Improved algebraic fibrings

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 $0⩽p⩽n$. 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 $\mathrm{FP}(\mathbb{Q})$.

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

Cited by in corpus (2)

• The Hanna Neumann Conjecture for graphs of free groups with cyclic edge groups
• Groups with exotic finiteness properties from complex Morse theory