Asymptotic properties of coverings in negative curvature

Sambusetti, Andrea¹

¹ Dipartimento di Matematica G Castelnuovo, Università “La Sapienza”, P.le Aldo Moro 5, 00185 Roma, Italy

Geometry & topology, Tome 12 (2008) no. 1, pp. 617-637.

Résumé

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.

Détail

DOI : 10.2140/gt.2008.12.617

Keywords: growth, entropy, systole, negative curvature, covering, geodesic, spectrum

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