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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.

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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