Generalized Baumslag–Solitar groups and geometric homomorphisms

A group is called a generalized Baumslag–Solitar group, or GBS-group, if it is the fundamental group of a graph of groups with infinite cyclic vertex and edge groups. A GBS-group is said to be GBS-simple if it has no proper, non-cyclic geometric quotients, i.e., quotients other than Z which arise from geometric homomorphisms. The main result gives a complete description of the GBS-groups which are GBS-simple. This is achieved by constructing a set of natural examples of geometric homomorphisms.