The $R_\infty$-nilpotency degree of some lollipop GBS groups

Abstract

We say that a group $G$ has the $R_{\infty}$ property if the number $R(φ)$ of twisted conjugacy classes is infinite for any automorphism $φ$ of $G$. The $R_{\infty}$-nilpotency degree is the least integer $c$ such that $G/γ_{c+1}(G)$ has property $R_{\infty}$; in case this integer does not exist, we say that $G$ has infinite $R_\infty$-nilpotency degree. In this paper, we determine the $R_{\infty}$-nilpotency degree of some lollipop GBS groups by using different approaches, such as Sylow subgroups, normal forms and presentation of nilpotent quotients, and diagrams of exact sequences. We also compute the $R_\infty$-nilpotency degree of all 2-generated lollipop GBS groups and of all GBS groups whose associated graph is a tree.