Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bounded ideal triangulations of infinite Riemann surfaces

Casey Whitney, Dragomir Saric

TL;DR

The paper develops a bounded-geometry, triangulation-based framework to parametrize Teichmüller spaces of infinite-type Riemann surfaces via shear coordinates on bounded ideal triangulations, proving a real-analytic diffeomorphism onto the image in $\ell^{\infty}$ and extending to a holomorphic map in a neighborhood inside the quasi-Fuchsian space. It connects these coordinates to quasisymmetric boundary maps, provides explicit planar examples of bounded triangulations (including parabolic and non-parabolic flute surfaces), and situates the construction within the larger structure of $T(X)$ inside $QF(X)$ using holomorphic motions. It also gives topological criteria for when mapping classes admit quasiconformal representatives in terms of finite intersections of triangulations, thereby linking geometric coordinates with big mapping class group actions. Overall, the work broadens analytic, triangulation-based coordinates to a wide class of infinite-type surfaces and clarifies their analytic structure and mapping-class implications.

Abstract

We introduce a notion of a bounded ideal triangulation of an infinite Riemann surface and parametrize Teichmüller spaces of infinite surfaces which allow bounded triangulations. We prove that our parametrization is real-analytic. Riemann surfaces with bounded geometry and countably many punctures belong to the class of surfaces with bounded ideal triangulations. In comparison, the Fenchel-Nielsen parametrization for surfaces with bounded geometry is not known, while the Fenchel-Nielsen parametrization for surfaces with bounded pants decompositions is known as a homeomorphism but it is not known whether it is real-analytic

Bounded ideal triangulations of infinite Riemann surfaces

TL;DR

The paper develops a bounded-geometry, triangulation-based framework to parametrize Teichmüller spaces of infinite-type Riemann surfaces via shear coordinates on bounded ideal triangulations, proving a real-analytic diffeomorphism onto the image in and extending to a holomorphic map in a neighborhood inside the quasi-Fuchsian space. It connects these coordinates to quasisymmetric boundary maps, provides explicit planar examples of bounded triangulations (including parabolic and non-parabolic flute surfaces), and situates the construction within the larger structure of inside using holomorphic motions. It also gives topological criteria for when mapping classes admit quasiconformal representatives in terms of finite intersections of triangulations, thereby linking geometric coordinates with big mapping class group actions. Overall, the work broadens analytic, triangulation-based coordinates to a wide class of infinite-type surfaces and clarifies their analytic structure and mapping-class implications.

Abstract

We introduce a notion of a bounded ideal triangulation of an infinite Riemann surface and parametrize Teichmüller spaces of infinite surfaces which allow bounded triangulations. We prove that our parametrization is real-analytic. Riemann surfaces with bounded geometry and countably many punctures belong to the class of surfaces with bounded ideal triangulations. In comparison, the Fenchel-Nielsen parametrization for surfaces with bounded geometry is not known, while the Fenchel-Nielsen parametrization for surfaces with bounded pants decompositions is known as a homeomorphism but it is not known whether it is real-analytic

Paper Structure

This paper contains 14 sections, 18 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. A class of planar Riemann surfaces with bounded geometry
  3. The shear coordinates on the Farey triangulation and quasisymmetric maps
  4. The shear coordinates for infinite surfaces and some examples
  5. A Parabolic Flute Surface with Bounded Ideal Triangulation
  6. Non-parabolic Flute Surface with Bounded Ideal Triangulation
  7. The manifold structure on the Teichmüller and quasi-Fuchsian spaces
  8. The shear coordinates parametrizations of the Teichmüller space of Riemann surfaces with bounded ideal triangulations
  9. The smoothness of shear coordinates
  10. The complexification of shears
  11. The map from $U_{[f^{\mu_0}]}(\varepsilon_0)$ to the complex shear coordinates is holomorphic
  12. The geometry of $\mathbb{H}^3$
  13. The map from complex shear coordinates to $QF(X)$
  14. The mapping class groups and triangulations

Key Result

Proposition 1.1

The Riemann surface $X$ obtained by erasing points in the upper half-plane $\mathbb{H}^2$ along the orbit of the index two orientation-preserving subgroup of the $(2,4,8)$ hyperbolic triangle group is not parabolic and has a bounded ideal triangulation.

Theorems & Definitions (30)

  • Proposition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1
  • Definition 4.1
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • ...and 20 more