The $R_{\infty }$ property for nilpotent quotients of generalized solvable Baumslag-Solitar groups

Abstract

We say a group $G$ has property $R_{\infty }$ if the number $R(\phi )$ of twisted conjugacy classes is infinite for every automorphism $\phi$ of $G$. For such groups, the $R_{\infty }$-nilpotency degree is the least integer $c$ such that $G/{\gamma }_{c+1}(G)$ has property $R_{\infty }$. In this work, we compute the $R_{\infty }$-nilpotency degree of all Generalized Solvable Baumslag-Solitar groups ${\Gamma }_n$. Moreover, we compute the lower central series of ${\Gamma }_n$, write the nilpotent quotients ${\Gamma }_{n,c}={\Gamma }_n/{\gamma }_{c+1}({\Gamma }_n)$ as semidirect products of finitely generated abelian groups and classify which integer invertible matrices can be extended to automorphisms of ${\Gamma }_{n,c}$.

11 pages