Commensurability Among Deligne-Mostow Monodromy Groups

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{ö}ller and McMullen on non-arithmetic Deligne–Mostow lattices.