Let be a subgroup of an arithmetic lattice in . The quotient 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 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).
Cite this article
Michael Magee, Quantitative spectral gap for thin groups of hyperbolic isometries. J. Eur. Math. Soc. 17 (2015), no. 1, pp. 151–187DOI 10.4171/JEMS/500