On homology of Torelli groups

The mapping class groups of oriented surfaces are important examples of groups whose properties are closely related to geometry and topology of moduli spaces, topology of 3-manifolds, automorphisms of free groups. The mapping class group of a closed oriented surface contains two important subgroups, the Torelli group, which consists of all mapping classes that act trivially on the homology of the surface, and the Johnson kernel, which is the subgroup generated by all Dehn twists about separating curves. The talk will be devoted to results on homology of these two subgroups. Namely, we will show that the $k$-dimensional homology group of the genus g Torelli group is not finitely generated, provided that k is in range from $2g-3$ and $3g-5$ (the case $3g-5$ was previously known by a result of Bestvina, Bux, and Margalit), and the $(2g-3)$-dimensional homology group the genus g Johnson kernel is also not finitely generated. The proof is based on a detailed study of the spectral sequences associated with the actions of these groups on the complex of cycles constructed by Bestvina, Bux, and Margalit.