Separability criteria for loops via the Goldman bracket

TL;DR

This paper develops algebraic criteria using the Goldman bracket to determine when two loops on a surface can be represented disjointly, extending existing methods with hyperbolic geometry and analyzing the Goldman Lie algebra's center.

Contribution

It introduces explicit algebraic criteria for loop disjointness via the Goldman bracket and extends previous methods to new surface cases like pairs of pants.

Findings

Criteria for disjoint loops on surfaces using Goldman bracket

Extension of Kabiraj's method with hyperbolic geometry

Identification of the Goldman Lie algebra center for pairs of pants

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.