The ICC property in Groups and Dynamics

A group is said to have the infinite conjugacy class (ICC) property if every non-identity element has an infinite conjugacy class. In this talk I will survey some ideas in geometric group theory, random walks and harmonic functions on groups, and topological dynamics  and show how the ICC property sheds light on these three seemingly distinct areas. In particular I will discuss when a group has only  constant bounded harmonic functions, when every proximal dynamical system has a fixed point, and what this all has to do with the growth of a group. No prior knowledge of geometric group theory, random walks and harmonic functions on groups or topological dynamics will be assumed. This is based on joint works with Yair Hartman, Omer Tamuz, and Pooya Vahidi Ferdowsi.