A basic question about any group or space is what its (rational) cohomology groups look like. For the mapping class group of an oriented surface of genus g, these groups are fully understood at the low and high degrees. We show that for certain of its finite index, torsion-free subgroups, there is a vast amount of cohomology at degree 4g − 5. This has consequences for the coherent cohomological dimension of the moduli space of curves, which I will outline. This is joint work with Andy Putman.