Multicurves and regular functions on the representation variety of a surface in SU(2)

Abstract

Given a compact surface $\Sigma$, we consider the representation space

We show that the trace functions associated to multicurves on $\Sigma$ are linearly independent as functions on $\mathcal{M}(\Sigma )$. The proof relies on the Fourier decomposition of the trace functions with respect to a torus action on $\mathcal{M}(\Sigma )$ associated to a pants decomposition of $\Sigma$. Consequently the space of trace functions is isomorphic to the Kauffman skein algebra at $A=-1$ of the thickened surface.