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

TL;DR

This paper defines a notion of length for closed geodesics in GHMC AdS 3-manifolds, showing their count grows exponentially with a rate linked to hyperbolic surface exponents, extending key results from quasi-Fuchsian contexts.

Contribution

It introduces a new definition of geodesic length in GHMC AdS manifolds and establishes exponential growth rates connected to hyperbolic surface exponents, extending classical results.

Findings

Number of closed geodesics of length less than R grows exponentially with R.

Exponential growth rate is related to the critical exponent of hyperbolic surfaces.

Results extend Bowen, Sanders, and McMullen's theorems to the GHMC AdS setting.

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 Teichm\"uller space.