Asymptotics for pseudo-Anosov elements in Teichmuller lattices

TL;DR

This paper proves that as the radius grows, the proportion of non-pseudo-Anosov elements in Teichmuller lattices diminishes to zero, highlighting the dominance of pseudo-Anosov elements in large-scale structures.

Contribution

It establishes the asymptotic scarcity of non-pseudo-Anosov elements in Teichmuller lattices, extending understanding of their distribution in geometric group theory.

Findings

Proportion of non-pseudo-Anosov points tends to zero as radius increases.

Elements with bounded translation length form a negligible subset in large balls.

Pseudo-Anosov elements dominate the asymptotic distribution in Teichmuller lattices.

Abstract

A Teichmuller lattice is the orbit of a point in Teichmuller space under the action of the mapping class group. We show that the proportion of lattice points in a ball of radius r which are not pseudo-Anosov tends to zero as r tends to infinity. In fact, we show that if R is a subset of the mapping class group, whose elements have an upper bound on their translation length on the complex of curves, then proportion of lattice points in the ball of radius r which lie in R tends to zero as r tends to infinity.