Work

On Cheeger constants of hyperbolic surfaces

Year: 2025

Type: article

Abstract: 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 $2/\pi \approx 0.63$ 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.

Source: Inventiones mathematicae

DOI: 10.1007/s00222-025-01361-w

Authors: Thomas Budzinski, Nicolas Curien, Bram Petri