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On the Burau representation modulo a small prime. D. Cooper D. D. Long \Lambda

Summary: On the Burau representation modulo a small prime.
D. Cooper D. D. Long \Lambda
1 Introduction
Despite the work of many authors, the group theoretic image of linear representations of the braids
groups remains mysterious in most cases. The first nontrivial example, the Burau representation is
not at all well understood. This representation
fi n : Bn ! GL(n \Gamma 1; Z[t; t \Gamma1 ])
is known not to be faithful for n 6 ([6] and [7]) but the nature of the image group and in
particular, a presentation for the image group has not been found. In [3], we simplified the problem
by composing fi n with the map which reduces coefficients modulo 2. In this way, we were able to
give a presentation for the image of the simplified representation fi
4\Omega Z 2 . (Throughout this paper
we use the notation Z p for the finite field with p elements.) Of course, the motivation for this
approach comes from the classical problem of whether the representation fi 4 is faithful and to this
end we pose the question:
Question 1.1 Is there any prime p for which the representation
4\Omega Z p : B 4 ! GL(3; Z p [t; t \Gamma1 ])
is faithful?
It is a consequence of some results of this note that the representation is not faithful in the case


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics