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As is well known, the conjugacy classes of the free group $F_2$ are parametrised by cyclically reduced words, up to cyclic permutation. In particular, it's easy to tell whether two elements of $F_2$ are conjugate.

What about the automorphism classes of $F_2$? For $u,v\in F_2$ write $u\sim v$ if there is an automorphism of $F_2$ mapping $u$ to $v$. Is there a similarly simple description of a representative from each $\sim$ class?

For example, if $F_2 = \langle x,y\rangle$ then $xyxyxy\sim xxx$ while $xyxyxyx\sim x$.

In particular, is it easy to tell whether $u\sim v$ for general words $u$ and $v$?

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    $\begingroup$ You can do this using the Whitehead Algorithm. More generally, it can be decided whether there is an automorphism mapping one $k$-tuple of elements onto another. $\endgroup$
    – Derek Holt
    Commented Mar 30, 2013 at 16:32

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"Let $w_1$ and $w_2$ be elements of a free group $F$. Then it is decidable whether there is an automorphism of $F$ carrying $w_1$ into $w_2$."

(R.C.Lyndon, P.E.Schupp, Combinatorial Group Theory, Chapter I, Prop.4.19)

Is this an answer on your question? I think it's hard to get something more specific.

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  • $\begingroup$ Thank you. This is probably exactly the answer I'm looking for. I will find the reference. $\endgroup$
    – Sean Eberhard
    Commented Mar 30, 2013 at 16:31
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    $\begingroup$ Complexity of the Whitehead algorithm has been intensively studied. See, for example Kapovich, Ilya. Clusters, currents, and Whitehead's algorithm. Experiment. Math. 16 (2007), no. 1, 67–76. $\endgroup$
    – user6976
    Commented Mar 30, 2013 at 17:18
  • $\begingroup$ @Sean Eberhar: I have soft copy of this book. If you need, e-mail me. $\endgroup$
    – Boris Novikov
    Commented Mar 30, 2013 at 17:18
  • $\begingroup$ Just to add, that the case of $F_2$ (free group of rank 2) in particular, is special. The automorphisms have been studied more extensively and there are a lot of results. $\endgroup$
    – Alexey Kvashchuk
    Commented Mar 30, 2013 at 17:48
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    $\begingroup$ Yes. B. Khan studied extensively automorphic conjugacy classes in $F_2$. systemic-inquiry.com/math/autF2/paper.pdf To get a better idea of what automorphisms of $F_2$ look like see eudml.org/doc/163568 See also sci.ccny.cuny.edu/~shpil/orbit.ps $\endgroup$
    – Alexey Kvashchuk
    Commented Mar 31, 2013 at 2:11

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