On the finiteness of the non-abelian tensor product of groups

Abstract

In this paper we provide sufficient conditions for the non-abelian tensor product G⊗H to be polycyclic/polycyclic-by-finite in terms of involved groups and derivative subgroups (cf. Theorem 1.1); we also give sufficient conditions for the (local) finiteness of the non-abelian tensor product of groups (cf. Theorems 1.2–1.5). Furthermore, we deduce similar results for some related constructions associated to the non-abelian tensor products, such as the Schur multiplier of a pair of groups M(G,N), the non-abelian q-tensor product M⊗qN, and homotopy pushout (cf. Section 5).