The homological torsion of PSL_2 of the imaginary quadratic integers
The homological torsion of PSL_2 of the imaginary quadratic integers
复制标题
虚数二次整数 PSL_2 的同调扭转
DOI：
10.1090/s0002-9947-2012-05690-x
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发表时间：
2011
影响因子：
1.3
通讯作者：
Alexander D. Rahm
中科院分区：
文献类型：
--
作者：
Alexander D. Rahm
用Q（sqrt{-m}）表示，其中m是无平方的正整数，是虚二次数域，用A表示它的整数环。比安奇群是群SL_2（A）。我们揭示了一个对应的同调扭转的比安奇群和新的几何不变量，这是有效的计算感谢他们的行动双曲空间。我们揭示了一种新的技术，扭转subcomplex减少，以获得这些不变量。我们用它来显式计算比安奇群的整群同调。此外，这种对应关系便于计算比安奇群的等变K-同调。利用Baum/Connes猜想，得到了比安奇群的约化C ~*-代数的K-理论，并证明了该猜想的正确性.
Denote by Q(sqrt{-m}), with m a square-free positive integer, an imaginary quadratic number field, and by A its ring of integers. The Bianchi groups are the groups SL_2(A). We reveal a correspondence between the homological torsion of the Bianchi groups and new geometric invariants, which are effectively computable thanks to their action on hyperbolic space. We expose a novel technique, the torsion subcomplex reduction, to obtain these invariants. We use it to explicitly compute the integral group homology of the Bianchi groups. Furthermore, this correspondence facilitates the computation of the equivariant K-homology of the Bianchi groups. By the Baum/Connes conjecture, which is verified by the Bianchi groups, we obtain the K-theory of their reduced C*-algebras in terms of isomorphic images of their equivariant K-homology.