The \(R_{\infty}\) property for nilpotent quotients of surface groups (Q2833180)

If \(\varphi\) is an endomorphism of a group \(G\), an equivalence relation \(\sim\) may be defined in \(G\) by putting \(x\sim y\) if and only if there exists \(z\in G\) such that \(x=zy\varphi(z)^{-1}\). The equivalence classes determined by \(\sim\) are called \textit{twisted conjugacy classes} (or \textit{Reidemeister classes}), and the number of these classes is denoted by \(R(\varphi)\). A group \(G\) is said to have the \textit{\(R_\infty\)-property} if \(R(\varphi)=\infty\) for each automorphism \(\varphi\) of \(G\). For instance, it is known that all non-elementary Gromov hyperbolic groups (and so in particular all free non-abelian groups of finite rank) have the \(R_\infty\)-property.NEWLINENEWLINEIf \(G\) is a group with the \(R_\infty\)-property, the \textit{\(R_\infty\)-nilpotency degree} of \(G\) is the smallest positive integer \(c\) such that \(G/\gamma_{c+1}(G)\) has the \(R_\infty\)-property, if such a number exists, and \(\infty\) otherwise. The \textit{\(R_\infty\)-solvability degree} of \(G\) is defined in a similar way, replacing the terms of the lower central series by those of the derived series of \(G\).NEWLINENEWLINEIn the paper under review, the authors study the \(R_\infty\)-nilpotency and the \(R_\infty\)-solvability degrees of the fundamental group of a closed surface.