Improved homological stability for certain general linear groups

TL;DR

This paper establishes improved homological stability results for general linear groups over integers, Gaussian integers, and Eisenstein integers, with specific stability slopes depending on the coefficient ring used.

Contribution

It provides new stability slopes for these groups with different coefficient rings, advancing understanding of their homological properties.

Findings

Homological stability of slope 1 over racket; coefficients in racket;

Homological stability of slope 2/3 over racket; coefficients in racket;

Applicable to general linear groups over integers, Gaussian, and Eisenstein integers.

Abstract

We prove that the general linear groups of the integers, Gaussian integers, and Eisenstein integers satisfy homological stability of slope 1 when using $\mathbb{Z}[1/2]$-coefficients and of slope $2/3$ when using $\mathbb{Z}$-coefficients.