Strong conciseness of Engel words in profinite groups

Abstract

A group word 𝑤 is said to be strongly concise in a class 𝒞 of profinite groups if, for any group 𝐺 in 𝒞, either 𝑤 takes at least continuum many values in 𝐺 or the verbal subgroup 𝑤(𝐺) is finite. It is conjectured that all words are strongly concise in the class of all profinite groups. Earlier Detomi, Klopsch, and Shumyatsky proved this conjecture for multilinear commutator words, as well as for some other particular words. They also proved that every group word is strongly concise in the class of nilpotent profinite groups, as well as that 2-Engel words are strongly concise (but their approach does not seem to generalize to 𝑛-Engel words for 𝑛 > 2). In this paper, we prove that for any 𝑛, the 𝑛-Engel word [… [ [𝑥, 𝑦], 𝑦], … 𝑦] (where 𝑦 is repeated 𝑛 times) is strongly concise in the class of finitely generated profinite groups.

Commutators, centralizers, and strong conciseness in profinite groups

“…A group‐word w is strongly concise if the verbal subgroup $$w(G)$$ is finite in any profinite group G in which w takes less than $$2^{\aleph _0}$$ values. A number of recent results on strong conciseness of group‐words can be found in [3, 5, 22]. The concept of strong conciseness can be applied in a wider context: suppose $$\varphi (G)$$ is a subset that can be naturally defined in every profinite group G , then one can ask whether the subgroup generated by $$\varphi (G)$$ is finite whenever $$|\varphi (G)| < 2^{\aleph _0}$$.…”