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

TL;DR

This paper determines the exact nilpotency index at which Baumslag--Solitar groups retain the $R_ty$--property, deepening understanding of their automorphism and conjugacy class structure.

Contribution

It explicitly calculates the $R_ty$--nilpotency degree for all Baumslag--Solitar groups, a novel classification within their automorphism properties.

Findings

Identified the $R_ty$--nilpotency degree for Baumslag--Solitar groups.

Established conditions under which these groups have the $R_ty$--property.

Enhanced understanding of twisted conjugacy classes in nilpotent quotients.

Abstract

A group $G$ has the $R_{\infty}$--property if the number $R(\varphi)$ of twisted conjugacy classes is infinite for any automorphism $\varphi$ of $G$. For such a group $G$, the $R_{\infty}$--nilpotency index is the least integer $c$ such that $G/\gamma_{c+1}(G)$ still has the $R_\infty$--property. In this paper, we determine the $R_{\infty}$--nilpotency degree of all Baumslag--Solitar groups.