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

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