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A generalization of a theorem of Vogtmann. (English) Zbl 0615.20024
Let A be a Dedekind domain (resp. PID) and let $$G_ n$$ be the sequence of groups $$O_{n,n}(A)$$ or $$Sp_{2n}(A)$$, each embedded in the next in an obvious way. Then for any abelian group of coefficients M, the author proves that the map induced by inclusion from $$H_ i(G_ n;M)$$ to $$H_ i(G_{n+1};M)$$ is surjective for $$n\geq 2i+5$$ (resp. $$n\geq 2i+4)$$ and bijective for $$n\geq 2i+6$$ (resp. $$n\geq 2i+5)$$. The method is fairly standard: construct a highly connected poset on which the G’s act with stabilizers isomorphic to smaller G’s, and then use spectral sequential induction. Many of the constructions and lemmas are given for somewhat more general groups of automorphisms of hyperbolic modules and for A any finite algebra over a commutative ring with noetherian maximal spectrum.
Reviewer: A.Ash

##### MSC:
 20G10 Cohomology theory for linear algebraic groups 20G30 Linear algebraic groups over global fields and their integers
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##### References:
 [1] Bak, A., On modules with quadratic forms, Lecture notes in math., 108, 55-66, (1969) · Zbl 0192.37202 [2] Bak, A., K-theory of forms, () · Zbl 0192.37202 [3] Bass, H., Algebraic K-theory, (1968), Benjamin New York · Zbl 0174.30302 [4] Betley, S., Homological stability for O_{n,n} over a local ring, (1985), Preprint, Notre Dame [5] Charney, R., On the problem of homology stability for congruence subgroups, Comm. in algebra, 12, 47, 2081-2123, (1984) · Zbl 0542.20023 [6] Dwyer, W., Twisted homological stability for general linear groups, Ann. of math., 111, 239-251, (1980) · Zbl 0404.18012 [7] van der Kallen, W., Homology stability for linear groups, Invent. math., 60, 269-295, (1980) · Zbl 0415.18012 [8] Maazen, H., Homology stability for the general linear group, (), Utrecht [9] Quillen, D., Homotopy properties of the poset of nontrivial p-subgroups of a group, Advances in math., 28, 101-128, (1978) · Zbl 0388.55007 [10] Vogtmann, K., Spherical posets and homology stability for O_{n,n}, Topology, 20, 2, 119-132, (1981) · Zbl 0455.20031
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