Examples of retracts in free groups that are not the fixed subgroup of any automorphism. (Q1414689)

For a free group \(F_n\) of rank \(n\), the relationship between the family of subgroups fixed by an endomorphism of \(F_n\) and the family of subgroups fixed by an automorphism of \(F_n\) is studied. An infinite sequence of explicit examples of retracts of \(F_n\) is constructed (i.e., subgroups \(H\) whose identity map extends to a homomorphism from \(F_n\) to \(H\), so these are in particular fixed subgroups of an endomorphism of \(F_n\)) which are not fixed subgroups of any automorphism of \(F_n\).