Residually finite rationally solvable groups and virtual fibring

Abstract

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.

29 pages, no figures, v2. improved exposition following referee's comments. To appear in J. Amer. Math. Soc

References in corpus (1)

• Amenability of groups is characterized by Myhill's Theorem

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