On Cheeger constants of hyperbolic surfaces

Thomas Budzinski; Nicolas Curien; Bram Petri

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arXiv · 2207.00469

On Cheeger constants of hyperbolic surfaces

Thomas Budzinski·Nicolas Curien·Bram Petri

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Abstract

It is a well-known result due to Bollobas that the maximal Cheeger constant of large $d$-regular graphs cannot be close to the Cheeger constant of the $d$-regular tree. We prove analogously that the Cheeger constant of closed hyperbolic surfaces of large genus is bounded from above by $2/\pi \approx 0.63...$ 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.

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Thomas Budzinski, Nicolas Curien, Bram Petri. 2022-07-01. On Cheeger constants of hyperbolic surfaces. https://arxiv.org/abs/2207.00469

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