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ISSN 1088-6826(e) ISSN 0002-9939(p)

On genericity and weight in the free group

Author(s): Anand Pillay
Journal: Proc. Amer. Math. Soc. 137 (2009), 3911-3917.
MSC (2000): Primary 03C45; Secondary 20F67
Posted: June 18, 2009
MathSciNet review: 2529900
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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:

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.

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Additional Information:

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

DOI: 10.1090/S0002-9939-09-09956-0
PII: S 0002-9939(09)09956-0
Received by editor(s): December 9, 2008,
Received by editor(s) in revised form: March 3, 2009
Posted: 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
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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