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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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The Lie bracket of undirected closed curves on a surface
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by Moira Chas and Arpan Kabiraj PDF
Trans. Amer. Math. Soc. 375 (2022), 2365-2386 Request permission

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.

References
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Additional Information
  • Moira Chas
  • Affiliation: Stony Brook Mathematics Department and Institute for Mathematical Sciences, Stony Brook, New York
  • MR Author ID: 601049
  • Email: moira.chas@stonybrook.edu
  • Arpan Kabiraj
  • Affiliation: Department of Mathematics, Indian Institute of Technology Palakkad, India
  • MR Author ID: 1186361
  • Email: arpaninto@iitpkd.ac.in
  • Received by editor(s): December 16, 2020
  • Received by editor(s) in revised form: June 14, 2021
  • Published electronically: November 5, 2021
  • Additional Notes: The first author was partially supported by the NSF.
    The second author was partially supported by the DST, India: INSPIRE Faculty fellowship.
  • © Copyright 2021 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 2365-2386
  • MSC (2020): Primary 57K20; Secondary 57M50
  • DOI: https://doi.org/10.1090/tran/8541
  • MathSciNet review: 4391721