We compare some natural triangulations of the Teichmuller 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 Teichmuller 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.
Riemann surfaces with boundary and natural triangulations of the Teichmüller space / Mondello, Gabriele. - In: JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY. - ISSN 1435-9855. - STAMPA. - 13:3(2011), pp. 635-684. [10.4171/jems/263]
Riemann surfaces with boundary and natural triangulations of the Teichmüller space
MONDELLO, GABRIELE
2011
Abstract
We compare some natural triangulations of the Teichmuller 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 Teichmuller 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.