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

Abstract

We prove that the mapping torus group $\FN \rtimes_{\alpha} \Z$ of any automorphism $\alpha$ of a free group $\FN$ of finite rank $n \geq 2$ is weakly hyperbolic relative to the canonical (up to conjugation) family $\mathcal H(\alpha)$ of subgroups of $\FN$ which consists of (and contains representatives of all) conjugacy classes that grow polynomially under iteration of $\alpha$. Furthermore, we show that $\FN \rtimes_{\alpha} \Z$ is strongly hyperbolic relative to the mapping torus of the family $\mathcal H(\alpha)$. As an application, we use a result of Drutu-Sapir to deduce that $\FN \rtimes_{\alpha} \Z$ has Rapic Decay.