Critical exponents of discrete groups and $L^2$-spectrum

Let $G$ be a noncompact semisimple Lie group and $\Gamma$ an arbitrary discrete, torsion-free subgroup of $G$. Let $\lambda_0(M)$ be the bottom of the spectrum of the Laplace-Beltrami operator on the locally symmetric space $M=\Gamma\backslash X$, and let $\delta(\Gamma)$ be the exponent of growth of $\Gamma$. If $G$ has rank $1$, then these quantities are related by a well-known formula due to Elstrodt, Patterson, Sullivan and Corlette. In this note we generalize that relation to the higher rank case by estimating $\lambda_0(M)$ from above and below by quadratic polynomials in $\delta(\Gamma)$. As an application we prove a rigiditiy property of lattices.