  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.
