On profinite groups with commutators covered by countably many cosets

Abstract

Let w be a group-word. Suppose that the set of all w-values in a profinite group G is contained in a union of countably many cosets of subgroups. We are concerned with the question to what extent the structure of the verbal subgroup w(G) depends on the properties of the subgroups. We prove the following theorem.Let C be a class of groups closed under taking subgroups, quotients, and such that in any group the product of finitely many normal C-subgroups is again a C-subgroup. If w is a multilinear commutator and G is a profinite group such that the set of all wvalues is contained in a union of countably many cosets g i G i , where each G i is in C, then the verbal subgroup w(G) is virtually-C.This strengthens several known results.

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.…”

“…More work is needed to confirm the Strong Conciseness Conjecture for multilinear commutator words; this involves combinatorial techniques that were developed in [2,4,5] specifically for handling such words. In this context we gratefully acknowledge contributions of Marta Morigi that arose from discussions with her.…”

Strong conciseness in profinite groups

“…Proposition 2.1 enables us to strengthen this result: we showwithout 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 [3,4] specifically for handling multilinear commutator words.…”

“…. , c + 1} such that ε ∈ γ j (Ω), and define a length function on E by associating to ε the length(ε) = (c + 1 − k(ε), deg(ε)) ∈ W.For instance, if c = 8 then (ε 1 ) = (8 + 1 − 6, 4) =(3,4).…”