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| Format: | Preprint |
| Published: |
2021
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| Online Access: | https://arxiv.org/abs/2112.00397 |
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| _version_ | 1866916487240876032 |
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| author | Fisher, Sam P. |
| author_facet | Fisher, Sam P. |
| contents | 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 $\mathrm{FP}(\mathbb{Q})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_00397 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Improved algebraic fibrings Fisher, Sam P. Group Theory 20F65 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 $\mathrm{FP}(\mathbb{Q})$. |
| title | Improved algebraic fibrings |
| topic | Group Theory 20F65 |
| url | https://arxiv.org/abs/2112.00397 |