Approximations of the Riley slice
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arXiv
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| Format: | Preprint |
| Published: |
2021
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| _version_ | 1866912450681503744 |
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| author | Elzenaar, Alex Martin, Gaven Schillewaert, Jeroen |
| author_facet | Elzenaar, Alex Martin, Gaven Schillewaert, Jeroen |
| contents | Adapting the ideas of L. Keen and C. Series used in their study of the Riley slice of Schottky groups generated by two parabolics, we explicitly identify `half-space' neighbourhoods of pleating rays which lie completely in the Riley slice. This gives a provable method to determine if a point is in the Riley slice or not. We also discuss the family of Farey polynomials which determine the rational pleating rays and their root set which determines the Riley slice; this leads to a dynamical systems interpretation of the slice. Adapting these methods to the case of Schottky groups generated by two elliptic elements in subsequent work facilitates the programme to identify all the finitely many arithmetic generalised triangle groups and their kin. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2111_03230 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Approximations of the Riley slice Elzenaar, Alex Martin, Gaven Schillewaert, Jeroen Geometric Topology Complex Variables Group Theory 32G15, 30F40, 57N05 Adapting the ideas of L. Keen and C. Series used in their study of the Riley slice of Schottky groups generated by two parabolics, we explicitly identify `half-space' neighbourhoods of pleating rays which lie completely in the Riley slice. This gives a provable method to determine if a point is in the Riley slice or not. We also discuss the family of Farey polynomials which determine the rational pleating rays and their root set which determines the Riley slice; this leads to a dynamical systems interpretation of the slice. Adapting these methods to the case of Schottky groups generated by two elliptic elements in subsequent work facilitates the programme to identify all the finitely many arithmetic generalised triangle groups and their kin. |
| title | Approximations of the Riley slice |
| topic | Geometric Topology Complex Variables Group Theory 32G15, 30F40, 57N05 |
| url | https://arxiv.org/abs/2111.03230 |