On automorphism-fixed subgroups of a free group
Bestvina and Handel proved that each 1-auto-fixed subgroup, i.e. the subgroup of fixed elements of one automorphism of the free group \(F\) of rank \(n\) has rank at most \(n\). This result has been generalized to endo-fixed subgroups of \(F\) (the elements fixed by each element of a set of endomorphisms). The present paper raises the question whether each auto-fixed subgroup of \(F\) is a 1-auto-fixed subgroup. Its main result implies that each auto-fixed subgroup of \(F_n\) is a free factor of some 1-auto-fixed subgroup (and consequently has rank at most \(n\)). On the other hand, it is shown that there exist free factors of 1-auto-fixed subgroups of \(F\) which are not auto-fixed (or endo-fixed). It is also proved that, for a maximum-rank 1-auto-fixed subgroup \(H\) of \(F\) (i.e. of rank \(n\)), the group of automorphisms of \(F\) which fix every element of \(H\) is free Abelian of rank at most \(n-1\). The results of the paper apply also to endomorphisms.
scientific article; zbMATH DE number 1512981