BNSR invariants and ${\ell }^2$-homology

Abstract

We prove that if the $n$th ${\ell }^2$-Betti number of a group is non-zero then its $n$th BNSR invariant over $\mathbb{Q}$ is empty, under suitable finiteness conditions. We apply this to answer questions of Friedl--Vidussi and Llosa Isenrich--Py about aspherical Kähler manifolds, to verify some cases of the Singer Conjecture, and to compute certain BNSR invariants of poly-free and poly-surface groups.

31 pages, comments welcome!