Abstract:
We prove that |An|⩾cn⋅|A|[(n+1)/2] for any finite subset A of a free group if A contains at least two noncommuting elements, where the cn>0 are constants not depending on A. Simple examples
show that the order of these estimates is best possible for each n>0.
Bibliography: 5 titles.
Keywords:
free group, relations in a free group, subsets of a free group.
\Bibitem{Saf11}
\by S.~R.~Safin
\paper Powers of sets in free groups
\jour Sb. Math.
\yr 2011
\vol 202
\issue 11
\pages 1661--1666
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This publication is cited in the following 12 articles:
Renxing Wan, Wenyuan Yang, “Uniform exponential growth for groups with proper product actions on hyperbolic spaces”, Journal of Algebra, 676 (2025), 189
Coulon R., Steenbock M., “Product Set Growth in Burnside Groups”, J. Ecole Polytech.-Math., 9 (2022), 463–504
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Yu-Miao Cui, Yue-Ping Jiang, Wen-Yuan Yang, “Lower bound on growth of non-elementary subgroups in relatively hyperbolic groups”, Journal of Group Theory, 2022
V. S. Atabekyan, V. G. Mikaelyan, “O proizvedenii podmnozhestv v pereodicheskikh gruppakh”, Proceedings of NAS RA. Mathematics, 2022, 12
Delzant T., Steenbock M., “Product Set Growth in Groups and Hyperbolic Geometry”, J. Topol., 13:3 (2020), 1183–1215
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Emmanuel Breuillard, The IMA Volumes in Mathematics and its Applications, 159, Recent Trends in Combinatorics, 2016, 369
J. O. Button, “Growth in infinite groups of infinite subsets”, Algebra Colloq., 22:2 (2015), 333–348
J. O. Button, “Explicit Helfgott type growth in free products and in limit groups”, J. Algebra, 389 (2013), 61–77
Emmanuel Breuillard, Ben Green, Terence Tao, Bolyai Society Mathematical Studies, 25, Erdős Centennial, 2013, 129
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