Counting orbits of certain infinitely generated non-sharp discontinuous groups for the anti-de Sitter space

• Kazuki Kannaka ^[1]
  1. [1] RIKEN Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS),Japan
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 1, 2024, págs. 1-31
• Idioma: inglés
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• Resumen

  • Inspired by an example of Guéritaud and Kassel (Geom Topol 21(2):693–840, 2017), we construct a family of infinitely generated discontinuous groups for the 3- dimensional anti-de Sitter space AdS3. These groups are not necessarily sharp (a kind of “strong” proper discontinuity condition introduced by Kassel and Kobayashi (Adv Math 287:123–236, 2016), and we give its criterion. Moreover, we find upper and lower bounds of the counting N(R) of a -orbit contained in a pseudo-ball B(R) as the radius R tends to infinity.We then find a non-sharp discontinuous group for which there exist infinitely many L2-eigenvalues of the Laplacian on the noncompact anti-de Sitter manifold \AdS3, by applying the method established by Kassel–Kobayashi. We also prove that for any increasing function f , there exists a discontinuous group for AdS3 such that the counting N(R) of a-orbit is larger than f (R)for a sufficiently large R.