Irreducible Sp-representations and subgroup distortion in the mapping class group (with N. Broaddus and B. Farb) - [pdf] [ps]
preprint 2007, submitted.

Abstract : We prove that various subgroups of the mapping class group $\Mod(\Sigma)$ of a surface $\Sigma$ are at least exponentially distorted. Examples include the Torelli group (answering a question of Hamenst\"adt), the ``point-pushing'' and surface braid subgroups, and the Lagrangian subgroup. For surfaces $\Sigma$ with boundary, we prove that every finitely generated, normal subgroup of $\Mod(\Sigma)$ contained in the Torelli group is at least exponentially distorted. We introduce a method for obtaining lower bounds on distortion. The key is to find certain representations containing some partially hyperbolic matrix, which we reduce via linear algebra to the problem of finding irreducible representations. We then apply this method together with classical symplectic representation theory (e.g. Weyl's classification of $\Sp$-invariant tensors), Morita's development of the ``higher Johnson homomorphisms'', and an extension of Johnson theory to arbitrary subgroups of $\HH_1(\Sigma;\Z)$.