Residually finite rationally solvable groups and virtual fibring
Dawid Kielak

TL;DR
This paper proves that finitely generated RFRS groups are virtually fibred if and only if their first $L^2$-Betti number vanishes, extending Agol's results for 3-manifold groups with a new proof.
Contribution
It establishes a new criterion linking $L^2$-Betti numbers to virtual fibering in RFRS groups, generalizing Agol's 3-manifold results.
Findings
RFRS groups are virtually fibred if their first $L^2$-Betti number is zero
Provides a new proof of Agol's theorem for 3-manifold groups
Extends the understanding of virtual properties in group theory
Abstract
We show that a finitely generated residually finite rationally solvable (or RFRS) group is virtually fibred, in the sense that it admits a virtual surjection to with a finitely generated kernel, if and only if the first -Betti number of vanishes. This generalises (and gives a new proof of) the analogous result of Ian Agol for fundamental groups of -manifolds.
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