Deformations of Fuchsian AdS representations are Quasi-Fuchsian

2013/01/18 by Barbot, Thierry · 2 citations
#Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1301.4309

Abstract

Let Γ be a finitely generated group, and let \opRep(Γ, \SO(2,n)) be the moduli space of representations of Γ into \SO(2,n) (n ≥ 2). An element ρ: Γ→ \SO(2,n) of \opRep(Γ, \SO(2,n)) is quasi-Fuchsian if it is faithful, discrete, preserves an acausal subset in the conformal boundary \Ein_n of the anti-de Sitter space; and if the associated globally hyperbolic anti-de Sitter space is spatially compact - a particular case is the case of Fuchsian representations, i.e. composition of a faithfull, discrete and cocompact representation ρ_f: Γ→ \SO(1,n) and the inclusion \SO(1,n) ⊂ \SO(2,n). In \citemerigot we proved that quasi-Fuchsian representations are precisely representations which are Anosov as defined in \citelabourie. In the present paper, we prove that quasi-Fuchsian representations form a connected component of \opRep(Γ, \SO(2,n)). This is an almost direct corollary of the following result: let Γ be the fundamental group of a globally hyperbolic spacetime locally modeled on \AdS_n, and let ρ: Γ→ \SO₀(2,n) be the holonomy representation. Then, if Γ is Gromov hyperbolic, the ρ(Γ)-invariant achronal limit set in \Ein_n is acausal.

Cited by

• Equiaffine immersions and pseudo-Riemannian space forms
• Geometric components of representation spaces via robust families of submanifolds

Related

• Anosov AdS representations are quasi-Fuchsian
• Quasi-Fuchsian AdS representations are Anosov
• Anosov representations as holonomies of globally hyperbolic spatially compact conformally flat spacetimes
• Convergence of quasi-Fuchsian groups using critical exponent
• Regularity of limit sets of AdS quasi-Fuchsian groups