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)

 
 

 

Definably simple groups in o-minimal structures


Authors: Y. Peterzil, A. Pillay and S. Starchenko
Journal: Trans. Amer. Math. Soc. 352 (2000), 4397-4419
MSC (2000): Primary 03C64, 22E15, 20G20; Secondary 12J15
DOI: https://doi.org/10.1090/S0002-9947-00-02593-9
Published electronically: February 24, 2000
MathSciNet review: 1707202
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract:

Let $\mathbb{G} =\langle G, \cdot\rangle$ be a group definable in an o-minimal structure $\mathcal{M}$. A subset $H$ of $G$ is $\mathbb{G} $-definable if $H$ is definable in the structure $\langle G,\cdot\rangle$(while definable means definable in the structure $\mathcal{M}$). Assume $\mathbb{G} $ has no $\mathbb{G} $-definable proper subgroup of finite index. In this paper we prove that if $\mathbb{G} $has no nontrivial abelian normal subgroup, then $\mathbb{G} $ is the direct product of $\mathbb{G} $-definable subgroups $H_1,\ldots,H_k$ such that each $H_i$ is definably isomorphic to a semialgebraic linear group over a definable real closed field. As a corollary we obtain an o-minimal analogue of Cherlin's conjecture.


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

  • 1. L. van den Dries, Tame topology and o-minimal structures, London Math. Soc. Lecture Note Ser., vol. 248, Cambridge Univ. Press, Cambridge, 1998. MR 99j:03001
  • 2. L. van den Dries, C. Miller, Geometric Categories and o-minimal structures, Duke Math. J. 84(1996), pp. 497-540. MR 97i:32008
  • 3. J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, 1972. MR 48:2197
  • 4. J. Knight, A. Pillay and C. Steinhorn, Definable sets in ordered structures II, Transactions of the Amer. Math. Soc. 295 (1986), pp. 593-605. MR 88b:03050b
  • 5. J. Loveys and Y. Peterzil, Linear o-minimal structures, Israel J. Math. 81 (1993), pp. 1-30. MR 94i:03075
  • 6. A. L. Onishchik, E. B.  Vinberg, Lie Groups and Algebraic Groups, Springer-Verlag, 1990. MR 91g:22001
  • 7. M. Otero, Y. Peterzil and A. Pillay Groups and rings definable in o-minimal expansions of real closed fields, Bull. London Math. Soc. 28 (1996), pp. 7-14. MR 96i:12006
  • 8. A. Pillay and C. Steinhorn, Definable sets in ordered structures I, Transactions of the Amer. Math. Soc. 295 (1986), pp. 565-592. MR 88b:03059
  • 9. Y. Peterzil, A. Pillay and S. Starchenko, Simple algebraic groups over real closed fields, Trans. Amer. Math. Soc. 352 (2000), 4421-4450.
  • 10. Y. Peterzil and S. Starchenko, A trichotomy theorem for o-minimal structures, Proc. London Math. Soc. 77 (1998), pp. 481-523. CMP 99:01
  • 11. A. Pillay, On groups and fields definable in o-minimal structures, Journal of Pure Applied Algebra 53 (1988), pp. 239-255. MR 89i:03069
  • 12. B. Poizat, Groupes Stables, Nur al-Mantiq Wal-Ma'rifah, 1987. MR 89b:03056
  • 13. A. W. Strzebonski, Euler characteristic in semialgebraic and other o-minimal groups, J. Pure Appl. Algebra 96 (1994), no. 2, pp. 173-201. MR 95j:03067

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 03C64, 22E15, 20G20, 12J15

Retrieve articles in all journals with MSC (2000): 03C64, 22E15, 20G20, 12J15


Additional Information

Y. Peterzil
Affiliation: Department of Mathematics and Computer Science, Haifa University, Haifa, Israel
Email: kobi@mathcs2.haifa.ac.il

A. Pillay
Affiliation: Department of Mathemetics, University of Illinois at Urbana-Champaign, 1409 W. Green St., Urbana, Illinois 61801
Email: pillay@math.uiuc.edu

S. Starchenko
Affiliation: Department of Mathemetics, University of Notre Dame, Room 370, CCMB, Notre Dame, Indiana 46556
Email: starchenko.1@nd.edu

DOI: https://doi.org/10.1090/S0002-9947-00-02593-9
Received by editor(s): February 25, 1998
Published electronically: February 24, 2000
Additional Notes: The second and the third authors were partially supported by NSF
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society