Homological stability for classical groups
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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 In addition, we include an exposition of Quillen's unpublished slope 1 stability argument for the general linear groups over fields other than $F_2$, and use it to recover also the improved range of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.
Originalsprog | Engelsk |
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Tidsskrift | Transactions of the American Mathematical Society |
Vol/bind | 373 |
Udgave nummer | 7 |
Sider (fra-til) | 4807-4861 |
ISSN | 0002-9947 |
DOI | |
Status | Udgivet - 2020 |
Links
- http://arxiv.org/pdf/1812.08742v2
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