Strong conciseness and equationally Noetherian groups
Iker de las Heras, Andoni Zozaya
TL;DR
The paper investigates when a word $w$ is strongly concise in profinite contexts by linking conciseness to equationally Noetherian properties. It proves that if a profinite group $G$ contains a dense equationally Noetherian subgroup, then the finiteness of the set of $w$-values $|w\{G\}|<2^{\aleph_0}$ forces the verbal subgroup $w(G)$ to be finite. This yields strong conciseness for classes including profinite linear groups and pro-$\mathcal C$ completions of residually $\mathcal C$ linear groups, as well as pro-$\mathcal C$ completions of virtually abelian-by-polycyclic groups, thereby extending known conciseness results. The work also develops crucial connections between verbal topologies, marginal subgroups, and equational Noetherianity, offering a framework that unifies several profinite and completion scenarios with algebraic geometry over groups.
Abstract
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.