By the work of Brunella and McQuillan, it is known that smooth foliated surfaces of general type with only canonical singularities admit a unique canonical model. It is then natural to wonder if these canonical models have a good moduli theory and, in particular, if they admit a moduli functor.In this talk, I will show that the canonical models and their minimal partial du Val resolutions are bounded.

Lattices in SU(2,1) can be viewed in several different ways: via their geometry as holomorphic complex hyperbolic isometries, as monodromy groups of hypergeometric functions, via algebraic geometry as ball quotients and (sometimes) using arithmeticity. In this talk I will describe these different points of view using examples constructed by Deligne and Mostow and by Deraux, Paupert and myself.