Critical exponents of discrete groups and 𝐿²–spectrum

Abstract

Abstract. 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 Γ. 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.

Schrödinger equations on locally symmetric spaces

“…Note that Leuzinger [26] proved that if G has no compact factors, its center is trivial and δ (Γ) ≤ ρ m , then λ 0 = |ρ| 2 . In this case, η Γ = 0 in (7).…”

Bottom of the $L^2$ spectrum of the Laplacian on locally symmetric spaces

“…Proof of Theorem 2.9 and Corollary 2.10. Let us resume the approach in [7, Section 4] and [15,Section 3]. It consists in studying the convergence of the positive series…”

Estimates for the Heat Kernel on Differential Forms on Riemannian Symmetric Spaces and Applications

“…In the case of functions, i.e ℓ = 0, some estimates for β 0 (Γ\G/K) have been proved by E. Leuzinger [21]. More precisely, let D( , ) be the Riemannian distance on G/K induced by the Killing form.…”

“…Theorem (E. Leuzinger, Section 1 of [21]). Let G be a semisimple Lie group without compact factors and with trivial center.…”