Finite presentability of twisted Brin-Thompson groups
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913585708400640 |
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| author | Zaremsky, Matthew C. B. |
| author_facet | Zaremsky, Matthew C. B. |
| contents | Given a group $G$ acting faithfully on a set $S$, we characterize precisely when the twisted Brin-Thompson group $SV_G$ is finitely presented. The answer is that $SV_G$ is finitely presented if and only if we have the following: $G$ is finitely presented, the action of $G$ on $S$ has finitely many orbits of two-element subsets of $S$, and the stabilizer in $G$ of any element of $S$ is finitely generated. Since twisted Brin-Thompson groups are simple, a consequence is that any subgroup of a group admitting an action as above satisfies the Boone-Higman conjecture. In the course of proving this, we also establish a sufficient condition for a group acting cocompactly on a simply connected complex to be finitely presented, even if certain edge stabilizers are not finitely generated, which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_18354 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finite presentability of twisted Brin-Thompson groups Zaremsky, Matthew C. B. Group Theory Given a group $G$ acting faithfully on a set $S$, we characterize precisely when the twisted Brin-Thompson group $SV_G$ is finitely presented. The answer is that $SV_G$ is finitely presented if and only if we have the following: $G$ is finitely presented, the action of $G$ on $S$ has finitely many orbits of two-element subsets of $S$, and the stabilizer in $G$ of any element of $S$ is finitely generated. Since twisted Brin-Thompson groups are simple, a consequence is that any subgroup of a group admitting an action as above satisfies the Boone-Higman conjecture. In the course of proving this, we also establish a sufficient condition for a group acting cocompactly on a simply connected complex to be finitely presented, even if certain edge stabilizers are not finitely generated, which may be of independent interest. |
| title | Finite presentability of twisted Brin-Thompson groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2405.18354 |