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)=#{γ∈Γ:d(x,γy)} for a Riemannian symmetric space X obtained from a semisimple Lie group of real rank one and a discontinuous group Γ of motions in X, such that Γ∖X has finite volume. We show that as r→∞ , for each ε>0 . The constant 2ρ 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 φj∈L2(Γ∖X) of the Laplacian, such that the eigenvalues ρ2−ν2j are less than 4nρ2/(n+1)2 .