Twisted Conjugacy Growth of the Generalised Heisenberg Groups

We determine the twisted conjugacy growth function for automorphisms on generalised Heisenberg groups. In particular we demonstrate for these groups that up to a natural equivalence this function is either given by $n^k$ or $n^k\ln (n)$ for some constant $k$. We give a precise description for which of these functions we obtain in terms of dimensions and ranks on the upper and lower central series. In particular we obtain that the twisted conjugacy growth is always bounded by the conjugacy growth.