Quantitative spectral gap for thin groups of hyperbolic isometries

Magee, Michael

Abstract

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).

Citation

Magee, M. (2015). Quantitative spectral gap for thin groups of hyperbolic isometries. Journal of the European Mathematical Society, 17(1), 151-187. https://doi.org/10.4171/jems/500

• Journal Article Type: Article
• Online Publication Date: Feb 5, 2015
• Publication Date: Jan 1, 2015
• Deposit Date: Sep 7, 2017
• Publicly Available Date: Oct 24, 2017
• Journal: Journal of the European Mathematical Society
• Print ISSN: 1435-9855
• Electronic ISSN: 1435-9863
• Publisher: EMS Press
• Peer Reviewed: Peer Reviewed
• Volume: 17
• Issue: 1
• Pages: 151-187
• DOI: https://doi.org/10.4171/jems/500
• Related Public URLs: https://arxiv.org/abs/1112.2004