Eigenvalues of congruence covers of geometrically finite hyperbolic manifolds

2013/02/12 by Hee Oh, Oh, Hee
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #Number Theory (math.NT) #Spectral Theory (math.SP) #math.GT #math.NT #math.SP

paper · pdf · doi:10.48550/arxiv.1302.2950

10 pages

arxiv created 2013/02/12 · arxiv updated 2013/02/14

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 λ₀ 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) Hⁿ smaller than lambda₀ is at most of order [Gamma:Gamma(q)]^c for some c=c(λ₀)>0.

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