Infinite volume and atoms at the bottom of the spectrum

Abstract

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 $\mathrm{Vol}(\Gamma \backslash X)=\infty$ if and only if no positive Laplace eigenfunction belongs to $L^2(\Gamma \backslash X)$, 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 $X$.

14 pages (1 figure), To appear in Comptes Rendus - Série Mathématique