Separability criteria for loops via the Goldman bracket

Abstract

We provide some explicit algebraic criteria in terms of the Goldman bracket to decide whether two free homotopy classes of loops on an oriented surface admit disjoint representatives. We extend Kabiraj's method using the hyperbolic geometry of surfaces to prove these criteria. As an application, we show that the center of the Goldman Lie algebra of a pair of pants is generated by the class of the constant loop together with the classes of loops that wind multiple times around a single puncture or boundary component. This case was not covered by Kabiraj, since a pair of pants is not filled by simple closed curves.