Augmented Teichmüller spaces and orbifolds

We study complex analytic properties of the augmented Teichmüller spaces T̄_g,n obtained by adding to the classical Teichmüller spaces T_g,n points corresponding to Riemann surfaces with nodal singularities. Unlike T_g,n, the space T̄_g,n is not a complex manifold (it is not even locally compact). We prove, however, that the quotient of the augmented Teichmüller space by any finite index subgroup of the Teichmüller modular group has a canonical structure of a complex orbifold. Using this structure, we construct natural maps from T̄ to stacks of admissible coverings of stable Riemann surfaces. This result is important for understanding the cup-product in stringy orbifold cohomology. We also establish some new technical results from the general theory of orbifolds which may be of independent interest.