On finiteness of some verbal subgroups in profinite groups

Abstract

Given a group word w and a group G, the set of w-values in G is denoted by Gw and the verbal subgroup w(G) is the one generated by Gw. In the present paper we consider profinite groups admitting a word w such that the cardinality of Gw is less than 2ℵ0 and w(G) is generated by finitely many w-values. For several families of words w we show that under these assumptions w(G) must be finite. Our results are related to the concept of conciseness of group words.