[Paper Review] Anosov AdS representations are quasi-Fuchsian

Quentin Mérigot|arXiv (Cornell University)|Oct 2, 2007

Geometry and complex manifolds6 references3 citations

TL;DR

This paper proves that Anosov representations of a cocompact lattice Γ ⊂ SO₀(1,n) into SO₀(2,n) are quasi-Fuchsian, meaning they are discrete, faithful, and preserve an acausal (n−1)-sphere in the boundary of anti-de Sitter space. The proof relies on the Anosov dynamics of the geodesic flow and the existence of equivariant, hyperbolic equivariant boundary maps that induce a topological embedding of the boundary of hyperbolic space into the Einstein boundary of AdS space.

ABSTRACT

Let Gamma be a cocompact lattice in SO(1,n). A representation rho: Gamma o SO(2,n) is quasi-Fuchsian if it is faithfull, discrete, and preserves an acausal subset in the boundary of anti-de Sitter space - a particular case is the case of Fuchsian representations, ie. composition of the inclusions of Gamma in SO(1,n) and of SO(1,n) in SO(2,n). We prove that if a representation is Anosov in the sense of Labourie then it is also quasi-Fuchsian. We also show that Fuchsian representations are Anosov : the fact that all quasi-Fuchsian representations are Anosov will be proved in a second part by T. Barbot. The study involves the geometry of locally anti-de Sitter spaces: quasi-Fuchsian representations are holonomy representations of globally hyperbolic spacetimes diffeomorphic to the product R imes Gamma\H^n and locally modeled on the anti-de Sitter space.

Motivation & Objective

• To establish that Anosov representations into SO₀(2,n) are quasi-Fuchsian, i.e., discrete, faithful, and preserve an acausal (n−1)-sphere in the boundary of anti-de Sitter space.
• To demonstrate that Fuchsian representations (as embeddings of SO₀(1,n) into SO₀(2,n)) are Anosov, completing a foundational link between rigidity and dynamical systems in AdS geometry.
• To show that the Anosov property implies the existence of a ρ-equivariant, hyperbolic boundary map from the unit tangent bundle of hyperbolic space to the boundary of AdS space.
• To establish that the image of the boundary map is a topological acausal sphere, which characterizes quasi-Fuchsian representations in the AdS setting.

Proposed method

• Utilizes the Anosov dynamics of the geodesic flow φᵗ on T¹ℍⁿ, which is lifted to a flow on a flat bundle over the quotient manifold Γ\T¹ℍⁿ.
• Constructs a family of ρ-equivariant metrics gˣ on the boundary Einₙ of anti-de Sitter space, varying continuously with x ∈ ℍⁿ, that exhibit exponential growth along geodesic rays.
• Defines a ρ-equivariant boundary map (ℓ₊, ℓ₋): T¹ℍⁿ → ∂AdSₙ₊₁ × ∂AdSₙ₊₁ \\(diagonal), using the endpoints of geodesics in ℍⁿ.
• Proves that the image of this map is a φₜ-invariant hyperbolic set in the flat bundle, using the exponential growth of the metrics gˣ.
• Deforms the representation ρ₀ (Fuchsian) to nearby representations and uses structural stability of hyperbolic sets to show the existence of a continuous section, lifting to a ρ-equivariant boundary map.
• Establishes that the boundary maps ℓ₊ and ℓ₋ are related by ℓ₊ = ℓ₋ ∘ α, where α reverses the direction of the tangent vector, implying they have the same image and are homeomorphisms onto an acausal sphere.

Experimental results

Research questions

• RQ1 Are Anosov representations into SO₀(2,n) necessarily quasi-Fuchsian, i.e., do they preserve an acausal (n−1)-sphere in the boundary of anti-de Sitter space?
• RQ2 Does the Anosov property of a representation imply the existence of a ρ-equivariant, hyperbolic boundary map with values in the Einstein boundary of AdSₙ₊₁?
• RQ3 Is the image of the boundary map ℓ₊: ∂ℍⁿ → Einₙ a topological (n−1)-sphere, and is it acausal?
• RQ4 Can the structural stability of hyperbolic sets in the flat bundle over Γ\T¹ℍⁿ be used to extend the existence of the boundary map to small deformations of Fuchsian representations?
• RQ5 Does the Anosov condition ensure that the boundary map induces a homeomorphism from ∂ℍⁿ to a topological sphere in Einₙ, thereby characterizing quasi-Fuchsian representations?

Key findings

• Anosov representations ρ: Γ → SO₀(2,n) are quasi-Fuchsian, meaning they are discrete, faithful, and preserve an acausal (n−1)-sphere in the boundary of anti-de Sitter space.
• The Anosov property implies the existence of a ρ-equivariant boundary map (ℓ₊, ℓ₋): T¹ℍⁿ → ∂AdSₙ₊₁ × ∂AdSₙ₊₁ \\(diagonal), which lifts to a continuous section of the flat bundle Eρ.
• The image of the boundary map ℓ₊: ∂ℍⁿ → Einₙ is a topological (n−1)-sphere, and ℓ₊ and ℓ₋ have the same image, which is acausal.
• The boundary maps ℓ₊ and ℓ₋ satisfy ℓ₊ = ℓ₋ ∘ α, where α reverses the direction of the tangent vector, ensuring symmetry and injectivity of the map.
• The existence of a ρ-equivariant family of metrics gˣ on Einₙ with exponential growth along geodesic rays ensures that the image of the boundary map is a hyperbolic set for the lifted flow φₜρ.
• Fuchsian representations are shown to be Anosov, completing the equivalence between Fuchsian and Anosov representations in this context, with the converse (all quasi-Fuchsian are Anosov) established in a follow-up work by Barbot.