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Journal of Group Theory

Editor-in-Chief: Parker, Christopher W. / Wilson, John S.

Managing Editor: Khukhro, Evgenii I. / Kramer, Linus

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An algorithm that decides translation equivalence in a free group of rank two

Citation Information: Journal of Group Theory. Volume 10, Issue 4, Pages 561–569, ISSN (Online) 1435-4446, ISSN (Print) 1433-5883, DOI: https://doi.org/10.1515/JGT.2007.043, July 2007

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Let F 2 be a free group of rank 2. We prove that there is an algorithm that decides whether or not, for two given elements u, v of F 2, u and v are translation equivalent in F 2, that is, whether or not u and v have the property that the cyclic length of (u) equals the cyclic length of (v) for every automorphism of F 2. This gives an affirmative solution to problem F38a in the online version (http://www.grouptheory.info) of [G. Baumslag, A. G. Myasnikov and V. Shpilrain. Open problems in combinatorial group theory, 2nd edn. In Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), Contemp. Math. 296 (American Mathematical Society, 2002), pp. 1–38.] for the case of F 2.

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