Linear Independence of Generalized Poincaré 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 $\mathrm{AdS}^{3}$, and $\square$ 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. 287 (2016), 123–236, arXiv:1209.4075], which are defined by the $\Gamma$-average of certain eigenfunctions on $\mathrm{AdS}^{3}$. We prove that the multiplicities of $L^{2}$-eigenvalues of the hyperbolic Laplacian $\square$ on $\Gamma\backslash\mathrm{AdS}^{3}$ 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.