# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1905.04435