# Riemann surfaces with boundary and natural triangulations of the Teichmüller space

Journal of the European Mathematical Society (2011)

- Volume: 013, Issue: 3, page 635-684
- ISSN: 1435-9855

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topMondello, 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 -

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