$3$D Farey graph, lambda lengths and $S{L}_2$-tilings

We explore a three-dimensional counterpart of the Farey tessellation and its relations to Penner’s lambda lengths and $S{L}_2$-tilings. In particular, we prove a three-dimensional version of Ptolemy relation, and generalise results of Ian Short to classify tame $S{L}_2$-tilings over Eisenstein integers in terms of pairs of paths in the 3D Farey graph.