On Cheeger constants of hyperbolic surfaces

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 / π ≈ 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.