This survey reveals insights into complexity of Whitehead's automorphism problem and automorphic orbits in free groups.
In this survey, we describe recent progress on asymptotic properties of various automorphic orbits in free groups. In particular, we address the problem of counting potentially positive elements of a given length. We also discuss complexity (worst-case, average-case, and generic-case) of Whitehead's automorphism problem and relevant properties of automorphic orbits, including orbit-blocking words.
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Vladimir Shpilrain (2025) studied this question.
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