-
A uniform spectral gap for congruence covers of a hyperbolic manifold
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 135, Number 4, August 2013
- pp. 1067-1085
- 10.1353/ajm.2013.0039
- Article
- Additional Information
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.