Small volume 3-manifolds constructible from string Coxeter groups of rank 4

Abstract

It is well known that various compact 3-manifolds of small volume can be constructed via the determination of torsion-free subgroups of minimum possible index in certain rank 4 string Coxeter groups. Examples include the Poincaré homology sphere and the Weber-Seifert and Gieseking manifolds, obtainable from the [3,3,5], [5,3,5] and [3,3,6] Coxeter groups respectively. Constructions were developed in some notes by Milnor on computing volumes in the late 1970s, and in papers by Lorimer (1992) and Everitt (2004) using the identification of faces of a Platonic solid. The notes by Milnor dealt only with hyperbolic cases, and did not resolve all of them, and also a subsequent paper by Lorimer (2002) unsuccessfully attempted to deal with the case of the [4,3,5] Coxeter group. We complete and extend these pieces of work by determining the smallest volume Euclidean or hyperbolic 3-manifolds constructible from torsion-free subgroups of minimum possible index in all of the infinite [p,q,r] Coxeter groups for which p,q,r≥3 and 1/p+1/q≥1/2 and 1/q+1/r≥1/2.