The Number of Pseudo-Anosov Elements in the Mapping Class Group of a Four-Holed Sphere

TL;DR

This paper analyzes the distribution of pseudo-Anosov elements in the pure mapping class group of a four-holed sphere, showing that their proportion approaches one as the element length increases.

Contribution

It provides explicit growth series for reducible and pseudo-Anosov elements and proves the asymptotic dominance of pseudo-Anosov elements in large metric balls.

Findings

The ratio of pseudo-Anosov elements approaches one as the radius increases.

Explicit growth functions for different element types are derived.

Pseudo-Anosov elements become overwhelmingly prevalent in large groups.

Abstract

We compute the growth series and the growth functions of reducible and pseudo-Anosov elements of the pure mapping class group of the sphere with four holes with respect to a certain generating set. We prove that the ratio of the number of pseudo-Anosov elements to that of all elements in a ball with center at the identity tends to one as the radius of the ball tends to infinity.