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 $H_1(Sigma;mathbb{Z})$.
Aramayona and Leininger have provided a finite rigid subset $mathfrak{X}(Sigma)$ of the curve complex $mathscr{C}(Sigma)$ of a surface $Sigma = Sigma^n_g$, characterized by the fact that any simplicial injection $mathfrak{X}(Sigma) to mathscr{C}(Sigma)$ is induced by a unique element of the mapping class group $mathrm{Mod}(Sigma)$. In this paper we prove that, in the case of the sphere with $ngeq 5$ marked points, the reduced homology class of the finite rigid set of Aramayona and Leininger is a $mathrm{Mod}(Sigma)$-module generator for the reduced homology of the curve complex $mathscr{C}(Sigma)$, answering in the affirmative a question posed by Aramayona and Leininger. For the surface $Sigma = Sigma_g^n$ with $ggeq 3$ and $nin {0,1}$ we find that the finite rigid set $mathfrak{X}(Sigma)$ of Aramayona and Leininger contains a proper subcomplex $X(Sigma)$ whose reduced homology class is a $mathrm{Mod}(Sigma)$-module generator for the reduced homology of $mathscr{C}(Sigma)$ but which is not itself rigid.
By the work of Harer, the reduced homology of the complex of curves is a fundamental cohomological object associated to all torsion free finite index subgroups of the mapping class group. We call this homology group the Steinberg module of the mapping class group. It was previously known that the curve complex has the homotopy type of a bouquet of spheres. Here, we give the first explicit homologically nontrivial sphere in the curve complex and show that under the action of the mapping class group, the orbit of this homology class generates the reduced homology of the curve complex.
We bound the value of the Casson invariant of any integral homology 3-sphere $M$ by a constant times the distance-squared to the identity, measured in any word metric on the Torelli group $T$, of the element of $T$ associated to any Heegaard splitting of $M$. We construct examples which show this bound is asymptotically sharp.
Johnson has defined a surjective homomorphism from the Torelli subgroup of the mapping class group of the surface of genus $g$ with one boundary component to $wedge^3 H$, the third exterior product of the homology of the surface. Morita then extended Johnsons homomorphism to a homomorphism from the entire mapping class group to ${1/2} wedge^3 H semi sp(H)$. This Johnson-Morita homomorphism is not surjective, but its image is finite index in ${1/2} wedge^3 H semi sp(H)$. Here we give a description of the exact image of Moritas homomorphism. Further, we compute the image of the handlebody subgroup of the mapping class group under the same map.
