Infinitely presented simple groups separated by homological finiteness properties
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912623157575680 |
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| author | Isenrich, Claudio Llosa Schesler, Eduard Wu, Xiaolei |
| author_facet | Isenrich, Claudio Llosa Schesler, Eduard Wu, Xiaolei |
| contents | Given a finitely generated linear group $G$ over $\mathbb{Q}$, we construct a simple group $Γ$ that has the same finiteness properties as $G$ and admits $G$ as a quasi-retract. As an application, we construct a simple group of type $\mathrm{FP}_{\infty}$ that is not finitely presented. Moreover we show that for every $n \in \mathbb{N}$ there is a simple group of type $\mathrm{FP}_n$ that is neither finitely presented nor of type $\mathrm{FP}_{n+1}$. Since our simple groups arise as Röver--Nekrashevych groups, this answers a question of Zaremsky. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_01952 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Infinitely presented simple groups separated by homological finiteness properties Isenrich, Claudio Llosa Schesler, Eduard Wu, Xiaolei Group Theory 20E32, 20E08 Given a finitely generated linear group $G$ over $\mathbb{Q}$, we construct a simple group $Γ$ that has the same finiteness properties as $G$ and admits $G$ as a quasi-retract. As an application, we construct a simple group of type $\mathrm{FP}_{\infty}$ that is not finitely presented. Moreover we show that for every $n \in \mathbb{N}$ there is a simple group of type $\mathrm{FP}_n$ that is neither finitely presented nor of type $\mathrm{FP}_{n+1}$. Since our simple groups arise as Röver--Nekrashevych groups, this answers a question of Zaremsky. |
| title | Infinitely presented simple groups separated by homological finiteness properties |
| topic | Group Theory 20E32, 20E08 |
| url | https://arxiv.org/abs/2510.01952 |