On countable coverings of word values in profinite groups

Abstract

Abstract. We prove the following results. Let w be a multilinear commutator word. If G is a profinite group in which all w-values are contained in a union of countably many periodic subgroups, then the verbal subgroup w(G) is locally finite. If G is a profinite group in which all w-values are contained in a union of countably many subgroups of finite rank, then the verbal subgroup w(G) has finite rank as well. As a byproduct of the techniques developed in the paper we also prove that if G is a virtually soluble profinite group in which all w-values have finite order, then w(G) is locally finite and has finite exponent.

On conciseness of words in profinite groups

“…Proposition 2.1 enables us to strengthen this result: we show -without recourse to the Continuum Hypothesis (or Martin's Axiom) -that every multilinear commutator word is strongly concise. For this we employ combinatorial techniques that were developed in [2,4,5] specifically for handling multilinear commutator words.…”

“…Then w[i] = w(G) (see [2,Proposition 7]) and, in particular, w(G) is abelian and periodic. Moreover, we have G w ⊆ G w , so the image of the continuous map 4.…”

Profinite groups with restricted centralizers of commutators

“…There is a conjecture stating that for any word w and any profinite group G in which all w-values have finite order, the verbal subgroup w(G) is locally finite. The conjecture is known to be correct in a number of particular cases (see [28,17,3]). In Section 5 we obtain another result in this direction.…”

“…The next section contains a collection of mostly well-known auxiliary lemmas which are used throughout the paper. In Section 3 we describe combinatorial techniques developed in [8,3,4] for handling multilinear commutator words. We also prove some new lemmas which are necessary for the purposes of the present article.…”

On profinite groups with commutators covered by countably many cosets

“…The reader can consult the articles [2,3,4,6,7,12] for results on countable coverings of word-values by subgroups. One of the results obtained in [7] is that if w is a multilinear commutator and G is a profinite group, then w(G) is finite-by-nilpotent if and only if the set of w-values in G is covered by countably many finite-by-nilpotent subgroups (see Section 2 for the definition of multilinear commutator).…”