Stable betti numbers of (Partial) toroidal compactifications of the moduli space of Abelian varieties

We present an algorithm for explicitly computing the number of generators of the stable cohomology algebra of any rationally smooth partial toroidal compactification ofA _g satisfying certain additivity and finiteness properties, in terms of the combinatorics of the corresponding toric fans. In particular, the algorithm determines the stable cohomology of the matroidal partial compactification A ^Matr _g , in terms of simple regular matroids that are irreducible with respect to the 1-sum operation, and their automorphism groups. The algorithm also applies to compute the stable Betti numbers in close to top degree for the perfect cone toroidal compactification APerf g. This suggests the existence of an algebra structure on H ^top−k _stable (A ^Perf _g , Q).