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

Abstract

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.

References in corpus (1)

• Spectral analysis on pseudo-Riemannian locally symmetric spaces