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Residually finite rationally solvable groups and virtual fibring


Author: Dawid Kielak
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
Published electronically: December 24, 2019
MathSciNet review: 4073866
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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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Additional Information

Dawid Kielak
Affiliation: Fakultät für Mathematik, Universität Bielefeld, Postfach 100131 D-33501 Bielefeld, Germany
Email: dkielak@math.uni-bielefeld.de

DOI: https://doi.org/10.1090/jams/936
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 \href{https://www.spp2026.de/}‘Geometry at Infinity’ of the German Science Foundation (DFG)
Article copyright: © Copyright 2019 American Mathematical Society