Asymptotic properties of coverings in negative curvature

We show that the universal covering (X) over tilde of any compact, negatively curved manifold X-0 has an exponential growth rate which is strictly greater than the exponential growth rate of any other normal covering X -> X-0. Moreover, we give an explicit formula estimating the difference between omega((X) over tilde) and omega(X) in terms of the systole of X and of other elementary geometric parameters of the base space X-0. Then we discuss some applications of this formula to periodic geodesics, to the bottom of the spectrum and to the critical exponent of normal coverings.