Botany of irreducible automorphisms of free groups. (Q433549)

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.NEWLINENEWLINE Before quoting their main result some definitions and terminology are needed.NEWLINENEWLINE The group of outer automorphisms \(\text{Out}(F_N)\) acts on the outer space \(CV_N\) and its boundary \(\partial CV_N\) [see \textit{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.NEWLINENEWLINE 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.NEWLINENEWLINE 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 \textit{G. Levitt} and \textit{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.NEWLINENEWLINE For a tree \(T\) in \(\partial CV_N\) with dense orbits in [\textit{T. Coulbois} and \textit{A. Hilion}, ``Rips induction: index of the dual lamination of an \(\mathbb R\)-tree'', \url{arXiv:1002.0972}] are summarized the above properties and is given the definition.NEWLINENEWLINE The tree \(T\) is:NEWLINENEWLINE --- a surface tree if it is both geometric and of surface type;NEWLINENEWLINE --- Levitt if it is geometric and of Levitt type;NEWLINENEWLINE --- pseudo-surface if it is not geometric and of surface type;NEWLINENEWLINE --- pseudo-Levitt if it is not geometric and of Levitt type.NEWLINENEWLINE The following theorem is the main result of this paper.NEWLINENEWLINE 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:{\parindent=5mm\begin{itemize}\item[1.] The trees \(T_\Phi\) and \(T_{\Phi^{-1}}\) are surface trees. Equivalently, \(\Phi\) is geometric.\item[2.] The tree \(T_\Phi\) is Levitt and the tree \(T_{\Phi^{-1}}\) is pseudo-surface. Equivalently, \(\Phi\) is parageometric.\item[3.] The tree \(T_{\Phi^{-1}}\) is Levitt, and the tree \(T_\Phi\) is pseudo-surface. Equivalently, \(\Phi^{-1}\) is parageometric.\item[4.] The trees \(T_\Phi\) and \(T_{\Phi^{-1}}\) are pseudo-Levitt.NEWLINENEWLINE\end{itemize}} Another interesting result in the paper, based on the property that iwip automorphisms can be represented by (absolute) train-track maps, is theNEWLINENEWLINE Theorem: Let \(\Phi\in\text{Out}(F_N)\) be an iwip outer automorphism. The attracting tree \(T_\Phi\) is indecomposable.NEWLINENEWLINE For the indecomposability of a tree see \textit{V. Guirardel} [Ann. Inst. Fourier 58, No. 1, 159-211 (2008; Zbl 1187.20020)].