Complex hyperbolic lattices and moduli spaces of flat surfaces

This work studies the Deligne-Mostow lattices in PU(2,1). These were introduced by Deligne and Mostow in several works, using monodromy of hypergeometric functions. The same lattices were rediscovered by Thurston using a geometric construction, which consists of studying possible configurations of cone points on a sphere of area 1 when the cone angles are prescribed. This space has a complex hyperbolic structure and certain automorphisms of the sphere which swap pairs of cone points, generate a lattice for some choice of initial cone angles (more precisely, the Deligne-Mostow lattices). Among these, we will consider the ones in PU(2,1). We use Thurston's approach to study the metric completion of this space, which is obtained by making pairs of cone points coalesce. Following the works of Parker and Boadi-Parker, we build a polyhedron. Using the Poincaré polyhedron theorem, we prove that the polyhedron we find is indeed a fundamental domain. Moreover, we give presentations for all Deligne-Mostow lattices in PU(2,1), calculate their volumes and show that they are coherent with the known commensurability theorems.