Research project, long version

The setting of my research is the still rather unexplored area of discrete groups of isometries of complex hyperbolic space, in particular that of lattices in PU(2; 1). This space is one of the two occurrences (with real hyperbolic space) of a rank 1 symmetric space in which Margulis ' super-rigidity result does not apply. But, as opposed to the real hyperbolic case, very little is known about these groups, beginning with interesting examples. Recall that there exist in the real case re ection groups (see the constructions of Vinberg, Makarov in small dimensions), and that there are examples of non-arithmetic lattices in all dimensions, by the construction of Gromov{Piatetskii-Shapiro. In the case of PU(n; 1), all these questions remain wide open, even in the smallest dimensions. This is mostly due to the fact that there are no totally geodesic real hypersurfaces in complex hy-perbolic geometry, and in particular no natural notion of polyhedra or re ection groups as in the real hyperbolic (or Euclidean) case. This situation makes it dicult to construct not only discrete subgroups of PU(n; 1), but also fundamental polyhedra for such groups. Possible substitutes in H n C are complex re ections (or C-re ections) which are holomorphic isometries xing pointwise a totally geodesic complex hypersurface (a copy of H n