Resolvent and lattice points on symmetric spaces of strictly negative curvature

Abstract

We study the asymptotics of the lattice point counting function \(N(x,y;r)=\#\{\gamma\in\Gamma\,:\,d(x,\gamma y)\}\) for a Riemannian symmetric space X obtained from a semisimple Lie group of real rank one and a discontinuous group \(\Gamma\) of motions in X, such that \(\Gamma\backslash X\) has finite volume. We show that \(\) as \(r\rightarrow\infty\), for each \(\varepsilon>0\). The constant \(2\rho\) corresponds to the sum of the positive roots of the Lie group associated to X, and n = dimX. The sum in the main term runs over a system of orthonormal eigenfunctions \(\varphi_j\in L^2(\Gamma\backslash X)\) of the Laplacian, such that the eigenvalues \(\rho^2-\nu_j^2\) are less than \(4n\rho^2/(n+1)^2\).