$3$D Farey graph, lambda lengths and $SL_2$-tilings

TL;DR

This paper introduces a 3D Farey graph and establishes its connections to lambda lengths and SL_2-tilings, including a 3D Ptolemy relation and classification of tame tilings over Eisenstein integers.

Contribution

It extends Farey tessellation concepts into three dimensions and generalizes classification results for SL_2-tilings using the 3D Farey graph.

Findings

Proves a 3D version of the Ptolemy relation.

Classifies tame SL_2-tilings over Eisenstein integers.

Establishes relations between 3D Farey graph, lambda lengths, and SL_2-tilings.

Abstract

We explore a three-dimensional counterpart of the Farey tessellation and its relations to Penner's lambda lengths and $SL_2$-tilings. In particular, we prove a three-dimensional version of Ptolemy relation, and generalise results of Ian Short to classify tame $SL_2$-tilings over Eisenstein integers in terms of pairs of paths in the 3D Farey graph.