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 Kg,1 of the mapping class group of a genus g surface with one marked point is finite-dimensional. Building on work of Dimca-Papadima [4], we use symplectic representation theory to show that, for all g>3, the completion of H1(Kg,1,Q) with respect to the augmentation ideal in the rational group algebra of Z2g 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.