Let be a finite graph, and let be an irreducible train track map whose mapping torus has word-hyperbolic fundamental group . Then acts freely and cocompactly on a cube complex. Hence, if is a finite-rank free group and if is an irreducible monomorphism so that is word-hyperbolic, then acts freely and cocompactly on a cube complex. This holds, in particular, if is an irreducible automorphism with word-hyperbolic.
"Cubulating hyperbolic free-by-cyclic groups: The irreducible case." Duke Math. J. 165 (9) 1753 - 1813, 15 June 2016. https://doi.org/10.1215/00127094-3450752