## Asymptotics for conjugacy classes of free groups

#### George Kenison (University of Warwick)

Let $$G$$ be a finite connected metric graph such that the degree of each vertex is at least 3.  The fundamental group of $$G$$ is a free group $$F$$ and the universal cover of $$G$$ is a tree $$\mathcal{T}$$. We are interested in asymptotics related to the action of $$F$$ on $$\mathcal{T}$$; in particular, when the elements of $$F$$ are restricted to a non-trivial conjugacy class.  Time permitting, we will discuss two results: an equidistribution limit and a central limit asymptotic.  This is joint work with Richard Sharp.