Finiteness properties of the Johnson subgroups

The main goal of this note is to provide evidence that the first rational homology of the Johnson subgroup $K_{g,1}$ of the mapping class group of a genus g surface with one marked point is finite-dimensional. Building on work of Dimca-Papadima, we use symplectic representation theory to show that, for all $g > 3$, the completion of $H_1(K_{g,1},\mathbb{Q})$ with respect to the augmentation ideal in the rational group algebra of $\mathbb{Z}^{2g}$ is finite-dimensional. We also show that the terms of the Johnson filtration of the mapping class group have infinite-dimensional rational homology in some degrees in almost all genera, generalizing a result of Akita.