1 results
Improved algebraic fibrings
- Part of
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- Journal:
- Compositio Mathematica / Volume 160 / Issue 9 / September 2024
- Published online by Cambridge University Press:
- 13 September 2024, pp. 2203-2227
- Print publication:
- September 2024
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- Article
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We show that a virtually residually finite rationally solvable (RFRS) group
$G$ of type 
$\mathtt {FP}_n(\mathbb {Q})$ virtually algebraically fibres with kernel of type 
$\mathtt {FP}_n(\mathbb {Q})$ if and only if the first 
$n$ 
$\ell ^2$-Betti numbers of 
$G$ vanish, that is, 
$b_p^{(2)}(G) = 0$ for 
$0 \leqslant p \leqslant n$. This confirms a conjecture of Kielak. 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 amenable virtually RFRS groups of type 
$\mathtt {FP}(\mathbb {Q})$ are virtually Abelian. It then follows that if 
$G$ is a virtually RFRS group of type 
$\mathtt {FP}(\mathbb {Q})$ such that 
$\mathbb {Z} G$ is Noetherian, then 
$G$ is virtually Abelian. This confirms a conjecture of Baer for the class of virtually RFRS groups of type 
$\mathtt {FP}(\mathbb {Q})$, which includes (for instance) the class of virtually compact special groups.
