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

## TL;DR

This paper computes the cohomology and Eisenstein cohomology of $ ext{SL}_3( ext{Z})$ and $ ext{GL}_3( ext{Z})$ with various coefficients, providing explicit descriptions and insights into boundary cohomology and ghost classes.

## Contribution

It provides explicit calculations of cohomology and Eisenstein cohomology for these arithmetic groups with arbitrary highest weight coefficients, extending previous understanding.

## Key findings

- Cohomology spaces are explicitly computed for various coefficients.
- Eisenstein cohomology coincides with group cohomology when coefficients are not self dual.
- Analysis of boundary cohomology and ghost classes is included.

## 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.