Riemann surfaces with boundary and natural triangulations of the Teichmueller space

TL;DR

This paper explores natural triangulations of Teichmüller spaces for hyperbolic surfaces with boundary, establishing homeomorphisms and interpolations between different triangulation methods, and analyzing associated differentials.

Contribution

It constructs a family of triangulations interpolating between Penner-Bowditch-Epstein and Harer-Mumford-Thurston approaches, and relates Strebel differentials to Schwarzian and Hopf differentials.

Findings

Grafting semi-infinite cylinders is a homeomorphism.

Constructed interpolating triangulations between known models.

Strebel differentials are approximated by Schwarzian and Hopf differentials.

Abstract

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 triangulations of the Teichm\"uller 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.