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Let G be a noncompact semisimple Lie group and Γ an arbitrary discrete, torsion-free subgroup of G. Let λ 0 (M) be the bottom of the spectrum of the Laplace-Beltrami operator on the locally symmetric space M = Γ\X, and let δ(Γ) be the exponent of growth of r. 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 λ 0 (M) from above and below by quadratic polynomials in δ(Γ). As an application we prove a rigiditiy property of lattices.

Critical exponents of discrete groups and 𝐿²–spectrum