Vol. 15, No. 3, 2022

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

Laura Monk

Vol. 15 (2022), No. 3, 727–752

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 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 proved using the Selberg trace formula. It implies the convergence of spectral measures to the spectral measure of as g + and a uniform Weyl law as b +. We deduce a bound on the number of small eigenvalues and the multiplicity of any eigenvalue.

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hyperbolic surfaces, eigenvalues of the Laplacian, Selberg trace formula, Benjamini–Schramm convergence, moduli spaces, Weil–Petersson volume
Mathematical Subject Classification 2010
Primary: 58J50, 32G15
Received: 31 January 2020
Revised: 17 September 2020
Accepted: 30 October 2020
Published: 10 June 2022
Laura Monk
Université de Strasbourg, CNRS, IRMA UMR 7501