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

Marché, Julien¹ ; Simon, Christopher-Lloyd²

¹ Sorbonne Université, Paris, France
² Laboratoire Paul Painlevé UMR CNRS, Université de Lille, Lille, France

Geometry & topology, Tome 28 (2024) no. 2, pp. 593-625.

Résumé

We study the space of measured laminations $\mathrm{ML}$ 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 $\mathrm{ML}$. Finally, we identify this symplectic space with previous constructions due to Thurston and Bonahon.

Détail

DOI : 10.2140/gt.2024.28.593

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

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