  Consider the kernel Mag_g of the Magnus representation of the Torelli group
and the kernel Bur_n of the Burau representation of the braid group. We prove
that for g >= 2 and for n >= 6 the groups Mag_g and Bur_n have infinite rank
first homology. As a consequence we conclude that neither group has any finite
generating set. The method of proof in each case consists of producing a kind
of "Johnson-type" homomorphism to an infinite rank abelian group, and proving
the image has infinite rank. For the case of Bur_n, we do this with the
assistance of a computer calculation.
