Local quaternionic rigidity for complex hyperbolic lattices

Abstract

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 $\rho :\Gamma ⟶G$ into a simple real Lie group $G$ essentially extends to a Lie morphism ${\rho }_L:L⟶G$ (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 $\rho :\Gamma \stackrel{i}{\hookrightarrow }L⟶G$ 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 $SU(n,1)\subset Sp(n,1)\subset SU(2n,2)\subset SO(4n,4))$.

24 pages

References in corpus (2)

• Bounded Cohomology and Deformation Rigidity in Complex Hyperbolic Geometry
• Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type