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arXiv · 2411.09682

Commensurability Among Deligne-Mostow Monodromy Groups

Chenglong Yu·Zhiwei Zheng
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Abstract

This paper gives the commensurability classification of Deligne--Mostow ball quotients and shows that the 104 Deligne--Mostow lattices form 38 commensurability classes. First, we find commensurability relations among Deligne--Mostow monodromy groups, which are not necessarily discrete. This generalizes previous work by Sauter and Deligne--Mostow in dimension two. In this part, we consider certain projective surfaces with two fibrations over the projective line, which induce two sets of Deligne--Mostow data. Correspondences between moduli spaces provide a geometric realization of commensurability relations. Secondly, we obtain commensurability invariants from conformal classes of Hermitian forms and toroidal boundary divisors. This completes the commensurability classification of Deligne--Mostow lattices and provides an alternative approach to the results of Kappes--M{\"o}ller and McMullen on non-arithmetic Deligne--Mostow lattices.

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Chenglong Yu, Zhiwei Zheng. 2024-11-14. Commensurability Among Deligne-Mostow Monodromy Groups. https://arxiv.org/abs/2411.09682

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