Moduli spaces of flat Riemannian metrics on orbifolds

Abstract

We study moduli spaces of flat metrics on closed Riemannian orbifolds admitting such metrics. We show that for such orbifolds $\mathcal{O}$, the Teichm\"uller space of flat metrics $\mathcal{T}_{\text{flat}}(\mathcal{O})$ serves as a classifying space for proper actions of the mapping class group, $\operatorname{MCG}(\mathcal{O})$, in the appropriate category. We show that the moduli space of flat metrics, $\mathcal{M}_{\text{flat}}(\mathcal{O}) = \mathcal{T}_{\text{flat}}(\mathcal{O}) / \operatorname{MCG}(\mathcal{O})$, is itself a very good orbifold, and, under a technical assumption, describe this orbifold algebraically up to commensurability.