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Abstract. We prove the following results. Let w be a multilinear commutator word. If G is a profinite group in which all w-values are contained in a union of countably many periodic subgroups, then the verbal subgroup w(G) is locally finite. If G is a profinite group in which all w-values are contained in a union of countably many subgroups of finite rank, then the verbal subgroup w(G) has finite rank as well. As a byproduct of the techniques developed in the paper we also prove that if G is a virtually soluble profinite group in which all w-values have finite order, then w(G) is locally finite and has finite exponent.

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, .

A group G has restricted centralizers if for each g in G the centralizer C G (g) either is finite or has finite index in G. A theorem of Shalev states that a profinite group with restricted centralizers is abelian-by-finite. In the present article we handle profinite groups with restricted centralizers of word-values. We show that if w is a multilinear commutator word and G a profinite group with restricted centralizers of w-values, then the verbal subgroup w(G) is abelian-by-finite.

A BFC-group is a group in which all conjugacy classes are finite with bounded size. In 1954 B. H. Neumann discovered that if G is a BFC-group then the derived group G ′ is finite. Let w = w(x 1 , . . . , x n ) be a multilinear commutator. We study groups in which the conjugacy classes containing w-values are finite of bounded order. Let G be a group and let w(G) be the verbal subgroup of G generated by all w-values. We prove that if |x G | ≤ m for every w-value x, then the derived subgroup of w(G) is finite of order bounded by a function of m and n. If |x w(G) | ≤ m for every w-value x, then [w(w(G)), w(G)] is finite of order bounded by a function of m and n.