A palindromization map for the free group
  We define a self-map Pal: F_2 --> F_2 of the free group on two generators a, b, using automorphisms of F_2 that form a group isomorphic to the braid group B_3. The map Pal restricts to de Luca's right iterated palindromic closure on the submonoid generated by a, b, and is continuous for the profinite topology on F_2. The values of Pal are palindromes and coincide with the elements g of F_2 such that abg is conjugate to bag. 
Christian Kassel, Christophe Reutenauer