Abstract : 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 $d$.

Keywords : automorphic forms; cohomology of arithmetic groups; Hecke operators; Galois representations