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

TL;DR

This paper provides evidence supporting Ivanov's conjecture by proving that certain elements are torsion in the abelianization of finite-index subgroups of the mapping class group and showing the finiteness of the abelianization when these subgroups contain a large portion of the Johnson kernel.

Contribution

It proves that powers of Dehn twists are torsion in the abelianization and establishes finiteness of the abelianization when the subgroup contains a large part of the Johnson kernel, supporting Ivanov's conjecture.

Findings

Powers of Dehn twists are torsion in the abelianization.

The abelianization is finite if the subgroup contains a large part of the Johnson kernel.

Supports Ivanov's conjecture on the finiteness of abelianizations.

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 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.