Words of Engel type are concise in residually finite groups

TL;DR

This paper investigates the conciseness of generalized Engel words in residually finite groups, showing that these words are concise within this class, thus contributing to the understanding of verbal subgroup properties.

Contribution

The paper proves that various generalizations of Engel words are concise in residually finite groups, extending previous results and addressing open questions in group theory.

Findings

Generalized Engel words are concise in residually finite groups

Addresses an open problem related to Hall's question in residually finite groups

Provides new insights into verbal subgroup properties in specific group classes

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 said to be concise if w(G) is finite whenever the set of w-values in G is finite. In the sixties P. Hall asked whether every word is concise but later Ivanov answered this question in the negative. On the other hand, Hall's question remains wide open in the class of residually finite groups. In the present article we show that various generalizations of the Engel word are concise in residually finite groups.