Decorated Teichmüller spaces, singular space-times and polyhedral surfaces

The Teichmüller space parametrizes the complete hyperbolic metrics on a given surface up to equivalence. In 1987, Penner introduced a so-called decorated Teichmüller space (a fiber bundle over the usual Teichmüller space) in order to obtain a stratification of the Teichmüller space. His construction can be re-explained using flat space-times and BTZ-extensions of flat space-times. This leads to correspondences between complete hyperbolic surfaces, Cauchy-compact space-times with BTZ and compact singular Euclidean surfaces. As by products, we obtain a parametrization of Cauchy-compact flat space-times with BTZ and a the existence of polyhedral Cauchy-surfaces in every such space-time.