Volume 12, issue 1 (2008)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Asymptotic properties of coverings in negative curvature

Andrea Sambusetti

Geometry & Topology 12 (2008) 617–637
Abstract

We show that the universal covering X̃ of any compact, negatively curved manifold X0 has an exponential growth rate which is strictly greater than the exponential growth rate of any other normal covering X X0. Moreover, we give an explicit formula estimating the difference between ω(X̃) and ω(X) in terms of the systole of X and of other elementary geometric parameters of the base space X0. 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/