Riemann surfaces with boundary and natural triangulations of the Teichmüller space

Mondello, Gabriele. "Riemann surfaces with boundary and natural triangulations of the Teichmüller space." Journal of the European Mathematical Society 013.3 (2011): 635-684. <http://eudml.org/doc/277198>.

@article{Mondello2011,
abstract = {We compare some natural triangulations of the Teichmüller 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.},
author = {Mondello, Gabriele},
journal = {Journal of the European Mathematical Society},
keywords = {Riemann surfaces; hyperbolic surfaces; arc complex; Teichmüller space; triangulations; Strebel differentials; Riemann surfaces with boundary; Weil-Petersson metric; augmented Teichmüller space; Weil-Petersson form; triangulation of Teichmüller spaces},
language = {eng},
number = {3},
pages = {635-684},
publisher = {European Mathematical Society Publishing House},
title = {Riemann surfaces with boundary and natural triangulations of the Teichmüller space},
url = {http://eudml.org/doc/277198},
volume = {013},
year = {2011},
}

TY - JOUR
AU - Mondello, Gabriele
TI - Riemann surfaces with boundary and natural triangulations of the Teichmüller space
JO - Journal of the European Mathematical Society
PY - 2011
PB - European Mathematical Society Publishing House
VL - 013
IS - 3
SP - 635
EP - 684
AB - We compare some natural triangulations of the Teichmüller 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.
LA - eng
KW - Riemann surfaces; hyperbolic surfaces; arc complex; Teichmüller space; triangulations; Strebel differentials; Riemann surfaces with boundary; Weil-Petersson metric; augmented Teichmüller space; Weil-Petersson form; triangulation of Teichmüller spaces
UR - http://eudml.org/doc/277198
ER -