Let ΛΛ be a subgroup of an arithmetic lattice in SO(n+1,1)SO(n+1,1). The quotient Hn+1/ΛHn+1/Λ 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 ΛΛ 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).
Magee, M. (2015). Quantitative spectral gap for thin groups of hyperbolic isometries. Journal of the European Mathematical Society, 17(1), 151-187. https://doi.org/10.4171/jems/500