Abstract
We study endomorphisms of a free group of finite rank by means of their action on specific sets of elements. In particular, we prove that every endomorphism of the free group of rank 2 which preserves an automorphic orbit (i.e., acts "like an automorphism" on one particular orbit), is itself an automorphism. Then, we consider special elements, defined by means of homological properties of the corresponding one-relator group. These elements are shown to make "nice" orbits under the action of the automorphism group of a free group, and also to have other interesting properties that bring together group theory, topology, and homological algebra.
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