Infinite volume and atoms at the bottom of the spectrum

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 ) = ∞ 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 .

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