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Homology of \(O(n)\) and \(O^1(1,n)\) made discrete: An application of edgewise subdivision. (English) Zbl 0910.20026
Let \(O(n)\) be the group of real orthogonal \(n\times n\)-matrices and \(O^1(1,n)\) the group of isometries of hyperbolic \(n\)-space. It is proved that the standard injection of \(O(n)\) into \(O^1(1,n)\) induces an isomorphism on homology in degrees \(\leq n-1\), where homology means group homology for the corresponding groups made discrete. This result establishes a conjecture of C. H. Sah [Appendix A in Comment. Math. Helv. 61, 308-347 (1986; Zbl 0607.57025)]. It has a number of consequences: It shows that the statement of the Friedlander-Milnor conjecture for \(O(n)\) is equivalent to that for \(O^1(1,n)\) in the stable range. Secondly it reduces the calculation of the scissors congruence group for the 3-sphere to a well known problem in algebraic K-theory; it implies in particular that this group is a rational vector space and gives necessary and sufficient conditions for determining the scissors congruence classes of spherical polyhedra with vertices whose coordinates are algebraic integers. Among the tools is edgewise subdivision which has apparently not been used in this context before.

MSC:
20G10 Cohomology theory for linear algebraic groups
20J05 Homological methods in group theory
22E99 Lie groups
52B45 Dissections and valuations (Hilbert’s third problem, etc.)
57T10 Homology and cohomology of Lie groups
55R45 Homology and homotopy of \(B\mathrm{O}\) and \(B\mathrm{U}\); Bott periodicity
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