Translation Equivalent Elements in Free Groups

Abstract. Let Fn be a free group of rank n ≥ 2. Two elements g, h in Fn are said to be translation equivalent in Fn if the cyclic length of φ(g) equals the cyclic length of φ(h) for every automorphism φ of Fn. 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 Fn whenever g, h ∈ Fn are translation equivalent in Fn, which hereby gives an affirmative solution to problem F38b in the online version (http://www.grouptheory.info) of [1].