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

Andrew Putman

Arxiv ID: 0812.0017•Last updated: 6/8/2020

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.

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