On several problems about automorphisms of the free group of rank two

Let Fn be a free group of rank n generated by x1,…,xn. In this paper we discuss three algorithmic problems related to automorphisms of F2.A word u=u(x1,…,xn) of Fn is called positive if no negative exponents of xi occur in u. A word u in Fn is called potentially positive if ϕ(u) is positive for some automorphism ϕ of Fn. We prove that there is an algorithm to decide whether or not a given word in F2 is potentially positive, which gives an affirmative solution to problem F34a in [G. Baumslag, A.G. Myasnikov, V. Shpilrain, Open problems in combinatorial group theory, second ed., in: Contemp. Math., vol. 296, 2002, pp. 1–38, online version: http://www.grouptheory.info] for the case of F2.Two elements u and v in Fn are said to be boundedly translation equivalent if the ratio of the cyclic lengths of ϕ(u) and ϕ(v) is bounded away from 0 and from ∞ for every automorphism ϕ of Fn. We provide an algorithm to determine whether or not two given elements of F2 are boundedly translation equivalent, thus answering question F38c in the online version of [G. Baumslag, A.G. Myasnikov, V. Shpilrain, Open problems in combinatorial group theory, second ed., in: Contemp. Math., vol. 296, 2002, pp. 1–38, online version: http://www.grouptheory.info] for the case of F2.We also provide an algorithm to decide whether or not a given finitely generated subgroup of F2 is the fixed point group of some automorphism of F2, which settles problem F1b in [G. Baumslag, A.G. Myasnikov, V. Shpilrain, Open problems in combinatorial group theory, second ed., in: Contemp. Math., vol. 296, 2002, pp. 1–38, online version: http://www.grouptheory.info] in the affirmative for the case of F2.