Department of Mathematics, Columbia University, New York, NY 10027.
Proc Natl Acad Sci U S A. 1986 Aug;83(15):5364-6. doi: 10.1073/pnas.83.15.5364.
Among other results, we rationally calculate the algebraic K-theory of any discrete cocompact subgroup of a Lie group G, where G is either O(n, 1), U(n, 1), Sp(n, 1), or F(4), in terms of the homology of the double coset space Gamma\G/K, where K is a maximal cocompact subgroup of G. We obtain the formula K(n)(ZGamma) [unk] [unk] congruent with unk (infinity)H(i)(Gamma\G/K; unk), where unk is a stratified system of Q vector spaces over Gamma\G/K and the vector space unk(GammagK) corresponding to the double coset GammagK is isomorphic to K(J)(Z(Gamma [unk] gKg(-1))) [unk] Q. Note Gamma [unk] gKg(-1) is a finite subgroup of Gamma. Earlier, a similar formula for discrete cocompact subgroups Gamma of the group of rigid motions of Euclidean space was conjectured by F. T. Farrell and W. C. Hsiang and proven by F. Quinn.
在其他结果中，我们根据双陪集空间 Gamma\G/K 的同调理论（其中 K 是 G 的最大紧子群），合理地计算了李群 G 中任何离散紧子群的代数 K-理论，其中 G 是 O(n, 1)、U(n, 1)、Sp(n, 1) 或 F(4)。我们得到公式 K(n)(ZGamma) [unk] [unk] 同余于 unk (infinity)H(i)(Gamma\G/K; unk)，其中 unk 是 Gamma\G/K 上的 Q 向量空间的分层系统，对应于双陪集 GammagK 的向量空间 unk(GammagK) 同构于 K(J)(Z(Gamma [unk] gKg(-1))) [unk] Q。注意 Gamma [unk] gKg(-1) 是 Gamma 的有限子群。早些时候，F. T. Farrell 和 W. C. Hsiang 对欧几里得空间刚体运动群的离散紧子群 Gamma 提出了一个类似的公式，并由 F. Quinn 证明。
