Bounding the residual finiteness of free groups
HTML articles powered by AMS MathViewer
- by Martin Kassabov and Francesco Matucci PDF
- Proc. Amer. Math. Soc. 139 (2011), 2281-2286 Request permission
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
- K. Bou-Rabee and D. B. McReynolds. Asymptotic growth and least common multiples in groups. Preprint, \verb+arXiv:math.GR/0907.3681v1+.
- Khalid Bou-Rabee, Quantifying residual finiteness, J. Algebra 323 (2010), no. 3, 729–737. MR 2574859, DOI 10.1016/j.jalgebra.2009.10.008
- 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, DOI 10.1007/s11202-009-0067-7
- Uzy Hadad. On the shortest identity in finite simple groups of lie type. Journal of Group Theory. To appear, \verb+arXiv:math.GR/0808.0622v1+.
- Marcel Herzog and Gil Kaplan, Large cyclic subgroups contain non-trivial normal subgroups, J. Group Theory 4 (2001), no. 3, 247–253. MR 1839997, DOI 10.1515/jgth.2001.022
- Alexander Lubotzky and Dan Segal, Subgroup growth, Progress in Mathematics, vol. 212, Birkhäuser Verlag, Basel, 2003. MR 1978431, DOI 10.1007/978-3-0348-8965-0
- 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
- 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
- Igor Rivin. Geodesics with one self-intersection, and other stories. Preprint, \verb+arXiv:math.GT/+ \verb+0901.2543v3+.
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
- MR Author ID: 788744
- Email: fm6w@virginia.edu
- 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
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 2281-2286
- MSC (2010): Primary 20F69; Secondary 20E05, 20E07, 20E26
- DOI: https://doi.org/10.1090/S0002-9939-2011-10967-5
- MathSciNet review: 2784792