JournalsjemsVol. 17, No. 1pp. 151–187

Quantitative spectral gap for thin groups of hyperbolic isometries

  • Michael Magee

    University of California at Santa Cruz, USA
Quantitative spectral gap for thin groups of hyperbolic isometries cover
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Abstract

Let Λ\Lambda be a subgroup of an arithmetic lattice in SO(n+1,1)\mathrm{SO}(n+1 , 1). The quotient Hn+1/Λ\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 Λ\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).

Cite this article

Michael Magee, Quantitative spectral gap for thin groups of hyperbolic isometries. J. Eur. Math. Soc. 17 (2015), no. 1, pp. 151–187

DOI 10.4171/JEMS/500