The non-abelian tensor square of residually finite groups

Abstract

Let m, n be positive integers and p a prime. We denote by $\nu (G)$ an extension of the non-abelian tensor square $G \otimes G$ by $G \times G$. We prove that if G is a residually finite group satisfying some non-trivial identity $f \equiv ~1$ and for every $x,y \in G$ there exists a p-power $q=q(x,y)$ such that $[x,y^{\varphi }]^q = 1$, then the derived subgroup $\nu (G)'$ is locally finite (Theorem A). Moreover, we show that if G is a residually finite group in which for every $x,y \in G$ there exists a p-power $q=q(x,y)$ dividing $p^m$ such that $[x,y^{\varphi }]^q$ is left n-Engel, then the non-abelian tensor square $G \otimes G$ is locally virtually nilpotent (Theorem B).