Equidistribution of families of expanding horospheres on moduli spaces of hyperbolic surfaces

Given a simple closed curve gamma on a connected, oriented, closed surface S of negative Euler characteristic, Mirzakhani showed that the set of points in the moduli space of hyperbolic structures on S having a simple closed geodesic of length L of the same topological type as gamma equidistributes with respect to a natural probability measure as L ->infinity. We prove several generalizations of Mirzakhani's result and discuss some of the technical aspects ommited in her original work. The dynamics of the earthquake flow play a fundamental role in the arguments in this paper.