The kernel of \(\mathrm{Burau}(4)\otimes\mathbb Z_p\) is all pseudo-Anosov. (Q816310)

The authors show that Burau representations of 4-braids with mod \(p\) integer coefficients are faithful for periodic and reducible braids so that their kernel is a normal subgroup of the 4-braid group that is nontrivial for \(p=2, 3\) by earlier results by Cooper and Long and consists of only non-Brunnian pseudo-Anosov braids. Since the kernel of the ordinary Burau representation of 4-braids is smaller, it has the same property. Up to power and conjugation, reducible 4-braids have one of two possible reduction systems and the authors give a set of generators for each case and analyze the Burau matrices of words on these generators and conclude that the Burau representation with mod \(p\) integer coefficients is faithful for reducible 4-braids.