Equivariant group presentations and the second homology group of the Torelli group

We develop a theory of equivariant group presentations and relate them to the second homology group of a group. Our main application says that the second homology group of the Torelli subgroup of the mapping class group is finitely generated as a \(\mathbb {Z}[{{\,\mathrm{Sp}\,}}_{2g}(\mathbb {Z})]\)-module.