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

### Laurent Charles

Université Pierre et Marie Curie VI, Paris, France### Julien Marché

École Polytechnique, Palaiseau, France

## Abstract

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

$\mathcal{M}(\Sigma)= \operatorname{Hom}(\pi_1(\Sigma),\operatorname{SU}(2))/\operatorname{SU}(2).$

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.

## Cite this article

Laurent Charles, Julien Marché, Multicurves and regular functions on the representation variety of a surface in SU(2). Comment. Math. Helv. 87 (2012), no. 2, pp. 409–431

DOI 10.4171/CMH/258