JournalscmhVol. 87, No. 2pp. 409–431

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

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

M(Σ)=Hom(π1(Σ),SU(2))/SU(2).\mathcal{M}(\Sigma)= \operatorname{Hom}(\pi_1(\Sigma),\operatorname{SU}(2))/\operatorname{SU}(2).

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