Infinite generation of the kernels of the Magnus and Burau representations

Church, Thomas; Farb, Benson

doi:10.2140/agt.2010.10.837

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Infinite generation of the kernels of the Magnus and Burau representations

Thomas Church and Benson Farb

Algebraic & Geometric Topology 10 (2010) 837–851

DOI: 10.2140/agt.2010.10.837

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 $g \geq 2$ and for $n \geq 6$ 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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Keywords

Magnus representation, Burau representation

Mathematical Subject Classification 2000

Primary: 20F34, 20F36, 57M07

References

Publication

Received: 28 October 2009

Accepted: 15 January 2010

Published: 7 April 2010

Authors

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

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

Volume 10, issue 2 (2010)