The lower algebraic \(K\)-theory of \(\Gamma_{3}\)
Let \(\Gamma_3=O^+(3,1)\cap GL(4,\mathbb Z)\). The main theorem computes the lower algebraic K-theory of \(\Gamma_3\) as follows: \(\text{Wh}(\Gamma_3)=0\), \(\tilde K_0(\mathbb Z \Gamma_3)\cong \mathbb Z/4 \oplus\mathbb Z/4\), \(K_{-1}(\mathbb Z \Gamma_3)\cong \mathbb Z \oplus\mathbb Z\), \(K_{n}(\mathbb Z \Gamma_3)=0\) for \(n<-1\). The computation is based on the fact that \(\Gamma_3\) is a group for which the isomorphism conjecture of \textit{F. T. Farrell} and \textit{L. E. Jones} [J. Am. Math. Soc. 6, 249--297 (1993; Zbl 0798.57018)] is known to hold. This reduces the computation to an assembly of results for virtually cyclic subgroups. In fact there will be a further reduction to finite subgroups using algebraic techniques of \textit{J. F. Davis} and \textit{W. Lück} [K-Theory 15, 201--252 (1998; Zbl 0921.19003)]. The author explains all this and then proceeds to the actual computation, which is quite involved. To learn about all the virtually cyclic subgroups of \(\Gamma_3\) she relies heavily on the fact that \(\Gamma_3\) is a hyperbolic reflection group. Many earlier \(K\)-theory computations for group rings are needed before one can start assembling. The reduction to an assembly over finite subgroups also needs some explicit checks, valid only in the low degrees needed in the theorem. This is a very instructive case study.
hyperbolic \(n\)-simplex reflection groups
scientific article; zbMATH DE number 2153602
scientific article