Lower algebraic $K$-theory of hyperbolic 3-simplex reflection groups

Abstract

A hyperbolic 3-simplex reflection group is a Coxeter group arising as a lattice in O^+(3,1) , with fundamental domain a geodesic simplex in ℍ^3 (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.