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

Anna Felikson; Oleg Karpenkov; Khrystyna Serhiyenko; Pavel Tumarkin

paper

$3$ D Farey graph, lambda lengths and $S L_{2}$ -tilings

arXiv:2306.17118

Abstract

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.

32 pages

3D Farey graph, lambda lengths and SL2​-tilings

Abstract

$3$ D Farey graph, lambda lengths and $S L_{2}$ -tilings