Counting closed geodesics in globally hyperbolic maximal compact AdS 3-manifolds

2015/03/31 by Glorieux, Olivier
#FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1503.09067

Abstract

We propose a definition for the length of closed geodesics in a globally hyperbolic maximal compact (GHMC) Anti-De Sitter manifold. We then prove that the number of closed geodesics of length less than R grows exponentially fast with R and the exponential growth rate is related to the critical exponent associated to the two hyperbolic surfaces coming from Mess parametrization. We get an equivalent of three results for quasi-Fuchsian manifolds in the GHMC setting : R. Bowen's rigidity theorem of critical exponent, A. Sanders' isolation theorem and C. McMullen's examples lightening the behaviour of this exponent when the surfaces range over Teichmüller space.

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