JournalsjemsVol. 24, No. 9pp. 3059–3108

Effective counting of simple closed geodesics on hyperbolic surfaces

  • Alex Eskin

    University of Chicago, USA
  • Maryam Mirzakhani

    Stanford University, USA
  • Amir Mohammadi

    University of California, San Diego, La Jolla, USA
Effective counting of simple closed geodesics on hyperbolic surfaces cover
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Abstract

We prove a quantitative estimate, with a power saving error term, for the number of simple closed geodesics of length at most LL on a compact surface equipped with a Riemannian metric of negative curvature. The proof relies on the exponential mixing rate for the Teichmüller geodesic flow.

Cite this article

Alex Eskin, Maryam Mirzakhani, Amir Mohammadi, Effective counting of simple closed geodesics on hyperbolic surfaces. J. Eur. Math. Soc. 24 (2022), no. 9, pp. 3059–3108

DOI 10.4171/JEMS/1144