Bounded ideal triangulations of infinite Riemann surfaces

Abstract

We introduce a notion of a bounded ideal triangulation of an infinite Riemann surface and parameterize Teichmüller spaces of infinite surfaces which allow bounded triangulations. We prove that our parameterization 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 parameterization for surfaces with bounded geometry is not known, while the Fenchel–Nielsen parameterization for surfaces with bounded pants decompositions is known as a homeomorphism but it is not known whether it is real-analytic.