Conciseness of compact $R$-analytic groups

Abstract

We prove that every word is strongly concise in the class of compact R-analytic groups. IntroductionA word in k-variables is an element w(x 1 , . . . , x k ) of the free group F (x 1 , . . . , x k ), and, given any group G, it inherently defines a map w : G (k) → G. The image of that map, which will be denoted by w{G}, is the set of word-values of w, and it is typically not a subgroup of G. However, we can naturally associate two subgroups to w: the verbal subgroup defined as w(G) = w{G} , and the marginal subgroup, that is,A word is said to be concise in G if w{G} being finite implies that w(G) is finite. More generally, w is concise in a class C of groups if it is concise in all the groups G ∈ C, and w is simply called concise if it is concise in any group. P. Hall initially conjectured that all words were concise, but Ivanov [8] proved that conjecture to be false. Nevertheless, as Jaikin-Zapirain highlighted in [11], P. Hall's conjecture is still open for profinite groups: Profinite Conciseness Conjecture. Every word is concise in the class of profinite groups.