Slim curves, limit sets and spherical CR uniformisations of 3-manifolds

.Joint work with A. Guilloux and E. Falbel.
A classical way of constructing  group actions on the complex hyperbolic plane is to start with a subgroup of PO(2,1) and deform it in PU(2,1). Geometrically, this correspond to starting with an action of a Fuchsian group on a real totally geodesic subspace of the complex hyperbolic plane, and deform it to obtain a complex hyperbolic quasi-Fuchsian group. We describe here an invariant, called slimness, which quantifies how far a group is from preserving such a real subspace.