We prove that various subgroups of the mapping class group of a surface 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 .
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Nathan Broaddus, Benson Farb, Andrew Putman, Irreducible Sp-representations and subgroup distortion in the mapping class group. Comment. Math. Helv. 86 (2011), no. 3, pp. 537–556DOI 10.4171/CMH/233