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

We prove that if the $n$th ${\mathrm{\ell }}^2$-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.