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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 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

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