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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A faithfulness criterion
for the Gassner representation
of the pure braid group


Author: Mohammad N. Abdulrahim
Journal: Proc. Amer. Math. Soc. 125 (1997), 1249-1257
MSC (1991): Primary 20F36
MathSciNet review: 1389500
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Abstract: This work is directed towards the open question of the faithfulness of the reduced Gassner representation of the pure braid group, $P_{n}(n > 3)$. Long and Paton proved that if a Burau matrix $M$ has ones on the diagonal and zeros below the diagonal then $M$ is the identity matrix. In this paper, a generalization of Long and Paton's result will be proved. Our main theorem is that if the trace of the image of an element of $P_{n}$ under the reduced Gassner representation is $n-1$, then this element lies in the kernel of this representation. Then, as a corollary, we prove that an analogue of the main theorem holds true for the Burau representation of the braid group.


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Additional Information

Mohammad N. Abdulrahim
Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802
Email: mna@math.psu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-97-03827-6
PII: S 0002-9939(97)03827-6
Keywords: Braid group, Burau representation, Gassner representation
Received by editor(s): October 31, 1995
Additional Notes: The results in this paper are written under the direction of Professor Edward Formanek whose help and encouragement are greatly appreciated.
Communicated by: Ronald M. Solomon
Article copyright: © Copyright 1997 American Mathematical Society