Counting orbits of certain infinitely generated non-sharp discontinuous groups for the anti-de Sitter space

Abstract

Generalizing an example of Guéritaud-Kassel ([Geom.Topol.2017]), we construct a family of infinitely generated groups Γ acting isometrically and properly discontinuously on the 3-dimensional anti-de Sitter space AdS 3 . These groups are "nonsharp" in the sense of Kassel-Kobayashi ([Adv.Math.2016]). Moreover, we estimate the number of elements in Γ-orbits contained in the "pseudo-ball" B(R). As its application, we construct L 2 -eigenfunctions of the Laplace-Beltrami operator on a non-sharp Lorentzian manifold Γ\AdS 3 , using the method established by Kassel-Kobayashi. In particular, we prove that the set of L 2 -eigenvalues of the Laplace-Beltrami operator on a non-sharp Lorentzian manifold Γ\AdS 3 is infinite. We also prove the following: Given an increasing function f : R → R >0 , there exists a discontinuous group Γ such that the number of elements in a certain Γ-orbit contained in B(R) is asymptotically larger than f (R).

Linear Independence of Generalized Poincaré Series for Anti-de Sitter 3-Manifolds

“…9] where they illustrate their general theory for reductive symmetric spaces X = G/H in details in the special setting where X = AdS 3 . See also [7].…”