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<p>It is a well-known result due to Bollobás that the maximal Cheeger constant of large <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2025_1361_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>d</mi> </math></EquationSource> <EquationSource Format="TEX">$d$</EquationSource> </InlineEquation>-regular graphs cannot be close to the Cheeger constant of the <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2025_1361_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>d</mi> </math></EquationSource> <EquationSource Format="TEX">$d$</EquationSource> </InlineEquation>-regular tree. We prove analogously that the Cheeger constant of closed hyperbolic surfaces of large genus is bounded from above by <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2025_1361_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="80" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mn>2</mn> <mo stretchy="false">/</mo> <mi>π</mi> <mo>≈</mo> <mn>0.63</mn> </math></EquationSource> <EquationSource Format="TEX">$2/\pi \approx 0.63$</EquationSource> </InlineEquation>... which is strictly less than the Cheeger constant of the hyperbolic plane. The proof uses a random construction based on a Poisson–Voronoi tessellation of the surface with a vanishing intensity.</p>

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On Cheeger constants of hyperbolic surfaces

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月1日

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  • Thomas Budzinski,
  • Nicolas Curien,
  • Bram Petri

摘要

It is a well-known result due to Bollobás that the maximal Cheeger constant of large d $d$ -regular graphs cannot be close to the Cheeger constant of the d $d$ -regular tree. We prove analogously that the Cheeger constant of closed hyperbolic surfaces of large genus is bounded from above by 2 / π ≈ 0.63 $2/\pi \approx 0.63$ ... which is strictly less than the Cheeger constant of the hyperbolic plane. The proof uses a random construction based on a Poisson–Voronoi tessellation of the surface with a vanishing intensity.