Counting mapping class group orbits under shearing coordinates

TL;DR

This paper investigates the asymptotic count of mapping class group orbits in Teichmüller space using shearing coordinates, extending Mirzakhani's work on hyperbolic structures and orbit counting.

Contribution

It provides new asymptotic formulas for orbit counts in Teichmüller space based on shearing coordinates, connecting geometric parametrizations with orbit enumeration.

Findings

Asymptotic growth rate of orbit counts established

Connection between shearing coordinates and orbit enumeration demonstrated

Extends Mirzakhani's results to new coordinate systems

Abstract

Let $S_{g,n}$ be an oriented surface of genus $g$ with $n$ punctures, where $2g-2+n>0$ and $n>0$. Any ideal triangulation of $S_{g,n}$ induces a global parametrization of the Teichm\"uller space $\mathcal{T}_{g,n}$ called the shearing coordinates. We study the asymptotics of the number of the mapping class group orbits with respect to the standard Euclidean norm of the shearing coordinates. The result is based on the works of Mirzakhani.