On Rationality of Verbal Subsets In a Group
  Let $F$ be a free non-abelian group. We show that for any group word $w$ the set $w[F]$ of all values of $w$ in $F$ is rational in $F$ if and only if $w[F] = 1$ or $w[F] = F.$ We generalize this to a wide class of free products of groups. 
A. Myasnikov and V. Roman'kov
math.GR