Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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. David Fried and William M. Goldman, Three-dimensional affine crystallographic groups, Adv. in Math. 47 (1983), no. 1, 1–49. MR 689763, 10.1016/0001-8708(83)90053-1
  • 3. G. Hochschild, The structure of Lie groups, Holden-Day, Inc., San Francisco-London-Amsterdam, 1965. MR 0207883
  • 4. G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991. MR 1090825
  • 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, New York-Heidelberg, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68. MR 0507234
  • 7. Dave Witte, Superrigidity of lattices in solvable Lie groups, Invent. Math. 122 (1995), no. 1, 147–193. MR 1354957, 10.1007/BF01231442
  • 8. D. Witte, Archimedean superrigidity of solvable $S$-arithmetic groups, J. Algebra 187 (1997) 268-288.
  • 9. D. P. Želobenko, Compact Lie groups and their representations, American Mathematical Society, Providence, R.I., 1973. Translated from the Russian by Israel Program for Scientific Translations; Translations of Mathematical Monographs, Vol. 40. MR 0473098

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 22E40, 22E25, 22E27, 22G05

Retrieve articles in all journals with MSC (1991): 22E40, 22E25, 22E27, 22G05


Additional Information

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

DOI: http://dx.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