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 $g\ge3$ (respectively, $n\ge3$). During the last 15 years, these questions were answered positively for $g\ge4$ and $n\ge4$, 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.
Categories: math.GR math.GT