Local quaternionic rigidity for complex hyperbolic lattices

TL;DR

This paper establishes that cocompact lattices in SU(n,1) exhibit local rigidity when embedded into certain quaternionic and related Lie groups, despite not being infinitesimally rigid, advancing understanding of their deformation properties.

Contribution

The paper proves that cocompact lattices in SU(n,1) are essentially locally rigid in specific quaternionic and orthogonal Lie groups, a novel result in the context of their deformation theory.

Findings

Cocompact lattices in SU(n,1) are essentially locally rigid in Sp(n,1), SU(2n,2), and SO(4n,4).

These lattices are not infinitesimally rigid but still exhibit local rigidity.

The result applies to the natural sequence of embeddings among these groups.

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 $\rho: \Gamma \longrightarrow G$ 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 $SU(n,1) \subset Sp(n,1) \subset SU(2n,2) \subset SO(4n,4))$.