Geodesic
Riemann surfaces with boundary and natural triangulations of the Teichmüller space
Journal of the European Mathematical Society, Tome 13 (2011) no. 3, pp. 635-684 Cet article a éte moissonné depuis la source EMS Press

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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.
DOI : 10.4171/jems/263
Classification : 30-XX, 00-XX
Keywords: Riemann surfaces, hyperbolic surfaces, arc complex, Teichmüller space, triangulations, Strebel differentials 
@article{JEMS_2011_13_3_a3,
     author = {Gabriele Mondello},
     title = {Riemann surfaces with boundary and natural triangulations of the {Teichm\"uller} space},
     journal = {Journal of the European Mathematical Society},
     pages = {635--684},
     year = {2011},
     volume = {13},
     number = {3},
     doi = {10.4171/jems/263},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/263/}
}
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Gabriele Mondello. Riemann surfaces with boundary and natural triangulations of the Teichmüller space. Journal of the European Mathematical Society, Tome 13 (2011) no. 3, pp. 635-684. doi: 10.4171/jems/263

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