Lower algebraic K-theory of certain reflection groups

For a finite volume geodesic polyhedron P in hyperbolic 3-space, with the property that all interior angles between incident faces are integral submultiples of Pi, there is a naturally associated Coxeter group generated by reflections in the faces. Furthermore, this Coxeter group is a lattice inside the isometry group of hyperbolic 3-space, with fundamental domain the original polyhedron P. In this paper, we provide a procedure for computing the lower algebraic K-theory of the integral group ring of such Coxeter lattices in terms of the geometry of the polyhedron P. As an ingredient in the computation, we explicitly calculate some of the lower K-groups of the dihedral groups and the product of dihedral groups with the cyclic group of order two.

💡 Research Summary

The paper investigates the lower algebraic K‑theory of integral group rings associated with a special class of hyperbolic Coxeter groups. These groups arise as reflection groups generated by the faces of a finite‑volume hyperbolic polyhedron P ⊂ ℍ³ under the condition that every dihedral angle of P is an integral submultiple of π. In this situation the reflections generate a discrete lattice Γ in Isom(ℍ³) whose fundamental domain is precisely P, and Γ is a Coxeter group whose Coxeter matrix is read off directly from the angles of P.

The main goal is to compute K₀(ℤ