Residually finite rationally solvable groups and virtual fibring

Kielak, Dawid

doi:10.1090/jams/936

Residually finite rationally solvable groups and virtual fibring
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by Dawid Kielak;

J. Amer. Math. Soc. 33 (2020), 451-486

DOI: https://doi.org/10.1090/jams/936

Published electronically: December 24, 2019

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Abstract:

We show that a non-trivial 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.

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