Asymptotic properties of coverings in negative curvature

Abstract

We show that the universal covering $\tilde{X}$ 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 $X\to X_0$. Moreover, we give an explicit formula estimating the difference between $\omega \left(\tilde{X}\right)$ and $\omega \left(X\right)$ 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.