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Improved algebraic fibrings

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  13 September 2024

Sam P. Fisher Open the ORCID record for Sam P. Fisher [Opens in a new window]
Sam P. Fisher*
Affiliation:
Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK sam.fisher@maths.ox.ac.uk
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Abstract

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.

Keywords

RFRS groupsalgebraic fibringfiniteness propertiesHughes-free division rings

MSC classification

Primary: 20F65: Geometric group theory

Information

Type
Research Article
Information
Compositio Mathematica , Volume 160 , Issue 9 , September 2024 , pp. 2203 - 2227
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Copyright
© The Author(s), 2024. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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