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

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.