Cones and ping-pong in three dimensions
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2021
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| _version_ | 1866929281639120896 |
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| author | Frieden, Gabriel Gélinas, Félix Soucy, Étienne |
| author_facet | Frieden, Gabriel Gélinas, Félix Soucy, Étienne |
| contents | We study the hypergeometric group in ${\rm GL}_3(\mathbb{C})$ with parameters $α= (\frac{1}{4}, \frac{1}{2}, \frac{3}{4})$ and $β= (0,0,0)$. We give a new proof that this group is isomorphic to the free product $\mathbb{Z}/4\mathbb{Z} * \mathbb{Z}/2\mathbb{Z}$ by exhibiting a ping-pong table. Our table is determined by a simplicial cone in $\mathbb{R}^3$, and we prove that this is the unique simplicial cone (up to sign) for which our construction produces a valid ping-pong table. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2111_14692 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Cones and ping-pong in three dimensions Frieden, Gabriel Gélinas, Félix Soucy, Étienne Group Theory We study the hypergeometric group in ${\rm GL}_3(\mathbb{C})$ with parameters $α= (\frac{1}{4}, \frac{1}{2}, \frac{3}{4})$ and $β= (0,0,0)$. We give a new proof that this group is isomorphic to the free product $\mathbb{Z}/4\mathbb{Z} * \mathbb{Z}/2\mathbb{Z}$ by exhibiting a ping-pong table. Our table is determined by a simplicial cone in $\mathbb{R}^3$, and we prove that this is the unique simplicial cone (up to sign) for which our construction produces a valid ping-pong table. |
| title | Cones and ping-pong in three dimensions |
| topic | Group Theory |
| url | https://arxiv.org/abs/2111.14692 |