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Let $F_n$ be a free group of rank $n$. Let $w\in F_n$ be cyclically reduced.

  1. What can be said about the element(s) of minimal length from the $\textit{ncl}(w)$ (normal closure of $w$ in $F_n$)? Under what conditions is it a trivial normal root of $w$ (i.e. is conjugate to $w^{\pm 1}$) ?

  2. Let $v, w\in F_n$ be cyclically reduced. What can be said about the element(s) of minimal length in $\textit{ncl}(w)\cap \textit{ncl}(v)$? Under what conditions is it a conjugate of $\[ v, w \]$?

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    $\begingroup$ I think there are no good answers. Sometimes there are no non-trivial normal roots, e.g., when $w$ is a proper power (by Newman's theorem); sometimes such normal roots exist, e.g., $[x,y]\in \langle\langle xy^{2013}\rangle\rangle$. $\endgroup$
    – Anton Klyachko
    Commented Apr 11, 2013 at 10:18
  • $\begingroup$ thanks Anton. I found this paper, which is very relevant: journals.cambridge.org/… $\endgroup$
    – Alexey Kvashchuk
    Commented Apr 11, 2013 at 16:04
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    $\begingroup$ sorry, the link doesn't seem to work. The paper is "On a question of Remeslennikov" by James McCool. $\endgroup$
    – Alexey Kvashchuk
    Commented Apr 11, 2013 at 16:08

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