Farey graphs and geodesic expansions of complex continued fractions
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| Format: | Preprint |
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2026
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| author | Nakada, Hitoshi Natsui, Rie Thuswaldner, Jörg |
| author_facet | Nakada, Hitoshi Natsui, Rie Thuswaldner, Jörg |
| contents | We discuss complex Farey graphs for the Euclidean imaginary quadratic number fields $\mathbb Q(\sqrt{-d})$, $d\in\{1, 2, 3, 7, 11\}$. We study hyperbolic versions of A. Schmidt's Farey polygons living in $3$-dimensional hyperbolic space $\mathbb{H}^3$. Using these Farey polygons we recover tessellations of the hyperbolic plane $\mathbb{H}^2$ that are defined by the action of the Hecke groups $H_4$ and $H_6$ and have been studied earlier by I. Short and M. Walker. Moreover, hyperbolic Farey polygons allow us to define polyhedra that induce Farey tessellations of $\mathbb{H}^3$ by the action of certain Bianchi groups. Using complex Farey graphs we consider geodesic complex continued fraction expansions. Our method provides a different and more general approach as the one from the discussion by M. Hockman. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_28468 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Farey graphs and geodesic expansions of complex continued fractions Nakada, Hitoshi Natsui, Rie Thuswaldner, Jörg Number Theory 11J70, 20G20, 51M10, 51M20, 52C22 We discuss complex Farey graphs for the Euclidean imaginary quadratic number fields $\mathbb Q(\sqrt{-d})$, $d\in\{1, 2, 3, 7, 11\}$. We study hyperbolic versions of A. Schmidt's Farey polygons living in $3$-dimensional hyperbolic space $\mathbb{H}^3$. Using these Farey polygons we recover tessellations of the hyperbolic plane $\mathbb{H}^2$ that are defined by the action of the Hecke groups $H_4$ and $H_6$ and have been studied earlier by I. Short and M. Walker. Moreover, hyperbolic Farey polygons allow us to define polyhedra that induce Farey tessellations of $\mathbb{H}^3$ by the action of certain Bianchi groups. Using complex Farey graphs we consider geodesic complex continued fraction expansions. Our method provides a different and more general approach as the one from the discussion by M. Hockman. |
| title | Farey graphs and geodesic expansions of complex continued fractions |
| topic | Number Theory 11J70, 20G20, 51M10, 51M20, 52C22 |
| url | https://arxiv.org/abs/2603.28468 |