A presentation for the image of Burau(4)| Z2

Let Bn denote the n-strand braid group. We recall that this admits a representation fin : Bn ---+ GLnl (Z[ t , t l ] ) the (reduced) Burau representation [1]. Although open for a long time, it is now known that this representation is not faithful for n > 6, (See [8, 5]) and it is an old result that it is faithful for n = 3. (See [7]) Despite these counterexamples, there is no understanding of the nature o f the image groups in the nonfaithful cases nor any kind of intrinsic characterisation o f braids which lie in the kernel. The two cases n = 4, 5 remain open. Resolution o f the case n = 4 is an important open problem, firstly for the implications for the automorphism group of a free group of rank 2 and secondly as a test case for the faithfulness of the Jones representation, [4]. In the case n = 4 the only summand which could be faithful is the Burau summand. There is a map c~ : GLn_l (Z[ t , t t ] ) ~ GL,_I(Z2[t , t -~]) given by reducing coefficients modulo two and thus a simplified representation fin | Z2. Using the ideas contained in [8] or [5], it is not difficult to show that this representation continues to be faithful in the case n = 3, and it was observed in [5] that it is not faithful for n = 5. The main result of this paper is that we shall give a complete description o f the image group in the case n = 4; this appears to be the first explicit description of the image group for any infinite linear representation o f a braid group with n > 4. An especially intriguing aspect is the picture which emerges o f the complex which carries the group which contains something rather analogous to an "geometrically infinite" end in the language of hyperbolic geometry [9]. It is