Algebraic K-theory of hyperbolic 3-simplex reflection groups

TL;DR

This paper computes the lower algebraic K-theory of the integral group rings for all hyperbolic 3-simplex reflection groups, which are specific Coxeter groups acting as lattices in hyperbolic space.

Contribution

It provides a complete and explicit calculation of the lower algebraic K-theory for all known hyperbolic 3-simplex reflection groups, filling a gap in the understanding of their algebraic properties.

Findings

Explicit K-theory groups for all 9 cocompact groups

Explicit K-theory groups for all 23 non-cocompact groups

Comprehensive classification-based computations

Abstract

A hyperbolic 3-simplex reflection group is a Coxeter group arising as a lattice in the isometry group of hyperbolic 3-space, with fundamental domain a geodesic simplex (possibly with some ideal vertices). The classification of these groups is known, and there are exactly 9 cocompact examples, and 23 non-cocompact examples. We provide a complete computation of the lower algebraic K-theory of the integral group ring of all the hyperbolic 3-simplex reflection groups.