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

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Abstract | References | Similar Articles | Additional Information

Let be a group definable in an o-minimal structure . A subset of is -definable if is definable in the structure (while *definable* means definable in the structure ). Assume has no -definable proper subgroup of finite index. In this paper we prove that if has no nontrivial abelian normal subgroup, then is the direct product of -definable subgroups such that each 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.

**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**

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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