  The Product Conjecture for the homological Bieri-Neumann-Strebel-Renz
invariants is proved over a field. Under certain hypotheses the Product
Conjecture is shown to also hold over Z, even though D. Schuetz 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 to derive new information about
subgroups of Thompson's group F, namely that F has subgroups F_m which are not
of type F_{m+1}.
