# zbMATH — the first resource for mathematics

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
Full Text: