THE LIE BRACKET OF UNDIRECTED CLOSED CURVES ON A SURFACE

Moira Chas, Arpan Kabiraj

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

A Lie bracket defined on the linear span of the free homotopy classes of undirected closed curves on surfaces 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 holds 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. These three results hold for surfaces with or without boundary. We also give a brief and more illuminating proof of a known result, namely, the TWG bracket counts intersection between a simple closed curve and any other closed curve. 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 computer experiments were performed on curves on surfaces with boundary. The main tools are basic hyperbolic geometry and Thurston’s earthquake theory.

Original languageEnglish
Pages (from-to)2365-2386
Number of pages22
JournalTransactions of the American Mathematical Society
Volume375
Issue number4
DOIs
StatePublished - 1 Jan 2022
Externally publishedYes

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics

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