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

Let F, be a free group of rank n generated by x(1),..., x(n). In this paper we discuss three algorithmic problems related to automorphisms of F(2). A word u = u(x(1),...,x(n)) of F(n) is called positive if no negative exponents of xi occur in u. A word u in F, is called potentially positive if phi(u) is positive for some automorphism phi of F(n). We prove that there is an algorithm to decide whether or not a given word in F(2) 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 F(2). Two elements u and v in F(n) are said to be boundedly translation equivalent if the ratio of the cyclic lengths of phi(u) and phi(v) is bounded away from 0 and from infinity for every automorphism 0 of F(n). We provide an algorithm to determine whether or not two given elements of F(2) 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 F(2). We also provide an algorithm to decide whether or not a given finitely generated subgroup of F(2) is the fixed point group of some automorphism of F(2), 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 F(2). (c) 2008 Elsevier Inc. All rights reserved.