Quantitative spectral gap for thin groups of hyperbolic isometries

Abstract

Let $\Lambda$ be a subgroup of an arithmetic lattice in $\mathrm{SO}(n+1,1)$. The quotient ${\mathbb{H}}^{n+1}/\Lambda$ has a natural family of congruence covers corresponding to ideals in a 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).