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