Homological stability for classical groups

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 \mathbb {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 -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 In addition, we include an exposition of Quillen’s unpublished slope 1 stability argument for the general linear groups over fields other than F 2 \mathbb {F}_2 , and use it to recover also the improved range of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.