Geodesic
Quantitative spectral gap for thin groups of hyperbolic isometries
Journal of the European Mathematical Society, Tome 17 (2015) no. 1, pp. 151-187.

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Let Λ be a subgroup of an arithmetic lattice in SO(n+1,1). The quotient Hn+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 Λ 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).
DOI : 10.4171/jems/500
Classification : 11-XX, 00-XX, 22-XX
Keywords: Spectral gap, thin groups 
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     title = {Quantitative spectral gap for thin groups of hyperbolic isometries},
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     year = {2015},
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Michael Magee. Quantitative spectral gap for thin groups of hyperbolic isometries. Journal of the European Mathematical Society, Tome 17 (2015) no. 1, pp. 151-187. doi : 10.4171/jems/500. http://geodesic.mathdoc.fr/articles/10.4171/jems/500/

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