Finite generation of abelianizations of the genus 3 Johnson kernel and the commutator subgroup of the Torelli group for $\mathrm{Out}(F_3)$

Abstract

Let ${\Sigma }_g^b$ be a compact oriented surface of genus $g$ with $b$ boundary components, where $b\in \{0,1\}$. The Johnson kernel ${\mathcal{K}}_g^b$ is the subgroup of the mapping class group $\mathrm{Mod}({\Sigma }_g^b)$ generated by Dehn twists about separating simple closed curves. Let $F_n$ be a free group with $n$ generators. The Torelli group for $\mathrm{Out}(F_n)$ is the subgroup ${\mathrm{IO}}_n\subset \mathrm{Out}(F_n)$ consisting of all outer automorphisms that act trivially on the abelianization of $F_n$. Long standing questions are whether the groups ${\mathcal{K}}_g^b$ and $[{\mathrm{IO}}_n,{\mathrm{IO}}_n]$ or their abelianizations $({\mathcal{K}}_g^b{)}^{\mathrm{ab}}$ and $[{\mathrm{IO}}_n,{\mathrm{IO}}_n{]}^{\mathrm{ab}}$ are finitely generated for $g\ge 3$ (respectively, $n\ge 3$). During the last 15 years, these questions were answered positively for $g\ge 4$ and $n\ge 4$, respectively. Nevertheless, the cases of $g=3$ and $n=3$ remained completely unsettled. In this paper, we prove that the abelianizations $({\mathcal{K}}_3^b{)}^{\mathrm{ab}}$ and $[{\mathrm{IO}}_3,{\mathrm{IO}}_3{]}^{\mathrm{ab}}$ are finitely generated. Our approach is based on a new general sufficient condition for a module over a Laurent polynomial ring to be finitely generated as an abelian group.

37 pages, 6 figures, proof of Theorem D simplified