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arXiv · 0707.0822

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

Francois Gautero·Martin Lustig
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Abstract

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

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Francois Gautero, Martin Lustig. 2008-10-26. The mapping-torus of a free group automorphism is hyperbolic relative to the canonical subgroups of polynomial growth. https://arxiv.org/abs/0707.0822

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