Braid Groups are Linear Groups

✍ Scribed by S. Bachmuth

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 396 KB
• Volume: 121
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1996.0046

✦ Synopsis

to the memories of wilhelm magnus and horace mochizuki

1. Introduction

Let B n designate the braid group on n strings (Artin [1]). In 1936 Burau [7] gave a matrix representation of B n and in 1961 Gassner [8] generalized Burau's representation. B 2 is infinite cyclic and hence is a linear group. Gassner [8] and many authors since have shown that B 3 is also a linear group. But it has remained an open problem as to whether any other braid groups are linear. In this paper we show that B n is linear for all n by proving Theorem A. The Gassner representation of B n is faithful for every n. This is proved after using reductions of Magnus and Lipschutz which we now explain. Let R be the kernel of the map of B n onto the mapping class group of the n-punctured sphere. (R is labelled T n in 9 3.7, Theorem N9 of Magnus, Karrass 6 Solitar [11].) In 1934, Magnus [10] showed that R is a direct product of a free group R n&1 of rank n&1 and the infinite cyclic center of B n . Using these results of Magnus, Lipschutz [9] showed that the kernel of the Gassner representation of B n is contained in R n&1 , provided that the Gassner representation of B n&1 is faithful. Although it is not needed here, Birman [6, Theorem 3.17] generalized Lipschutz' Theorem and showed that the kernel of the Gassner representation lies in the second derived group of R n&1 . We will show that the R n are faithfully represented for all n, thereby inductively establishing the faithfulness of the Gassner representation for all n. Specifically we prove Theorem B. The image of R n under the Gassner representation is a free group of rank n freely generated by the image of the generators of R n .

Let G(n, t) be the image of R n in the Gassner representation. G(n, t) is the group generated by the n_n matrices M i =(a jk ), i=1, ..., n where the article no. 0046 50