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 F 2 (F\ 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 .