Torelli groups are subgroups of mappings class groups of surfaces that in a sense measure the difference between the moduli spaces of curves and principally polarized abelian varieties. In contrast to mapping class groups, very little is known about the homology of these groups, even in a stable range. I will present a new result on this homology, that computes a large quotient in the stable range, as well as explain how this came about by trying to compute tautological classes in the cohomology of the moduli space of curves with twisted coefficients.

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