A uniform spectral gap for congruence covers of a hyperbolic manifold

TL;DR

This paper establishes a uniform spectral gap bound for congruence covers of hyperbolic manifolds arising from non-congruence lattices, extending known results from congruence lattices and contributing to spectral theory in geometric group theory.

Contribution

It provides the first effective uniform spectral gap bound for congruence subgroups of non-congruence lattices in hyperbolic geometry.

Findings

Proves a uniform spectral gap bound for non-congruence lattices

Extends Ramanujan-type bounds to a broader class of lattices

Enhances understanding of spectral properties of hyperbolic manifolds

Abstract

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.