Mathematics – Group Theory
Scientific paper
2007-07-05
Mathematics
Group Theory
40 pages, no figure. Differences with respect to the first version: there is now an Appendix about $\beta$-train tracks, writt
Scientific paper
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.
Gautero François
Lustig Martin
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