Augmented Teichmuller Spaces and Orbifolds

TL;DR

This paper explores the complex-analytic structure of augmented Teichmuller spaces, showing they form complex orbifolds when quotiented by certain groups, and connects these structures to orbifold cohomology and admissible coverings.

Contribution

It demonstrates that quotients of augmented Teichmuller spaces are complex orbifolds and constructs maps to stacks of admissible coverings, advancing understanding of their geometric and cohomological properties.

Findings

Quotients of ATS by finite index subgroups are complex orbifolds.

Constructed natural maps from ATS to stacks of admissible coverings.

Established new technical results in orbifold theory.

Abstract

We study complex-analytic properties of the augmented Teichmuller spaces ATS introduced by Lipman Bers. These spaces are obtained by adding to the classical Teichmuller space TS the points corresponding to nodal Riemann surfaces. Unlike TS, the space ATS is not a complex manifold (it is not even locally compact). We prove however that the quotient of ATS by any finite index subgroup of the Teichmuller modular group has a canonical structure of a complex orbifold. Using this structure we construct natural maps from ATS 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.