Quantitative spectral gap for thin groups of hyperbolic isometries

2011/12/09 by Michael Magee, Magee, Michael · 1 citation
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #Spectral Theory (math.SP) #math.NT #math.SP

paper · pdf · doi:10.48550/arxiv.1112.2004

arxiv created 2013/10/10 · arxiv updated 2013/10/14

Abstract

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 primes in some 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).

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