Botany of irreducible automorphisms of free groups.

*(English)*Zbl 1259.20031Let \(F_N\) 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:

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)].

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.

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)].

Reviewer: Dimitrios Varsos (Athenai)

##### 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 |