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.