We prove a quantitative estimate, with a power saving error term, for the number of simple closed geodesics of length at most 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. 3059–3108DOI 10.4171/JEMS/1144