Superrigid subgroups of solvable Lie groups

Abstract

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