Sigma invariants of direct products of groups

Abstract

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 ${\mathrm{FP}}_{n+1}$.