Examples of retracts in free groups that are not the fixed subgroup of any automorphism

Let n be an arbitrary cardinal, and let Fn be a free group of rank n. The fixed subgroup of an endomorphism ψ of Fn is the subgroup of elements in Fn fixed by ψ. In this paper, the relationship between the family of fixed subgroups of endomorphisms of Fn and the family of fixed subgroups of automorphisms of Fn will be studied. We prove that these two families of subgroups do not coincide for n ≥ 3, by showing an infinite sequence of explicit examples of retracts of Fn –and so, fixed subgroups of endomorphisms of Fn – which are not fixed subgroups of any automorphism of Fn. Key words: Free group, automorphism, 1-auto-fixed subgroup, retract.