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| Format: | Preprint |
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2024
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| Online-Zugang: | https://arxiv.org/abs/2407.07703 |
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| _version_ | 1866915239045365760 |
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| author | Wu, Fan Wu, Xiaolei Zhao, Mengfei Zhou, Zixiang |
| author_facet | Wu, Fan Wu, Xiaolei Zhao, Mengfei Zhou, Zixiang |
| contents | We show that the \sϕ-labeled Thompson groups and the twisted Brin--Thompson groups are boundedly acyclic. This allows us to prove several new embedding results for groups. First, every group of type $F_n$ embeds quasi-isometrically into a boundedly acyclic group of type $F_n$ that has no proper finite index subgroups. This improves a result of Bridson and a theorem of Fournier-Facio--Löh--Moraschini. Second, every group of type $F_n$ embeds quasi-isometrically into a $5$-uniformly perfect group of type $F_n$. Third, using Belk--Zaremsky's construction of twisted Brin--Thompson groups, we show that every finitely generated group embeds quasi-isometrically into a finitely generated boundedly acyclic simple group. We also partially answer some questions of Brothier and Tanushevski regarding the finiteness property of $ϕ$-labeled Thompson group $V_ϕ(G)$ and $F_ϕ(G)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_07703 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Embedding groups into boundedly acyclic groups Wu, Fan Wu, Xiaolei Zhao, Mengfei Zhou, Zixiang Group Theory K-Theory and Homology 57M07, 21J06 We show that the \sϕ-labeled Thompson groups and the twisted Brin--Thompson groups are boundedly acyclic. This allows us to prove several new embedding results for groups. First, every group of type $F_n$ embeds quasi-isometrically into a boundedly acyclic group of type $F_n$ that has no proper finite index subgroups. This improves a result of Bridson and a theorem of Fournier-Facio--Löh--Moraschini. Second, every group of type $F_n$ embeds quasi-isometrically into a $5$-uniformly perfect group of type $F_n$. Third, using Belk--Zaremsky's construction of twisted Brin--Thompson groups, we show that every finitely generated group embeds quasi-isometrically into a finitely generated boundedly acyclic simple group. We also partially answer some questions of Brothier and Tanushevski regarding the finiteness property of $ϕ$-labeled Thompson group $V_ϕ(G)$ and $F_ϕ(G)$. |
| title | Embedding groups into boundedly acyclic groups |
| topic | Group Theory K-Theory and Homology 57M07, 21J06 |
| url | https://arxiv.org/abs/2407.07703 |