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Proceedings of the American Mathematical Society

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ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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Superrigid subgroups of solvable Lie groups
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by Dave Witte PDF
Proc. Amer. Math. Soc. 125 (1997), 3433-3438 Request permission

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 $\alpha \colon \Gamma \to \mathrm {GL}_n(\mathbb {R})$ 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
  • Email: dwitte@math.okstate.edu
  • Received by editor(s): June 21, 1996
  • Communicated by: Roe Goodman
  • © Copyright 1997 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 125 (1997), 3433-3438
  • MSC (1991): Primary 22E40; Secondary 22E25, 22E27, 22G05
  • DOI: https://doi.org/10.1090/S0002-9939-97-04147-6
  • MathSciNet review: 1423339