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Parageometric outer automorphisms of free groups. (English) Zbl 1120.20042
The ‘expansion factor’ of an (outer) automorphism $$\varphi$$ of a finitely generated group $$G$$ is defined as $\lambda(\varphi)=\sup_{c\in \mathcal C}(\limsup_{n\to+\infty}\|\varphi^n(c)\|^{1/n}),$ where $$\mathcal C$$ denotes the set of conjugacy classes of $$G$$ and $$\| c \|$$ the smallest word length of a representative of $$c\in\mathcal C$$. If $$\varphi$$ is a pseudo-Anosov element in the mapping class group of a surface (the outer automorphism group of its fundamental group) then there is a unique $$\varphi$$-invariant geodesic in Teichmüller space consisting of the points in Teichmüller space which minimize the translation distance under $$\varphi$$, and this translation distance equals $$\log\lambda(\varphi)$$. In the present paper, the question is considered whether this has an analogue for a fully irreducible outer automorphism $$\varphi$$ of a free group $$F_r$$.
The main result of the paper implies that the answer is no for ‘parageometric’ fully irreducible outer automorphisms $$\varphi\in\text{Out}(F_r)$$, by showing that $$\lambda(\varphi)>\lambda(\varphi^{-1})$$ for parageometric automorphisms. Here an outer automorphism $$\varphi$$ is parageometric if the attracting fixed point of $$\varphi$$ in the boundary of outer space is a geometric $$\mathbb{R}$$-tree with respect to the action of $$F_r$$, but that $$\varphi$$ itself is not geometric in that it is not represented by a homeomorphism of a bounded surface. As corollaries (proved independently by V. Guirardel, [Ann. Sci. Éc. Norm. Supér. (4) 38, No. 6, 847-888 (2005; Zbl 1110.20019)]), the inverse of a parageometric outer automorphism is neither geometric nor parageometric, and a fully irreducible outer automorphism is geometric if and only if its attracting and repelling fixed points in the boundary of outer space are geometric $$\mathbb{R}$$-trees.

##### MSC:
 20F65 Geometric group theory 20E05 Free nonabelian groups 20F34 Fundamental groups and their automorphisms (group-theoretic aspects) 20E36 Automorphisms of infinite groups 57M07 Topological methods in group theory 37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010) 37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
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