Homological stability for classical groups

Sprehn, David; Wahl, Nathalie

doi:10.1090/tran/8030

Homological stability for classical groups
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by David Sprehn and Nathalie Wahl PDF

Trans. Amer. Math. Soc. 373 (2020), 4807-4861 Request permission

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 $\mathbb {F}_2$, 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 $-1$ 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In addition, we include an exposition of Quillen’s unpublished slope 1 stability argument for the general linear groups over fields other than $\mathbb {F}_2$, and use it to recover also the improved range of Galatius–Kupers–Randal-Williams in the case of finite fields, at the characteristic.

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