Critical exponents of discrete groups and 𝐿²–spectrum

Abstract

Let G G be a noncompact semisimple Lie group and Γ \Gamma an arbitrary discrete, torsion-free subgroup of G G . Let λ 0 ( M ) \lambda _0(M) be the bottom of the spectrum of the Laplace-Beltrami operator on the locally symmetric space M = Γ ∖ X M=\Gamma \backslash X , and let δ ( Γ ) \delta (\Gamma ) be the exponent of growth of Γ \Gamma . If G G has rank 1 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 ) \lambda _0(M) from above and below by quadratic polynomials in δ ( Γ ) \delta (\Gamma ) . As an application we prove a rigiditiy property of lattices.