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We compare some natural triangulations of the Teichm\"uller space ofhyperbolic surfaces with geodesic boundary and of some bordifications. We adaptScannell-Wolf's proof to show that grafting semi-infinite cylinders at the endsof hyperbolic surfaces with fixed boundary lengths is a homeomorphism. Thisway, we construct a family of triangulations of the Teichm\"uller space ofpunctures surfaces that interpolates between Penner-Bowditch-Epstein's (usingthe spine construction) and Harer-Mumford-Thurston's (using Strebel'sdifferentials). Finally, we show (adapting arguments of Dumas) that on a fixedpunctured surface, when the triangulation approaches HMT's, the associatedStrebel differential is well-approximated by the Schwarzian of the associatedprojective structure and by the Hopf differential of the collapsing map.

Riemann surfaces with boundary and natural triangulations of the Teichmueller space