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Farey graphs and geodesic expansions of complex continued fractions

Hitoshi Nakada, Rie Natsui, Jörg Thuswaldner

Abstract

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.

Farey graphs and geodesic expansions of complex continued fractions

Abstract

We discuss complex Farey graphs for the Euclidean imaginary quadratic number fields , . We study hyperbolic versions of A. Schmidt's Farey polygons living in -dimensional hyperbolic space . Using these Farey polygons we recover tessellations of the hyperbolic plane that are defined by the action of the Hecke groups and 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 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.

Paper Structure

This paper contains 6 sections, 24 theorems, 51 equations, 14 figures.

Table of Contents

  1. The generalized setting in the imaginary quadratic case
  2. Properties of the Farey graph
  3. The subgraph of $\mathcal{E}_d$ on the walls and Hecke groups
  4. The Farey tessellation of $\mathbb H^{3}$
  5. Geodesic complex continued fraction algorithms
  6. Large partial quotients imply geodesicness

Key Result

proposition 1

For $d\in\{1,2,3,7,11\}$ let $\tfrac{p_{k}}{q_{k}} \in \mathbb Q(\sqrt{-d}) \cup \{\infty\}$ with $\gcd(p_{k}, q_{k})=1$ for $k\in\{1,2\}$. Then $\tfrac{p_{1}}{q_{1}}\to \tfrac{p_{2}}{q_{2}}$ is an edge of $\mathcal{E}_{d}$ if and only if $|p_{1}q_{2} - p_{2}q_{1}| = 1$

Figures (14)

  • Figure 1: The classical Farey tessellation of $\mathbb{H}^2$.
  • Figure 2: Ford circles (black) and their interplay with the Farey graph (shaded).
  • Figure 3: A part of the Farey graph $\mathcal{E}_2$ with its edges drawn as geodesics in $\mathbb{H}^3$ (left) and its interplay with Ford spheres (right).
  • Figure 4: The region $U_3$.
  • Figure 5: The Farey quadrangle $\square$ and the Farey hexagon $\hexagon$. The numbers on the dashed lines between cusps $\frac{p}{q}$ and $\frac{r}{s}$ are the values of $|ps-qr|$ (for the hexagon, not all connecting lines are drawn, however, the corresponding values of $|ps-qr|$ are symmetric as can be confirmed by a short direct calculation).
  • ...and 9 more figures

Theorems & Definitions (57)

  • remark 1
  • definition 1: Generalization of Farey Graph
  • remark 2
  • proposition 1
  • proof
  • definition 2: Ford sphere
  • definition 3: Farey triangle
  • definition 4: Farey quadrangle
  • definition 5: Farey hexagon
  • definition 6: standard cell and wall
  • ...and 47 more