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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
DOI: https://doi.org/10.1090/S0002-9939-97-04147-6
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).


References [Enhancements On Off] (What's this?)

  • 1. N. Bourbaki, ``Lie Groups and Lie Algebras, Part I,'' Addison-Wesley, Reading, MA, 1975.
  • 2. D. Fried and W. Goldman, Three-dimensional affine crystallographic groups, Adv. Math 47 (1983) 1-49. MR 84d:20047
  • 3. G. Hochschild, ``The Structure of Lie Groups,'' Holden-Day, San Francisco, 1965. MR 34:7696
  • 4. G. A. Margulis, ``Discrete Subgroups of Semisimple Lie Groups,'' Springer-Verlag, Berlin/New York, 1991. MR 92h:22021
  • 5. V. Platonov, A. Rapinchuk, ``Algebraic Groups and Number Theory,'' Academic Press, Boston, 1994.
  • 6. M. S. Raghunathan, ``Discrete Subgroups of Lie Groups,'' Springer-Verlag, Berlin/New York, 1972. MR 58:22394a
  • 7. D. Witte, Superrigidity of lattices in solvable Lie groups, Invent. Math. 122 (1995) 147-193. MR 96k:22024
  • 8. D. Witte, Archimedean superrigidity of solvable $S$-arithmetic groups, J. Algebra 187 (1997) 268-288.
  • 9. D. P. Zelobenko, ``Compact Lie Groups and their Representations'', American Mathematical Society, Providence, R. I., 1973. MR 57:12776b

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Additional Information

Dave Witte
Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078
Email: dwitte@math.okstate.edu

DOI: https://doi.org/10.1090/S0002-9939-97-04147-6
Received by editor(s): June 21, 1996
Communicated by: Roe Goodman
Article copyright: © Copyright 1997 American Mathematical Society

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