Finiteness conditions for the n-fold tensor product of groups

Let G be a finitely generated group. We prove that the n-fold tensor product G(circle times n) is finite (respectively, polycyclic) if and only if G is finite (respectively, polycyclic). Further, assuming that G is finitely presented, we show that G(circle times n) is finitely presented if and only if gamma(n)(G) is finitely presented. We also examine some finiteness conditions for the non-abelian tensor product of groups.