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Free group automorphisms with many fixed points at infinity. (English) Zbl 1140.20027
Boileau, Michel (ed.) et al., The Zieschang Gedenkschrift. Coventry: Geometry & Topology Publications. Geometry and Topology Monographs 14, 321-333 (2008).
Let \(F_n\) be a free group of finite rank \(n\geq 2\) with a free generating set of elements \(a_1,\dots,a_n\). Let \(\partial F_n\) be the Gromov boundary of \(F_n\). Every automorphism \(\alpha\in\operatorname{Aut}(F_n)\) induces a homeomorphism \(\partial\alpha\in\text{Homeo}(\partial F_n)\). Any fixed point of \(\partial\alpha\) is either attracting or repelling, or it belongs to \(\partial\text{Fix}(\alpha)\subseteq\partial F_n\).
An automorphism \(\alpha_n\in\operatorname{Aut}(F_n)\) is constructed for every \(n\geq 3\) which is defined by the map: \[ \alpha_n: a_1\to a_1a_2\cdots a_n,\quad a_2\to a_2a_1a_2,\quad a_3\to a_3a_1a_2a_3,\quad\dots,\quad a_n\to a_na_1a_2a_3\cdots a_n. \] It is shown that \(\alpha_n\) is an irreducible automorphism with irreducible powers such that \(\text{Fix}(\alpha_n)=1\), and \(\alpha_n\) has exactly \(2n-1\) attractive as well as \(2n\) repelling fixed points on \(\partial F_n\). This proof combined with recent results by V. Guirardel answers a question posed by G. Levitt.
For the entire collection see [Zbl 1135.00012].

MSC:
20E36 Automorphisms of infinite groups
57M05 Fundamental group, presentations, free differential calculus
20E05 Free nonabelian groups
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