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.