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 AdS 3 , and the Laplacian which is a second-order hyperbolic differential operator. We study linear independence of a family of generalized Poincaré series introduced by Kassel-Kobayashi [Adv. Math. 2016], which are defined by the Γ-average of certain eigenfunctions on AdS 3 . We prove that the multiplicities of L 2 -eigenvalues of the hyperbolic Laplacian on Γ\AdS 3 are unbounded when Γ is finitely generated. Moreover, we prove that the multiplicities of stable L 2 -eigenvalues for compact anti-de Sitter 3-manifolds are unbounded.