Abstract:
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.
