Spectra of Lorentzian Quasi-Fuchsian Manifolds

A three-dimensional quasi-Fuchsian Lorentzian manifold M is a globally hyperbolic spacetime diffeomorphic to Σ × (-1,1) Σ × ( - 1 , 1 ) for a closed orientable surface Σ Σ of genus ≥ 2 ≥ 2 . It is the quotient M=Γ Ω _Γ M = Γ \ Ω Γ of an open set Ω _Γ ⊂ AdS₃ Ω Γ ⊂ AdS 3 by a discrete group Γ Γ of isometries of AdS₃ AdS 3 which is a particular example of an Anosov representation of π ₁(Σ ) π 1 ( Σ ) . We first show that the spacelike geodesic flow of M is Axiom A, has a discrete Ruelle resonance spectrum with associated (co-)resonant states, and that the Poincaré series for Γ Γ extend meromorphically to C C . This is then used to prove that there is a natural notion of resolvent of the pseudo-Riemannian Laplacian □ of M , which is meromorphic on C C with poles of finite rank, defining a notion of quantum resonances and quantum resonant states related to the Ruelle resonances and (co-)resonant states by a quantum-classical correspondence. This initiates the spectral study of convex co-compact pseudo-Riemannian locally symmetric spaces.