The mapping-torus of a free group automorphism is hyperbolic relative to the canonical subgroups of polynomial growth

We prove that the mapping torus group $\text{FN}{⋊}_{\alpha }\mathbb{Z}$ of any automorphism $\alpha$ of a free group $\text{FN}$ of finite rank $n\ge 2$ is weakly hyperbolic relative to the canonical (up to conjugation) family $\mathcal H(\alpha)$ofsubgroupsof$\FN$ which consists of (and contains representatives of all) conjugacy classes that grow polynomially under iteration of $\alpha$. Furthermore, we show that $\text{FN}{⋊}_{\alpha }\mathbb{Z}$ is strongly hyperbolic relative to the mapping torus of the family $\mathcal H(\alpha)$\mathrm{.}Asanapplication,weusearesultofDrutu-Sapirtodeducethat$\FN \rtimes_{\alpha} \Z$ has Rapic Decay.