Improved algebraic fibrings

We show that a virtually RFRS group $G$ of type $\mathrm{FP}_n(\mathbb{Q})$ virtually algebraically fibres with kernel of type $\mathrm{FP}_n(\mathbb{Q})$ if and only if the first $n$ $\ell^2$-Betti numbers of $G$ vanish, that is, $b_p^{(2)}(G) = 0$ for $0 \leqslant p \leqslant n$. We also offer a variant of this result over other fields, in particular in positive characteristic. As an application of the main result, we show that virtually amenable RFRS groups of type $\mathrm{FP}(\mathbb{Q})$ are polycyclic-by-finite. It then follows that if $G$ is a virtually RFRS group of type $\mathrm{FP}(\mathbb{Q})$ such that $\mathbb{Z}G$ is Noetherian, then $G$ is polycyclic-by-finite. This answers a longstanding conjecture of Baer for virtually RFRS groups of type $\mathrm{FP}(\mathbb{Q})$.
Comment: Version accepted for publication at Compositio Mathematica. An addendum (which will appear in the author's thesis, not in the published version of this article) has been added, giving a simple proof that virtually locally indicable groups of finite cohomological dimension satisfy Baer's Conjecture. This was proven in the previous version under the stronger assumptions RFRS and finite type