Let \(P\) be a geodesic polyhedron in hyperbolic \(3\)-space \({\mathbb H}^3\), having the property that every pair of incident faces intersects at an angle \(\pi/m_{ij}\) (\(m_{ij} \geq 2\) an integer). Clearly then each of the faces of \(P\) can be extended to a hyperplane, so there can be obtained a naturally associated Coxeter group \(\Gamma_P\) generated by reflections in these hyperplanes, which may be viewed as a lattice in the isometry group \(O^+(3, 1)\) of \({\mathbb H}^3\), with fundamental domain this \(P\). In this paper the authors propose a general method for computing the lower algebraic \(K\)-theory of such a Coxeter group \(\Gamma_P\), namely \(Wh(\Gamma_P)\) for \(* = 1\), \(\widetilde{K}_0({\mathbb Z}\Gamma_P)\) for \(* = 0\) and \(K_*({\mathbb Z}\Gamma_P)\) for \(* < 0\) (practically it suffices to consider only the case \(* = -1\) since \(K_*({\mathbb Z}\Gamma_P)=0\) for \(* < -1\) which is already known).
This is done by trying to extend the method developed in a recent work [J.-F. Lafont and I. J. Ortiz, Comment. Math. Helv. 84, No. 2, 297–337 (2009; Zbl 1172.19001)] of the first and third authors to any such \(P\), in which there are given explicit computations for such \(K\)-theory of all the 32 hyperbolic \(3\)-simplices in \({\mathbb H}^3\). Indeed the key to the authors’ approach is observing the splitting formula \[ K_*({\mathbb Z}\Gamma_P)\cong H_*^{\Gamma_P}(E_{\mathcal FIN}(\Gamma_P); {\mathbb K}{\mathbb Z}^{-\infty})\oplus\bigoplus_V H_*^{V}(E_{\mathcal FIN}(V)\to *) \] established in the quoted work above. Here the group in the first term denotes a specific equivariant homology of \(E_{\mathcal FIN}(\Gamma_P)\), a model for the classifying space for proper actions, and those in the second term denote cokernels of certain (assembly) maps associated to virtually cyclic subgroups \(V\) with specific geometric properties. By virtue of this splitting formula the computation of \(K_*({\mathbb Z}\Gamma_P)\) can be assigned to examine these two terms on the right hand side. The first term is discussed in Sections 3 and 4, in which a spectral sequence that allows one to compute this homology group is analyzed, and the second term is discussed in Section 5. The results obtained from these analyses of the two terms lead to the conclusion that the \(K_*({\mathbb Z}\Gamma_P)\) for \(* \leq 1\) can be directly determined from the geometry of \(P\). In the last section (Section 6) there are given two types of examples of computation, which is useful to help one understand the procedure for computing the lower algebraic \(K\)-theory of \(\Gamma_P\) based on the method presented here.