Conjugacy growth in groups has been studied, from a geometric perspective, for many decades. Initially, the growth of conjugacy classes naturally occurred while counting closed geodesics (up to free homotopy) on complete Riemannian manifolds, as formulas for the number of such geodesics give, via quasi-isometry, good estimates for the number of conjugacy classes in the manifolds’ fundamental groups. More recently, the study of conjugacy growth has expanded to groups of all flavours, ranging from nilpotent to linear to acting on cube complexes, and beyond.

In this talk I will give an overview of what is known about conjugacy growth and the formal series associated with it in infinite discrete groups. I will highlight how the rationality (or rather lack thereof) of these series is connected to both the algebraic and the geometric nature of groups such as (relatively) hyperbolic or nilpotent, and how tools from analytic combinatorics can be employed in this context.