Infinite volume and atoms at the bottom of the spectrum

Sam Edwards; Mikolaj Fraczyk; Minju Lee; Hee Oh

doi:10.5802/crmath.586

Article de recherche - Théorie spectrale

Infinite volume and atoms at the bottom of the spectrum
[Volume infini et atomes au bas du spectre]

Sam Edwards ¹ ; Mikolaj Fraczyk ^2,³ ; Minju Lee ² ; Hee Oh ⁴

¹ Department of Mathematical Sciences, Durham University, Lower Mountjoy, DH1 3LE Durham, United Kingdom
² Mathematics department, University of Chicago, Chicago, IL 60637, USA
³ Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland
⁴ Mathematics department, Yale university, New Haven, CT 06520, USA

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1873-1880.

Résumé
Abstract

Soit $G$ un groupe algébrique réel simple de rang supérieur, ou plus généralement un groupe algébrique réel semi-simple sans facteurs de rang un et $X$ l’espace symétrique riemannien associé. Pour tout sous-groupe discret dense de Zariski $Γ < G$ , on prouve que $Vol (Γ ∖ X) = \infty$ si et seulement si aucune fonction propre de Laplacien positive appartient à $L^{2} (Γ ∖ X)$ , ou de manière équivalente, le bas du spectre $L^{2}$ n’est pas un atome de la mesure spectrale du Laplacien négatif. Cela contraste avec la situation de rang un où l’intégrabilité au carré de la fonction propre de base est déterminée par la taille de l’exposant critique par rapport à l’entropie volumique de $X$ .

Let $G$ be a higher rank simple real algebraic group, or more generally, any semisimple real algebraic group with no rank one factors and $X$ the associated Riemannian symmetric space. For any Zariski dense discrete subgroup $Γ < G$ , we prove that $Vol (Γ ∖ X) = \infty$ if and only if no positive Laplace eigenfunction belongs to $L^{2} (Γ ∖ X)$ , or equivalently, the bottom of the $L^{2}$ -spectrum is not an atom of the spectral measure of the negative Laplacian. This contrasts with the rank one situation where the square-integrability of the base eigenfunction is determined by the size of the critical exponent relative to the volume entropy of $X$ .

Reçu le : 2023-08-02
Révisé le : 2023-11-16
Accepté le : 2023-11-17
Publié le : 2024-12-03

DOI : 10.5802/crmath.586

Classification : 22F30, 43A85
Keywords: Laplace eigenfunction, locally symmetric manifolds, infinite volume, Patterson–Sullivan measure
Mots-clés : Fonctions propres de l’opérateur de Laplace–Beltrami, espaces localement symétriques, volume infini, mesures de Patterson–Sullivan

Affiliations des auteurs :

Sam Edwards ¹ ; Mikolaj Fraczyk ^{2, 3} ; Minju Lee ² ; Hee Oh ⁴

¹ Department of Mathematical Sciences, Durham University, Lower Mountjoy, DH1 3LE Durham, United Kingdom
² Mathematics department, University of Chicago, Chicago, IL 60637, USA
³ Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland
⁴ Mathematics department, Yale university, New Haven, CT 06520, USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Infinite volume and atoms at the bottom of the spectrum},
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Sam Edwards; Mikolaj Fraczyk; Minju Lee; Hee Oh. Infinite volume and atoms at the bottom of the spectrum. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1873-1880. doi : 10.5802/crmath.586. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.586/

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