Valuations on the character variety: Newton polytopes and Residual Poisson Bracket

TL;DR

This paper explores the structure of measured laminations on surfaces through algebraic and geometric tools, introducing Newton polytopes for functions on character varieties and linking Poisson brackets to symplectic structures.

Contribution

It introduces Newton polytopes for algebraic functions on character varieties and connects the Goldman Poisson bracket to symplectic structures on measured laminations.

Findings

Trace functions have unit coefficients at extremal points of their Newton polytope.

The Goldman Poisson bracket induces a symplectic structure on the valuative model for ML.

The symplectic space aligns with Thurston and Bonahon's constructions.

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.