Coordinates for Teichmüller spaces of b-groups with torsion

Abstract

In this paper we will use b-groups to construct coordinates for the Teichmüller spaces of 2-orbifolds. The main technical tool is the parametrization of triangle groups, which allows us to compute explicitly formulæ for generators of b-groups uniformizing orbifolds. In this way, we obtain a technique to pass from the abstract objects of deformation spaces to concrete calculations with Möbius transformations. We explore this computational character of our work by finding the expressions of certain classical isomorphisms between Teichmüller spaces.