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 $v$?



Automorphism classes of the free group
"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.


