Geodesic
Linear Independence of Generalized Poincaré Series for Anti-de Sitter $3$-Manifolds
Symmetry, integrability and geometry: methods and applications, Tome 17 (2021) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: anti-de Sitter $3$-manifold, Laplacian, stable $L^2$-eigenvalue. 
@article{SIGMA_2021_17_a41,
     author = {Kazuki Kannaka},
     title = {Linear {Independence} of {Generalized} {Poincar\'e} {Series} for {Anti-de} {Sitter} $3${-Manifolds}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2021},
     volume = {17},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a41/}
}
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Kazuki Kannaka. Linear Independence of Generalized Poincaré Series for Anti-de Sitter $3$-Manifolds. Symmetry, integrability and geometry: methods and applications, Tome 17 (2021). http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a41/

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