The topological IHX relation, pure braids, and the Torelli group (Q1847912)

Let \(T_{g,1}\) denote the Torelli group of a surface of genus \(g\) with \(1\) boundary component, and let \(P(g)\) denote the pure braid group on \(g\) strands. Recall that the lower central series of a group \(G\) is given by \(G_1=G\), and inductively by \(G_{n+1}=[G,G_n]\). The authors prove that if \(g\geq 3\), then for all \(n\geq 1\), \(P(g)_{n+1}\subset(T_{g,1})_n\). Then they explicitly give the relation obtained from the Jacobi relation via the map \(P(g)_3\to(T_{g,1})_2\).