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 $\FN \rtimes_α \Z$ of any automorphism $\alpha$ of a free group $\FN$ of finite rank $n\ge 2$ 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 $\FN \rtimes_α \Z$ 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 $\FN \rtimes_α \Z$ has Rapic Decay.

arXiv comment40 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 added

References in corpus (3)

• The Recognition Theorem for Out(F_n)
• Free-group automorphisms, train tracks and the beaded decomposition
• Geodesics in trees of hyperbolic and relatively hyperbolic groups

Cited by in corpus (11)

• Cubulating hyperbolic free-by-cyclic groups: the general case
• Mapping tori of free group automorphisms, and the Bieri-Neumann-Strebel invariant of graphs of groups
• Free-by-cyclic groups, automorphisms and actions on nearly canonical trees
• Acylindrical hyperbolicity of automorphism groups of infinitely-ended groups
• Relative hyperbolicity of hyperbolic-by-cyclic groups
• Unsolvability of the isomorphism problem for [free abelian]-by-free groups
• Some new CAT(0) free-by-cyclic groups
• Reducing the conjugacy problem for relatively hyperbolic automorphisms to peripheral components
• Relatively irreducible free subroups in Out($\mathbb{F}$)
• Train track maps for graphs of groups
• Poisson boundary of groups acting on real trees