Sigma invariants of direct products of groups

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. Schütz 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 F_n which are not of type FPn + 1 .