#### Volume 12, issue 1 (2008)

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Asymptotic properties of coverings in negative curvature

### Andrea Sambusetti

Geometry & Topology 12 (2008) 617–637
##### Abstract

We show that the universal covering $\stackrel{̃}{X}$ 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 $\omega \left(\stackrel{̃}{X}\right)$ 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.

##### Keywords
growth, entropy, systole, negative curvature, covering, geodesic, spectrum
##### Mathematical Subject Classification 2000
Primary: 53C23
Secondary: 53C21, 53C22, 20F67, 20F69