Lower algebraic K-theory of certain reflection groups

TL;DR

This paper develops a geometric method to compute the lower algebraic K-theory of Coxeter lattices derived from hyperbolic polyhedra with specific angle properties, linking geometry with algebraic invariants.

Contribution

It introduces a procedure to calculate lower algebraic K-theory of integral group rings for certain hyperbolic Coxeter groups based on polyhedral geometry.

Findings

Computed lower K-groups for dihedral and related groups.

Established a geometric framework for algebraic K-theory calculations.

Linked polyhedral angles to algebraic invariants of Coxeter groups.

Abstract

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.