Let V be a fi nite graph and let ∅ : V → V be an irreducible train track map whose mapping torus has word-hyperbolic fundamental group G. Then G acts freely and cocompactly on a CAT(0) cube complex. Hence, if F is a finite-rank free group and Φ : F → F an irreducible monomorphism so that G = F∗ᵩ is word-hyperbolic, then G acts freely and cocompactly on a CAT(0) cube complex. This holds in particular if Φ is an irreducible automorphism with G = F ⋊ᵩ Z word-hyperbolic.