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)

 
 

 

On genericity and weight in the free group


Author: Anand Pillay
Journal: Proc. Amer. Math. Soc. 137 (2009), 3911-3917
MSC (2000): Primary 03C45; Secondary 20F67
DOI: https://doi.org/10.1090/S0002-9939-09-09956-0
Published electronically: June 18, 2009
MathSciNet review: 2529900
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the generic type of the (theory of the) free group $ F_{n}$ on $ n\geq 2$ generators has infinite weight, strengthening the well-known result that these free groups are not superstable. A preliminary result, possibly of independent interest, is that the realizations in $ F_{n}$ of the generic type are precisely the primitives.


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

  • 1. D. E. Cohen, Combinatorial Group Theory: A Topological Approach, London Math. Soc. Student Texts 14, Cambridge University Press, 1989. MR 1020297 (91d:20001)
  • 2. O. Kharlampovich and A. Myasnikov, Elementary theory of free nonabelian groups, J. Algebra 302 (2006), 451-552. MR 2293770 (2008e:20033)
  • 3. R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Springer-Verlag, 1977. MR 0577064 (58:28182)
  • 4. D. Marker, Model Theory: An Introduction, Springer, 2002. MR 1924282 (2003e:03060)
  • 5. A. Nies, Aspects of free groups, Journal of Algebra 263 (2003), 119-125. MR 1974081 (2004g:20033)
  • 6. C. Perin, Plongements éleméntaires dans un groupe hyperbolique sans torsion, Ph.D. thesis, Caen, October 2008.
  • 7. A. Pillay, Geometric Stability Theory, Oxford University Press, 1996. MR 1429864 (98a:03049)
  • 8. A. Pillay, Forking in the free group, Journal of the Inst. of Math. Jussieu 7 (2008), 375-389. MR 2400726
  • 9. B. Poizat, Groupes stables, avec types génériques réguliers, Journal of Symbolic Logic 48 (1983), 339-355. MR 704088 (85e:03082)
  • 10. Z. Sela, Diophantine geometry over groups, VI: The elementary theory of a free group, Geom. Funct. Anal. 16 (2006), 707-730. MR 2238945 (2007j:20063)
  • 11. Z. Sela, Diophantine geometry over groups, VIII: Stability (arXiv:math/0609096v1).
  • 12. S. Shelah, Strongly dependent theories (arXiv:math/0504197v2 ), to appear.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 03C45, 20F67

Retrieve articles in all journals with MSC (2000): 03C45, 20F67


Additional Information

Anand Pillay
Affiliation: School of Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom
Email: pillay@maths.leeds.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-09-09956-0
Received by editor(s): December 9, 2008
Received by editor(s) in revised form: March 3, 2009
Published electronically: June 18, 2009
Additional Notes: This work was supported by Marie Curie Chair EXC 024052 as well as EPSRC grant EP/F009712/1
Communicated by: Julia Knight
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society