A conjugacy class counting in Teichmuller space

TL;DR

This paper derives asymptotic formulas for counting conjugacy classes of pseudo-Anosov homeomorphisms in Teichmuller space, focusing on their axes intersecting large balls, thus advancing understanding of their distribution.

Contribution

It provides new asymptotic estimates for the number of conjugate pseudo-Anosov homeomorphisms with axes intersecting large regions in Teichmuller space.

Findings

Asymptotic formulas for conjugacy class counts as radius R grows

Quantitative understanding of pseudo-Anosov distribution in Teichmuller space

Extension of counting techniques to geometric structures in surface theory

Abstract

Let $\gamma$ be a pseudo-Anosov homeomorphism and $X$ an element of the Teichmuller space of a genus $g$ surface. In this paper, we find asymptotics for the number of pseudo-Anosov homeomorphisms that are conjugate to $\gamma$ and the axis of their action on Teichmuller space intersects the ball of radius $R$ centered at $X$, as $R$ tends to infinity.