A uniform spectral gap for congruence covers of a hyperbolic manifold

Let $G$ be $\SO(n,1)$ or $\SU(n,1)$ and let $Γ\subset G$ 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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