We 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. The divisor is determined by the eigenvalues and scattering poles of the Laplacian on X = ;nH n+1 together with the Euler characteristic of X compactified to a manifold with boundary.Ifn is even, the singularities of the zeta funciton associated to the Euler characteristic of X are identified using work of Bunke and Olbrich.

Abstract. On an asymptotically hyperbolic manifold (Xn+1, g), Mazzeo and Melrose have constructed the meromorphic extension of the resolvent R(λ): = (∆g − λ(n − λ)) −1 for the Laplacian. However, there are special points on 1 (n − N) that they did not deal with. We 2 show that the points of n − N are at most some poles of finite multiplicity, and that the same 2 property holds for the points of n+1 − N if and only if the metric is ‘even’. On the other 2 hand, there exist some metrics for which R(λ) has an essential singularity on n+1 − N and 2 these cases are generic. At last, to illustrate them, we give some examples with a sequence of poles of R(λ) approaching an essential singularity.

this paper is to show how the methods of Sjostrand for proving the geometric bounds for the density of resonances [28] apply to the case of convex co-compact hyperbolic surfaces. We prove that the exponent in the Weyl estimate for the number of resonances in subconic neighbourhoods of the continuous spectrum is related to the dimension of the limit set of the corresponding Kleinian group. Figure 1. Tesselation by the Schottky group generated by inversions in three symmetrically placed circles each cutting the unit circle in an 110

Scattering is defined on compact manifolds with boundary which are equipped with an asymptotically hyperbolic metric, g: A model form is established for such metrics close to the boundary. It is shown

Abstract. For a class of manifolds X that includes quotients of real hyperbolic (n + 1)-dimensional space by a convex co-compact discrete group, we show that the resonances of the meromorphically continued resolvent kernel for the Laplacian on X coincide, with multiplicities, with the poles of the meromorphically continued scattering operator for X. In order to carry out the proof, we use Shmuel Agmon’s perturbation theory of resonances to show that both resolvent resonances and scattering poles are simple for generic potential perturbations. 1.

Abstract. For convex co-compact hyperbolic quotients Γ\H n+1 we obtain a formula relating the 0-trace of the wave operator with the resonances and some conformal invariants of the boundary, generalizing a formula of Guillopé and Zworski in dimension 2. Then, by writing this 0-trace with the length spectrum, we prove precise asymptotics of the number of closed geodesics with an effective, exponentially small error term when the dimension of the limit set of Γ is greater than n 2. 1.

We construct a determinant of the Laplacian for infinite-area surfaces which are hyperbolic near infinity and without cusps. In the case of a convex co-compact hyperbolic metric, the determinant can be related to the Selberg zeta function and thus shown to be an entire function of order two with zeros at the eigenvalues and resonances of the Laplacian. In the hyperbolic near infinity case the determinant is analyzed through the zeta-regularized relative determinant for a conformal metric perturbation. We establish that this relative determinant is a ratio of entire functions of order two with divisor corresponding to eigenvalues and resonances of the perturbed and unperturbed metrics. These results are applied to the problem of compactness in the smooth topology for the class of metrics with a given set of eigenvalues and resonances.

Abstract. For hyperbolic Riemann surfaces of finite geometry, we study Selberg’s zeta function and its relation to the relative scattering phase and the resonances of the Laplacian. As an application we show that the conjugacy class of a finitely generated, torsion-free, discrete subgroup of SL(2, R) is determined by its trace spectrum up to finitely many possibilities, thus generalizing results of McKean [19] and Müller [22] to groups which are not necessarily cofinite. 1.

. We exhibit pairs of infinite-volume, hyperbolic three-manifolds that have the same scattering poles and conformally equivalent boundaries, but which are not isometric. The examples are constructed using Schottky groups and the Sunada construction. In memory of Evsey Dyn'kin 1. Introduction The purpose of this paper is to explore the spectral geometry of the scattering operator by exhibiting examples of infinite volume hyperbolic three-manifolds that are `isoscattering' in a sense we will make precise, but have distinct geometries. To do so we will work with convex co-compact hyperbolic manifolds associated to Schottky groups of hyperbolic isometries. This class of manifolds, described in greater detail in what follows, may be thought of as interiors X of handlebodies X equipped with a metric that puts the boundary at metric infinity. The interior carries a hyperbolic structure, and the boundary, @ X , carries an induced conformal structure as a Riemann surface. Associated to the ...

Abstract. For convex co-compact hyperbolic quotients X = Γ\H n+1, we analyze the long-time asymptotic of the solution of the wave equation u(t) with smooth compactly supported initial data f = (f0, f1). We show that, if the Hausdorff dimension δ of the limit set is less than n/2, then u(t) = Cδ(f)e (δ − n 2)t /Γ(δ − n/2 + 1) + e (δ − n 2)t R(t) where Cδ(f) ∈ C ∞ (X) and ||R(t)| | = O(t − ∞). We explain, in terms of conformal theory of the conformal infinity of X, the special cases δ ∈ n/2 − N where the leading asymptotic term vanishes. In a second part, we show for all ǫ> 0 the existence of an infinite number of resonances (and thus zeros of Selberg zeta function) in the strip {−nδ − ǫ < Re(λ) < δ}. As a byproduct we obtain a lower bound on the remainder R(t) for generic initial data f. 1.