The SL(2;C)-character variety X of a group pi is an affine algebraic variety defined as the quotient of the space of representations Hom(pi,SL(2;C)) by the conjugacy action of SL(2;C) at the target.

When pi is the fundamental group of a closed oriented surface S of genus > 1, the variety X admits a symplectic structure. Moreover its algebra of functions C[X] admits a natural linear basis indexed by multicurves in S, those are the homotopy classes of 1-dimensional submanifolds in S with no trivial components. This privileged basis from the topological viewpoint is stable neither under multiplication nor under Poisson brackets.

In collaboration with Julien Marché, we introduce a special class of valuations on X which behave monomially with respect to this linear basis and study their properties, individually and as a whole. This enables us to define the Newton polytopes of functions in C[X], and study the structure coefficients for the multiplication as well as the Poisson bracket between elements in the multicurve basis.

In this talk, we will emphasise the relations between the topology of S, the algebra of C[X], and the geometry of X.