A quantitative study of orbit counting and discrete spectrum for anti-de Sitter 3-manifolds

Abstract

Let $\Gamma$ be a discontinuous group for the 3-dimensional anti-de Sitter space $\mathrm{AdS}^{3}:=\mathrm{SO}_{0}(2,2)/\mathrm{SO}_{0}(2,1)$. In this article, we discuss a growth rate of the counting of $\Gamma$-orbits at infinity and the discrete spectrum of the hyperbolic Laplacian of the complete anti-de Sitter manifold $\Gamma\backslash\mathrm{AdS}^{3}$.