Growth Of Dehn Twist and Pseudo-Anosov Conjugacy Classes in Teichmüller Space

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 this thesis, we first 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 of all multi-twists lattice points intersecting a closed ball of radius $R$ grows coarsely at least at the rate of $R \cdot e^{\frac{h}{2}R}$. Furthermore, we show for any pseudo-Anosov mapping class $f$, there exists a power $n$, such that the number of lattice points of the $f^n$ conjugacy class intersecting a closed ball of radius $R$ is coarsely asymptotic to $e^{\frac{h}{2}R}$.