On residually finite groups satisfying an Engel type identity

Abstract

Let $n,q$ be positive integers. We show that if $G$ is a finitely generated residually finite group satisfying the identity $[x{,}_n{y}^q]\equiv 1,$ then there exists a function $f(n)$ such that $G$ has a nilpotent subgroup of finite index of class at most $f(n)$. We also extend this result to locally graded groups.