Translation equivalent elements in free groups

Abstract

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