Finiteness properties of Subgroups of Houghton Groups of full Hirsch length
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Cox, Charles Kropholler, Peter Martino, Armando |
| author_facet | Cox, Charles Kropholler, Peter Martino, Armando |
| contents | In the 1980's K.S. Brown proved that the Houghton group $H_n$ is of type $\operatorname{F}_{n-1}$ but not $\operatorname{FP}_n$. We show that, provided $n\ge3$, the same conclusion holds for all subgroups $G$ of $H_n$ that are 'large' in the sense that there is an epimorphism $G\twoheadrightarrow\mathbb{Z}^{n-1}$.
Our research leads naturally to the study of generalised permutational wreath products in which the base of the wreath product is a direct product of finite groups which are allowed to vary in isomorphism type from one orbit to another. Such generalised wreath products arise naturally amongst the large subgroups of Houghton groups and are accommodated by a generalised Jordan--Wielandt theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07816 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finiteness properties of Subgroups of Houghton Groups of full Hirsch length Cox, Charles Kropholler, Peter Martino, Armando Group Theory 18G10, 20J05 In the 1980's K.S. Brown proved that the Houghton group $H_n$ is of type $\operatorname{F}_{n-1}$ but not $\operatorname{FP}_n$. We show that, provided $n\ge3$, the same conclusion holds for all subgroups $G$ of $H_n$ that are 'large' in the sense that there is an epimorphism $G\twoheadrightarrow\mathbb{Z}^{n-1}$. Our research leads naturally to the study of generalised permutational wreath products in which the base of the wreath product is a direct product of finite groups which are allowed to vary in isomorphism type from one orbit to another. Such generalised wreath products arise naturally amongst the large subgroups of Houghton groups and are accommodated by a generalised Jordan--Wielandt theorem. |
| title | Finiteness properties of Subgroups of Houghton Groups of full Hirsch length |
| topic | Group Theory 18G10, 20J05 |
| url | https://arxiv.org/abs/2508.07816 |