2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/172305We prove that the mapping torus group $\FN \rtimes_α \Z$ of any automorphism $α$ of a free group $\FN$ of finite rank $n \geq 2$ is weakly hyperbolic relative to the canonical (up to conjugation) family $\mathcal H(α)$ of subgroups of $\FN$ which consists of (and contains representatives of all) conjugacy classes that grow polynomially under iteration of $α$. Furthermore, we show that $\FN \rtimes_α \Z$ is strongly hyperbolic relative to the mapping torus of the family $\mathcal H(α)$. As an application, we use a result of Drutu-Sapir to deduce that $\FN \rtimes_α \Z$ has Rapic Decay.40 pages, no figure. Differences with respect to the first version: there is now an Appendix about $β$-train tracks, written by the second author. A Corollary about Rapid Decay for free-by-cyclic groups has been addedGroup TheoryGeometric Topology20F65text