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

SIGMA 17 (2021), 042, 15 pages      arXiv:2005.03308

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

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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  1. Benoist Y., Actions propres sur les espaces homogènes réductifs, Ann. of Math. 144 (1996), 315-347.
  2. Bourbaki N., Integration. II. Chapters 7-9, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2004.
  3. Calabi E., Markus L., Relativistic space forms, Ann. of Math. 75 (1962), 63-76.
  4. Fox J., Strichartz R.S., Unexpected spectral asymptotics for wave equations on certain compact spacetimes, J. Anal. Math. 136 (2018), 209-251, arXiv:1407.2517.
  5. Goldman W.M., Nonstandard Lorentz space forms, J. Differential Geom. 21 (1985), 301-308.
  6. Guéritaud F., Kassel F., Maximally stretched laminations on geometrically finite hyperbolic manifolds, Geom. Topol. 21 (2017), 693-840, arXiv:1307.0250.
  7. Kannaka K., Counting orbits of certain infinitely generated non-sharp discontinuous groups for the anti-de Sitter space, arXiv:1907.09303.
  8. Kassel F., Quotients compacts d'espaces homogènes réels ou $p$-adiques, Ph.D. Thesis, Université Paris-Sud, 2009.
  9. Kassel F., Deformation of proper actions on reductive homogeneous spaces, Math. Ann. 353 (2012), 599-632, arXiv:0911.4247.
  10. Kassel F., Kobayashi T., Poincaré series for non-Riemannian locally symmetric spaces, Adv. Math. 287 (2016), 123-236, arXiv:1209.4075.
  11. Kassel F., Kobayashi T., Spectral analysis on standard locally homogeneous spaces, arXiv:1912.12601.
  12. Kassel F., Kobayashi T., Spectral analysis on pseudo-Riemannian locally symmetric spaces, Proc. Japan Acad. Ser. A Math. Sci. 96 (2020), 69-74, arXiv:2001.03292.
  13. Kassel F., Kobayashi T., Analyticity of Poincaré series on standard non-Riemannian locally symmetric spaces, in preparation.
  14. Klingler B., Complétude des variétés lorentziennes à courbure constante, Math. Ann. 306 (1996), 353-370.
  15. Kobayashi T., Proper action on a homogeneous space of reductive type, Math. Ann. 285 (1989), 249-263.
  16. Kobayashi T., Criterion for proper actions on homogeneous spaces of reductive groups, J. Lie Theory 6 (1996), 147-163.
  17. Kobayashi T., Deformation of compact Clifford-Klein forms of indefinite-Riemannian homogeneous manifolds, Math. Ann. 310 (1998), 395-409.
  18. Kobayashi T., Discontinuous groups for non-Riemannian homogeneous spaces, in Mathematics Unlimited - 2001 and Beyond, Springer, Berlin, 2001, 723-747.
  19. Kobayashi T., Intrinsic sound of anti-de Sitter manifolds, in Lie Theory and its Applications in Physics, Springer Proc. Math. Stat., Vol. 191, Springer, Singapore, 2016, 83-99, arXiv:1609.05986.
  20. Kobayashi T., Nasrin S., Deformation of properly discontinuous actions of ${\mathbb Z}^k$ on ${\mathbb R}^{k+1}$, Internat. J. Math. 17 (2006), 1175-1193, arXiv:math.DG/0603318.
  21. Kulkarni R.S., Raymond F., $3$-dimensional Lorentz space-forms and Seifert fiber spaces, J. Differential Geom. 21 (1985), 231-268.
  22. Rudin W., Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991.
  23. Salein F., Variétés anti-de Sitter de dimension 3 exotiques, Ann. Inst. Fourier (Grenoble) 50 (2000), 257-284.

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