Sigma theory and twisted conjugacy classes.
For a group \(G\) and an automorphism \(\varphi\in\Aut(G)\), the \(\varphi\)-conjugacy class of an element \(x\) of \(G\) is the set \(\{g^{-1}xg^\varphi\mid g\in G\}\) and the Reidemeister number \(R_G(\varphi)\) is the number of \(\varphi\)-conjugacy classes in \(G\). In the paper under review the authors prove that for large classes of groups \(G\) there is a subgroup \(\Gamma\) of finite index in \(\Aut(G)\) such that \(R_G(\varphi)=\infty\) for all \(\varphi\in\Gamma\). This includes all finitely generated nonpolycyclic groups that fall into one of the following classes: (i) nilpotent-by-Abelian groups of type \(\mathrm{FP}_\infty\); (ii) groups \(G\) with \(G/G''\) of finite Prüfer rank; (iii) groups of type \(\mathrm{FP}_2\) without free nonabelian subgroups and with nonpolycyclic maximal metabelian quotient; (iv) some direct products of groups; (v) the pure symmetric automorphism group of free groups of finite rank; (vi) Houghton groups \(H_n\) with \(n\geq 2\); (vii) \(1\)-ended nonabelian nonsurface limit groups. In many interesting cases (such as generalized Thompson groups and their finite direct products) it is proved that \(\Gamma=\Aut(G)\) (i.e. \(G\) has the \(R_\infty\)-property).
scientific article; zbMATH DE number 5809042