Growth Rate Of Dehn Twist Lattice Points In Teichm\"{u}ller Space

TL;DR

This paper investigates the growth rates of Dehn twist and multi-twist lattice points in Teichmüller space, revealing they grow at rates significantly slower than the total mapping class group lattice points, with precise asymptotic estimates.

Contribution

It provides the first coarse asymptotic estimates for Dehn twist and multi-twist lattice points in Teichmüller space, contrasting with known results for the full mapping class group.

Findings

Dehn twist lattice points grow roughly as e^{(h/2) R}

Multi-twist lattice points grow at least as R * e^{(h/2) R}

Total mapping class group points grow as e^{h R}

Abstract

Athreya, Bufetov, Eskin and Mirzakhani have shown the number of mapping class group lattice points intersecting a closed ball of radius $R$ in Teichm\"{u}ller space is asymptotic to $e^{hR}$, where $h$ is the dimension of the Teichm\"{u}ller space. In contrast we show the number of Dehn twist lattice points intersecting a closed ball of radius $R$ is coarsely asymptotic to $e^{\frac{h}{2}R}$. Moreover, we show the number multi-twist lattice points intersecting a closed ball of radius $R$ grows coarsely at least at the rate of $R \cdot e^{\frac{h}{2}R}$.