Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 042, 15 pages      arXiv:2005.03308      https://doi.org/10.3842/SIGMA.2021.042

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

Kazuki Kannaka
RIKEN iTHEMS, Wako, Saitama 351-0198, Japan

Received May 13, 2020, in final form April 13, 2021; Published online April 23, 2021

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.

Key words: anti-de Sitter 3-manifold; Laplacian; stable $L^2$-eigenvalue.

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