Counting closed geodesics in Moduli space

TL;DR

This paper determines the growth rate of the number of closed geodesics in Moduli space and pseudo-Anosov elements of the mapping class group with bounded translation length, as the length bound tends to infinity.

Contribution

It provides the asymptotic count of closed geodesics and pseudo-Anosov elements in Moduli space, advancing understanding of their distribution and growth.

Findings

Asymptotic formula for the number of closed geodesics as R approaches infinity

Asymptotic growth rate of pseudo-Anosov elements in the mapping class group

Quantitative understanding of geodesic distribution in Moduli space

Abstract

We compute the asymptotics, as R tends to infinity, of the number of closed geodesics in Moduli space of length at most R, or equivalently the number of pseudo-Anosov elements of the mapping class group of translation length at most R.