Riemann surfaces with boundary and natural triangulations of the Teichmüller space

Abstract

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.

Riemann surfaces, ribbon graphs and combinatorial classes

“…Thus, the construction above gives a point w sp ∈ |A(S, X)| × (0, ∞). It is easy to check (see [Mon06b] or [Mon06a]) that w sp converges to the w sp defined above when the hyperbolic surface with boundary converges to a decorated surface with cusps in T (S, X). Thus, the Γ(S, X)-equivariant map…”

“…Proposition 4.11([Mon06a]). The map Ψ extends to a Γ(S, X)-equivariant homeomorphismΨ : T (S, X) × ∆ X × [0, ∞] −→ |A • (S, X)| × [0, ∞]and Ψ ∞ coincides with Harer-Mumford-Thurston's Ψ JS .…”

“…This point of view is particularly useful to understand singular surfaces (see also [BE88], [Kon92], [Loo95], [Pen03], [Zvo02], [ACGH] and [Mon06a]). The object dual to a weighted system of arcs in A ∞ (S, X) is a collection of data that we called an (S, X)-marked "enriched" ribbon graph.…”

On the asymptotic behavior of complex earthquakes and Teichmüller disks

“…For the second statement of the theorem, consider the conformal limit Y ∞ of the Teichmüller disk D γ (see Lemma 3.13). By Theorem 5.4 of [13], there is a conformally equivalent "infinitely-grafted" surface X ∞ . We then have a conformal map g : X ∞ → Y ∞ as in (4.1) and can construct the quasiconformal maps exactly as above.…”

A criterion of convergence in the augmented Teichmüller space

“…We shall use this criterion in [9] to prove (roughly speaking) that the modulus of a Riemann surface, which is constructed by gluing flat and hyperbolic tiles along a graph, varies continuously with respect to the length parameters of the graph and the thickness of the hyperbolic tiles.…”