On geometry and combinatorics of van Kampen diagrams

Abstract

The subject of this work is application of combinatorial group theory to the problem of constructing groups with prescribed properties. It is shown how certain groups can be presented by generators and defining relations, thus proving their existence. Several existence theorems proved in this paper are based on one approach: van Kampen diagrams over group presentations are used to derive algebraic properties of the groups from combinatorial properties of their presentations. The focus of this paper is on boundnely generated and boundedly simple groups. It is proved that there exist an infinite simple boundedly generated group, a torsion-free group with a finite regular file basis and with a free non-cyclic subgroup, and a boundedly simple finitely generated group with a free non-cyclic subgroup. In particular, a question of Vasiliy Bludov, which has been open since 1995, is settled. The question was whether every torsion-free group with a finite regular file basis has to be virtually polycyclic, and this question is answered negatively by providing a counterexample. The groups in question, or rather their presentations, are constructed by imposing relations that force the group to be boundedly simple, or boundedly generated, or have a ``regular file basis,' accordingly, while in the same time choosing those relations so that certain small-cancellation-type conditions are satisfied. These conditions are used to establish other properties of the groups.