Coordinates for the moduli space of flat $PSL(2,\mathbb{R})$-connections

Let $\mathcal{M}$ be the moduli space of irreducible flat $PSL(2,\mathbb{R})$ connections on a punctured surface of finite type with parabolic holonomies around punctures. By using a notion of \emph{admissibility} of an ideal arc, $\mathcal{M}$ is covered by dense open subsets associated to ideal triangulations of the surface. A principal bundle over $\mathcal{M}$ is constructed which, when restricted to the Teichmüller component of $\mathcal{M}$, is isomorphic to the decorated Teichmüller space of Penner. The construction gives a generalization to $\mathcal{M}$ of Penner’s coordinates for the Teichmüller space.

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Published 1 January 2005