A group is coherent if its finitely generated subgroups are finitely presented. Two well-known classes of coherent groups are (virtually) free-by-cyclic and three-manifold groups. Examples of coherent groups outside of these classes are hard to come by: small cancellation groups are not necessarily coherent, free-by-free groups are almost never coherent and there are no known examples of coherent hyperbolic groups of cohomological dimension at least four. The aim of this talk will be to explain how one can characterise virtually free-by-cyclic groups among hyperbolic and virtually compact special groups in terms of cohomological dimension and L^2-Betti numbers. Using this characterisation, I will then explain how many groups of cohomological dimension two that are known to be coherent actually possess the seemingly stronger property of being virtually free-by-cyclic. In particular, I will show that one-relator groups with torsion are virtually free-by-cyclic, resolving a conjecture of Baumslag.

(Joint work with Dawid Kielak).