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

Abstract

AbstractIn this article, several cohomology spaces associated to the arithmetic groups $$\mathrm {SL}_3({\mathbb {Z}})$$SL3(Z) and $$\mathrm {GL}_3({\mathbb {Z}})$$GL3(Z) with coefficients in any highest weight representation $${\mathcal {M}}_\lambda $$Mλ have been computed, where $$\lambda $$λ denotes their highest weight. Consequently, we obtain detailed information of their Eisenstein cohomology with coefficients in $${\mathcal {M}}_\lambda $$Mλ. When $${\mathcal {M}}_\lambda $$Mλ is not self dual, the Eisenstein cohomology coincides with the cohomology of the underlying arithmetic group with coefficients in $${\mathcal {M}}_\lambda $$Mλ. 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 $${\mathcal {M}}_\lambda $$Mλ. At the end, we employ our study to discuss the existence of ghost classes.