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Braid groups are linear. (English) Zbl 0988.20021

D. Krammer [Invent. Math. 142, No. 3, 451-486 (2000; see the review Zbl 0988.20023 below)] proved that a representation of the braid groups \(B_n\) is faithful in the case \(n=4\). The representation Krammer used is essentially the same as one used by R. J. Lawrence [Commun. Math. Phys. 135, No. 1, 141-191 (1990; Zbl 0716.20022)]. The author calls this representation the Lawrence-Krammer representation.
In the paper the author proves by topological methods that the Lawrence-Krammer representation is faithful for all \(n\).

MSC:

20F36 Braid groups; Artin groups
20C15 Ordinary representations and characters
57M07 Topological methods in group theory
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References:

[1] Stephen Bigelow, The Burau representation is not faithful for \(n=5\), Geometry and Topology 3 (1999), 397-404. CMP 2000:05 · Zbl 0942.20017
[2] Joan S. Birman, Braids, links, and mapping class groups, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. Annals of Mathematics Studies, No. 82. Joan S. Birman, Erratum: ”Braids, links, and mapping class groups” (Ann. of Math. Studies, No. 82, Princeton Univ. Press, Princeton, N. J., 1974), Princeton University Press, Princeton, N. J.; University of Tokyo Press, Toyko, 1975. Based on lecture notes by James Cannon. · Zbl 0297.57001
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[13] Vladimir Turaev, Faithful Linear Representations of the Braid Groups, arXiv: math.GT/0006202. · Zbl 1050.20026
[14] Matthew G. Zinno, On Krammer’s Representation of the Braid Group, arXiv: math.RT/0002136. · Zbl 1042.20023
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