Effective counting of simple closed geodesics on hyperbolic surfaces

TL;DR

This paper provides a precise estimate, including an error term, for counting simple closed geodesics of bounded length on negatively curved surfaces, utilizing the exponential mixing of the Teichmüller flow.

Contribution

It introduces a new quantitative estimate with a power saving error term for geodesic counting on hyperbolic surfaces, based on exponential mixing properties.

Findings

Established a power saving error term in geodesic counting

Linked geodesic counting to exponential mixing of Teichmüller flow

Enhanced understanding of geodesic distribution on hyperbolic surfaces

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 Teichm\"{u}ller geodesic flow.