A recipe for short-word pseudo-Anosovs

Given any generating set of any subgroup $G$ of the mapping class group of a surface, we find an element $f$ with word length bounded by a constant $K$ depending only on the surface, and with the property that the minimal subsurface supporting a power of $f$ is as large as possible for elements of $G$. In particular, if $G$ contains a pseudo-Anosov map, we find one of word length at most $K$. We also find new examples of convex cocompact free subgroups of the mapping class group.