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\ 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\|< 2₀ . 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\ is finite whenever |w\|< 2₀ . 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.