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

Abstract

Let Γ be a discrete group acting properly discontinuously and isometrically on the three-dimensional anti-de Sitter space AdS³, and □ the Laplacian, which is a second-order hyperbolic differential operator. We study the linear independence of a family of generalized Poincaré series introduced by Kassel-Kobayashi [Adv. Math. 287 (2016), 123-236, arXiv:1209.4075], which are defined by the Γ-average of certain eigenfunctions on AdS³. We prove that the multiplicities of 𝐿²-eigenvalues of the hyperbolic Laplacian □ on Γ∖AdS³ are unbounded when Γ is finitely generated. Moreover, we prove that the multiplicities of stable 𝐿²-eigenvalues for compact anti-de Sitter 3-manifolds are unbounded.