The Lie bracket of undirected curves on a surface

TL;DR

This paper provides a geometric definition and analysis of the Thurston-Wolpert-Goldman Lie bracket on undirected curves on surfaces, revealing its algebraic structure, centers, and intersection properties through hyperbolic geometry and earthquake theory.

Contribution

It introduces a local geometric definition of the TWG bracket and proves key algebraic properties, including the structure of its center and the relation to curve intersections.

Findings

The center of the TWG bracket is generated by trivial and boundary-parallel loops.

The TWG bracket of two non-central curves is always a linear combination of non-central curves.

Zero TWG bracket implies disjoint representatives for the curves.

Abstract

A Lie bracket defined on the linear span of the free homotopy classes of undirected closed curves was discovered in stages passing through Thurston's earthquake deformations, Wolpert's corresponding calculations with Hamiltonian vector fields and Goldman's algebraic treatment of the latter leading to a Lie bracket on the span of directed closed curves. The purpose of this work is to deepen the understanding of the former Lie bracket which will be referred to as the Thurston-Wolpert-Goldman Lie bracket or, briefly, the TWG bracket. We give a local direct geometric definition of the TWG bracket and use this geometric point of view to prove three results: firstly, the center of the TWG-bracket is the Lie sub algebra generated by the class of the trivial loop and the classes of loops parallel to boundary components or punctures; secondly the analogous result hold for the centers of the universal enveloping algebra and of the symmetric algebra determined by the TWG Lie algebra; and thirdly, in terms of the natural basis, the TWG bracket of two non-central curves is always a linear combination of non-central curves. We also give a brief and more illuminating proof of a known result, namely, the TWG bracket counts intersection. We conclude by discussing substantial computer evidence suggesting an unexpected and strong conjectural statement relating the intersection structure of curves and the TWG bracket, namely, if the TWG bracket of two distinct undirected curves is zero then these curve classes have disjoint representatives. The main tools are basic hyperbolic geometry and Thurston's earthquake theory.