Conjugacy Growth in Polycyclic Groups

TL;DR

This paper investigates the conjugacy growth function in virtually polycyclic groups, establishing that it is either exponential or polynomially bounded, with polynomial bounds characterizing virtually nilpotent groups.

Contribution

It proves a dichotomy for conjugacy growth in virtually polycyclic groups, linking polynomial bounds to virtual nilpotency using Lie group embeddings.

Findings

Conjugacy growth is either exponential or polynomial in virtually polycyclic groups.

Polynomial conjugacy growth characterizes virtually nilpotent groups.

The proof leverages Lie group embeddings and properties of exponential radicals.

Abstract

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.