On Coxeter’s families of group presentations

In 1939 Coxeter published three infinite families of group presentations. He studied their properties, in particular determining when groups defined by members of the families are infinite and the structure of finite ones. Eight presentations remained for which the finiteness question was unsettled. We show that two of these eight presentations define finite groups (for which we give comprehensive proofs and provide detailed structural information) and that two of the presentations define infinite groups. Our results rely on substantial amounts of computer calculations, in particular on coset enumeration to prove finiteness and on computation of automatic structures using Knuth-Bendix rewriting to prove infiniteness.