Rigidity of Kleinian groups via self-joinings

• 成果类型: Article
• 署名作者: Kim, Dongryul M.; Oh, Hee
• 署名单位: Yale University; Korea Institute for Advanced Study (KIAS)
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-023-01213-5
• 发表日期: 2023
• 页码: 937-948
• 关键词: sets; rank

摘要

Let Gamma < PSL2( C) similar or equal to Isom(+) (H-3) be a finitely generated non-Fuchsian Kleinian group whose ordinary set Omega = S-2 - Lambda has at least two components. Let rho : Gamma -> PSL2( C) be a faithful discrete non-Fuchsian representation with boundary map f : Lambda -> S-2 on the limit set. In this paper, we obtain a new rigidity theorem: if f is conformal on Lambda, in the sense that f maps every circular slice of Lambda into a circle, then f extends to a Mobius transformation g on S-2 and rho is the conjugation by g. Moreover, unless rho is a conjugation, the set of circles C such that f (C boolean AND Lambda) is contained in a circle has empty interior in the space of all circles meeting Lambda. This answers a question asked by McMullen on the rigidity of maps Lambda -> S-2 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 Gamma with the higher rank dynamics of the self-joining (id x rho)(Gamma) < PSL2(C) x PSL2(C).