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

Jitendra Bajpai, Günter Harder, Ivan Horozov, Matias Moya Giusti

Published 2018-12-10Version 1

In this article, several cohomology spaces associated to the arithmetic groups $\mr{SL}_3(\Z)$ and $\mr{GL}_3(\Z)$ with coefficients in any highest weight representation $\m_\lambda$ have been computed, where $\lambda$ denotes their highest weight. Consequently, we obtain detailed information of their Eisenstein cohomology with coefficients in $\m_\lambda$. When $\m_\lambda$ is not self dual, the Eisenstein cohomology coincides with the cohomology of the underlying arithmetic group with coefficients in $\m_\lambda$. 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 $\m_\lambda$. At the end, we employ our study to discuss the existence of ghost classes.