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

DOI:
https://doi.org/10.1090/S0002-9939-97-03827-6

MathSciNet review:
1389500

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This work is directed towards the open question of the faithfulness of the reduced Gassner representation of the pure braid group, . Long and Paton proved that if a Burau matrix has ones on the diagonal and zeros below the diagonal then 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 under the reduced Gassner representation is , 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.

**[1]**M. Abdulrahim,*The Gassner Representation of the Pure Braid Group*, Doctoral Thesis, The Pennsylvania State University, 1995.**[2]**J.S. Birman,*Braids, Links and Mapping Class Groups*, Vol. 82 of Annals of Mathematical Studies, Princeton University Press, New Jersey, 1975. MR**51:11477****[3]**E. Formanek,*Braid group representations of low degree*, Proc. London Math. Soc. (3)**73**(1996), no. 2, 279-322. CMP**96:15****[4]**V.L. Hansen,*Braids and Coverings*, London Mathematical Society. Cambridge University Press, 1989. MR**94g:57004****[5]**D.D. Long,*On The Linear Representation of Braid Groups*, Transactions of the American Mathematical Society**311 (2)**(1989), 535-560. MR**89h:20054****[6]**D.D. Long and M. Paton,*The Burau representation is not faithful for*, Topology**32(2)**(1992), 439-447. MR**94c:20071****[7]**C.C. Squier,*The Burau representation is unitary*, Proceedings of the American Mathematical Society**90(2)**(1984), 199-202. MR**85b:20056****[8]**N. Stoltzfus, Private Communications, 1995.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
20F36

Retrieve articles in all journals with MSC (1991): 20F36

Additional Information

**Mohammad N. Abdulrahim**

Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802

Email:
mna@math.psu.edu

DOI:
https://doi.org/10.1090/S0002-9939-97-03827-6

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