Triangulations and volume form on moduli spaces of flat surfaces

TL;DR

This paper introduces a volume form on the moduli space of flat surfaces with conical singularities, extending known structures and demonstrating consistency with existing volume forms in classical cases.

Contribution

It defines a new volume form on moduli spaces of flat surfaces with conical singularities using geodesic triangulations, generalizing previous volume forms.

Findings

The volume form agrees with classical ones up to a constant.

The approach uses geodesic triangulations to define the volume form.

The moduli space considered is a deformation of translation surface moduli space.

Abstract

In this paper, we are interested in flat metric structures with conical singularities on surfaces which are obtained by deforming translation surface structures. The moduli space of such flat metric structures can be viewed as some deformation of the moduli space of translation surfaces. Using geodesic triangulations, we define a volume form on this moduli space, and show that, in the well-known cases, this volume form agrees with usual ones, up to a multiplicative constant.