Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Cartan-decomposition subgroups of $\operatorname{SO}(2,n)$


Authors: Hee Oh and Dave Witte Morris
Journal: Trans. Amer. Math. Soc. 356 (2004), 1-38
MSC (2000): Primary 22E46; Secondary 20G20, 22E15, 57S20
DOI: https://doi.org/10.1090/S0002-9947-03-03428-7
Published electronically: August 25, 2003
MathSciNet review: 2020022
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For $G = \operatorname{SL} (3,\mathord{\mathbb{R} })$ and $G = \operatorname{SO}(2,n)$, we give explicit, practical conditions that determine whether or not a closed, connected subgroup $H$of $G$ has the property that there exists a compact subset $C$ of $G$with $CHC = G$. To do this, we fix a Cartan decomposition $G = K A^+ K$of $G$, and then carry out an approximate calculation of $(KHK) \cap A^+$for each closed, connected subgroup $H$ of $G$.


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

  • [Ben] Y. Benoist, Actions propres sur les espaces homogènes réductifs, Ann. Math. 144 (1996) 315-347. MR 97j:22023
  • [Bor] A. Borel, Linear Algebraic Groups, 2nd ed., Springer-Verlag, New York, 1991. MR 92d:20001
  • [BT] A. Borel and J. Tits, Groupes réductifs, Publ. Math. IHES 27 (1965) 55-150. MR 34:7527
  • [Hoc] G. Hochschild, The Structure of Lie Groups, Holden-Day, San Francisco, 1965. MR 34:7696
  • [Hm1] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972. MR 48:2197
  • [Hm2] J. E. Humphreys, Linear Algebraic Groups, Springer-Verlag, New York, 1975. MR 53:633
  • [Jac] N. Jacobson, Lie Algebras, Dover, New York, 1962. MR 26:1345
  • [KPS] J. Knight, A. Pillay and C. Steinhorn, Definable sets in ordered structures II, Trans. Amer. Math. Soc. 295 (1986) 593-605. MR 88b:03050b
  • [Kb1] T. Kobayashi, On discontinuous groups acting on homogeneous spaces with non-compact isotropy groups, J. Geom. Physics 12 (1993) 133-144. MR 94g:22025
  • [Kb2] T. Kobayashi, Criterion of proper actions on homogeneous spaces of reductive groups, J. Lie Th. 6 (1996) 147-163. MR 98a:57048
  • [Kb3] T. Kobayashi, Discontinuous groups and Clifford-Klein forms of pseudo-Riemannian homogeneous manifolds, in: B. Ørsted and H. Schlichtkrull, eds., Algebraic and Analytic Methods in Representation Theory, Academic Press, New York, 1997, pp. 99-165. MR 97g:53061
  • [Kos] B. Kostant, On convexity, the Weyl group, and the Iwasawa decomposition, Ann. Sc. ENS. 6 (1973) 413-455. MR 51:806
  • [Kul] R. Kulkarni, Proper actions and pseudo-Riemannian space forms, Adv. Math. 40 (1981) 10-51. MR 84b:53047
  • [OW1] H. Oh and D. Witte, New examples of compact Clifford-Klein forms of homogeneous spaces of $\operatorname{SO}(2,n)$, Internat. Math. Res. Notices 2000, no. 5, 235-251. MR 2001c:53071
  • [OW2] H. Oh and D. Witte, Compact Clifford-Klein forms of homogeneous spaces of $\operatorname{SO}(2,n)$ (preprint).
  • [PS] A. Pillay and C. Steinhorn, Definable sets in ordered structures I, Trans. Amer. Math. Soc. 295 (1986) 565-592. MR 88b:03050a
  • [Pog] D. Poguntke, Dense Lie group homomorphisms, J. Algebra 169 (1994) 625-647. MR 95m:22003
  • [Rag] M. S. Raghunathan, Discrete Subgroups of Lie Groups, Springer-Verlag, New York, 1972. MR 58:22394a
  • [vdD] L. P. D. van den Dries, Tame Topology and O-minimal Structures, Cambridge U. Press, Cambridge, 1998. MR 99j:03001
  • [Var] V. S. Varadarajan, Lie Groups, Lie Algebras, and their Representations, Springer, New York, 1984. MR 85e:22001
  • [W1] A. Wilkies, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J. Amer. Math. Soc. 9 (1996) 1051-1094. MR 98j:03052
  • [W2] A. Wilkies, O-Minimality, Documenta Mathematica, Extra Volume ICM I (1998) 457-460. http://www.mathematik.uni-bielefeld.de/documenta/xvol-icm/99/Wilkie.MAN.html MR 2000b:03115
  • [Wit] D. Witte, Superrigidity of lattices in solvable Lie groups, Invent. Math. 122 (1995) 147-193. MR 96k:22024
  • [Zim] R. J. Zimmer, Ergodic Theory and Semisimple Groups, Birkhäuser, Boston, 1984. MR 86j:22014

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 22E46, 20G20, 22E15, 57S20

Retrieve articles in all journals with MSC (2000): 22E46, 20G20, 22E15, 57S20


Additional Information

Hee Oh
Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078
Address at time of publication: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email: heeoh@math.princeton.edu

Dave Witte Morris
Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078
Address at time of publication: Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta, Canada T1K 3M4
Email: dwitte@math.okstate.edu, dmorris@cs.uleth.ca

DOI: https://doi.org/10.1090/S0002-9947-03-03428-7
Received by editor(s): February 4, 1999
Received by editor(s) in revised form: March 4, 1999, and November 6, 1999
Published electronically: August 25, 2003
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society