Separability criteria for loops via the Goldman bracket

In this talk, we give algebraic criteria using the Goldman bracket to determine whether two free homotopy classes of loops on an oriented surface have disjoint representatives. As an application, we determine the center of the Goldman Lie algebra of a pair of pants. We extend Kabiraj's method, which was originally limited to oriented surfaces filled by simple closed geodesics, and show that in this case, the center is generated by the class of loops homotopic to a point, and the classes of loops winding multiple times around a single puncture or boundary component.