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  to show that for all n Thompson’s group F contains subgroups of type Fn which are not of type FPn + 1.
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Robert Bieri, Ross Geoghegan, Sigma invariants of direct products of groups. Groups Geom. Dyn. 4 (2010), no. 2, pp. 251–261DOI 10.4171/GGD/82