On words that are concise in residually finite groups

Abstract

Abstract. A group-word w is called concise if whenever the set of wvalues in a group G is finite it always follows that the verbal subgroup w(G) is finite. More generally, a word w is said to be concise in a class of groups X if whenever the set of w-values is finite for a group G ∈ X, it always follows that w(G) is finite. P. Hall asked whether every word is concise. Due to Ivanov the answer to this problem is known to be negative. Dan Segal asked whether every word is concise in the class of residually finite groups. In this direction we prove that if w is a multilinear commutator and q is a prime-power, then the word w q is indeed concise in the class of residually finite groups. Further, we show that in the case where w = γ k the word w q is boundedly concise in the class of residually finite groups. It remains unknown whether the word w q is actually concise in the class of all groups.

On countable coverings of word values in profinite groups

“…Thus, we assume that G is virtually soluble. By Lemma 2.4 of [3] the derived series of G has only finitely many terms, that is, G (i) = G (i+1) for some positive integer i. It is sufficient to prove that G (i) is locally finite, because then factoring it out we get the case where G is soluble and the result is immediate from Lemma 8.…”

On rational and concise words

“…Segal [12, p. 15] or A. Jaikin-Zapirain [6]). It was shown in [1] that if w is a multilinear commutator word and n is a prime-power, then the word w n is concise in the class of residually finite groups. We say that a word w is boundedly concise in a class of groups X if for every integer m there exists a number ν = ν(X, w, m) such that whenever |G w | ≤ m for a group G ∈ X it always follows that |w(G)| ≤ ν. Fernández-Alcober and Morigi [2] showed that every word which is concise in the class of all groups is actually boundedly concise.…”

“…Moreover they showed that whenever w is a multilinear commutator word having at most m values in a group G, one has |w(G)| ≤ (m − 1) (m−1) . It was shown in [1] that if w = γ k is the kth lower central word and n a prime-power, then the word w n is boundedly concise in the class of residually finite groups. Recall that the word γ k is defined inductively by the formulae…”

Words of Engel type are concise in residually finite groups

“…In recent years, several words were shown to be concise in residually finite groups while their conciseness in the class of all groups remains unknown. In particular, it was shown in [2] that if w is a multilinear commutator word and n is a primepower, then the word w n is concise in the class of residually finite groups. Further examples of words that are concise in residually finite groups were given in [13].…”