A group is free-by-cyclic if it is an extension of a free group by a cyclic group. Knowing that a group is virtually free-by-cyclic is often quite useful; it implies that the group is coherent and that it is cohomologically good in the sense of Serre. In this talk we will give a homological characterisation of when a finitely generated RFRS group is virtually free-by-cylic and discuss some generalisations.
