2024/01/28 by Pavel Shumyatsky, Shumyatsky, Pavel
Computer Science · Mathematics · #Advanced Algebra and Logic #FOS: Mathematics #Group Theory (math.GR) #Mathematics and Applications #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2401.15707
openalex publication_date 2024/01/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 2ℵ0 values. In this paper we show that if the word w takes less than 2ℵ0 values in a profinite group G then w(w(G)) is finite.