# Words of Engel type are concise in residually finite groups. Part II

## TL;DR

This paper investigates the conciseness of specific group words in residually finite groups, establishing new results about the finiteness and boundedness of verbal subgroups generated by these words.

## Contribution

It proves that the words [w^q,_n y] are concise and [w,_n y] are boundedly concise in residually finite groups, advancing understanding of word conciseness.

## Key findings

- [w^q,_n y] is concise in residually finite groups
- [w,_n y] is boundedly concise in residually finite groups
- Advances knowledge on the behavior of multilinear commutator words

## Abstract

Given a group-word w and a group G, the verbal subgroup w(G) is the one generated by all w-values in G. The word w is called concise if w(G) is finite whenever the set of w-values in G is finite. It is an open question whether every word is concise in residually finite groups. Let w=w(x_1,..,x_k) be a multilinear commutator word, n a positive integer and q a prime power. In the present article we show that the word [w^q,_n y] is concise in residually finite groups while the word [w,_n y] is boundedly concise in residually finite groups.