On Borel's stable range of the twisted cohomology of $\mathrm{GL}(n,\mathbb{Z})$

TL;DR

This paper refines Borel's stable range for the twisted cohomology of the general linear group over integers, using adapted methods to extend the stability results to a broader class of representations.

Contribution

It improves the known stable range for the cohomology of $ ext{GL}(n,bZ)$ with algebraic coefficients by adapting existing methods to a wider class of representations.

Findings

Computed an improved stable range for cohomology

Extended Borel's stability results to more representations

Enhanced understanding of the cohomological stability of $ ext{GL}(n,bZ)$

Abstract

Borel's stability and vanishing theorem gives the stable cohomology of $\mathrm{GL}(n,\mathbb{Z})$ with coefficients in algebraic $\mathrm{GL}(n,\mathbb{Z})$-representations. We compute the improved stable range that Borel remarked about. In order to further improve Borel's stable range, we adapt the method of Kupers-Miller-Patzt to any algebraic $\mathrm{GL}(n,\mathbb{Z})$-representations.