Homology of classical Lie groups made discrete. III
[For Parts I and II see Comment. Math. Helv. 61, 308-347 (1986; Zbl 0607.57025) and J. Algebra 113, No.1, 215-260 (1988; Zbl 0657.55022), respectively.] In this paper the author continues his interesting discussions of the low-degree homology of the classical groups and its connections with algebraic K-theory and with scissors congruences. Three of his principal results, in simplified form, are as follows: Theorem 3.0. \(H_ 3(SL(2,D))\to H_ 2(SL(3,D))\) is injective. Theorem 4.1. For any infinite field F, \(K_ 3(F)^{ind}\cong H_ 0(F^{\times},H_ 3(SL(2,F)))\) mod \({\mathcal C}_ 6\), where \(K_ 3(F)\cong K_ 3(F)^{ind}\coprod K^ M_ 3(F)mod {\mathcal C}_ 2\). \({\mathcal C}_ n\) denotes the Serre class of torsion abelian groups annihilated by suitable powers of n, where \(K^ M_*\) is Milnor K- theory and the indecomposable part of Algebraic K-theory, \(K_ 3(F)^{ind}\) is defined as \(K_ 3(F)/im K^ M_ 3(F).\) Theorem 4.16. The scissors congruence group \({\mathcal P}{\mathcal H}^ 3\) in hyperbolic 3-space is a \({\mathbb{Q}}\)-vector space (i.e., uniquely divisible).
scissors congruence group in hyperbolic 3-space
scientific article; zbMATH DE number 4121049
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