Coverings of commutators in profinite groups

TL;DR

This survey explores how the structure of verbal subgroups in profinite groups is influenced by coverings of word-values, especially commutators, by various types of subgroups, and relates these findings to Hall's problem on word conciseness.

Contribution

It compiles recent results on coverings of word-values in profinite groups, highlighting the impact on the structure of verbal subgroups and connections to longstanding problems.

Findings

Finite and countable coverings influence subgroup structure.

Results relate coverings to properties like nilpotency and solubility.

Connections made to Hall's problem on word conciseness.

Abstract

Let $w$ be a group-word. Suppose that the set of all $w$-values in a profinite group $G$ is contained in a union of countably many subgroups. It is natural to ask in what way the structure of the verbal subgroup $w(G)$ depends on the properties of the covering subgroups. The present article is a survey of recent results related to that question. In particular we survey results on finite and countable coverings of word-values (mostly commutators) by procyclic, abelian, nilpotent, and soluble subgroups, as well as subgroups with finiteness conditions. The last section of the paper is devoted to relation of the described results with Hall's problem on conciseness of group-words.