Fonctions Zêta de Selberg et Surfaces de Géométrie Finie

Abstract

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Inverse scattering on asymptotically hyperbolic manifolds

“…In this section we extend some of the results of [12] and [13], obtained in the case of Riemann surfaces, to asymptotically hyperbolic manifolds. We show that the kernel of the Poisson operator is a multiple of the Eisenstein function and, as in [12], we obtain a formula for the scattering matrix in terms of the resolvent. (Similar results have been established by Borthwick in [7] for ℜζ = n/2.)…”

“…This is the analogue of Definition 2.2 of [12]. For simplicity we will work with the definition given by (4.2) and so we fix a product decomposition X ∼ ∂X × [0, ǫ) of X near ∂X.…”

“…However the proof in the general case does not seem to be available in the literature. The proof of several particular cases have been given, see for example [16], [12], [13] and [35] and references given there. The case ℜζ = n 2 is done in [7].…”

“…Our approach is heavily influenced by the work of Guillopé and Zworski, [12], [13] and [14]. In particular, we compute the scattering matrix as a boundary value of the resolvent.…”

Group Cohomology and the Singularities of the Selberg Zeta Function Associated to a Kleinian Group

“…In [38] (see also [42]) it was shown that the infinite product converges for Re(s) > δ Γ , and that the Selberg zeta function has a meromorphic continuation to all of C. In the special case of surfaces this was also proved in [17]. Partial results concerning the logarithmic derivative of the Selberg zeta function have been obtained in [34] for δ Γ < 0 and in [41] in the general case.…”