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.

中文翻译：

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剩余有限有理可解群和虚拟纤维化

我们证明了有限生成的剩余有限合理可解（或 RFRS）群 $G$ 是虚拟纤维化的，因为它允许对 $\mathbb{Z}$ 的虚拟投影具有有限生成的核，当且仅当$G$ 的第一个 $L^2$-Betti 数消失。这概括了（并给出了新的证明）Ian Agol 对 $3$-流形的基本群的类似结果。