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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2022
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| Accès en ligne: | https://arxiv.org/abs/2206.12631 |
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| _version_ | 1866910344410038272 |
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| author | Belk, James Bleak, Collin Quick, Martyn Skipper, Rachel |
| author_facet | Belk, James Bleak, Collin Quick, Martyn Skipper, Rachel |
| contents | We introduce the concept of a type system~$\Part$, that is, a partition on the set of finite words over the alphabet~$\{0,1\}$ compatible with the partial action of Thompson's group~$V$, and associate a subgroup~$\Stab{V}{\Part}$ of~$V$. We classify the finite simple type systems and show that the stabilizers of various simple type systems, including all finite simple type systems, are maximal subgroups of~$V$. We also find an uncountable family of pairwise non-isomorphic maximal subgroups of~$V$. These maximal subgroups occur as stabilizers of infinite simple type systems and have not been described in previous literature: specifically, they do not arise as stabilizers in $V$ of finite sets of points in Cantor space. Finally, we show that two natural conditions on subgroups of $V$ (both related to primitivity) are each satisfied only by $V$ itself, giving new ways to recognise when a subgroup of $V$ is not actually proper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_12631 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Type systems and maximal subgroups of Thompson's group $V$ Belk, James Bleak, Collin Quick, Martyn Skipper, Rachel Group Theory 20E28, 20E32, 20F65 We introduce the concept of a type system~$\Part$, that is, a partition on the set of finite words over the alphabet~$\{0,1\}$ compatible with the partial action of Thompson's group~$V$, and associate a subgroup~$\Stab{V}{\Part}$ of~$V$. We classify the finite simple type systems and show that the stabilizers of various simple type systems, including all finite simple type systems, are maximal subgroups of~$V$. We also find an uncountable family of pairwise non-isomorphic maximal subgroups of~$V$. These maximal subgroups occur as stabilizers of infinite simple type systems and have not been described in previous literature: specifically, they do not arise as stabilizers in $V$ of finite sets of points in Cantor space. Finally, we show that two natural conditions on subgroups of $V$ (both related to primitivity) are each satisfied only by $V$ itself, giving new ways to recognise when a subgroup of $V$ is not actually proper. |
| title | Type systems and maximal subgroups of Thompson's group $V$ |
| topic | Group Theory 20E28, 20E32, 20F65 |
| url | https://arxiv.org/abs/2206.12631 |