Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-26T12:38:38.386Z Has data issue: false hasContentIssue false
  • English
  • Français

Stallings graphs, algebraic extensions and primitive elements in F2

Published online by Cambridge University Press:  21 March 2014

ORI PARZANCHEVSKI  and
DORON PUDER
ORI PARZANCHEVSKI
Affiliation:
Einstein Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel. e-mails: [email protected], [email protected]
DORON PUDER
Affiliation:
Einstein Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel. e-mails: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the free group of rank two from the point of view of Stallings core graphs. The first half of the paper examines primitive elements in this group, giving new and self-contained proofs for various known results about them. In particular, this includes the classification of bases of this group. The second half of the paper is devoted to constructing a counterexample to a conjecture by Miasnikov, Ventura and Weil, which seeks to characterize algebraic extensions in free groups in terms of Stallings graphs.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 157 , Issue 1 , July 2014 , pp. 1 - 11
Copyright
Copyright © Cambridge Philosophical Society 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[CMZ81]Cohen, M., Metzler, W. and Zimmermann, A.What does a basis of F(a, b) look like? Math. Ann., 257 (4) (1981), 435–445.Google Scholar
[Coh72]Cohn, H.Markoff forms and primitive words. Math. Ann., 196 (1) (1972), 8–22.CrossRefGoogle Scholar
[KM02]Kapovich, I. and Myasnikov, A.Stallings foldings and subgroups of free groups. J. Algebra, 248 (2) (2002), 608–668.Google Scholar
[MVW07]Miasnikov, A., Ventura, E. and Weil, P.Algebraic extensions in free groups. In Arzhantseva, G.N., Bartholdi, L., Burillo, J. and Ventura, E., editors, Geometric Group Theory Trends Math. (Birkhauser, 2007), pages 225–253.CrossRefGoogle Scholar
[Nie17]Nielsen, J.Die isomorphismen der allgemeinen, unendlichen gruppe mit zwei erzeugenden. Math. Ann., 78 (1) (1917), 385–397.Google Scholar
[OZ81]Osborne, R. P. and Zieschang, H.Primitives in the free group on two generators. Invent. Math., 63 (1) (1981), 17–24.Google Scholar
[PP14]Puder, D. and Parzanchevski, O. Measure preserving words are primitive. To appear in: J. Amer. Math. Soc., arXiv:1202.3269.Google Scholar
[Pud13]Puder, D.Primitive words, free factors and measure preservation. Israel J. Math., (2013), doi:10.1007/s11856-013-0055-2.Google Scholar
[Sta83]Stallings, J. R.Topology of finite graphs. Invent. Math., 71 (3) (1983), 551–565.Google Scholar
[Tak51]Takahasi, M.Note on chain conditions in free groups. Osaka Math. J., 3 (2) (1951), 221–225.Google Scholar