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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 $H_1(Sigma;mathbb{Z})$.
We calculate the abelianizations of the level $L$ subgroup of the genus $g$ mapping class group and the level $L$ congruence subgroup of the $2g times 2g$ symplectic group for $L$ odd and $g geq 3$.
We show that the mapping class group acts properly on the space of maximal representations of the fundamental group of a closed Riemann surface into G when G = Sp(2n,R), SU(n,n), SO*(2n) or Spin(2,n).
Let $M_n$ be the connect sum of $n$ copies of $S^2 times S^1$. A classical theorem of Laudenbach says that the mapping class group $text{Mod}(M_n)$ is an extension of $text{Out}(F_n)$ by a group $(mathbb{Z}/2)^n$ generated by sphere twists. We prove
For some $g geq 3$, let $Gamma$ be a finite index subgroup of the mapping class group of a genus $g$ surface (possibly with boundary components and punctures). An old conjecture of Ivanov says that the abelianization of $Gamma$ should be finite. In t
We construct a minimal generating set of the level 2 mapping class group of a nonorientable surface of genus $g$, and determine its abelianization for $gge4$.