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Generalized primitive elements of a free group

Vladimir Shpilrain
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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 elements of a different nature, defined by means of homological properties of the corresponding one-relator group. These elements (``generalized primitive elements"), interesting in their own right, can also be used for distinguishing automorphisms among arbitrary endomorphisms.

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Vladimir Shpilrain. 1996-05-17. Generalized primitive elements of a free group. https://arxiv.org/abs/math/9605207

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