All Finite Generalized Tetrahedron Groups

Abstract

A generalized tetrahedron groups is defined to be a group admitting a presentation <x, y, z | x^l = y^m = z^n = W^{p}_{1}(x,y) = W^{q}_{2}(y,z) = W^{r}_{z}(x,z) = 1> where l,m,n,p,q,r >= 2, each W_i(a,b) is a cyclically reduced word involving both a and b. These groups appear in many contexts, not least as fundamental groups of certain hyperbolic orbifolds or as subgroups of generalized triangle groups. In this paper, we build on previous work to give a complete classification of all finite generalized tetrahedron groups.