Geodesic
On verbal subgroups in finite and profinite groups
Algebra and discrete mathematics, Tome 14 (2012) no. 1, pp. 1-13.

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Let $w$ be a multilinear commutator word. In the present paper we describe recent results that show that if $G$ is a profinite group in which all $w$-values are contained in a union of finitely (or in some cases countably) many subgroups with a prescribed property, then the verbal subgroup $w(G)$ has the same property as well. In particular, we show this in the case where the subgroups are periodic or of finite rank.
Keywords: Profinite groups, verbal subgroups, coverings, multilinear commutators. 
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Cristina Acciarri; Pavel Shumyatsky. On verbal subgroups in finite and profinite groups. Algebra and discrete mathematics, Tome 14 (2012) no. 1, pp. 1-13. http://geodesic.mathdoc.fr/item/ADM_2012_14_1_a1/

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