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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 Teichm├╝ller 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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