Eigenvalues of Congruence Covers of Geometrically Finite Hyperbolic Manifolds

Let G = SO(n, 1)◦ for n ≥ 2 and Γ a geometrically finite Zariski dense subgroup of G which is contained in an arithmetic subgroup of G. Denoting by Γ(q) the principal congruence subgroup of Γ of level q, and fixing a positive number λ0 strictly smaller than (n − 1)/4, we show that, as q → ∞ along primes, the number of Laplacian eigenvalues of the congruence cover Γ(q)\H smaller than λ0 is at most of order [Γ : Γ(q)] for some c = c(λ0) > 0.