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

Putman, Andrew

doi:10.1090/S0002-9939-09-10124-7

A note on the abelianizations of finite-index subgroups of the mapping class group
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Proc. Amer. Math. Soc. 138 (2010), 753-758 Request permission

Abstract:

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 the paper 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.

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