Teichmüller space and the mapping class group of the twice punctured torus

Abstract

We introduce coordinate systems to the Teichmüller space of the twice-punctured torus and give matrix representations for the points of Teichmüller space. The coordinate systems allow representation of the mapping class group of the twice punctured torus as a group of rational transformations and provide several applications to the mapping class group and also to Kleinian groups.