| Résume||In recent years, there have been great advances in understanding when the reduced C*-algebra of a discrete group is simple and when it has the unique trace property. A central role in these discoveries has been played by Furstenberg's notion of boundary action (specially the so-called Furstenberg boundary).
For quasi-regular representations in general, the situation is much different, with Haagerup and Olesen showing that there is such a representations \pi of Thompson's group V such that C*_\pi(V) is the Cuntz algebra O_2 (hence simple and without traces).
In this talk, we will review the notions above, and present applications of boundary actions to the study of invariant random subgroups, traces and C*-simplicity of quasi-regular and Koopman representations. We will be specially interested in representations of Thompson's groups. This is joint work with Mehrdad Kalantar.|