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)

Request Permissions   Purchase Content 
 

 

Bounding the residual finiteness of free groups


Authors: Martin Kassabov and Francesco Matucci
Journal: Proc. Amer. Math. Soc. 139 (2011), 2281-2286
MSC (2010): Primary 20F69; Secondary 20E05, 20E07, 20E26
Published electronically: February 25, 2011
MathSciNet review: 2784792
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We find a lower bound to the size of finite groups detecting a given word in the free group. More precisely we construct a word $ w_n$ of length $ n$ in non-abelian free groups with the property that $ w_n$ is the identity on all finite quotients of size $ \sim n^{2/3}$ or less. This improves on a previous result of Bou-Rabee and McReynolds quantifying the lower bound of the residual finiteness of free groups.


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

  • 1. K. Bou-Rabee and D. B. McReynolds.
    Asymptotic growth and least common multiples in groups.
    Preprint,
    arXiv:math.GR/0907.3681v1.
  • 2. Khalid Bou-Rabee, Quantifying residual finiteness, J. Algebra 323 (2010), no. 3, 729–737. MR 2574859, 10.1016/j.jalgebra.2009.10.008
  • 3. N. V. Buskin, Efficient separability in free groups, Sibirsk. Mat. Zh. 50 (2009), no. 4, 765–771 (Russian, with Russian summary); English transl., Sib. Math. J. 50 (2009), no. 4, 603–608. MR 2583614, 10.1007/s11202-009-0067-7
  • 4. Uzy Hadad.
    On the shortest identity in finite simple groups of lie type.
    Journal of Group Theory.
    To appear, arXiv:math.GR/0808.0622v1.
  • 5. Marcel Herzog and Gil Kaplan, Large cyclic subgroups contain non-trivial normal subgroups, J. Group Theory 4 (2001), no. 3, 247–253. MR 1839997, 10.1515/jgth.2001.022
  • 6. Alexander Lubotzky and Dan Segal, Subgroup growth, Progress in Mathematics, vol. 212, Birkhäuser Verlag, Basel, 2003. MR 1978431
  • 7. Andrea Lucchini, On the order of transitive permutation groups with cyclic point-stabilizer, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 9 (1998), no. 4, 241–243 (1999) (English, with English and Italian summaries). MR 1722784
  • 8. V. D. Mazurov and E. I. Khukhro (eds.), The Kourovka notebook, Sixteenth edition, Russian Academy of Sciences Siberian Division, Institute of Mathematics, Novosibirsk, 2006. Unsolved problems in group theory; Including archive of solved problems. MR 2263886
  • 9. Igor Rivin.
    Geodesics with one self-intersection, and other stories.
    Preprint,
    arXiv:math.GT/
    0901.2543v3.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 20F69, 20E05, 20E07, 20E26

Retrieve articles in all journals with MSC (2010): 20F69, 20E05, 20E07, 20E26


Additional Information

Martin Kassabov
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853
Address at time of publication: School of Mathematics, University of Southampton, University Road, Southampton, SO17 1BJ, United Kingdom
Email: kassabov@math.cornell.edu

Francesco Matucci
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
Email: fm6w@virginia.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10967-5
Keywords: Free group, residually finite group, identities in a group
Received by editor(s): March 3, 2010
Published electronically: February 25, 2011
Additional Notes: The first author was partially funded by National Science Foundation grants DMS 0600244, 0635607 and 0900932.
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.