A uniform spectral gap for congruence covers of a hyperbolic manifold

Details A uniform spectral gap for congruence covers of a hyperbolic manifold A uniform spectral gap for congruence covers of a hyperbolic manifold

2010-10-05

arxiv.org/abs/1010.1010v2

Mathematics

Number Theory

15 pages

Scientific paper

Let $G$ be $\SO(n,1)$ or $\SU(n,1)$ and let $\Gamma\subset G$ denote an arithmetic lattice. The hyperbolic manifold $\Gamma\backslash \calH$ comes with a natural family of covers, coming from the congruence subgroups of $\Gamma$. In many applications, it is useful to have a bound for the spectral gap that is uniform for this family. When $\Gamma$ 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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