Strong conciseness and equationally Noetherian groups

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 \(|w\{G\}|< 2^{\aleph _0}\). 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 \(|w\{G\}|< 2^{\aleph _0}\). Consequently, we conclude that every word is strongly concise in the classes of profinite linear groups, pro-\(\mathcal {C}\) completions of residually \(\mathcal {C}\) linear groups and pro-\(\mathcal {C}\) completions of virtually abelian-by-polycyclic groups, thereby extending well-known conciseness properties of these classes of groups.