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 $\mathrm{ FP}(\mathbb{ Q})$.
Categories: math.GR