The Tits alternative for non‐spherical triangles of groups

Triangles of groups have been introduced by Gersten and Stallings. They are, roughly speaking, a generalization of the amalgamated free product of two groups and occur in the framework of Corson diagrams. First, we prove an intersection theorem for Corson diagrams. Then, we focus on triangles of groups. It has been shown by Howie and Kopteva that the colimit of a hyperbolic triangle of groups contains a non‐abelian free subgroup. We give two natural conditions, each of which ensures that the colimit of a non‐spherical triangle of groups either contains a non‐abelian free subgroup or is virtually solvable.