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

Monk, Laura

doi:10.2140/apde.2022.15.727

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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

DOI: 10.2140/apde.2022.15.727

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

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Keywords

hyperbolic surfaces, eigenvalues of the Laplacian, Selberg trace formula, Benjamini–Schramm convergence, moduli spaces, Weil–Petersson volume

Mathematical Subject Classification 2010

Primary: 58J50, 32G15

Milestones

Received: 31 January 2020

Revised: 17 September 2020

Accepted: 30 October 2020

Published: 10 June 2022

Authors

Laura Monk
Université de Strasbourg, CNRS, IRMA UMR 7501 Strasbourg France

Vol. 15, No. 3, 2022