Abstract
We prove a quantitative estimate, with a power saving error term, for the number of simple closed geodesics of length at most $L$ 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.
Highlights
Let g ≥ 2, and let S be a compact Riemann surface of genus g
Problems related to the asymptotic growth rate of the number of closed geodesics on M have been long studied
The growth rate of the number of closed geodesics on a negatively curved compact manifold was studied by Margulis, [Mar]
Summary
Let g ≥ 2, and let S be a compact Riemann surface of genus g. The growth rate of the number of closed geodesics on a negatively curved compact manifold was studied by Margulis, [Mar]. His proof, which is different from the above mentioned works, is based on the mixing property of the Margulis measure for the geodesic flow. The proof of Theorem 1.1 is based on the study of a related counting problem in the space of geodesic measured laminations on S, `a la Mirzakhani. The source of the polynomially effective error term in Theorem 1.1 is the exponential mixing property of the Teichmuller geodesic flow proved by Avila, Gouezel, and Yoccoz, [AGY, AR, AG] We combine this estimate with ideas developed by Margulis in his PhD thesis, [Mar], to prove the following theorem which is of independent interest — see Theorem 7.1 for a more general statement. But not least, we thank the anonymous referee for their careful reading and several helpful comments
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