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

### Eloisa Detomi

Università di Padova, Italy### Marta Morigi

Università di Bologna, Italy### Pavel Shumyatsky

Universidade de Brasília, Brazil

## Abstract

This work is a natural follow-up of the article [5]. 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 $w=w(x_1,\ldots,x_k)$ be a multilinear commutator word, $n$ a positive integer and $q$ a prime power. In the present article we show that the word $[w^q,_ny]$ is concise in residually finite groups (Theorem 1.2) while the word $[w,_ny]$ is boundedly concise in residually finite groups (Theorem 1.1).

## Cite this article

Eloisa Detomi, Marta Morigi, Pavel Shumyatsky, Words of Engel type are concise in residually finite groups. Part II. Groups Geom. Dyn. 14 (2020), no. 3, pp. 991–1005

DOI 10.4171/GGD/571