Relativity of the spectrum and discrete groups on hyperbolic spaces

In [13], Guillopé and Zworski prove a Poisson summation formula for the scattering resonances of a hyperbolic surface X with finite geometry, that is, X consisting of a compact manifold with boundary together with a finite number of cusps and funnels. This Poisson summation formula implies a polynomial lower bound on the counting function for scattering resonances and lower bounds for the counting function of scattering resonances in strips. Our purpose here is to generalize both the Poisson summation formula and the lower bounds on resonances to convex cocompact hyperbolic manifolds in any dimension. The result of [13] relies on the analysis of [12], which depends crucially on the special geometry of two-dimensional hyperbolic surfaces; our analysis uses the results of Bunke and Olbrich [4] and Patterson and Perry [26] on the divisor of Selberg's zeta function, which are valid in any dimension, but only for hyperbolic manifolds without cusps. The Poisson summation formula will be shown to be a simple consequence of the main result in [26], and the lower bounds on resonances will follow precisely as in [12, 13] once the Poisson formula is established. To state our results, we recall some basic facts about the spectral and scattering theory of convex cocompact hyperbolic manifolds and the main result of [26] about the zeta function. If Γ is a discrete group of isometries acting on H n+1 , the group Γ is called convex cocompact if there is a finite-sided fundamental domain for Γ which does

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The divisor of Selberg's zeta function for Kleinian groups

Patterson¹,

Perry²

2001

Duke Math. J.

132

174

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W e compute the divisor of Selberg's zeta function for convex co-compact, torsion-free discrete groups ; acting on a real hyperbolic space of dimension n + 1. T h e divisor is determined by the eigenvalues and scattering poles of the Laplacian on X = ;nH n+1 together with the Euler characteristic of X compacti ed to a manifold with boundary. I f n is even, the singularities of the zeta funciton associated to the Euler characteristic of X are identi ed using wo r k o f B u n k e and Olbrich. Contents 1.

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Relativity of the spectrum and discrete groups on hyperbolic spaces

Abstract: Abstract. We give a simple proof of the analytic continuation of the resolvent kernel for a convex cocompact Kleinian group.

Cited by 3 publications

References 11 publications

The Ubiquitous Heat Kernel

The Ubiquitous Heat Kernel

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The divisor of Selberg's zeta function for Kleinian groups

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