Hyperbolic hyperbolic-by-cyclic groups are cubulable

We show that the mapping torus of a hyperbolic group by a hyperbolic automorphism is cubulable. Alongthe way, we give an alternate proof of Hagen and Wise's theorem that hyperbolic free-by-cyclic groups arecubulable, and extend to the case with torsion Brinkmann's thesis that a torsion-free hyperbolic-by-cyclicgroup is hyperbolic if and only if it does not containZ2-subgroups.