Eigenvalues of congruence covers of geometrically finite hyperbolic manifolds

TL;DR

This paper investigates the growth of Laplacian eigenvalues on congruence covers of geometrically finite hyperbolic manifolds, establishing bounds on the number of eigenvalues below a certain threshold as the level increases.

Contribution

It provides new bounds on the number of Laplacian eigenvalues below a fixed threshold for congruence covers of hyperbolic manifolds, extending understanding of spectral properties in this setting.

Findings

Number of eigenvalues below lambda_0 grows at most polynomially with the index of the subgroup.

Results hold for primes q tending to infinity.

Applicable to geometrically finite Zariski dense subgroups contained in arithmetic groups.

Abstract

Let G=SO(n,1) and Gamma a geometrically finite Zariski dense subgroup of G which is contained in an arithmetic subgroup of G. Denoting by Gamma(q) the principal congruence subgroup of Gamma of level q, and fixing a positive number \lambda_0 strictly smaller than (n-1)^2/4, we show that, as q tends to infinity along primes, the number of Laplacian eigenvalues of the congruence cover Gamma(q)\ H^n smaller than lambda_0 is at most of order [Gamma:Gamma(q)]^c for some c=c(\lambda_0)>0.