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

TL;DR

This paper demonstrates that several subgroups of the mapping class group are exponentially distorted, using representation theory and extended Johnson theory to establish lower bounds on their distortion.

Contribution

It introduces new methods to compute subgroup distortion in the mapping class group, including extending Johnson theory and applying representation theory.

Findings

Torelli group is exponentially distorted

Point-pushing and surface braid subgroups are exponentially distorted

Lagrangian subgroup is exponentially distorted

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 Hamenstadt), 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 $H_1(\Sigma;\mathbb{Z})$.