Lower Algebraic K-theory of Certain Reflection Groups

For P ⊂ H3 a finite volume geodesic polyhedron, with the property that all interior angles between incident faces are of the form π/mij (mij ≥ 2 an integer), there is a naturally associated Coxeter group ΓP . Furthermore, this Coxeter group is a lattice inside the semi-simple Lie group O+(3, 1) = Isom(H3), with fundamental domain the original polyhedron P . In this paper, we provide a procedure for computing the lower algebraic Ktheory of the integral group ring of such groups ΓP in terms of the geometry of the polyhedron P . As an ingredient in the computation, we explicitly calculate the K−1 and Wh of the groups Dn and Dn×Z2, and we also summarize what is known about the K̃0.