Conjugacy growth of free nilpotent groups (Q6633830)

Let \(G\) be a finitely generated group and let \(S\) be a finite (symmetric) set of generators. Let \(B_{G,S}(n)\) be the \(n\)-ball in the Cayley graph of \(G\) with respect to \(S\) (this is the set of words of length at most \(n\) in the generators from \(S\)). The conjugacy growth \(c_{G,S}(n)\) of \(G\) with respect to \(S\) measures the volume of \(B_{G,S}(n)\) up to conjugacy, that is, \(c_{G,S}(n)\) is the number of elements in the set \(\{\text{conjugacy classes in }G \text{ which intersect } B_{G,S}(n)\}\). The invariant \(c_{G,S}(n)\) was defined and studied by \textit{V. Guba} and \textit{M. Sapir} [Ill. J. Math. 54, No. 1, 301--313 (2010; Zbl 1234.20041)] and has since been studied in numerous papers mentioned in the references. Let \(\mathcal{N}_{c,d}\) be the \(d\)-generated \(c\)-step free nilpotent group. It was shown by Guba and Sapir [loc. cit.] that \(c_{\mathcal{N}_{2,2}} \sim n^2\log n\). In this paper it is shown that for \(d\geq 3\) we have \( c_{\mathcal{N}_{2,d}} \sim n^{d^2-d}\).