On Several Problems about Automorphisms of the Free Group of Rank Two

Let Fn be a free group of rank n. In this paper we discuss three algorithmic problems related to automorphisms of F2. A word u of Fn is called positive if u does not have negative exponents. 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 [1] 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 [1] for the case of F2. We further prove that there exists 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 [1] in the affirmative for the case of F2.