Hyperbolic hyperbolic-by-cyclic groups are cubulable

Abstract

We show that the mapping torus of a hyperbolic group by a hyperbolic automorphism is cubulable. Along the way, we (i) give an alternate proof of Hagen and Wise's theorem that hyperbolic free-by-cyclic groups are cubulable, and (ii) extend to the case with torsion Brinkmann's thesis that a torsion-free hyperbolic-by-cyclic group is hyperbolic if and only if it does not contain ${\mathbb{Z}}^2$-subgroups.

Incorporated referee suggestions. Final version, to appear in Geom. Topol