5
$\begingroup$

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 $v$?

$\endgroup$
1
  • 2
    $\begingroup$ 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
    Mar 30, 2013 at 16:32

1 Answer 1

5
$\begingroup$

"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.

$\endgroup$
6
  • $\begingroup$ Thank you. This is probably exactly the answer I'm looking for. I will find the reference. $\endgroup$ Mar 30, 2013 at 16:31
  • 1
    $\begingroup$ 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
    Mar 30, 2013 at 17:18
  • $\begingroup$ @Sean Eberhar: I have soft copy of this book. If you need, e-mail me. $\endgroup$ 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. $\endgroup$ Mar 30, 2013 at 17:48
  • 1
    $\begingroup$ 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 $\endgroup$ Mar 31, 2013 at 2:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.