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<p>We explore a three-dimensional counterpart of the Farey tessellation and its relations to Penner’s lambda lengths and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_997_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="30" /> </InlineMediaObject> <EquationSource Format="TEX">\(SL_2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>L</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>-tilings. In particular, we prove a three-dimensional version of the Ptolemy relation, and generalise results of Short to classify tame <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_997_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="30" /> </InlineMediaObject> <EquationSource Format="TEX">\(SL_2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>L</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>-tilings over Eisenstein integers in terms of pairs of paths in the 3D Farey graph.</p>

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3d Farey graph, lambda lengths and \(SL_2\)-tilings

  • 期刊论文

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  • 发表日期：

    2025年3月12日

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

摘要

We explore a three-dimensional counterpart of the Farey tessellation and its relations to Penner’s lambda lengths and \(SL_2\) S L 2 -tilings. In particular, we prove a three-dimensional version of the Ptolemy relation, and generalise results of Short to classify tame \(SL_2\) S L 2 -tilings over Eisenstein integers in terms of pairs of paths in the 3D Farey graph.