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Separating twists and the Magnus representation of the Torelli group

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Abstract

The Magnus representation of the Torelli subgroup of the mapping class group of a surface is a homomorphism \({r: \fancyscript{I}_{g,1} \to {\rm GL}_{2g}(\mathbb{Z}H)}\). 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 ker 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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Authors and Affiliations

  1. Department of Mathematics, University of Chicago, Chicago, IL, 60637, USA

    Thomas Church

  2. Mathematics Department, Princeton University, Princeton, NJ, 08544, USA

    Aaron Pixton

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  1. Thomas Church
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  2. Aaron Pixton
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Correspondence to Thomas Church.

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Church, T., Pixton, A. Separating twists and the Magnus representation of the Torelli group. Geom Dedicata 155, 177–190 (2011). https://doi.org/10.1007/s10711-011-9584-1

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  • DOI: https://doi.org/10.1007/s10711-011-9584-1

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