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.