  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 K\"ahler manifolds, to verify some cases of the
Singer Conjecture, and to compute certain BNSR invariants of poly-free and
poly-surface groups.
