The low-intensity Poisson-Voronoi tessellation of H2 × H2 with the L1 metric converges to an isometry-invariant ideal tessellation whose cell ends are unions of boundary circles, and equal-separation loci between corona points are unbounded almost surely.
On Cheeger constants of hyperbolic surfaces
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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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Ideal Poisson--Voronoi tessellations beyond hyperbolic spaces
The low-intensity Poisson-Voronoi tessellation of H2 × H2 with the L1 metric converges to an isometry-invariant ideal tessellation whose cell ends are unions of boundary circles, and equal-separation loci between corona points are unbounded almost surely.