On the Top Homology Group of the Johnson Kernel

The action of the mapping class group $\mathrm{Mod}_g$ of an oriented surface $\Sigma_g$ on the lower central series of $\pi_1(\Sigma_g)$ defines the descending filtration in $\mathrm{Mod}_g$ called the Johnson filtration. The first two terms of it are the Torelli group $\mathcal{I}_g$ and the Johnson kernel $\mathcal{K}_g$. By a fundamental result of Johnson (1985), $\mathcal{K}_g$ is the subgroup of $\mathrm{Mod}_g$ generated by all Dehn twists about separating curves. In 2007, Bestvina, Bux, and Margalit showed the group $\mathcal{K}_g$ has cohomological dimension $2g-3$. We prove that the top homology group $H_{2g-3}(\mathcal{K}_g)$ is not finitely generated. In fact, we show that it contains a free abelian subgroup of infinite rank, hence, the vector space $H_{2g-3}(\mathcal{K}_g,\mathbb{Q})$ is infinite-dimensional. Moreover, we prove that $H_{2g-3}(\mathcal{K}_g,\mathbb{Q})$ is not finitely generated as a module over the group ring $\mathbb{Q}[\mathcal{I}_g]$.