Deligne-Mostow lattices with three fold symmetry and cone metrics on the sphere

Deligne and Mostow constructed a class of lattices inPU(2,1)PU(2,1)using monodromy of hypergeometric functions. Thurston reinterpreted them in terms of cone metrics on the sphere. In this spirit we construct a fundamental domain for the lattices with three fold symmetry in the list of Deligne and Mostow. This is a generalisation of the works of Boadi and Parker and gives a different interpretation of the fundamental domain constructed by Deraux, Falbel, and Paupert.