A presentation for the image of Burau\((4)\otimes Z_2\)
Let \(\beta_n: B_n\to\text{GL}_{n-1} (\mathbb{Z}[t^{\pm 1}])\) be the Burau representation of the \(n\)-strand braid group \(B_n\). This representation is known to be faithful if \(n=3\), but \(\beta_n\) is not faithful for all \(n\geq 6\). The paper under review concentrates on the case \(n=4\). Reducing modulo 2 yields a representation \(\beta: B_4\to \text{GL}_3 (\mathbb{Z}_2 [t^{\pm 1}])\) and the authors derive two descriptions of the image of \(\beta\). First, a presentation of \(\text{im} (\beta)\) by generators and relations is given. As a by-product of the construction of this presentation, the image \(\text{im} (\beta)\) is identified with the group of isometries of a certain form \(J\). By using the standard valuation of the field \(\mathbb{Z}_2(t)\), the group \(\text{SL}_3 (\mathbb{Z}_2(t))\) acts on an Euclidean building \(\Delta\) of rank 3, and this gives an action of \(\text{im} (\beta)\) on the building \(\Delta\). Haefliger's theory of group complexes contains a result which identifies a group \(G\) acting in a certain way on a complex \(X\) with the underlying group of a group complex defined on the quotient \(X/G\). In order to use this result, the authors determine the complex \(B=\Delta/ \text{im} (\beta)\) and the stabilizers of \(B_4\) on the simplices of \(\Delta\). These computations occupy the main part of the paper, and the form \(J\), mentioned above, enters in a crucial way.
scientific article; zbMATH DE number 1026423
scientific article