Strong conciseness in profinite groups

Abstract

A group word w is said to be strongly concise in a class C of profinite groups if, for every group G in C such that w takes less than 2 ℵ 0 values in G, the verbal subgroup w(G) is finite. Detomi, Morigi and Shumyatsky established that multilinear commutator words-and the particular words x 2 and [x 2 , y]-have the property that the corresponding verbal subgroup is finite in a profinite group G whenever the word takes at most countably many values in G. They conjectured that, in fact, this should be true for every word. In particular, their conjecture included as open cases power words and Engel words. In the present paper, we take a new approach via parametrised words that leads to stronger results. First we prove that multilinear commutator words are strongly concise in the class of all profinite groups. Then we establish that every group word is strongly concise in the class of nilpotent profinite groups. From this we deduce, for instance, that, if w is one of the group words x 2 , x 3 , x 6 , [x 3 , y] or [x, y, y], then w is strongly concise in the class of all profinite groups. Indeed, the same conclusion can be reached for all words of the infinite families [x m , z1, .

Strong conciseness of coprime and anti-coprime commutators

“…The set of anti-coprime commutators generates the commutator subgroup G ′ of G. Hence if G ′ is finite, then so is this set. As a consequence of the results in [3], we have the following new characterization of finite-by-abelian groups.…”

“…In [3] the strong conciseness of several words was established. Among these are the simple commutator Proof of Theorem 1.2.…”

Strong conciseness of Engel words in profinite groups

“…As shown by Abért [1], an analogue of this theorem does not hold in the version where the union of closed subsets is taken over a set of cardinality less than 2 ℵ 0 (without the Continuum Hypothesis). But a certain special case of such a generalization was recently obtained in [3]. Proposition 2.5 ([3, Proposition 2.1]).…”

“…Since G is finitely generated, G/N is nilpotent. By [3,Theorem 1.2] the verbal subgroup [x, n y] | x, y ∈ G/N is finite, and since N is finite, the verbal subgroup [x, n y] | x, y ∈ G is also finite.…”

On finiteness of some verbal subgroups in profinite groups

“…For profinite groups a variation of the classical notion of conciseness arises quite naturally: following [3] we say that w is strongly concise in a class C of profinite groups if, for each G in C, already the bound |G w | < 2 ℵ 0 implies that w(G) is finite (a somewhat weaker notion of conciseness for profinite groups was considered in [4]). It was conjectured in [3] that every group word is strongly concise, and the conjecture was confirmed for several families of group words. In particular, in [3] the conjecture was confirmed for multilinear commutator words (see the next section for the relevant definition).…”