  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.
