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Laminations, trees, and irreducible automorphisms of free groups. (English) Zbl 0884.57002
Geom. Funct. Anal. 7, No. 2, 215-244 (1997); erratum 7, No. 6, 1143 (1997).
Suppose $$H$$ is a subgroup of $$\text{Out} (F_n)$$, the outer automorphism group of $$F_n$$, the free group of rank $$n$$, that contains an irreducible outer automorphism $$\varphi$$ of infinite order. The authors show that either $$H$$ contains $$F_2$$ or $$H$$ is virtually cyclic. They demonstrate the word hyperbolicity of semidirect products $$F_n\rtimes \mathbb{Z}$$ induced by infinite order irreducible elements of $$\text{Out} (F_n)$$ and generalize this to certain semidirect products $$F_n\rtimes F_2$$. They also show that if $$A$$ is a finitely generated subgroup of $$F_n$$ of infinite index, then the action of $$A$$ on $$T^+$$ is discrete, where $$T^+$$ is a $$\varphi$$-fixed real tree associated to $$\varphi$$.

MSC:
 57M07 Topological methods in group theory 20F28 Automorphism groups of groups 20E36 Automorphisms of infinite groups 20E05 Free nonabelian groups
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