## 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

### Keywords:

free groups; automorphisms; fixed subgroups
Full Text:

### References:

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