Geometry and Topology 12 (2008) 617–637

DOI: 10.2140/gt.2008.12.617

Abstract

We show that the universal covering ~X 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(~X) 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

Primary: 53C23

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

References

Publication

Received: 12 June 2006
Accepted: 6 December 2007
Published: 12 March 2008
Proposed: Jean-Pierre Otal
Seconded: Benson Farb, David Gabai

Authors