Residually finite rationally solvable groups and virtual fibring

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 $G$ 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.