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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
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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