Sigma invariants of direct products of groups

The Product Conjecture for the homological Bieri-Neumann-Strebel-Renz invari- ants is proved over a field. Under certain hypotheses the Product Conjecture is shown to also hold over Z, even though D. Schutz has recently shown that the Conjecture is false in general over Z. Our version over Z is applied in a joint paper with D. Kochloukova (5) to show that for all n Thompson's group F contains subgroups of type Fn which are not of type FPnC1.