Volume 10, issue 2 (2010)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Infinite generation of the kernels of the Magnus and Burau representations

Thomas Church and Benson Farb

Algebraic & Geometric Topology 10 (2010) 837–851
Abstract

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.

Additional material
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