Farey graphs and geodesic expansions of complex continued fractions

TL;DR

This paper explores complex Farey graphs and hyperbolic Farey polygons in relation to complex continued fractions, tessellations, and Bianchi groups, offering a new geometric approach to these mathematical structures.

Contribution

It introduces a novel geometric framework using Farey polygons and graphs in hyperbolic space to analyze complex continued fractions and related tessellations, extending previous methods.

Findings

Recovered tessellations of hyperbolic plane by Hecke groups

Defined polyhedra inducing Farey tessellations of hyperbolic 3-space

Provided a more general approach to geodesic complex continued fractions

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.