Infinite generation of the kernels of the Magnus and Burau representations

TL;DR

This paper proves that the kernels of the Magnus and Burau representations have infinite rank first homology, implying they are not finitely generated, using Johnson-type homomorphisms and computational methods.

Contribution

It establishes the infinite generation of the kernels of the Magnus and Burau representations for certain parameters, a new result in the study of these groups.

Findings

Mag_g and Bur_n have infinite rank first homology for g >= 2 and n >= 6.

Neither Mag_g nor Bur_n has a finite generating set.

The proof involves Johnson-type homomorphisms and computer-assisted calculations.

Abstract

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.