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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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