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Twisted conjugacy in free groups and Makanin's question

Valerij Bardakov·Leonid Bokut·Andrei Vesnin
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Abstract

We discuss the following question of G. Makanin from ``Kourovka notebook'': does there exist an algorithm to determine is for an arbitrary pair of words $U$ and $V$ of a free group $F_n$ and an arbitrary automorphism $ϕ\in Aut(F_n)$ the equation $ϕ(X) U = V X$ solvable in $F_n$? We give the affirmative answer in the case when an automorphism is virtually inner, i.e. some its non-zero power is an inner automorphism of $F_n$.

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Valerij Bardakov, Leonid Bokut, Andrei Vesnin. 2004-02-04. Twisted conjugacy in free groups and Makanin's question. https://arxiv.org/abs/math/0401349

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