A uniform spectral gap for congruence covers of a hyperbolic manifold

2010/10/05 by Dubi Kelmer, Kelmer, Dubi, Lior Silberman +1
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #Spectral Theory (math.SP) #math.NT #math.SP

paper · pdf · doi:10.48550/arxiv.1010.1010

15 pages

arxiv created 2010/11/16 · arxiv updated 2010/11/18

Abstract

Let G be \SO(n,1) or \SU(n,1) and let Γ⊂ G denote an arithmetic lattice. The hyperbolic manifold Γ\backslash \calH comes with a natural family of covers, coming from the congruence subgroups of Γ. In many applications, it is useful to have a bound for the spectral gap that is uniform for this family. When Γ is itself a congruence lattice, there are very good bounds coming from known results towards the Ramanujan conjectures. In this paper, we establish an effective bound that is uniform for congruence subgroups of a non-congruence lattice.

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