Boundary and Eisenstein Cohomology of ${\mathrm{SL}}_3(\mathbb{Z})$

Abstract

In this article, several cohomology spaces associated to the arithmetic groups ${\mathrm{SL}}_3(\mathbb{Z})$ and ${\mathrm{GL}}_3(\mathbb{Z})$ with coefficients in any highest weight representation ${\mathcal{M}}_{\lambda }$ have been computed, where $\lambda$ denotes their highest weight. Consequently, we obtain detailed information of their Eisenstein cohomology with coefficients in ${\mathcal{M}}_{\lambda }$. When ${\mathcal{M}}_{\lambda }$ is not self dual, the Eisenstein cohomology coincides with the cohomology of the underlying arithmetic group with coefficients in ${\mathcal{M}}_{\lambda }$. In particular, for such a large class of representations we can explicitly describe the cohomology of these two arithmetic groups. We accomplish this by studying the cohomology of the boundary of the Borel-Serre compactification and their Euler characteristic with coefficients in ${\mathcal{M}}_{\lambda }$. At the end, we employ our study to discuss the existence of ghost classes.

42 Pages, 1 Figure, 9 Tables, To Appear in Math. Annalen