Conjugacy growth in the higher Heisenberg groups

TL;DR

This paper provides asymptotic estimates for conjugacy growth in higher Heisenberg groups, revealing stability under commensurability and implications for the nature of conjugacy growth series.

Contribution

It introduces precise asymptotic estimates for conjugacy growth in higher Heisenberg groups and analyzes their stability and properties of growth series.

Findings

Asymptotic conjugacy growth estimates for higher Heisenberg groups

Stability of these estimates under commensurability

Conjugacy growth series may not be holonomic in some cases

Abstract

We calculate asymptotic estimates for the conjugacy growth function of finitely generated class 2 nilpotent groups whose derived subgroup is infinite cyclic, including the so-called higher Heisenberg groups. We prove that these asymptotics are stable when passing to commensurable groups, by understanding their twisted conjugacy growth. We also use these estimates to prove that, in certain cases, the conjugacy growth series cannot be a holonomic function.