2025/02/11 by Iker de las Heras, Heras, Iker de las, Andoni Zozaya +1 · 1 citation
Computer Science · Mathematics · #20E18 #20F10 #20F22 #20F70 #20H20 #Advanced Algebra and Logic #Advanced Topology and Set Theory #FOS: Mathematics #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.2502.07427
openalex publication_date 2025/02/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A word w is said to be concise in a class of groups if, for every G in that class such that the set of w-values w\G\ is finite, the verbal subgroup w(G) is also finite. In the context of profinite groups, the notion of strong conciseness imposes a more demanding condition on w, requiring that w(G) is finite whenever |w\G\|< 2ℵ0. We investigate the relation between these two properties and the notion of equationally Noetherian groups, by proving that in a profinite group G with a dense equationally Noetherian subgroup, w\G\ is finite whenever |w\G\|< 2ℵ0. Consequently, we conclude that every word is strongly concise in the classes of profinite linear groups, pro-C completions of residually C linear groups and pro-C completions of virtually abelian-by-polycyclic groups, thereby extending well-known conciseness properties of these classes of groups.