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).
Let À be a discontinuous group for the 3-dimensional anti-de Sitter space AdS 3 :¼ SO 0 ð2; 2Þ=SO 0 ð2; 1Þ. In this article, we discuss a growth rate of the counting of À-orbits at infinity and the discrete spectrum of the hyperbolic Laplacian of the complete anti-de Sitter manifold ÀnAdS 3 .
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.