Benjamini–Schramm convergence and spectra of random hyperbolic surfaces of high genus

Abstract

We study geometric and spectral properties of typical hyperbolic surfaces of high genus, excluding a set of small measure for the Weil-Petersson probability measure. We first prove Benjamini-Schramm convergence to the hyperbolic plane H as the genus g goes to infinity. An estimate for the number of eigenvalues in an interval [a,b] in terms of a, b and g is then proven using the Selberg trace formula. It implies the convergence of spectral measures to the spectral measure of H as g $\rightarrow$+$\infty$, and a uniform Weyl law as b $\rightarrow$+$\infty$. We deduce a bound on the number of small eigenvalues, and the multiplicity of any eigenvalue.