Separating twists and the Magnus representation of the Torelli group

TL;DR

This paper investigates the Magnus representation of the Torelli group, providing new criteria to distinguish when certain elements commute or generate free groups, enhancing understanding of the group's structure.

Contribution

It introduces a novel method using the trace of the Magnus representation to analyze the subgroup generated by separating multitwists.

Findings

The images of two positive separating multitwists either commute or generate a free group.

A characterization of when commuting or free group generation occurs.

Application of a new technique to study the Magnus representation's kernel.

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.