Representation stability for the first homology of congruence subgroups

Abstract

We study sequences of modular representations of the symplectic and special linear groups over finite fields arising from the first homology of congruence subgroups of mapping class groups and automorphism groups of free groups, as well as from the module of coinvariants for the abelianization of the Torelli group. In each case, we determine the composition factors and their multiplicities, and establish periodic representation stability in the sense of Church--Farb. We apply our results to study flat line bundles over the moduli space of curves with level 2 structure arising from spin structures on the underlying surface.