The Magnus representation of the Torelli group \(\mathcal{I}_{g,1}\) is not faithful for \(g \geq 2\) (Q2759044)

Let \(M_{g,1}\) be the mapping class group, i.e. the group of isotopy classes of orientation-preserving homeomorphisms \(\Sigma_{g,1} \to \Sigma_{g,1}\) where \(\Sigma_{g,1}\) is a closed, oriented surface of genus \(g\) with 1 boundary components. The Torelli group, \(T_{g,1}\), is the normal subgroup of \(M_{g,1}\) which acts trivially on the homology of \(\Sigma_{g,1}\). Fox's free differential calculus can be used to obtain a ``crossed homomorphism'' \(r:M_{g, 1} \to\text{GL} (2g;\mathbb{Z} [\Gamma_0])\) where \(\Gamma_0\) is the fundamental group of \(\Sigma_{g,1}\). Restricting this function to \(T_{g,1}\) and reducing the coefficients to \(\mathbb{Z}[H]\) where \(H=H_1 (\Sigma_{g,1}, \mathbb{Z})\) induced by abelianization yields the Magnus representation \(\overline r:T_{g,1} \to\text{GL}(2g;\mathbb{Z} [H])\). The representation is faithful if \(g=1\). In this paper, the author shows that if \(g\geq 2\) there is a nontrivial element in \(\ker (\overline r)\) which has the form \([\varphi_1, \varphi_2\varphi_1 \varphi_2^{-1}]\) where \(\varphi_1\) and \(\varphi_2\) are Dehn twists about two given simple, closed curves on \(\Sigma_{g,1}\).