Bounded ideal triangulations of infinite Riemann surfaces

TL;DR

This paper introduces bounded ideal triangulations for infinite Riemann surfaces, providing a real-analytic parametrization of their Teichmüller spaces, and compares it with existing Fenchel-Nielsen parametrizations.

Contribution

It defines bounded ideal triangulations for infinite surfaces and establishes a real-analytic parametrization of their Teichmüller spaces, expanding understanding of infinite Riemann surfaces.

Findings

Bounded ideal triangulations exist for certain infinite Riemann surfaces.

The parametrization of Teichmüller spaces via bounded ideal triangulations is real-analytic.

Surfaces with bounded geometry and countably many punctures admit bounded ideal triangulations.

Abstract

We introduce a notion of a bounded ideal triangulation of an infinite Riemann surface and parametrize Teichm\"uller 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