Irreducible Sp-representations and subgroup distortion in the mapping class group

Abstract

We prove that various subgroups of the mapping class group $\mathrm{Mod}(\Sigma )$ of a surface $\Sigma$ are at least exponentially distorted. Examples include the Torelli group (answering a question of Hamenstädt), the “point-pushing” and surface braid subgroups, and the Lagrangian subgroup. Our techniques include a method to compute lower bounds on distortion via representation theory and an extension of Johnson theory to arbitrary subgroups of ${\mathrm{H}}_1(\Sigma ;\mathbb{Z})$.