Cubulating hyperbolic free-by-cyclic groups: The irreducible case

Let V be a finite graph, and let φ : 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 if Φ : F → F is 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.