Central stability for the homology of congruence subgroups and the second homology of Torelli groups

Abstract

We prove a representation stability result for the second homology groups of Torelli subgroups of mapping class groups and automorphism groups of free groups. This strengthens the results of Boldsen–Hauge Dollerup and Day–Putman. We also prove a new representation stability result for the homology of certain congruence subgroups, partially improving upon the work of Putman–Sam. These results follow from a general theorem on syzygies of certain modules with finite polynomial degree.