Growth of Pseudo-Anosov Conjugacy Classes in Teichm\"{u}ller Space

TL;DR

This paper demonstrates that the growth rate of conjugacy classes of pseudo-Anosov elements in Teichmüller space is roughly half the exponential rate of the entire mapping class group, revealing new asymptotic behavior.

Contribution

It establishes a precise asymptotic growth rate for conjugacy classes of pseudo-Anosov elements, showing they grow at half the rate of the full mapping class group.

Findings

Conjugacy classes of pseudo-Anosov elements grow at rate e^{(h/2)R}.

Existence of a power n for each pseudo-Anosov element with this growth rate.

The result links the growth of conjugacy classes to the overall Teichmüller space volume growth.

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. 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 $e^{\frac{h}{2}R}$.