A trace on a group is a positive-definite conjugation-invariant function on it. In the past couple of decades, the study of traces has led to exciting connections to the rigidity, stability, and dynamics of groups. In this talk, I will explain these connections and focus on the topological structure of the space of traces of some groups. We will see the different behaviours of these spaces for free groups vs. higher-rank lattices. This is based on joint works with Arie Levit, Joav Orovitz and Itamar Vigdorovich.
