nLab augmented Teichmüller space

Idea

The augmented Teichmüller space (Bers 73) is Teichmüller space with also the points for the degenerate nodal curve admitted.

Accordingly, the orbifold quotient of the augmented Teichmüller space by the mapping class group is equivalently the Deligne-Mumford compactification of the moduli space of curves over the complex numbers (Hubbard-Koch 13, main theorem).

References

The concept was introduced in

• Lipman Bers, Spaces of degenerating Riemann surfaces. Discontinuous groups and Riemann surfaces, Ann. Math. Stud. 79, 43–55 (1973).

Discussion of orbifold structure is in

• Vladimir Hinich, Arkady Vaintrob, Augmented Teichmuller Spaces and Orbifolds (arXiv:0705.2859)

Discussion of the relation to the Deligne-Mumford compactification of the moduli stack of curves is in

• John Hubbard, Sarah Koch, An analytic construction of the Deligne-Mumford compactification of the moduli space of curves (arXiv:1301.0062)