\(R_\infty\)-property for finitely generated torsion-free \(2\)-step nilpotent groups of small Hirsch length (Q6659422)

Let \(G\) be a group and \(\varphi\) an endomorphism of \(G\). Two elements \(a, b \in G\) are said to be \(\varphi\)-conjugate if \(a = c b \varphi(c)^{-1}\) for some \(c \in G\). For a given \(\varphi\), being \(\varphi\)-conjugate is an equivalence relation; the number \(R(\varphi)\) of classes (which might be \(\infty\)) is called the Reidemeister number of \(\varphi\). The Reidemeister spectrum of \(G\) is \(\operatorname{Spec}_{R} (G) := \{R (\varphi) : \varphi \in \operatorname{Aut}(G) \}\). A group \(G\) is said to have the \(R_{\infty}\)-property if \(\operatorname{Spec}_{R} (G) = \{ \infty \}\). \par The main result (Theorem 5.1) of the paper under review is that all finitely generated, torsion-free, nilpotent groups of nilpotence class two and Hirsch length at most \(6\) admit an automorphism with finite Reidemeister number, while there are examples of Hirsch length \(7\) that have the \(R_{\infty}\)-property.