A uniform spectral gap for congruence covers of a hyperbolic manifold

Lior Silberman, Dubi Kelmer

Abstract: Let G be $\SO(n,1)$ or $\SU(n,1)$ and let Γ⊂G denote an arithmetic lattice. The hyperbolic manifold $\Gamma\backslash \calH$ 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.