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.Comment: 40 pages, no figure. Differences with respect to the first version: there is now an Appendix about $\beta$-train tracks, written by the second author. A Corollary about Rapid Decay for free-by-cyclic groups has been adde