Riemann surfaces with boundary and natural triangulations of the Teichmueller space

Understand

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 triangulations of the Teichm\"uller space of punctures surfaces that interpolates between Penner-Bowditch-Epstein's (using the spine construction) and Harer-Mumford-Thurston's (using Strebel's 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.