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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.

##### MSC:
 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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