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).

中文翻译：

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Engel类型的单词在残差有限的组中很简洁。第二部分

这项工作是对文章[5]的自然跟进。给定组词$ w $和组$ G $，言语子组$ w（G）$是$ G $中所有$ w $值生成的子组。如果$ w（G）$是有限的，则只要$ G $中$ w $值的集合是有限的，则将$ w $称为简洁。是否每个词都在残差有限的组中简洁是一个悬而未决的问题。设$ w = w（x_1，\ ldots，x_k）$为多线性换向器字，$ n $为正整数，$ q $为质数幂。在本文中，我们显示单词$ [w ^ q，_ny] $在残差有限组中是简洁的（定理1.2），而单词$ [w，_ny] $在残差有限组中是简洁的（定理1.1）。