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 $\alpha \colon \Gamma \to \mathrm {GL}_n(\mathbb {R})$ 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).

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

  • N. Bourbaki, “Lie Groups and Lie Algebras, Part I,” Addison-Wesley, Reading, MA, 1975.
  • David Fried and William M. Goldman, Three-dimensional affine crystallographic groups, Adv. in Math. 47 (1983), no. 1, 1–49. MR 689763, DOI
  • G. Hochschild, The structure of Lie groups, Holden-Day, Inc., San Francisco-London-Amsterdam, 1965. MR 0207883
  • 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
  • V. Platonov, A. Rapinchuk, “Algebraic Groups and Number Theory,” Academic Press, Boston, 1994.
  • M. S. Raghunathan, Discrete subgroups of Lie groups, Springer-Verlag, New York-Heidelberg, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68. MR 0507234
  • Dave Witte, Superrigidity of lattices in solvable Lie groups, Invent. Math. 122 (1995), no. 1, 147–193. MR 1354957, DOI
  • D. Witte, Archimedean superrigidity of solvable $S$-arithmetic groups, J. Algebra 187 (1997) 268–288.
  • D. P. Zhelobenko, Kompaktnye gruppy Li i ikh predstavleniya, Izdat. “Nauka”, Moscow, 1970 (Russian). MR 0473097

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

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