  We show that a finitely generated residually finite rationally solvable (or
RFRS) group $G$ is virtually fibred, in the sense that it admits a virtual
surjection to $\mathbb{Z}$ with a finitely generated kernel, if and only if the
first $L^2$-Betti number of $G$ vanishes. This generalises (and gives a new
proof of) the analogous result of Ian Agol for fundamental groups of
$3$-manifolds.
