The Tits alternative for generalized tetrahedron groups

A generalized tetrahedron group is the colimit of a triangle of groups whose vertex groups are generalized triangle groups and whose edge groups are finite cyclic. We prove an improved spelling theorem for generalized triangle groups which enables us to compute the precise Gersten-Stallings angles of this triangle of groups, and hence obtain a classification of generalized 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 non-abelian free subgroup. Finally, we apply these results to prove the Tits alternative for all generalized tetrahedron groups where the triangle is non-spherical: with three abelian-by-finite exceptions, every such group contains a non-abelian free subgroup. © de Gruyter 2006.