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Hyperbolic automorphisms of free groups. (English) Zbl 0970.20018
If $$F$$ is a finitely generated free group, then an automorphism $$\varphi$$ of $$F$$ is called hyperbolic (in the sense of Gromov) if there exist numbers $$M>0$$ and $$\lambda>1$$ such that $\lambda|g|\leq\max\{|\varphi^M(g)|,|\varphi^{-M}(g)|\}$ for all $$g\in F$$, where $$|g|$$ denotes the length of $$g$$ with respect to some fixed basis of $$F$$. An automorphism of $$F$$ is called atoroidal if it has no nontrivial periodic conjugacy classes. The main result of the paper is that an atoroidal automorphism $$\varphi$$ of a finitely generated free group is hyperbolic. This theorem was claimed before by M. Bestvina and M. Feighn [J. Differ. Geom. 35, No. 1, 85-102 (1992; Zbl 0746.57021)] and a proof was given for the special case of irreducible automorphisms by M. Bestvina, M. Feighn and M. Handel [Geom. Funct. Anal. 7, No. 2, 215-244 (1997; Zbl 0884.57002)]. Here a proof is given in the general case. The methods are geometric using train track techniques developed by M. Bestvina, M. Handel and M. Feighn.

##### MSC:
 20E36 Automorphisms of infinite groups 20E05 Free nonabelian groups 20F67 Hyperbolic groups and nonpositively curved groups 57M07 Topological methods in group theory
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