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

In this paper we provide sufficient conditions for the non-abelian tensor product $G\otimes H$ 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. Theorem 1.2, Theorem 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 $M{\otimes }^{q}N$, and homotopy pushout (cf. Section 5).