Large cone angles on a punctured sphere

Abstract: Do and Norbury found a so-called differential relation which relates the volume of the moduli space of singular surface with a cone point to that of a smooth surface obtained by forgetting the cone point. Their procedure is valid for cone angles less than $\pi$ by work of Tan, Wong and Zhang. We study the moduli space of a surface with a single cone point of angle ranging from $0$ to $2\pi$ using a coordinate system closely related to Penner's $\lambda$-lengths. We compute the action of the mapping class group in these coordinates, give an explicit expression for Wolpert's symplectic form and use this to justify Do and Norbury's approach using just hyperbolic geometry.