Translation equivalent elements in free groups

Let F-n be a free group of rank n >= 2. Two elements g, h in F-n are said to be translation equivalent in F-n if the cyclic length of phi(g) equals the cyclic length of phi(h) for every antomorphism phi of F-n. Let F(a, b) be the free group generated by {a, b} and let w(a, b) be an arbitrary word in F(a, b). We prove that w(g, h) and w(h, g) are translation equivalent in F-n whenever g, h is an element of F-n are translation equivalent in F-n, and thereby give an affirmative solution to problem F38b in the online version (http://www.grouptheory. info) of [1].