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

TL;DR

This paper investigates the $R_ fty$ property in generalized solvable Baumslag-Solitar groups, computing their nilpotency degree, structure of nilpotent quotients, and automorphism extensions.

Contribution

It determines the $R_ abla$-nilpotency degree for all such groups and characterizes automorphisms of their nilpotent quotients.

Findings

Computed the $R_ abla$-nilpotency degree for all $ ext{BS}$-type groups.

Described the lower central series and nilpotent quotients as semidirect products.

Classified extendable automorphisms via invertible matrices.

Abstract

We say a group $G$ has property $R_\infty$ if the number $R(\varphi)$ of twisted conjugacy classes is infinite for every automorphism $\varphi$ 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 $\Gamma_{n,c}$.