Separating twists and the Magnus representation of the Torelli group

Thomas Church; Aaron Pixton

doi:10.1007/s10711-011-9584-1

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arXiv · 0804.3633

Separating twists and the Magnus representation of the Torelli group

Thomas Church·Aaron Pixton

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Abstract

The Magnus representation of the Torelli subgroup of the mapping class group of a surface is a homomorphism r: I_{g,1} -> GL_{2g}(Z[H]). Here H is the first homology group of the surface. This representation is not faithful; in particular, Suzuki previously described precisely when the commutator of two Dehn twists about separating curves is in the kernel of r. Using the trace of the Magnus representation, we apply a new method of showing that two endomorphisms generate a free group to prove that the images of two positive separating multitwists under the Magnus representation either commute or generate a free group, and we characterize when each case occurs.

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BibTeX RIS

Thomas Church, Aaron Pixton. 2008-04-23. Separating twists and the Magnus representation of the Torelli group. https://doi.org/10.1007/s10711-011-9584-1

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