Let be a group-word. Suppose that the set of all -values in a profinite group is contained in a union of countably many subgroups. It is natural to ask in what way the structure of the verbal subgroup 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.
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Cristina Acciarri, Pavel Shumyatsky, Coverings of commutators in profinite groups. Rend. Sem. Mat. Univ. Padova 137 (2017), pp. 237–257DOI 10.4171/RSMUP/137-14