On the asymptotic properties of the Laplace operator and the Cheeger constant on graphs and surfaces

Andrzej Zuk

We present results concerning the asymptotic distribution of the eigenvalues of the Laplace operator for infinite families of finite graphs. We use probabilistic arguments to study the Cheeger constant of a Riemann surface. We show that the modular surfaces have Cheeger constants uniformly bounded away from the maximal value.