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.

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