Finiteness conditions for the box-tensor product of groups and related constructions

We study finiteness conditions and closure properties for the box-tensor product G ⊠ H and related constructions. The content of this paper extends some results concerning the non-abelian tensor product of groups G ⊗ H . In particular, we deduce a quantitative version of the finiteness criterion for the non-abelian tensor product. Moreover, we obtain finiteness conditions for some functors that arise out of the non-abelian tensor square of groups, such as the second homology group H 2 ( G ) , the non-abelian exterior square G ∧ G and the second stable homotopy group of an Eilenberg-MacLane space π 2 S ( K ( G , 1 ) ) .