Residually finite rationally solvable groups and virtual fibring

Abstract

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