Infinite generation of the kernels of the Magnus and Burau representations

Church, Thomas¹ ; Farb, Benson¹

¹Department of Mathematics, 5734 S. University Ave., Chicago, IL 60637

Algebraic and Geometric Topology, Tome 10 (2010) no. 2, pp. 837-851

Résumé

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.

Détail

DOI : 10.2140/agt.2010.10.837

Keywords: Magnus representation, Burau representation

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