Moduli spaces of flat Riemannian metrics on orbifolds

TL;DR

This paper investigates the structure of moduli spaces of flat Riemannian metrics on orbifolds, establishing their properties as classifying spaces and describing their orbifold structure under certain conditions.

Contribution

It demonstrates that the Teichmüller space of flat metrics acts as a classifying space for the mapping class group and characterizes the moduli space as a good orbifold, with algebraic descriptions under specific assumptions.

Findings

Teichmüller space is a classifying space for proper actions of the mapping class group.

The moduli space of flat metrics is a very good orbifold.

Under a technical assumption, the orbifold can be described algebraically up to commensurability.

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.