On the cohomological dimension of kernels of maps to $\mathbb Z$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914266680918016 |
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| author | Fisher, Sam P. |
| author_facet | Fisher, Sam P. |
| contents | We prove that if $G$ is a finitely generated RFRS group of cohomological dimension $2$, then $G$ is virtually free-by-cyclic if and only if $b_2^{(2)}(G) = 0$. This answers a question of Wise and generalises and gives a new proof of a recent theorem of Kielak and Linton, where the same result is obtained under the additional hypotheses that $G$ is virtually compact special and hyperbolic. More generally, we show that if $G$ is a RFRS group of cohomological dimension $n$ and of type $\mathrm{FP}_{n-1}$, then $G$ admits a virtual map to $\mathbb Z$ with kernel of rational cohomological dimension $n-1$ if and only if $b_n^{(2)}(G) = 0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_18758 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the cohomological dimension of kernels of maps to $\mathbb Z$ Fisher, Sam P. Group Theory 20F65, 20J05 (Primary) 16S34 (Secondary) We prove that if $G$ is a finitely generated RFRS group of cohomological dimension $2$, then $G$ is virtually free-by-cyclic if and only if $b_2^{(2)}(G) = 0$. This answers a question of Wise and generalises and gives a new proof of a recent theorem of Kielak and Linton, where the same result is obtained under the additional hypotheses that $G$ is virtually compact special and hyperbolic. More generally, we show that if $G$ is a RFRS group of cohomological dimension $n$ and of type $\mathrm{FP}_{n-1}$, then $G$ admits a virtual map to $\mathbb Z$ with kernel of rational cohomological dimension $n-1$ if and only if $b_n^{(2)}(G) = 0$. |
| title | On the cohomological dimension of kernels of maps to $\mathbb Z$ |
| topic | Group Theory 20F65, 20J05 (Primary) 16S34 (Secondary) |
| url | https://arxiv.org/abs/2403.18758 |