Special groups with an elementary hierarchy are virtually free-by-$Z$

Abstract

Let $G$ be a group with a hierarchy whose edge groups at each level are cyclic. We show that when $G$ is also word-hyperbolic, then $G$ is virtually free-by-cyclic. In particular, this holds for groups splitting as graphs of free groups with cyclic edge groups unless they have Baumslag–Solitar subgroups that are not virtually products.