Infinite volume and atoms at the bottom of the spectrum

Edwards, Sam; Fraczyk, Mikolaj; Lee, Minju; Oh, Hee

Comptes Rendus. Mathématique, Tome 362 (2024) no. G13, pp. 1873-1880

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 $\Gamma <G$, we prove that $\mathrm{Vol}(\Gamma \setminus X)=\infty$ if and only if no positive Laplace eigenfunction belongs to $L^2(\Gamma \setminus X)$, 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 $X$.

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 $\Gamma <G$, on prouve que $\mathrm{Vol}(\Gamma \setminus X)=\infty$ si et seulement si aucune fonction propre de Laplacien positive appartient à $L^2(\Gamma \setminus X)$, 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 $X$.

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

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