Commensurability Relations Between Deligne-Mostow-Thurston and Ghazouani-Pirio Monodromy Groups

Chenglong Yu; Yiming Zhong

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arXiv · 2609.11482

Commensurability Relations Between Deligne-Mostow-Thurston and Ghazouani-Pirio Monodromy Groups

Chenglong Yu·Yiming Zhong

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Abstract

We classify commensurability relations between Deligne--Mostow--Thurston monodromy groups and the sixteen arithmetic Ghazouani--Pirio monodromy groups arising from moduli of flat cone metrics on the sphere and on the torus. In both settings, the monodromy group preserves a skew-Hermitian form coming from twisted homology. We use a degeneration method to compute their determinant classes in the genus one case. The defining CM fields and determinant classes of the skew-Hermitian forms determine the commensurability relations. In particular, each of the sixteen arithmetic Ghazouani--Pirio monodromy groups is commensurable with an arithmetic Deligne--Mostow--Thurston monodromy group.

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Chenglong Yu, Yiming Zhong. 2026-09-10. Commensurability Relations Between Deligne-Mostow-Thurston and Ghazouani-Pirio Monodromy Groups. https://arxiv.org/abs/2609.11482

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