Improved algebraic fibrings

TL;DR

This paper establishes a criterion linking the vanishing of the first n $ ext{l}^2$-Betti numbers of virtually RFRS groups of type $ ext{FP}_n( ext{Q})$ to their algebraic fibering properties, with implications for their structure.

Contribution

It proves a new equivalence between $ ext{l}^2$-Betti numbers and algebraic fibering for virtually RFRS groups, extending to positive characteristic fields, and resolves a longstanding conjecture for these groups.

Findings

Vanishing of first n $ ext{l}^2$-Betti numbers characterizes algebraic fibering.

Virtually amenable RFRS groups of type $ ext{FP}( ext{Q})$ are polycyclic-by-finite.

Virtually RFRS groups with Noetherian group rings are polycyclic-by-finite.

Abstract

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 $\mathrm{FP}(\mathbb{Q})$.