Strong conciseness and equationally Noetherian groups

TL;DR

This paper explores the relationship between strong conciseness and equationally Noetherian groups in profinite groups, showing that certain classes of groups exhibit strong conciseness properties for all words.

Contribution

It proves that in profinite groups with dense equationally Noetherian subgroups, the set of word-values is finite under specific conditions, extending known conciseness results.

Findings

Strong conciseness holds in profinite linear groups.

Pro-$c$ completions of residually $c$ linear groups are strongly concise.

Pro-$c$ completions of virtually abelian-by-polycyclic groups are strongly concise.

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.