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# Mathematical Research Letters

## Volume 12 (2005)

### Number 1

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

Pages: 23 – 36

DOI: http://dx.doi.org/10.4310/MRL.2005.v12.n1.a3

#### Author

#### Abstract

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.