Complex Hyperbolic Triangle Groups

Consider a group generated by three complex hyperbolic reflections that braid with lengths (2,4,4). With even braidings, there can be a variety of different generators' orders. We show that the (2,4,4) groups can be identified with Pasquinelli's groups, and thus, are commensurable with Deligne-Mostow groups. After we have the group structures, we consider a subgroup of the form (r,4,4;4) for the sake of geometric construction as we want to apply Deraux-Parker-Paupert algorithm on this group to construct its fundamental domain.