A note on the abelianizations of finite-index subgroups of the mapping class group

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 this note, we prove two theorems supporting this conjecture. For the first, let $T_x$ denote the Dehn twist about a simple closed curve $x$. For some $n \geq 1$, we have $T_x^n \in \Gamma$. We prove that $T_x^n$ is torsion in the abelianization of $\Gamma$. Our second result shows that the abelianization of $\Gamma$ is finite if $\Gamma$ contains a "large chunk" (in a certain technical sense) of the Johnson kernel, that is, the subgroup of the mapping class group generated by twists about separating curves. This generalizes work of Hain and Boggi.Comment: 6 pages, 1 figure; a few revisions; to appear in Proc. Amer. Math. So

Partial Torelli groups and homological stability

We prove a homological stability theorem for the subgroup of the mapping class group acting as the identity on some fixed portion of the first homology group of the surface. We also prove a similar theorem for the subgroup of the mapping class group preserving a fixed map from the fundamental group to a finite group, which can be viewed as a mapping class group version of a theorem of Ellenberg-Venkatesh-Westerland about braid groups. These results require studying various simplicial complexes formed by subsurfaces of the surface, generalizing work of Hatcher-Vogtmann.Comment: 58 pages, 59 figures; fixed some typo