| Résume | Let S be a topological surface with holes. Let M (S, L) be the moduli space parametrising hyperbolic structures on S with geodesic boundary, and a given set L of lengths of the boundary circles. It carries the Weil-Peterson volume form. The volumes of the spaces M (S, L) are finite. Mirzakhani proved remarkable recursion formulas for them, related to several areas of Mathematics. However if S is a surface P with polygonal boundary, e.g. just a polygon, similar volumes are infinite. We consider a variant of these moduli spaces, and show that they carry a canonical exponential volume form. We prove that the exponential volumes are finite, and satisfies unf |
