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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2510.16879 |
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| _version_ | 1866909856697417728 |
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| author | Palmer, Martin Wu, Xiaolei |
| author_facet | Palmer, Martin Wu, Xiaolei |
| contents | We show that labelled Thompson groups and twisted Brin--Thompson groups are all acyclic. This allows us to prove several new embedding results for groups. First, every group of type $F_n$ embeds quasi-isometrically as a subgroup of an acyclic group of type $F_n$ that has no proper finite-index subgroups. This improves results of Baumslag--Dyer--Heller ($n=1$) and Baumslag--Dyer--Miller ($n=2$) from the early 80s, as well as a more recent result of Bridson ($n=2$). Second, we show that every finitely generated group embeds quasi-isometrically as a subgroup of a $2$-generated, simple, acyclic group. Our results also allow us to produce, for each $n\geqslant 2$, the first known example of an acyclic group that is of type $F_n$ but not $F_{n+1}$. These examples can moreover be taken to be simple. Furthermore, our examples provide a rich source of universally boundedly acyclic groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_16879 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Embedding groups into acyclic groups Palmer, Martin Wu, Xiaolei Group Theory Algebraic Topology Dynamical Systems 20J06, 22A22 We show that labelled Thompson groups and twisted Brin--Thompson groups are all acyclic. This allows us to prove several new embedding results for groups. First, every group of type $F_n$ embeds quasi-isometrically as a subgroup of an acyclic group of type $F_n$ that has no proper finite-index subgroups. This improves results of Baumslag--Dyer--Heller ($n=1$) and Baumslag--Dyer--Miller ($n=2$) from the early 80s, as well as a more recent result of Bridson ($n=2$). Second, we show that every finitely generated group embeds quasi-isometrically as a subgroup of a $2$-generated, simple, acyclic group. Our results also allow us to produce, for each $n\geqslant 2$, the first known example of an acyclic group that is of type $F_n$ but not $F_{n+1}$. These examples can moreover be taken to be simple. Furthermore, our examples provide a rich source of universally boundedly acyclic groups. |
| title | Embedding groups into acyclic groups |
| topic | Group Theory Algebraic Topology Dynamical Systems 20J06, 22A22 |
| url | https://arxiv.org/abs/2510.16879 |