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 G w and the verbal subgroup w ( G ) is the one generated by G w . In the present paper we consider profinite groups admitting a word w such that the cardinality of G w is less than 2 ℵ 0 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.