$E_\infty$-cells and general linear groups of finite fields

TL;DR

This paper establishes new homological stability results for general linear groups over finite fields by constructing CW approximations in the category of $E_inite$-algebras, leveraging homology computations with the $E_1$-split Steinberg module.

Contribution

It introduces a novel approach using $E_inite$-algebra structures to analyze the homology of linear groups over finite fields, providing new stability results.

Findings

Homological stability for general linear groups over finite fields.

Construction of CW approximations in $E_inite$-algebras.

Homology computations with the $E_1$-split Steinberg module.

Abstract

We prove new homological stability results for general linear groups over finite fields. These results are obtained by constructing CW approximations to the classifying spaces of these groups, in the category of $E_\infty$-algebras, guided by computations of homology with coefficients in the $E_1$-split Steinberg module.