Some Computations of Non - Abelian Tensor Products of Groups

A generalised tensor product G 0 H of groups G, H has been introduced by R. Brown and J.-L. Loday in [3,4]. It arises in applications in homotopy theory of a generalised Van Kampen theorem. The reason why G 0 H does not necessarily reduce to GUh Oz Huh, the usual tensor product over Z of the abelianisations, is that it is assumed that G acts on H (on the left) and H acts on G (on the left), and these actions are taken into account in the definition of the tensor product. A group G acts on itself by conjugation (" g = hgh-') and so the tensor square GO G is always defined. Further, the commutator map G x G + G induces a homomorphism of groups K: G 0 G + G, sending g@ h to [g, h] =ghg-'h-l. We write J,(G) for Ker IC; its topological interest is the formula [3,4] qSK(G, l)=J,(G).