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

Let Gamma be a discrete group acting properly discontinuously and isometrically on the three-dimensional anti-de Sitter space AdS(3), and rectangle the Laplacian which is a second-order hyperbolic differential operator. We study linear independence of a family of generalized Poincare series introduced by Kassel-Kobayashi [Adv. Math. 287 (2016), 123{236, arXiv:1209.4075], which are defined by the Gamma-average of certain eigenfunctions on AdS(3). We prove that the multiplicities of L-2-eigenvalues of the hyperbolic Laplacian rectangle on Gamma\AdS(3) are unbounded when Gamma is finitely generated. Moreover, we prove that the multiplicities of stable L-2-eigenvalues for compact anti-de Sitter 3-manifolds are unbounded.