2022/08/11 by Kim, Dongryul M., Oh, Hee · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2208.05806
Let Γ<PSL2(ℂ)≃ Isom+(ℍ3) be a finitely generated non-Fuchsian Kleinian group whose ordinary set Ω=\mathbbS2-Λ has at least two components. Let ρ: Γ→ PSL2(ℂ) be a faithful discrete non-Fuchsian representation with boundary map f:Λ→ \mathbbS2 on the limit set. In this paper, we obtain a new rigidity theorem: if f is \it conformal on Λ, in the sense that f maps every circular slice of Λ into a circle, then f extends to a Möbius transformation g on \mathbbS2 and ρ is the conjugation by g. Moreover, unless ρ is a conjugation, the set of circles C such that f(C∩ Λ) is contained in a circle has empty interior in the space of all circles meeting Λ. This answers a question asked by McMullen on the rigidity of maps Λ→ \mathbbS2 sending vertices of every tetrahedron of zero-volume to vertices of a tetrahedron of zero-volume. The novelty of our proof is a new viewpoint of relating the rigidity of Γ with the higher rank dynamics of the self-joining (id × ρ)(Γ)<PSL2(ℂ)× PSL2(ℂ).