Superrigid subgroups of solvable Lie groups

Abstract

Let $\hat{I}“$ be a discrete subgroup of a simply connected, solvable Lie group~$G$, such that $\Ad_GΓ$ has the same Zariski closure as $\Ad G$. If $α\colon Γ\to \GL_n(\real)$ is any finite-dimensional representation of~$\hat{I}“$,we show that $\hat{I}\pm$ virtually extends to a continuous representation~$\ddot{I}ƒ$ of~$G$. Furthermore, the image of~$\ddot{I}ƒ$ is contained in the Zariski closure of the image of~$\hat{I}\pm$. When $\hat{I}“$ is not discrete, the same conclusions are true if we make the additional assumption that the closure of $[\hat{I}“,\hat{I}“]$ is a finite-index subgroup of $[G,G]\cap \hat{I}“$ (and $\hat{I}“$ is closed and $\hat{I}\pm$ is continuous).