On Cheeger constants of hyperbolic surfaces

Abstract It is a well-known result due to Bollobás that the maximal Cheeger constant of large $d$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>d</mml:mi> </mml:math> -regular graphs cannot be close to the Cheeger constant of the $d$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>d</mml:mi> </mml:math> -regular tree. We prove analogously that the Cheeger constant of closed hyperbolic surfaces of large genus is bounded from above by $2/\pi \approx 0.63$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>2</mml:mn> <mml:mo>/</mml:mo> <mml:mi>π</mml:mi> <mml:mo>≈</mml:mo> <mml:mn>0.63</mml:mn> </mml:math> ... 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.