Product between ultrafilters and applications to Connes' embedding problem

In this paper we want to apply the notion of product between ultrafilters to answer several questions which arise around Connes' embedding problem. We shall prove that an ultraproduct of hyperlinear groups is still hyperlinear and consequently the von Neumann algebra of the free group with uncountable many generators is embeddable into Rω. This follows also from a general construction that allows, starting from an hyperlinear group, to find a family of hyperlinear groups. We shall prove also that the cross product of an hyperlinear group via a profinite action is embeddable into Rω.