teichmüller space via geodesic currents. Invent. Math. 92, 139–162 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chas, M., Lalley, S.P.: Self-intersections in combinatorial topology: statistical structure. Invent. Math. 188, 429–463 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chas, M., Phillips, A.: Self-intersection numbers of curves on the punctured torus. Exp. Math. 19, 129–148 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chas, M., Phillips, A.: Self-intersection numbers of curves in the doubly punctured plane. Exp. Math. 21, 26–37 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cohen, M., Lustig, M.: Paths of geodesics and geometric intersection numbers I. In: Combinatorial Group Theory and Topology, Alta, Utah, 1984. Annals of Mathematics Studies, vol. 111, pp. 479–500. Princeton University Press, Princeton (1987)Google Scholar
  6. 6.
    de Graaf, M., Schrijver, A.: Making curves minimally crossing by Reidemeister moves. J. Comb. Theory Ser. B 70, 134–156 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Despré, V., Lazarus, F.: Computing the geometric intersection number of curves. Preprint (2016)Google Scholar
  8. 8.
    Erlandsson, V.: A remark on the word length in surface groups. Preprint (2016)Google Scholar
  9. 9.
    Erlandsson, V., Parlier, H., Souto, J.: Counting curves, and the stable length of currents. Preprint (2016)Google Scholar
  10. 10.
    Erlandsson, V., Souto, J.: Counting curves in hyperbolic surfaces. Geom. Funct. Anal. 26, 729–777 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fathi, A., Laudenbach, F., Poénaru, V.: Travaux de Thurston sur les surfaces, vol. 66. Astérisque, Société Mathématique de France, Paris (1979)Google Scholar
  12. 12.
    Hass, J., Scott, P.: Intersections of curves on surfaces. Isr. J. Math. 51, 90–120 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Imayoshi, Y., Taniguchi, M.: An Introduction to Teichmüller Spaces. Springer, Berlin (1992)CrossRefzbMATHGoogle Scholar
  14. 14.
    Mirzakhani, M.: Growth of the number of simple closed geodesics on hyperbolic surfaces. Ann. Math. 168, 97–125 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mirzakhani, M.: Counting mapping class group orbits on hyperbolic surfaces. Preprint (2016)Google Scholar
  16. 16.
    Reinhart, B.L.: Algorithms for Jordan curves on compact surfaces. Ann. Math. 75, 209–222 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Stillwell, J.: Geometry of Surfaces. Springer, Berlin (1992)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Moira Chas
    • 1
  • Curtis T. McMullen
    • 2
  • Anthony Phillips
    • 1
    Email author
  1. 1.Mathematics DepartmentStony Brook UniversityStony BrookUSA
  2. 2.Mathematics DepartmentHarvard UniversityCambridgeUSA

Personalised recommendations