On residually finite groups with Engel-like conditions

TL;DR

This paper proves that residually finite groups with certain Engel-like conditions on elements or products of specific values are locally virtually nilpotent, advancing understanding of their structure under these conditions.

Contribution

It establishes new conditions under which residually finite groups are locally virtually nilpotent, extending previous results to broader classes involving Engel-like properties.

Findings

Groups with elementwise Engel-like conditions are locally virtually nilpotent.

Products of a bounded number of multilinear commutator values with Engel-like conditions form locally virtually nilpotent subgroups.

The results apply to residually finite groups satisfying specific power and Engel conditions.

Abstract

Let $m,n$ be positive integers. Suppose that $G$ is a residually finite group in which for every element $x \in G$ there exists a positive integer $q=q(x) \leqslant m$ such that $x^q$ is $n$-Engel. We show that $G$ is locally virtually nilpotent. Further, let $w$ be a multilinear commutator and $G$ a residually finite group in which for every product of at most $896$ $w$-values $x$ there exists a positive integer $q=q(x)$ dividing $m$ such that $x^q$ is $n$-Engel. Then $w(G)$ is locally virtually nilpotent.