We will describe two projects. The first which is joint with Avner Ash and Paul Gunnells, concerns arithmetic subgroups $\Gamma$ of $G = SL_4(Z)$. We compute the cohomology of $\Gamma \setminus G/K$, focusing on the cuspidal degree $H^5$. We compute a range of Hecke operators on this cohomology. We fi Galois representations that appear to be attached to the Hecke eigenclasses, based on the operators we have computed. We have done this for both non-torsion and torsion classes. The second project, which is joint with Bob MacPherson, is an algorithm for computing the Hecke operators on the cohomology $H^d$ of $\Gamma$ in $SL_n(Z)$ for all $n$ and all $d$.[-]

In this talk, I will give an overview of known results on the stable cohomology of the automorphism groups of free groups with twisted coefficients. After explaining the notion of wheeled PROPs, I will describe a wheeled PROP structure on the stable cohomology of automorphism groups of free groups with some particular coefficients. I will explain how cohomology classes, constructed previously by Kawazumi, can be interpreted using this wheeled PROP structure and I will construct a morphism of wheeled PROPs from a PROP given in terms of functor homology and the wheeled PROP evoked previously. This is joint work with Nariya Kawazumi.[-]