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. In particular, Olshanskii gave an example of such a word, which we denote by \(w_o\). The problem whether every word is concise in the class of residually finite groups remains wide open. In this note, we observe that \(w_o\) is concise in residually finite groups. Moreover, we show that \(w_o\) is strongly concise in profinite groups, that is, \(w_o(G)\) is finite whenever G is a profinite group in which \(w_o\) takes less than \(2^{\aleph _0}\) values.
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This paper was written while P. Shumyatsky was visiting the Department of Mathematics of the University of the Basque Country. He expresses his sincere gratitude to the department for excellent hospitality. M. Pintonello was supported by the Spanish Government project PID2020-117281GB-I00, partially by FEDER funds, by the Basque Government project IT483-22, and grant FPI-2018 of the Spanish Government. He is extremely thankful to G. Fernández-Alcober and M. Casals for the useful suggestions.
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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
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DOI: https://doi.org/10.1007/s00013-023-01955-x