| Résume||If M is compact and 4-dimensional, one can then show that the conformal classes of almost-Kähler metrics sweep out an open
subset in the space of the conformal classes. This provides a natural tool for exploring difficult global problems in 4-dimensional conformal geometry. However, this technique certainly has its limitations. For example, if a 4-manifold admits scalar-flat Kähler metrics, these can be deformed into anti-self-dual almost-Kähler metrics, and these then sweep out an open set in the moduli space of anti-self-dual conformal structures. One might somehow hope that this subset would also turn out to be closed, and so sweep out entire connected components in the moduli space. Alas, however, this simply isn’t true! In this talk, I’ll explain recent joint work with Chris Bishop that constructs a large hierarchy of counter-examples by studying the limit sets of quasi-Fuchsian groups.