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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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Cite

BibTeXRIS
BibTeX
@misc{yu2026commensurability,
  title = {{Commensurability Relations Between Deligne-Mostow-Thurston and Ghazouani-Pirio Monodromy Groups}},
  author = {Chenglong Yu and Yiming Zhong},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11482},
  eprint = {2609.11482},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11482},
}
RIS
TY  - UNPB
TI  - Commensurability Relations Between Deligne-Mostow-Thurston and Ghazouani-Pirio Monodromy Groups
AU  - Chenglong Yu
AU  - Yiming Zhong
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.11482
UR  - https://arxiv.org/abs/2609.11482
AB  - 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.
ER  -