A conjugacy class in a finitely generated group has a natural notion of length: the length of a shortest word over the generators that represents an element of the conjugacy class. The conjugacy growth function is defined as the number of conjugacy classes of length at most n. Equivalently, it is the number of conjugacy classes intersecting the ball of radius n in the Cayley graph. This function first appeared in the guise of counting primitive closed geodesics on a Riemannian manifold (up to free homotopy). Up to bounded scaling and shifting, the asymptotic behaviour of the conjugacy growth function does not depend on the choice of finite generating set.
This project deals primarily with the asymptotics of conjugacy growth in groups of polynomial standard growth (i.e. virtually nilpotent groups). Little is known in this specific area, unlike the case for groups of non-positive curvature such as hyperbolic and relatively hyperbolic groups. Hull and Osin have shown that conjugacy growth fails to be even a commensurability invariant in general. On the other hand, it is a quasi-isometry invariant in the restricted case of virtually abelian groups. A theorem of Pansu implies that if G and H are finitely generated nilpotent groups which are quasi-isometric, then the torsion-free ranks of the quotients composing their lower central series must coincide. Thus in some sense the geometry of these groups depends only on their nilpotent structure. This leads to the conjecture that quasi-isometric nilpotent groups have equivalent conjugacy growth functions. This project aims to make progress towards this conjecture, by generalising work of Babenko and others to derive asymptotic estimates for the conjugacy growth of certain classes of nilpotent groups of step 2. A secondary goal is to understand the algebraic complexity of the associated generating function: the formal power series whose coefficients are the values of the conjugacy growth function.