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Superrigid subgroups of solvable Lie groups

Authors:	Dave Witte

Institution:	Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078

Abstract:	Let $\Gamma$ be a discrete subgroup of a simply connected, solvable Lie group , such that $\operatorname {Ad}_G\Gamma$ has the same Zariski closure as $\operatorname {Ad}G$ . If $\alpha \colon \Gamma \to \operatorname {GL}_n(\mathord {\mathbb R})$ is any finite-dimensional representation of $\Gamma$ , we show that $\alpha$ virtually extends to a continuous representation $\sigma$ of . 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).

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