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Intersections of automorphism fixed subgroups in the free group of rank three. (English) Zbl 1054.20009

Let \(F_n\) be a free group of finite rank \(n\). A subgroup \(H\) of \(F_n\) is called ‘1-auto-fixed’ when there exists an automorphism \(\varphi\) of \(F_n\) such that \(H=\text{fix\,}\varphi=\{x\in F_n:\varphi x=x\}\). By definition an ‘auto-fixed’ subgroup of \(F_n\) is an arbitrary intersection of 1-auto-fixed subgroups.
It is shown that the intersection \(H\cap K\) of each 1-auto-fixed subgroup \(H\) with any finitely generated subgroup \(K\) of \(F_n\) has rank not exceeding \(n\). Thus the author generalises the celebrated theorem of Bestvina-Handel showing that a 1-auto-fixed subgroup of \(F_n\) has rank at most \(n\).
It is also shown that in the case of the group \(F_3\) any auto-fixed subgroup is in fact 1-auto-fixed. A similar assertion is valid in the case of the group \(F_2\) and remains open for the groups \(F_n\) for \(n\geq 4\).

MSC:

20E05 Free nonabelian groups
20F28 Automorphism groups of groups
20E36 Automorphisms of infinite groups
20E07 Subgroup theorems; subgroup growth
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