Commentarii Mathematici Helvetici


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Volume 86, Issue 3, 2011, pp. 537–556
DOI: 10.4171/CMH/233

Published online: 2011-06-01

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

Nathan Broaddus[1], Benson Farb[2] and Andrew Putman[3]

(1) University of Chicago, USA
(2) University of Chicago, USA
(3) Rice University, Houston, USA

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 $\mathrm{H}_1(\Sigma;\mathbb{Z})$.

Keywords: Mapping class group, Torelli group, subgroup distortion, symplectic representation, Johnson homomorphism

Broaddus Nathan, Farb Benson, Putman Andrew: Irreducible Sp-representations and subgroup distortion in the mapping class group. Comment. Math. Helv. 86 (2011), 537-556. doi: 10.4171/CMH/233