  Let $\Lambda$ be a subgroup of an arithmetic lattice in SO(n+1,1). The
quotient $\mathbb{H}^{n+1} / \Lambda$ has a natural family of congruence covers
corresponding to primes in some ring of integers. We establish a super-strong
approximation result for Zariski-dense $\Lambda$ with some additional
regularity and thickness properties. Concretely, this asserts a quantitative
spectral gap for the Laplacian operators on the congruence covers. This
generalizes results of Sarnak and Xue (1991) and Gamburd (2002).
