3D Farey graph, lambda lengths and SL₂-tilings

2023/06/29 by Anna Felikson, Felikson, Anna, Oleg Karpenkov +5
Engineering · Mathematics · #05B99 #05E15 #11A55 #13F60 #51F15 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2306.17118

openalex publication_date 2023/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We explore a three-dimensional counterpart of the Farey tessellation and its relations to Penner's lambda lengths and SL₂-tilings. In particular, we prove a three-dimensional version of Ptolemy relation, and generalise results of Ian Short to classify tame SL₂-tilings over Eisenstein integers in terms of pairs of paths in the 3D Farey graph.

Related

• SL₂(\ℤ)-tilings of the torus, Coxeter-Conway friezes and\n Farey triangulations
• Wildest SL₂-tilings
• Geometry of multidimensional Farey summation algorithm and frieze patterns
• SL_k-Tilings and Paths in ℤ^k
• SL₂-Tilings Do Not Exist in Higher Dimensions (mostly)