Finiteness conditions for the n-fold tensor product of groups

Let [Formula: see text] be a finitely generated group. We prove that the [Formula: see text]-fold tensor product [Formula: see text] is finite (respectively, polycyclic) if and only if [Formula: see text] is finite (respectively, polycyclic). Further, assuming that [Formula: see text] is finitely presented, we show that [Formula: see text] is finitely presented if and only if [Formula: see text] is finitely presented. We also examine some finiteness conditions for the non-abelian tensor product of groups.