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 $F_2$, 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 In addition, we include an exposition of Quillen's unpublished slope 1 stability argument for the general linear groups over fields other than $F_2$, and use it to recover also the improved range of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.

v2: Revision. Now recovers the Galatius-Kupers-Randal-Williams improved stability range for general linear groups over finite fields

References in corpus (1)

• Homological vanishing for the Steinberg representation

Cited by in corpus (6)

• Apartment classes of integral symplectic groups
• A Quillen Stability Criterion for Bounded Cohomology
• Twisted homology stability of O_n for valuation rings
• On the Homology stability range for symplectic groups
• Homology of $\GL_n$ over infinite fields outside the stability range
• Periodicity in the cohomology of finite general linear groups via q-divided powers