Separating twists and the Magnus representation of the Torelli group
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.
Cite
@article{arxiv.0804.3633,
title = {Separating twists and the Magnus representation of the Torelli group},
author = {Thomas Church and Aaron Pixton},
journal= {arXiv preprint arXiv:0804.3633},
year = {2013}
}
Comments
15 pages. v2: Paper completely rewritten. Final version, to appear in Geometriae Dedicata