Separating twists and the Magnus representation of the Torelli group

Details Separating twists and the Magnus representation of the Torelli group Separating twists and the Magnus representation of the Torelli group

2008-04-23

arxiv.org/abs/0804.3633v2

Mathematics

Geometric Topology

15 pages. v2: Paper completely rewritten. Final version, to appear in Geometriae Dedicata

Scientific paper

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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