The planar $3$-colorable subgroup $\mathcal{E}$ of Thompson's group $F$ and its even part
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866916625045782528 |
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| author | Aiello, Valeriano Nagnibeda, Tatiana |
| author_facet | Aiello, Valeriano Nagnibeda, Tatiana |
| contents | We study the planar $3$-colorable subgroup $\mathcal{E}$ of Thompson's group $F$ and its even part $\mathcal{E}_{\rm EVEN}$. The latter is obtained by cutting $\mathcal{E}$ with a finite index subgroup of $F$ isomorphic to $F$, namely the rectangular subgroup $K_{(2,2)}$. We show that the even part $\mathcal{E}_{\rm EVEN}$ of the planar $3$-colorable subgroup admits a description in terms of stabilisers of suitable subsets of dyadic rationals. As a consequence $\mathcal{E}_{\rm EVEN}$ is closed in the sense of Golan and Sapir. We then study three quasi-regular representations associated with $\mathcal{E}_{\rm EVEN}$: two are shown to be irreducible and one to be reducible. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2212_12269 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The planar $3$-colorable subgroup $\mathcal{E}$ of Thompson's group $F$ and its even part Aiello, Valeriano Nagnibeda, Tatiana Group Theory We study the planar $3$-colorable subgroup $\mathcal{E}$ of Thompson's group $F$ and its even part $\mathcal{E}_{\rm EVEN}$. The latter is obtained by cutting $\mathcal{E}$ with a finite index subgroup of $F$ isomorphic to $F$, namely the rectangular subgroup $K_{(2,2)}$. We show that the even part $\mathcal{E}_{\rm EVEN}$ of the planar $3$-colorable subgroup admits a description in terms of stabilisers of suitable subsets of dyadic rationals. As a consequence $\mathcal{E}_{\rm EVEN}$ is closed in the sense of Golan and Sapir. We then study three quasi-regular representations associated with $\mathcal{E}_{\rm EVEN}$: two are shown to be irreducible and one to be reducible. |
| title | The planar $3$-colorable subgroup $\mathcal{E}$ of Thompson's group $F$ and its even part |
| topic | Group Theory |
| url | https://arxiv.org/abs/2212.12269 |