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arXiv · 2507.16017

Moduli spaces of flat Riemannian metrics on orbifolds

Karla García·Ingrid Amaranta Membrillo Solis·Motiejus Valiunas
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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.

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Karla García, Ingrid Amaranta Membrillo Solis, Motiejus Valiunas. 2025-07-21. Moduli spaces of flat Riemannian metrics on orbifolds. https://arxiv.org/abs/2507.16017

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