Effective mapping class group dynamics I: Counting lattice points in Teichm\"uller space

TL;DR

This paper provides a quantitative estimate with a power-saving error term for counting points in mapping class group orbits within Teichmüller space, improving asymptotic counting results.

Contribution

It introduces effective counting estimates with error bounds for mapping class group orbits in Teichmüller space, extending previous asymptotic results.

Findings

Power-saving error estimates for orbit counts

Effective sector and bisector counting results

Improved asymptotic counting precision

Abstract

We prove a quantitative estimate with a power saving error term for the number of points in a mapping class group orbit of Teichm\"uller space that lie within a Teichm\"uller metric ball of given center and large radius. Estimates of the same kind are also proved for sector and bisector counts. These estimates effectivize asymptotic counting results of Athreya, Bufetov, Eskin, and Mirzakhani.