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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
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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.

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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

Francesco Matucci
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904

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.