Resumen de Riemann surfaces with boundary and natural triangulations of the Teichmüller space
We compare some natural triangulations of the Teichm¨uller space of hyperbolic surfaces with geodesic boundary and of some bordifications. We adapt Scannell�Wolf�s proof to show that grafting semi-infinite cylinders at the ends of hyperbolic surfaces with fixed boundary lengths is a homeomorphism. This way, we construct a family of equivariant triangulations of the Teichmüller space of punctured surfaces that interpolates between Bowditch�Epstein�Penner�s (using the spine construction) and Harer�Mumford�Thurston�s (using Strebel differentials). Finally, we show (adapting arguments of Dumas) that on a fixed punctured surface, when the triangulation approaches HMT�s, the associated Strebel differential is well-approximated by the Schwarzian of the associated projective structure and by the Hopf differential of the collapsing map.
