Mathematische Zeitschrift

, Volume 291, Issue 3–4, pp 1175–1196 | Cite as

Almost simple geodesics on the triply-punctured sphere

  • Moira Chas
  • Curtis T. McMullen
  • Anthony PhillipsEmail author


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 



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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

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