3D Farey Graph, Lambda Lengths and SL₂-Tilings

Felikson, Anna; Karpenkov, Oleg; Serhiyenko, Khrystyna; Tumarkin, Pavel

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 the Ptolemy relation, and generalise results of Short to classify tame SL₂-tilings over Eisenstein integers in terms of pairs of paths in the 3D Farey graph.

Citation

Felikson, A., Karpenkov, O., Serhiyenko, K., & Tumarkin, P. (2025). 3D Farey Graph, Lambda Lengths and SL₂-Tilings. Geometriae Dedicata, 219, Article 33. https://doi.org/10.1007/s10711-025-00997-5

• Journal Article Type: Article
• Acceptance Date: Feb 25, 2025
• Online Publication Date: Mar 12, 2025
• Publication Date: 2025
• Deposit Date: Feb 25, 2025
• Publicly Available Date: Mar 14, 2025
• Journal: Geometriae Dedicata
• Print ISSN: 0046-5755
• Electronic ISSN: 1572-9168
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 219
• Article Number: 33
• DOI: https://doi.org/10.1007/s10711-025-00997-5
• Public URL: https://durham-repository.worktribe.com/output/3545964