Poisson structures on the Teichmueller space of hyperbolic surfaces with conical points

Abstract

In this paper two Poisson structures on the moduli space of hyperbolic surfaces with conical points are compared: the Weil-Petersson one and the ηcoming from the representation variety. We show that they are multiple of each other, if the angles do not exceed 2π. Moreover, we exhibit an explicit formula for ηin terms of hyperbolic lengths of a suitable system of arcs.

arXiv comment23 pages, 2 figures. Two mistakes in the description of the holonomy map are corrected. Exposition improved and more details added