  We produce a family of new, non arithmetic lattices in PU(2,1). All
previously known examples were commensurable with lattices constructed by
Picard, Mostow and Deligne-Mostow, and fell into 9 commensurability classes.
Our groups produce 5 new distinct commensurability classes. Most of the
techniques are completely general, and provide efficient geometric and
computational tools for constructing fundamental domains for discrete group
acting on the complex hyperbolic plane.
