International Seminar on Automorphic Forms

Harald Grobner (University of Vienna): On the cohomology of $SL(n,\mathbb Z)$ beyond 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 show new non-vanishing results in degrees n−1 and n. As a byproduct, we will also answer a question, recently asked by F. Brown for n=6 and explain a phenomenon for n=8, which has been considered by A. Ash. (This is joint work with N. Grbac.)

https://tu-darmstadt.zoom.us/j/68048280736