The Tits alternative for generalized tetrahedron groups

Abstract

A generalised tetrahedron group is the colimit of a triangle of groups whose vertex groups are generalised triangle groups and whose edge groups are finite cyclic. We prove an improved Spelling Theorem for generalised triangle groups which enables us to compute the precise Gersten-Stallings angles of this triangle of groups, and hence obtain a classification of generalised tetrahedron groups according to the curvature properties of the triangle. We also prove that the colimit of a negatively curved triangle of groups contains a nonabelian free subgroup. Finally, we apply these results to prove the Tits alternative for all generalised tetrahedron groups where the triangle is non-spherical: with three abelian-by-finite exceptions, every such group contains a nonabelian free subgroup.