Equidistribution of 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 $L\to \mathrm{\infty }$. 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.