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Superrigid subgroups of solvable Lie groups

Author: Dave Witte
Journal: Proc. Amer. Math. Soc. 125 (1997), 3433-3438
MSC (1991): Primary 22E40; Secondary 22E25, 22E27, 22G05
MathSciNet review: 1423339
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Abstract: Let $\Gamma $ be a discrete subgroup of a simply connected, solvable Lie group $G$, 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 $G$. 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).

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Additional Information

Dave Witte
Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078

Received by editor(s): June 21, 1996
Communicated by: Roe Goodman
Article copyright: © Copyright 1997 American Mathematical Society