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