Decorated enhanced Teichm\"uller spaces

TL;DR

This paper introduces a new deformation space for hyperbolic structures on surfaces, combining enhancements and decorations, and generalizes existing coordinate systems like shear and lambda-length coordinates.

Contribution

It defines a novel variation of Teichmüller space that unifies shear and lambda-length coordinates and introduces a compatible lamination space.

Findings

Parameterization of the new deformation space

Unification of shear and lambda-length coordinates

Introduction of a compatible lamination space

Abstract

In this paper, we introduce a new variation of the Teichm\"{u}ller space, namely the deformation space of hyperbolic structures on a surface with both enhancement and decoration. We construct the parameterization of this deformation space, which is a common generalization of the shear coordinates and the $\lambda$-length coordinates. Furthermore, we introduce the lamination space corresponding to this deformation space, and show the compatibility of the shear coordinates and the $\lambda$-length coordinates.