Riemann surfaces with boundary and natural triangulations of the Teichmueller space

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 triangulations of the Teichmüller 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.

arXiv comment45 pages, 2 figures - typos corrected, notation revised

References in corpus (3)

• A criterion of convergence in the augmented Teichmueller space
• Stable curves and screens on fatgraphs
• Triangulated Riemann surfaces with boundary and the Weil-Petersson Poisson structure

Cited by in corpus (1)

• Open-closed Hyperbolic String Vertices