A trace on a group is a positive-definite conjugation-invariant function on it. These traces correspond to tracial states on the group's maximal C*-algebra. 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 and see the different behaviors of these spaces for free groups vs. higher-rank lattices. Finally, we will see some applications to character limits of finite groups.

This is based on joint works with Arie Levit, Joav Orovitz and Itamar Vigdorovich.