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

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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: Bowen’s rigidity theorem of critical exponent, Sanders’ isolation theorem and McMullen’s examples lightening the behaviour of this exponent when the surfaces range over Teichmüller space.

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Notes

  1. A property P is satisfied by most curves on \(S_0\) if there is \(\eta >0\) such that \(\frac{{{\mathrm{Card}}}\{ c\in {\mathcal {C}}\, |\, \ell _0(c)\le R \} \cap P }{{{\mathrm{Card}}}\{ c\in {\mathcal {C}}\, |\, \ell _0(c)\le R \}} =o(e^{-\eta R } )\).

  2. We replace \( \{ c\in {\mathcal {C}}\, |\, \ell _0(c)\le R \}\) by \(\{ c\in {\mathcal {C}}\, |\, \ell _0(c)+\ell _n(c)\le R \}\) in the previous definition.

  3. With Monclair, we propose a Lorentzian proof of the previous result independent of the one of Bishop Steger in [15].

  4. This follows from a small modifications of the arguments in [22].

  5. (Mapping class group is defined by \(MCG = Diff(S)/Diff_0(S) \), where \(Diff_0(S)\) is the group of diffeomorphisms isotopic to the identity).

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Acknowledgements

This work is a part of my Ph.D. thesis and I am very grateful to Gilles Courtois for his support and advices. I would like to thank Maxime Wolff for his help about Teichmüller space, laminations, and Thurston compactification. I am also grateful to Jean-Marc Schlenker who suggested me the Anti-de Sitter interpretation of my work. Finally I would like to thank the referees for many useful comments.

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  1. Instituto Nacional de Matematica Pura e Aplicada, Rio de Janeiro, Brazil

    Olivier Glorieux

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Glorieux, O. Counting closed geodesics in globally hyperbolic maximal compact AdS 3-manifolds. Geom Dedicata 188, 63–101 (2017). https://doi.org/10.1007/s10711-016-0206-9

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