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

Abstract

Inspired by an example of Guéritaud and Kassel (Geom Topol 21(2):693–840, 2017), we construct a family of infinitely generated discontinuous groups \(\Gamma \) for the 3-dimensional anti-de Sitter space \(\textrm{AdS}^{3}\). 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 \(N_{\Gamma }(R)\) of a \(\Gamma \)-orbit contained in a pseudo-ball B(R) as the radius R tends to infinity. We then find a non-sharp discontinuous group \(\Gamma \) for which there exist infinitely many \(L^2\)-eigenvalues of the Laplacian on the noncompact anti-de Sitter manifold \(\Gamma \backslash \textrm{AdS}^{3}\), by applying the method established by Kassel–Kobayashi. We also prove that for any increasing function f, there exists a discontinuous group \(\Gamma \) for \(\textrm{AdS}^{3}\) such that the counting \(N_{\Gamma }(R)\) of a \(\Gamma \)-orbit is larger than f(R) for a sufficiently large R.