2009/03/22 by Инканг Ким, Inkang, Kim, Bruno Klingler +3
Mathematics · #14D07 #20G10 #20G20 #32L20 #53C24 #53C26 #53C35 #53C43 #53C55 #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.0903.3706
openalex publication_date 2009/03/22 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Let Γ\stackreli\hookrightarrow L be a lattice in the real simple Lie group L. If L is of rank at least 2 (respectively locally isomorphic to Sp(n,1)) any unbounded morphism ρ: Γ\longrightarrow G into a simple real Lie group G essentially extends to a Lie morphism ρL: L \longrightarrow G (Margulis's superrigidity theorem, respectively Corlette's theorem). In particular any such morphism is infinitesimally, thus locally, rigid. On the other hand, for L=SU(n,1), even morphisms of the form ρ: Γ\stackreli\hookrightarrow L \longrightarrow G are not infinitesimally rigid in general. Almost nothing is known about their local rigidity. In this paper we prove that any \em cocompact lattice Γ in SU(n,1) is essentially locally rigid (while in general not infinitesimally rigid) in the quaternionic groups Sp(n,1), SU(2n,2) or SO(4n,4) (for the natural sequence of embeddings SU(n,1) ⊂ Sp(n,1) ⊂ SU(2n,2) ⊂ SO(4n,4)).