Infinite generation of the kernels of the Magnus and Burau representations

Original · EN

Consider the kernel Magg of the Magnus representation of the Torelli group and the kernel Burₙ of the Burau representation of the braid group. We prove that for g >= 2 and for n >= 6 the groups Magg and Burₙ 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 Burₙ, we do this with the assistance of a computer calculation.