Quantitative spectral gap for thin groups of hyperbolic isometries

Michael Magee

Type: Article

Publication Date: 2015-02-05

Citations: 16

DOI: https://doi.org/10.4171/jems/500

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).

Locations

eScholarship (California Digital Library) - View - PDF
arXiv (Cornell University) - View - PDF
Durham Research Online (Durham University) - View - PDF
Journal of the European Mathematical Society - View - PDF

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