Quantitative spectral gap for thin groups of hyperbolic isometries | Zendy

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Quantitative spectral gap for thin groups of hyperbolic isometries

Author(s) -

Michael Magee

Publication year - 2015

Publication title -

journal of the european mathematical society

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 3.549

H-Index - 64

eISSN - 1435-9863

pISSN - 1435-9855

DOI - 10.4171/jems/500

Subject(s) - mathematics , pure mathematics , spectral gap , relatively hyperbolic group , mathematical analysis , hyperbolic manifold , hyperbolic function

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)

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