Asymptotic properties of coverings in negative curvature

Sambusetti, Andrea

doi:10.2140/gt.2008.12.617

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Asymptotic properties of coverings in negative curvature

Andrea Sambusetti

Geometry & Topology 12 (2008) 617–637

DOI: 10.2140/gt.2008.12.617

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 $ω (\tilde{X})$ and $ω (X)$ 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.

Keywords

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

Mathematical Subject Classification 2000

Primary: 53C23

Secondary: 53C21, 53C22, 20F67, 20F69

References

Publication

Received: 12 June 2006

Accepted: 6 December 2007

Published: 12 March 2008

Proposed: Jean-Pierre Otal

Seconded: Benson Farb, David Gabai

Authors

Andrea Sambusetti
Dipartimento di Matematica G Castelnuovo Università “La Sapienza” P.le Aldo Moro 5, 00185 Roma Italy
http://www.mat.uniroma1.it/people/sambusetti/

Volume 12, issue 1 (2008)