All Finite Generalized Tetrahedron Groups

A generalized tetrahedron group is defined to be a group admitting a presentation [Formula: see text] where l, m, n, p, q, r ≥ 2, each Wi(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.

The Tits Alternative for Spherical Generalized Tetrahedron Groups

A generalized tetrahedron group is defined to be a group admitting a presentation [Formula: see text] where l, m, n, p, q, r≥ 2, each Wi(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 show that the Tits alternative holds for a generalized tetrahedron group G whenever (p,q,r)≠ (2,2,2), that is, G contains a non-abelian free subgroup or is solvable-by-finite. The term Tits alternative comes from the respective property for finitely generated linear groups over a field (see [16]).