  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 $\operatorname{Vol}(\Gamma\backslash X)=\infty$ 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 $X$.
