Groups with bounded verbal conjugacy classes

Abstract

Let F be a free group and let w A F . For a group G, let G w denote the set of all w-values in G and wðGÞ the verbal subgroup of G corresponding to w. A word w is called boundedly concise if, for each group G such that jG w j c m, we have jwðGÞj c c for some integer c ¼ cðmÞ depending only on m. The main theorem of the paper says that if w is a boundedly concise word and G is a group such that jx Gw j c m for all x A G then jx wðGÞ j c d for all x A G and some integer d ¼ dðm; wÞ depending only on m and w.

Outer commutator words are uniformly concise

“…Now we can easily see that Theorem B is true for derived words. This fact is already proved in Lemma 3.3 of [1], and the proof we provide is essentially the same. We include it here for the sake of completeness, and because the use of Lemma 3.5 simplifies the presentation.…”

“…Finally, we derive Theorem A from Theorem B by adapting the argument given by Brazil, Krasilnikov and Shumyatsky in [1] for the case of derived words. We will need Dietzmann's Lemma, whose proof can we found in [5, 14.5.7].…”

“…The existence of such a series was proved in [1] for derived words, and this particular case is the starting point for our proof of Theorem B. However, dealing with an arbitrary outer commutator word ω is a much more delicate matter, since one has to keep control of the nesting of commutators in ω, and then there might be problems such as the commutator of two values of ω not being necessarily a value of ω (contrary to the case of derived words).…”

“…(Segal and Shalev, [6].) More recently, Brazil, Krasilnikov and Shumyatsky [1] have given explicit bounds for all lower central words and for all derived words; as a matter of fact, they find a single upper bound for this infinite family of words, namely (m!) m .…”

On words that are concise in residually finite groups

“…Our goal in the remaining part of this section is to prove that the verbal subgroup w(G) of a soluble-by-finite group G possesses a normal series with some very specific properties. Similar series were considered in [1] and [2]. We start with the case where G is perfect.…”