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Donghi Lee

Translation equivalent elements in free groups

Let F_n be a free group of rank n ≥ 2. 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 φ of 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 ∈ F_n are translation equivalent in F_n, and thereby give an affermative solution to problem F38b in the online version (http://www.grouptheory.info) of [G. Baumslag, A. G. Myasnikov and V. Shpilrain. Open problems in combinatorial group theory, 2nd edition. In Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), Contemp. Math. 296 (American Mathematical Society, 2002), pp. 1–38.].

Journal of Group Theory, Walter de Gruyter

Print ISSN: 1433-5883
Volume: 9, 11/2006
Pages: 809 - 814

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