Polynomial bounds on the number of resonances for some complete spaces of constant negative curvature near infinity

Let X be a conformally compact n-dimensional manifold with constant negative curvature -1 near infinity. The resolvent (Delta - s(n - 1 - s))-1, Re s > n - 1, of the Laplacian on X extends to a meromorphic family of operators on C and its poles are called resonances or scattering poles. If N-X(r) is the number of resonances in a disc of radius r we prove the following upper bound: N-X(r) less than or equal to Cr-n+1 + C.