An algorithm that decides translation equivalence in a free group of rank two

Let F 2 be a free group of rank 2. We prove that there is an algorithm that decides whether or not, for two given elements u, v of F 2, u and v are translation equivalent in F 2, that is, whether or not u and v have the property that the cyclic length of (u) equals the cyclic length of (v) for every automorphism of F 2. This gives an affirmative solution to problem F38a in the online version (http://www.grouptheory.info) of [G. Baumslag, A. G. Myasnikov and V. Shpilrain. Open problems in combinatorial group theory, 2nd edn. In Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), Contemp. Math. 296 (American Mathematical Society, 2002), pp. 1–38.] for the case of F 2.