Non-abelian tensor product of residually finite groups

Let G and H be groups that act compatibly on each other. We denote by Η(G, H) a certain extension of the non-abelian tensor product G⊗ H by G× H. Suppose that G is residually finite and the subgroup [G,H]=〈g-1gh∣g∈G,h∈H〉 satisfies some non-trivial identity f≡1. We prove that if p is a prime and every tensor has p-power order, then the non-abelian tensor product G⊗ H is locally finite. Further, we show that if n is a positive integer and every tensor is left n-Engel in Η(G, H) , then the non-abelian tensor product G⊗ H is locally nilpotent. The content of this paper extends some results concerning the non-abelian tensor square G⊗ G. © 2017, Instituto de Matemática e Estatística da Universidade de São Paulo.