Local quaternionic rigidity for complex hyperbolic lattices
Résumé
Let $\Gamma \stackrel{i}{\hookrightarrow} L$ 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 $\rho: \Gamma \longrightarrow G$ into a simple real Lie group $G$ essentially extends to a Lie morphism $\rho_L: L \longrightarrow G$ (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 $\rho : \Gamma \stackrel{i}{\hookrightarrow} L \longrightarrow G$ are not infinitesimally rigid in general. Almost nothing is known about their local rigidity. In this paper we prove that any {\em cocompact} lattice $\Gamma$ 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 $SU(n,1) \subset Sp(n,1) \subset SU(2n,2) \subset SO(4n,4))$.
Origine : Fichiers produits par l'(les) auteur(s)
Loading...