A recipe for short-word pseudo-Anosovs

TL;DR

This paper presents a method to construct short-word pseudo-Anosov elements in subgroups of the mapping class group, providing bounds on their word length and identifying large minimal subsurfaces, with applications to convex cocompact subgroups.

Contribution

Introduces a construction for short-word pseudo-Anosovs with bounds depending only on the surface, and finds new convex cocompact free subgroups.

Findings

Bounded word length for pseudo-Anosovs depending on surface

Existence of elements with maximal minimal subsurface support

New examples of convex cocompact free subgroups

Abstract

Given any generating set of any pseudo-Anosov-containing subgroup of the mapping class group of a surface, we construct a pseudo-Anosov with word length bounded by a constant depending only on the surface. More generally, in any subgroup G we find an element f with the property that the minimal subsurface supporting a power of f is as large as possible for elements of G; the same constant bounds the word length of f. Along the way we find new examples of convex cocompact free subgroups of the mapping class group.