There are several natural ways to endow the compactification of the space of marked Riemann surfaces $M_{g,n}$ ($2g+n\geq 3$), with a topology, which is defined using "differential geometric or analytic" notions, e.g. the topology inherited from the "augmented Teichmüller space". As far as I understand, most of these natural topologies coincide, since the marked Riemann surfaces (or the associated hyperbolic surfaces) are (sufficiently) rigid.

On the other hand there is the classical Deligne-Mumford compactification of $M_{g,n}$, the space of genus g nodal curves with n marked points, which inherits a natural topology as a projective variety.

What is a good source which shows that the compactification of $M_{g,n}$, endowed with any one of those "analytic" topologies, is homeomorphic to the Deligne-Mumford compactification?

Sometimes this result is attributed to:

Harvey, William; Chabauty spaces of discrete groups. Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973), pp. 239–246. Ann. of Math. Studies, No. 79, Princeton Univ. Press, Princeton, N.J., 1974.

But in my understanding this result is not really claimed or proved there, or if it is, then only in a quite implicit way. Can anybody point me to some more explicit reference?



See John Hubbard and Sarah Koch: "An analytic construction of the Deligne-Mumford compactification of the moduli space of curves".

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    $\begingroup$ Hubbard and Koch describe an analytic isomorphism, and mention in the introduction that the homeomorphism is proved in the Harvey paper cited above... $\endgroup$ – S. Carnahan Feb 7 '14 at 13:28

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