## Residually finite rationally solvable groups and virtual fibring

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Dawid Kielak
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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.## References

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## Additional Information

**Dawid Kielak**- Affiliation: Fakultät für Mathematik, Universität Bielefeld, Postfach 100131 D-33501 Bielefeld, Germany
- MR Author ID: 1027989
- ORCID: 0000-0002-5536-9070
- Email: dkielak@math.uni-bielefeld.de
- Received by editor(s): September 25, 2018
- Received by editor(s) in revised form: July 22, 2019, and August 27, 2019
- Published electronically: December 24, 2019
- Additional Notes: The author was supported by the grant KI 1853/3-1 within the Priority Programme 2026 ‘Geometry at Infinity’ of the German Science Foundation (DFG)
- © Copyright 2019 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**33**(2020), 451-486 - MSC (2010): Primary 20F65; Secondary 57M10, 20E26, 12E15, 16S35, 20J05
- DOI: https://doi.org/10.1090/jams/936
- MathSciNet review: 4073866