# Bounded Engel elements in residually finite groups

## TL;DR

This paper proves that certain residually finite groups with bounded Engel elements or specific word-values are locally virtually nilpotent or have bounded nilpotency class, respectively.

## Contribution

It establishes new conditions under which residually finite groups exhibit local virtual nilpotency or bounded nilpotency class based on Engel properties.

## Key findings

- Residually finite groups with bounded Engel elements are locally virtually nilpotent.
- Groups with all $w$-values being $n$-Engel have verbal subgroups with bounded nilpotency class.
- The results connect Engel conditions with structural properties of residually finite groups.

## Abstract

Let $q$ be a prime. Let $G$ be a residually finite group satisfying an identity. Suppose that for every $x \in G$ there exists a $q$-power $m=m(x)$ such that the element $x^m$ is a bounded Engel element. We prove that $G$ is locally virtually nilpotent. Further, let $d,n$ be positive integers and $w$ a non-commutator word. Assume that $G$ is a $d$-generator residually finite group in which all $w$-values are $n$-Engel. We show that the verbal subgroup $w(G)$ has $\{d,n,w\}$-bounded nilpotency class.