A uniform spectral gap for congruence covers of a hyperbolic manifold

Let $G$ be ${\rm SO}(n,1)$ or ${\rm SU}(n,1)$ and let $\Gamma\subset G$ denote an arithmetic lattice. The hyperbolic manifold $\Gamma\backslash{\cal H}$ 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.