Harald Grobner: On the cohomology of SL(n,Z) right outside the stable range

The cohomology of the group $SL(n,\mathbb Z)$, $n>1$, plays a fundamental role in geometry, topology and representation theory, while yielding many number theoretical applications: For instance, Borel used his description of $H^*(SL(n,\mathbb Z))$ to compute the algebraic K-theory of the integers; whereas the (non-)vanishing of $H^*(SL(n,\mathbb Z))$ tells a lot about the existence of certain automorphic forms. In this talk we will study the cohomology of $SL(n,\mathbb Z)$, „right outside“ of what one calls the stable range. More precisely, we will present new results on the internal automorphic structure of cohomology classes in degrees $n-1$ and $n$. As a by-product, we obtain new results of non-vanishing of cohomology in degree $n-1$. (This is joint work with N. Grbac.)