Infinite generation of the kernels of the Magnus and Burau representations

Details Infinite generation of the kernels of the Magnus and Burau representations Infinite generation of the kernels of the Magnus and Burau representations

2009-09-26

arxiv.org/abs/0909.4825v1

Mathematics

Group Theory

13 pages, 7 figures

Scientific paper

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.

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