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Irreducible automorphisms of \(F_n\) have north-south dynamics on compactified outer space. (English) Zbl 1034.20038
Let \(F_n\) be the non-Abelian free group of rank \(n\). In their study of the outer automorphism group \(\text{Out}(F_n)\) of \(F_n\), Culler and Vogtmann defined a moduli space \(CV_n\) of marked graphs, called ‘outer space’, which is finite dimensional, contractible and which has a spine which admits a discrete co-compact action with finite point stabilizers of \(\text{Out}(F_n)\). M. Bestvina and M. Handel, in their study of the automorphisms of \(F_n\), introduced in their paper [Ann. Math. (2) 135, No. 1, 1-51 (1992; Zbl 0757.57004)] an analogue of Thurston’s pseudo-Anosov maps, and they called these maps “irreducible automorphisms” of \(F_n\).
In the paper under review, the authors study the action of an irreducible automorphism on the closure \(\overline{CV_n}\) of \(CV_n\). They prove that if \(\alpha\in\operatorname{Aut}(F_n)\) is irreducible with irreducible powers, then its action on \(\overline{CV_n}\) has north-south dynamics. In other words, there exist two points \([T^+]\) and \([T^-]\) in \(\partial CV_n\) such that \(\alpha^p([T])\to[T^+]\) as \(p\to\infty\) for all \([T]\not=[T^-]\) and \(\alpha^{-p}([T])\to[T^-]\) as \(p\to\infty\) for all \([T]\not=[T^+]\). This property is an analog of a property of the action of a pseudo-Anasov mapping class on Thurston’s compactification of Teichmüller space.

20F65 Geometric group theory
20E05 Free nonabelian groups
20E36 Automorphisms of infinite groups
20E08 Groups acting on trees
57M07 Topological methods in group theory
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