  In this paper, we consider the conjugacy growth function of a group, which
counts the number of conjugacy classes which intersect a ball of radius $n$
centered at the identity. We prove that in the case of virtually polycyclic
groups, this function is either exponential or polynomially bounded, and is
polynomially bounded exactly when the group is virtually nilpotent. The proof
is fairly short, and makes use of the fact that any polycyclic group has a
subgroup of finite index which can be embedded as a lattice in a Lie group, as
well as exponential radical of Lie groups and Dirichlet's approximation
theorem.
