The Tits alternative for non-spherical triangles of groups

TL;DR

This paper investigates the algebraic structure of non-spherical triangles of groups, establishing conditions under which their colimits contain free subgroups or are virtually solvable, extending the understanding of their geometric and algebraic properties.

Contribution

It introduces two natural conditions that determine when the colimit of a non-spherical triangle of groups contains a free subgroup or is virtually solvable.

Findings

Colimit of hyperbolic triangle of groups contains a free subgroup

Conditions ensuring colimit is virtually solvable or contains free subgroup

Extension of Tits alternative to non-spherical triangles of groups

Abstract

Triangles of groups have been introduced by Gersten and Stallings. They are, roughly speaking, a generalisation 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.