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.
| Original language | English |
|---|---|
| Article number | 33 |
| Journal | Geometriae Dedicata |
| Volume | 219 |
| Issue number | 2 |
| DOIs | |
| State | Published - Apr 2025 |
Bibliographical note
Publisher Copyright:© The Author(s) 2025.
Funding
The work was initiated and partially done at the Isaac Newton Institute for Mathematical Sciences, Cambridge; we are grateful to the organizers of the program “Cluster algebras and representation theory”, and to the Institute for support and hospitality during the program; this work was supported by EPSRC grant no EP/R014604/1. We would like to thank Arthur Baragar, Sergey Fomin, Ivan Izmestiev, Valentin Ovsienko and Ian Whitehead for helpful discussions. We are grateful to the anonymous referee for numerous insightful comments and suggestions. Research was supported in part by the Leverhulme Trust research grant RPG-2019-153 (PT) and the National Science Foundation grant DMS-2054255 (KS).
| Funders | Funder number |
|---|---|
| UK Medical Research Council, Engineering and Physical Sciences Research Council | EP/R014604/1 |
| Leverhulme Trust | RPG-2019-153 |
| National Science Foundation Arctic Social Science Program | DMS-2054255 |
Keywords
- Farey graph
- Ford circle
- SL-tiling
- lambda length
ASJC Scopus subject areas
- Geometry and Topology
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