Abstract
A group-word w is called concise if the verbal subgroup w(G) is finite whenever w takes only finitely many values in a group G. It is known that there are words that are not concise. The problem whether every word is concise in the class of profinite groups remains wide open. Moreover, there is a conjecture that every word w is strongly concise in profinite groups, that is, w(G) is finite whenever G is a profinite group in which w takes less than \(2^{\aleph _0}\) values. In this paper we show that if the word w takes less than \(2^{\aleph _0}\) values in a profinite group G then w(w(G)) is finite.
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References
Acciarri, C., Shumyatsky, P.: On words that are concise in residually finite groups. J. Pure Appl. Algebra 218, 130–134 (2014)
Acciarri, C., Shumyatsky, P.: Varieties of groups and the problem on conciseness of words. arXiv:2308.02209 (2023)
Azevedo, J., Shumyatsky, P.: On finiteness of some verbal subgroups in profinite groups. J. Algebra 574, 573–583 (2021)
de las Heras, I., Pintonello, M., Shumyatsky, P.: Strong conciseness of coprime commutators in profinite groups. J. Algebra 633, 1–19 (2023)
Detomi, E.: A note on strong conciseness in virtually nilpotent profinite groups. Arch. Math. (Basel) 120(2), 115–121 (2023)
Detomi, E., Klopsch, B., Shumyatsky, P.: Strong conciseness in profinite groups. J. Lond. Math. Soc. 102(3), 977–993 (2020)
Detomi, E., Morigi, M., Shumyatsky, P.: Words of Engel type are concise in residually finite groups. Bull. Math. Sci. 9, 1950012 (2019)
Detomi, E., Morigi, M., Shumyatsky, P.: Words of Engel type are concise in residually finite groups. Part II. Groups Geom. Dyn. 14, 991–1005 (2020)
Detomi, E., Morigi, M., Shumyatsky, P.: On bounded conciseness of Engel-like words in residually finite groups. J. Algebra 521, 1–15 (2019)
Detomi, E., Morigi, M., Shumyatsky, P.: Bounding the order of a verbal subgroup in a residually finite group. Isr. J. Math. 253, 771–785 (2023)
Fernández-Alcober, G., Shumyatsky, P.: On bounded conciseness of words in residually finite groups. J. Algebra 500, 19–29 (2018)
Guralnick, R., Shumyatsky, P.: On rational and concise word. J. Algebra 429, 213–217 (2015)
Ivanov, S.P.: Hall’s conjecture on the finiteness of verbal subgroups. Soviet Math. (Iz. VUZ) 33, 59–70 (1989)
Jaikin-Zapirain, A.: On the verbal width of finitely generated pro-p groups. Rev. Mat. Iberoam. 168, 393–412 (2008)
Khukhro, E., Shumyatsky, P.: Strong conciseness of Engel words in profinite groups. Math. Nachr. 296(6), 2404–2416 (2023)
Pintonello, M., Shumyatsky, P.: On conciseness of the word in Olshanskii’s example. Arch. Math. 122, 241–247 (2024). https://doi.org/10.1007/s00013-023-01955-x
Ribes, L., Zalesskii, P.: Profinite Groups. Springer, Berlin (2010)
Robinson, D.J.S.: A Course in the Theory of Groups. Springer, New York (1996)
Segal, D.: Words: Notes on Verbal Width in Groups. Cambridge Univ. Press, Cambridge (2009)
Wilson, J.S.: Profinite Groups. Clarendon Press, Oxford (1998)
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Communicated by John S. Wilson.
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This research was supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), and Fundação de Apoio à Pesquisa do Distrito Federal (FAPDF), Brazil.
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Shumyatsky, P. On profinite groups admitting a word with only few values. Monatsh Math (2024). https://doi.org/10.1007/s00605-024-01967-x
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DOI: https://doi.org/10.1007/s00605-024-01967-x