Symplectic Geometry of Moduli Spaces of Hurwitz Covers

Abstract

We extend results by Mirzakhani in [Mir07] to moduli spaces of Hurwitz covers. In particular we obtain equations relating Weil-Petersson volumes of moduli spaces of Hurwitz covers, Hurwitz numbers and certain Hurwitz cycles on Deligne-Mumford space related to those Riemann surfaces admitting Hurwitz covers of a specified branching profile. We state the precise orbifold structure of the moduli space of Hurwitz covers by applying ideas and results from Robbin-Salamon in arXiv:math/0407090. Furthermore we prove compactness of the involved moduli spaces by applying SFT-compactness in the Cieliebak-Mohnke version from [CM05].