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We prove that the mapping torus group $\FN \rtimes_{\alpha} \Z$ of anyautomorphism $\alpha$ of a free group $\FN$ of finite rank $n \geq 2$ is weaklyhyperbolic relative to the canonical (up to conjugation) family $\mathcalH(\alpha)$ of subgroups of $\FN$ which consists of (and containsrepresentatives of all) conjugacy classes that grow polynomially underiteration of $\alpha$. Furthermore, we show that $\FN \rtimes_{\alpha} \Z$ isstrongly hyperbolic relative to the mapping torus of the family $\mathcalH(\alpha)$. As an application, we use a result of Drutu-Sapir to deduce that$\FN \rtimes_{\alpha} \Z$ has Rapic Decay.

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