Valuations on the character variety: Newton polytopes and residual Poisson bracket

Marché, Julien; Simon, Christopher-Lloyd

doi:10.2140/gt.2024.28.593

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Valuations on the character variety: Newton polytopes and residual Poisson bracket

Julien Marché and Christopher-Lloyd Simon

Geometry & Topology 28 (2024) 593–625

DOI: 10.2140/gt.2024.28.593

Abstract

We study the space of measured laminations $ML$ on a closed surface from the valuative point of view. We introduce and study a notion of Newton polytope for an algebraic function on the character variety. We prove, for instance, that trace functions have unit coefficients at the extremal points of their Newton polytope. Then we provide a definition of tangent space at a valuation and show how the Goldman Poisson bracket on the character variety induces a symplectic structure on this valuative model for $ML$ . Finally, we identify this symplectic space with previous constructions due to Thurston and Bonahon.

Keywords

surface group, measured lamination, character variety, valuation, Newton polytope, Poisson bracket, Goldman Poisson bracket, real tree, symplectic structure, skein algebra

Mathematical Subject Classification

Primary: 20E08, 53D30, 57K20, 57M60

References

Publication

Received: 28 May 2021

Revised: 22 May 2022

Accepted: 1 August 2022

Published: 13 March 2024

Proposed: Frances Kirwan

Seconded: Anna Wienhard, Benson Farb

Authors

Julien Marché
Sorbonne Université Paris France

Christopher-Lloyd Simon
Laboratoire Paul Painlevé UMR CNRS, Université de Lille Lille France

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Volume 28, issue 2 (2024)