Symplectic Geometry of Moduli Spaces of Hurwitz Covers

TL;DR

This paper extends Mirzakhani's results to moduli spaces of Hurwitz covers, establishing relations between Weil-Petersson volumes, Hurwitz numbers, and Hurwitz cycles, and analyzing their orbifold structure and compactness properties.

Contribution

It introduces new equations relating volumes and Hurwitz invariants, describes the orbifold structure of Hurwitz moduli spaces, and proves their compactness using SFT-compactness techniques.

Findings

Derived equations linking Weil-Petersson volumes and Hurwitz numbers.

Established the orbifold structure of Hurwitz moduli spaces.

Proved compactness of the moduli spaces using SFT-compactness.

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].