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We prove a representation stability result for the second homology groups ofTorelli subgroups of mapping class groups and automorphism groups of freegroups. This strengthens the results of Boldsen-Hauge Dollerup and Day-Putman.We also prove a new representation stability result for the homology of certaincongruence subgroups, partially improving upon the work of Putman-Sam. Theseresults follow from a general theorem on syzygies of certain modules withfinite polynomial degree.

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