On Cheeger constants of hyperbolic surfaces

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 $2/\pi \approx \mathrm{0.63...}$ 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.

16 pages, 2 figures

Cited by in corpus (1)

• Higher genus meanders and Masur-Veech volumes