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

Abstract

Let $V$ be a finite graph, and let $\varphi :V\to V$ be an irreducible train track map whose mapping torus has word-hyperbolic fundamental group $G$. Then $G$ acts freely and cocompactly on a $CAT\left(0\right)$ cube complex. Hence, if $F$ is a finite-rank free group and if $\Phi :F\to F$ is an irreducible monomorphism so that $G=F{\ast }_{\Phi }$ is word-hyperbolic, then $G$ acts freely and cocompactly on a $CAT\left(0\right)$ cube complex. This holds, in particular, if $\Phi$ is an irreducible automorphism with $G=F{⋊}_{\Phi }\mathbb{Z}$ word-hyperbolic.