Quantitative spectral gap for thin groups of hyperbolic isometries

TL;DR

This paper proves a quantitative spectral gap for Laplacian operators on congruence covers of hyperbolic manifolds associated with Zariski-dense subgroups, extending previous results by Sarnak, Xue, and Gamburd.

Contribution

It establishes a super-strong approximation and spectral gap result for Zariski-dense subgroups of arithmetic lattices in hyperbolic isometries, with specific regularity conditions.

Findings

Proves a spectral gap for Laplacians on congruence covers.

Generalizes previous spectral gap results to broader classes of groups.

Provides quantitative bounds for the spectral gap.

Abstract

Let $\Lambda$ be a subgroup of an arithmetic lattice in SO(n+1,1). The quotient $\mathbb{H}^{n+1} / \Lambda$ has a natural family of congruence covers corresponding to primes in some ring of integers. We establish a super-strong approximation result for Zariski-dense $\Lambda$ with some additional regularity and thickness properties. Concretely, this asserts a quantitative spectral gap for the Laplacian operators on the congruence covers. This generalizes results of Sarnak and Xue (1991) and Gamburd (2002).