A faithfulness criterion for the Gassner representation of the pure braid group
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- by Mohammad N. Abdulrahim PDF
- Proc. Amer. Math. Soc. 125 (1997), 1249-1257 Request permission
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.References
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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
- 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
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 1249-1257
- MSC (1991): Primary 20F36
- DOI: https://doi.org/10.1090/S0002-9939-97-03827-6
- MathSciNet review: 1389500