# Homological stability for classical groups

## TL;DR

This paper establishes improved homological stability ranges for classical groups such as symplectic, orthogonal, and unitary groups over various fields, extending previous results and including a detailed exposition of Quillen's stability argument.

## Contribution

It provides a new slope 1 stability range for classical groups over fields other than F_2, improving known bounds and applying to automorphism groups with degenerate forms.

## Key findings

- Improved stability range by a factor of 2 over finite fields
- Stability results for orthogonal and unitary groups over specific fields
- Exposition and application of Quillen's slope 1 stability argument

## Abstract

We prove a slope 1 stability range for the homology of the symplectic, orthogonal and unitary groups with respect to the hyperbolic form, over any fields other than $F_2$, improving the known range by a factor 2 in the case of finite fields. Our result more generally applies to the automorphism groups of vector spaces equipped with a possibly degenerate form (in the sense of Bak, Tits and Wall). For finite fields of odd characteristic, and more generally fields in which -1 is a sum of two squares, we deduce a stability range for the orthogonal groups with respect to the Euclidean form, and a corresponding result for the unitary groups. In addition, we include an exposition of Quillen's unpublished slope 1 stability argument for the general linear groups over fields other than $F_2$, and use it to recover also the improved range of Galatius-Kupers-Randal-Williams in the case of finite fields, at the characteristic.