Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-24T01:26:27.342Z Has data issue: false hasContentIssue false
  • English
  • Français

Homology stability for unitary groups over S-arithmetic rings

Published online by Cambridge University Press:  05 November 2010

G. Collinet
G. Collinet
Affiliation:
Institut de Recherche Mathématiques Avancées, UMR 7501 de l'Université de Strasbourg et du CNRS. [email protected]
Get access

Abstract

We prove that the homology of unitary groups over rings of S-integers in number fields stabilizes. Results of this kind are well known to follow from the high acyclicity of ad-hoc polyhedra. Given this, we exhibit two simple conditions on the arithmetic of hermitian forms over a ring A relatively to an anti-automorphism which, if they are satisfied, imply the stabilization of the homology of the corresponding unitary groups. When R is a ring of S-integers in a number field K, and A is a maximal R-order in an associative composition algebra F over K, we use the strong approximation theorem to show that both of these properties are satisfied. Finally we take a closer look at the case of On(ℤ[½]).

Keywords

Homology stabilityunitary groupsS-arithmetic groupsadjacency relationsstrong approximation
Type
Research Article
Information
Journal of K-Theory , Volume 8 , Issue 2 , October 2011 , pp. 293 - 322
Copyright
Copyright © ISOPP 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Brown. Brown, K.S.; Cohomology of Groups. GTM 87, Springer Verlag, 1982.Google Scholar
Ca2007. Cathelineau, J.-L.; Homology stability for orthogonal groups over algebraically closed fields. Ann. Sc. E.N.S. 40(3), 487–517, 2007.Google Scholar
Ch1979. Charney, R. M.; Homology stability for GLn over Dedekind domains. Bull. A.M.S. 1(2), 428–431, 1979.CrossRefGoogle Scholar
ATLAS. Conway, J. H., Parker, R. A., Norton, S. P.; The ATLAS of finite groups, 1985.Google Scholar
Dieudonné. Dieudonné, J.; La géométrie des groupes classiques. Ergebnisse der Math. 5, Springer Verlag 1971.Google Scholar
Fiedorowicz-Priddy. Fiedorowicz, Z., Priddy, S.; Homology of classical groups over finite fields and their associated infinite loop spaces, LNM. 674, Springer Verlag, 1978.Google Scholar
F. Frisch, W.; Thesis, available at http://www.uni-math.gwdg.de/preprint/meta/mg.2003.08.html.Google Scholar
HW. Hatcher, A., Wahl, N.; Stabilization for mapping class groups of 3-Manifolds, preprint available at http://arxiv.org/0709.2173v3.Google Scholar
HL. Henn, H.-W., Lannes, J.; in preparation.Google Scholar
J1962. Jacobowitz, R.; Hermitian forms over local fields, Amer. J. Math. 84, 441–465, 1962.CrossRefGoogle Scholar
K1957. Kneser, M.; Klassenzahlen definiter quadratischer Formen. Arch. Math. 8, 241–250, 1957.CrossRefGoogle Scholar
K1966. Kneser, M.; Strong Approximation. In Algebraic groups and discontinuous subgroups, Proc. Symp. Pure Math. 9, AMS, 1966.Google Scholar
MvdK2002. Mizraii, B., van der Kallen, W.; Homology stability for unitary groups. Doc. Math. 7, 143–166, 2002.Google Scholar
O'Meara. O'Meara, O.T.; Introduction to quadratic forms. Grundlehren der math. 117, Springer Verlag 1963.Google Scholar
Rosenberg. Rosenberg, J.; Algebraic K-theory and its applications. GTM 147, Springer Verlag, 1994.Google Scholar
S1964. Shimura, G., Arithmetic of unitary groups, Annals of Math. 79, 369–409, 1964.CrossRefGoogle Scholar
vdK1980. van der Kallen, W.; Homology stability for linear groups. Invent. Math. 60(3), 269–295, 1980.CrossRefGoogle Scholar
V1977. Vogtmann, K., Thesis, University of California, 1977.Google Scholar
V1982. Vogtmann, K., A Stiefel Complex for the orthogonal group of a field, Comment. Math. Helv. 57(1), 11–21, 1982.CrossRefGoogle Scholar