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Botany of irreducible automorphisms of free groups. (English) Zbl 1259.20031
Let $$F_N$$ be a free group of rank $$N$$. An outer automorphism $$\Phi$$ of $$F_N$$ is fully irreducible (iwip) if no positive power $$\Phi^n$$ fixes a proper free factor of $$F_N$$. In this paper the authors classify the fully irreducible outer automorphisms of a free group.
Before quoting their main result some definitions and terminology are needed.
The group of outer automorphisms $$\text{Out}(F_N)$$ acts on the outer space $$CV_N$$ and its boundary $$\partial CV_N$$ [see K. Vogtmann, Geom. Dedicata 94, 1-31 (2002; Zbl 1017.20035)]. An iwip outer automorphism $$\Phi$$ has a unique attracting fixed tree $$[T_\Phi]$$ and a unique repelling fixed tree $$[T_{\Phi^{-1}}]$$ in the boundary of outer space.
The free group $$F_N$$ may be realized as the fundamental group of a surface $$S$$ with boundary. If $$\Phi$$ comes from a pseudo-Anosov mapping class on $$S$$, then its limit trees $$T_\Phi$$ and $$T_{\Phi^{-1}}$$ are called surface trees and such an iwip outer automorphism $$\Phi$$ is called geometric. If $$\Phi$$ does not come from a pseudo-Anosov mapping class and if $$T_\Phi$$ is geometric then $$\Phi$$ is called parageometric.
For a tree $$T$$ in the boundary of outer space with dense orbits, the limit set $$\Omega\subseteq\overline T$$ ($$\overline T$$ is the metric completion of $$T$$) consists of points of $$\overline T$$ with at least two pre-images by the map $$\mathfrak D\colon\partial F_N\to\overline T\cup\partial T$$ introduced by G. Levitt and M. Lustig [J. Inst. Math. Jussieu 2, No. 1, 59-72 (2003; Zbl 1034.20038)]. If $$T\subseteq\Omega$$, the tree $$T$$ is called of surface type. If $$\Omega$$ is totally disconnected, the tree $$T$$ is called of Levitt type.
For a tree $$T$$ in $$\partial CV_N$$ with dense orbits in [T. Coulbois and A. Hilion, “Rips induction: index of the dual lamination of an $$\mathbb R$$-tree”, arXiv:1002.0972] are summarized the above properties and is given the definition.
The tree $$T$$ is:
— a surface tree if it is both geometric and of surface type;
— Levitt if it is geometric and of Levitt type;
— pseudo-surface if it is not geometric and of surface type;
— pseudo-Levitt if it is not geometric and of Levitt type.
The following theorem is the main result of this paper.
Theorem: Let $$\Phi$$ be an iwip outer automorphism of $$F_N$$. Let $$T_\Phi$$ and $$T_{\Phi^{-1}}$$ be its attracting and repelling trees. Then exactly one of the following occurs:
1.
The trees $$T_\Phi$$ and $$T_{\Phi^{-1}}$$ are surface trees. Equivalently, $$\Phi$$ is geometric.
2.
The tree $$T_\Phi$$ is Levitt and the tree $$T_{\Phi^{-1}}$$ is pseudo-surface. Equivalently, $$\Phi$$ is parageometric.
3.
The tree $$T_{\Phi^{-1}}$$ is Levitt, and the tree $$T_\Phi$$ is pseudo-surface. Equivalently, $$\Phi^{-1}$$ is parageometric.
4.
The trees $$T_\Phi$$ and $$T_{\Phi^{-1}}$$ are pseudo-Levitt.
Another interesting result in the paper, based on the property that iwip automorphisms can be represented by (absolute) train-track maps, is the
Theorem: Let $$\Phi\in\text{Out}(F_N)$$ be an iwip outer automorphism. The attracting tree $$T_\Phi$$ is indecomposable.
For the indecomposability of a tree see V. Guirardel [Ann. Inst. Fourier 58, No. 1, 159-211 (2008; Zbl 1187.20020)].

##### MSC:
 20E05 Free nonabelian groups 20E36 Automorphisms of infinite groups 20F65 Geometric group theory 20E08 Groups acting on trees 57R30 Foliations in differential topology; geometric theory 37B10 Symbolic dynamics
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